Machines, Mechanism and Robotics: Proceedings of iNaCoMM 2019 [1 ed.] 9811605491, 9789811605499
This volume includes select papers presented during the 4th International and 19th National Conference on Machines and M
331 54 68MB
English Pages 1881 [1781] Year 2021
Organization
Preface
Contents
Editors and Contributors
A Mechanical Contrivance for Acoustic Levitation and Mixing of Particles
1 Introduction
2 Numerical Modeling
2.1 Geometry of Standing Wave Acoustic Levitation System
2.2 Procedure
2.3 Finite Element Method and Convergence Study
3 Results and Discussion
4 Summary
References
Design and Validation of Flexure-Based Hinges for Space Deployable Antenna Reflector
1 Introduction
2 Design Configuration of Deployable Mesh Reflector
2.1 Description
2.2 Kinematics of Deployable Mesh Reflector
3 Design of Joints with Flexure Hinges
3.1 Geometry of Tape Flexures
3.2 Kinematics of Tape Flexures
3.3 Structural Analysis of Tape Flexures
3.4 Implementation of Compliant 5J Joint in Deployable Reflector
4 Advantages of Tape Flexures in Deployable Configuration
5 Conclusion
References
Effect of Implant Materials on Bone Remodelling Around Cemented Acetabular Cup
1 Introduction
2 Materials and Method
2.1 Material Properties of Cortical Bone, Cancellous Bone, Cement Mantle and Implant
2.2 Loading Boundary Conditions and Bone Remodelling Models
2.3 Interfacial Conditions
3 Results
4 Discussion
5 Conclusion
References
Influence of Ageing and High BMI on Lower Back Pain
1 Introduction
2 Methodology
3 Results
References
Design and Analysis of a Robotic Lizard Using Five-Bar Mechanisms
1 Introduction
2 Topological Design of the Robotic Lizard Mechanism
3 Position Analysis of the Robotic Lizard Mechanism
3.1 Coordiates of Linkages in the Mechanism
3.2 Workspace of Robot Parts
4 Prototype of the Robotic Lizard Mechanism
5 Conclusions
References
Development of an Automated Material Handling System Inside a Nuclear Containment Structure
1 Introduction
2 Design Considerations and Constraints
3 System Details
4 Conclusions
References
Nonlinear Modeling and Stability Analysis of Piezoelectric Energy Harvesting Mechanism Under Aeroelastic Vibration
1 Introduction
2 Theoretical Background
3 Mathematical Formulation
3.1 Nonlinear Distributed Parameter Model
4 Results and Discussion
5 Conclusions
References
Optimization of Surface Roughness of Laser Trepanned Hole in ZTA Plate
1 Introduction
2 Experimentation
3 Modeling and Optimization
4 Conclusions
References
PI Control-Based Modelling of Segway Using Bond Graph
1 Introduction
2 Bond Graph Modelling of Segway
2.1 Word Bond Graph of Segway
3 Bond Graph Modelling of Segway
3.1 Word Bond Graph of Segway
4 Bond Graph Simulation Result of Segway
5 Conclusions
References
Strategic Coordination and Navigation of Multiple Wheeled Robots
1 Introduction
2 Developed Cooperative Multi-agent Systems
3 Developed Motion Planning Approaches
3.1 Approach 1: Potential Field-Based Motion Planner
3.2 Approach 2: Manually Constructed Mamdani-Type Fuzzy Logic-Based Motion Planner
3.3 Automatically Evolved Genetic-Fuzzy Motion Planner
4 Implementation of Strategic Coordination
5 Results and Discussion
5.1 GA parametric Study and Identification of Best Rule Base of FLC
5.2 The Need for Coordination Schemes
5.3 Comparison of Traveling Time Taken by the Robots
6 Concluding Remarks
References
Spur Gear Mechanism for Accurate Angular Indexing and Locking of Angular Position by Using Additive Manufacturing
1 Introduction
2 Literature Review
3 Methodology
3.1 Machine, Materials and Software
3.2 Design of Modified Ratchet Mechanism
3.3 Design of Spur Gear
4 Analysis
5 Development of Spur Gear Mechanism
6 Results and Discussion
7 Conclusion
References
Fabrication of Solid Lubricant Coating and Its Optimization Using Response Surface Methodology
1 Introduction
2 Experimentation
3 Results and Discussion
3.1 Response Surface Methodology
3.2 Analysis of Variance (ANOVA)
3.3 FESEM Analysis
4 Conclusions
References
Computing and Verification of IPMC Parameters Through Equivalent Beam Theory
1 Introduction
2 Mathematical Model
2.1 Beam Model
3 Results
4 Conclusion
References
Characterization of Mechanical Properties of Different Agro-derived Reinforcements Reinforced in Aluminium Alloy (AA6061) Matrix Composite: A Review
1 Introduction
2 Metal Matrix Composites (MMCs)
3 Reinforcing Materials in MMCs
3.1 Synthetic Reinforcements
3.2 Agro-industrial Waste Derivatives as Reinforcements
3.3 Advantages and Disadvantages of Agro-Derived and Synthetic Reinforcements.
4 Production of Metal Matrix Composites
4.1 Solid Route
4.2 Liquid Route
5 Summary and Conclusions
References
Shoe-Based Energy Harvesting Using Ionic Polymer Metal Composites
1 Introduction
2 Experimental Setup
2.1 Fabrication of Double Curved-Shaped Structure
2.2 Working Mechanism of Energy Harvester
3 Result and Discussion
4 Conclusion
References
IoT-Based Health Monitoring System (IHMS)
1 Introduction
2 Related Work
3 Working of IHMS
4 Simulation Results and Analysis
5 Conclusion
References
Search and Reconnaissance Robot for Disaster Management
1 Introduction
2 Existing ATVs
2.1 Legged Robots
2.2 SuperDroid Robots
2.3 Honda’s 3E-D18
3 Design
3.1 Material
3.2 Specifications
3.3 Articulating Chassis
4 Electronics and Power Management
4.1 Power Distributions and Voltage Levels
4.2 Operating Time
4.3 Power Calculations
5 Locking Mechanism and Modularity
5.1 Modularity
5.2 Locking Mechanism
6 Assistive Autonomy and Wireless Communication
6.1 Modes for Climbing
6.2 Assistive Autonomy
6.3 Robotics Operating System (ROS) Framework
6.4 Video and Audio Feed
7 Conclusion and Future Prospects
References
Experimental Validation of Various Existing Impedance Models for Acoustic Liners
1 Introduction
2 Analytical Models
2.1 Modified Howe Model
2.2 Betts Impedance Model
2.3 Bauer Impedance Model
3 Experimental Setup
4 Results
5 Conclusion
References
Design and Modeling of Pipeline Inspection Robot (PIR) for Underground Pipelines
1 Introduction
2 Design of Pipeline Inspection Robot (PIR) for Underground Pipelines Inspection
3 Kinematics Modeling of Link Mechanism
4 Results and Discussion
5 Conclusion
References
Mechanism of Material Removal in Magneto Abrasive Flow Machining
1 Introduction
2 Experimental Work
2.1 Fabrication of Work Specimen
2.2 Finishing of Aluminum Tubes Using MAFM
2.3 Variables and Responses
3 Regression Analysis
4 Results and Discussion
4.1 Analysis of MR and Surface Roughness in MAFM
4.2 Optimization
4.3 SEM Examination
5 Conclusions
References
Optimization of Cutting Parameters of EN9 Steel with Plain Carbide Tool Using Response Surface Methodology
1 Introduction
2 Experimental Methods
2.1 Selection of Work Piece and Tool
2.2 Composition of EN9 Steel with Energy Dispersive Spectroscopy (EDS)
3 Experimental Design
3.1 Response Surface Methodology
3.2 Measurement of Responses
4 Results and Discussions
4.1 Analysis of Output Parameters with Plain Carbide Tool Analysis of Crater Wear
4.2 Predicted Versus Actual Values for Response Factors with Plain Carbide Tool
4.3 Effect of Machining Parameters on Response Variables with Plain Carbide Tool Crater Wear
4.4 Optimization of Cutting Conditions with Plain Carbide Tool
5 Conclusions
References
Design and Development of a Sit-to-Stand Assistive Device
1 Introduction
2 Determination of Natural STS Trajectory
3 Four-Position Motion Generation for STS Mechanism
4 STS Device Design and Fabrication
5 STS Experiments with Assistive Device
6 Conclusion and Future Work
7 Declaration
References
Effect of Structural Characteristics on Kinematics of Planar Kinematic Chains
1 Introduction
2 Methodology
2.1 Joint Value
2.2 Design Parameters
2.3 Chain Motion Characteristic Matrix (CMC)
3 Detection of Isomorphism
4 Characteristic of Kinematic Chains
5 Path Analysis
6 Summary
Appendix: Eight-link Single Degree of Freedom Kinematic Chains Figure
References
A Combined Experimental/Finite Element Model Analysis on Compressive Behavior of Tamarind Pod Shell Filler Reinforced Composites
1 Introduction
2 Experimental
2.1 Materials
2.2 Preparation of Composite Specimens
2.3 Density and Void Measurement
2.4 Compression Testing
2.5 Multiscale Material Modeling
2.6 Surface Treatment in Digimat-FE
2.7 Failure Indicator Model
3 Results and Discussions
3.1 Density and Void Content
3.2 Compressive Behavior
4 Conclusion
References
Workspace Evaluation of Robotino-XT Under Reconfiguration
1 Introduction
2 CBHA Description
3 Experimental Setup of Multi-section CBHA
4 Experimental Results
5 Conclusions and Future Scope
References
Assessment of Surface Water Quality Using Principal Component Analysis in the Yamuna River: A Case Study
1 Introduction
2 Literature Review
3 Material and Method
4 Result and Discussion
5 Conclusion
References
Performance Analysis of Gripper Assembly of an In-Vessel Fuel Handling Machine
1 Introduction
2 Description of Gripper Assembly
3 Analysis of Gripper Assembly
4 Results and Discussion
5 Design Improvements for Gripper Assembly with Larger Offset
6 Summary
References
Design and Development of a Remote Racking Mechanism for Switchgear
1 Introduction
2 Design of Remote Racking Mechanism
3 Remote Racking Mechanism Performance Estimation with FEA Analysis Simulation and Testing
3.1 Static Analysis
3.2 Modal Analysis
4 Conclusion
References
Iwan Model for Bolted Joint with Residual Macroslip Stiffness and Pinning
1 Introduction
2 Modeling of Backbone Curve
2.1 Iwan Model
2.2 Pinning Force
2.3 Backbone Curve
2.4 Comparison with Other Models
3 Conclusion
References
Kinematics Model of Bionic Manipulator by Using Elliptic Integral Approach
1 Introduction
2 System Configuration
3 Experimental Set-Up
4 Results and Discussion
5 Conclusions
References
A Review on the Effect of Biomechanical Aspects and the Type of Stability Fixation on the Bone Fracture Healing Process
1 Introduction
2 Biology of Bone Fracture Healing
3 Finite Element Analysis
4 Factors Affecting the Bone Healing Process
4.1 Simple and Complex Fracture
4.2 Open and Closed Reduction
4.3 Absolute and Relative Stability
4.4 Implants and Their Outcomes
4.5 Direction of Interfragmentary Movement
5 Discussion
6 Conclusion
References
Mechanical Behaviour of Special Type Seals Used in the FBR Application
1 Introduction
2 Seal Description
3 Numerical Analysis
3.1 Stage 1: Development of a Numerical Model
3.2 Stage 2: Optimisation Study
4 Conclusion
References
Realization of a Simple Mechanism to Simulate Core Subassembly Growth of FBR
1 Introduction
2 Different Methods Proposed to Simulate Subassembly Growth
3 Description of Test Set-up Based on Selected Mechanism
4 Tests on Lifting Tool Mechanism
5 Results and Discussion
6 Conclusion
References
Investigation of Multiple Stable States of Tensegrity Structure
1 Introduction
2 Stability of Tensegrity Structure
2.1 Monte Carlo-Based Algorithm and Simulation
2.2 Selection of Initial Configuration, Computation of Member Lengths, Member Forces and Nodal Forces
2.3 Steps of Simulation and Implementation in a Tensegrity Simplex
2.4 Implementation of Dynamic Relaxation (DR) Method
3 Conclusion
References
Design and Development of a Short-Wave Electric Infrared Heater of 215 kW Capacity
1 Introduction
2 Electric SWIR Heater
2.1 Background
2.2 SWIR Lamp
2.3 Electric IR Heater
3 Experiments on EIH
4 Conclusions
References
Kinematic and Dynamic Modeling of a Quadruped Robot
1 Introduction
2 Kinematic Mechanism of the Legged Robot
2.1 D-H Parameter Representation
2.2 Inverse Kinematics
3 Dynamic Motion and Joint Space Formulation
4 Simulation Results
5 Conclusions
References
Nonlinear Joint Stiffness Parameter Identification
1 Introduction
2 Theory: Nonlinear Joint Parameter Identification
3 Numerical Simulation
3.1 Joint Parameter Identification
4 Conclusion
References
Design and Development of Reaper for Harvesting Maize
1 Introduction
2 Problem Statement
3 Design of the Reaper
4 Detailed Design
4.1 Design of Traction System
4.2 Traction System Arrangement
4.3 Design of Reaper Mechanism
4.4 Design of Frame
5 Prototype of Maize Harvesting Reaper
6 Cost Analysis of Maize Harvesting
7 Conclusions and Future Directions
References
Transpose Jacobian Control of Flexible Joint Upper Limb Exoskeleton System
1 Introduction
2 System Dynamics
2.1 Device Specifications
2.2 Flexible Joint Exoskeleton Dynamics
3 Control Scheme
3.1 Rigid Dynamics Transpose Jacobian Control Scheme
3.2 Modified Dynamics Transpose Jacobian Control Scheme
4 Results and Discussion
5 Conclusion
References
Defect-Free Synthesis, Analysis and Optimization of Planar Lower Limb Assistive Device for Gait Rehabilitation
1 Introduction
2 Clinical Design Specifications
3 Caster Walker Gait Trainer (CGT) Design
4 Optimal Synthesis of Mechanism
5 Conclusions
References
Theoretical and Experimental Investigation of Friction in Hydraulic Actuators
1 Introduction
2 Friction Models
2.1 LuGre Model [1–3]
2.2 Modified LuGre Model [1]
3 Experimental Setup and Methodology
4 Results and Discussion
4.1 Identification of σ2, Vb, Fs, and Fc
4.2 Model Validations
5 Conclusion
References
Support Vector Classifier-Based Broken Rotor Bar Detection in Squirrel Cage Induction Motor
1 Introduction
2 Broken Rotor Bar Fault
3 Fast Fourier Transformation
4 Support Vector Machine
4.1 SVM Classifier for Broken Bar Detection
4.2 Data Acquisition and Implementation of SVM Classifier
4.3 Results
5 Conclusion
References
Kinematic and Dynamic Analysis of Primary FCS Circuits of Typical 25 Seater Transport Aircraft
1 Introduction
2 Problem Definition and Methodology
3 Geometric Modeling and Kinematic Simulation
4 Kinematic Simulation in MSC ADAMS
4.1 ADAMS Models and Kinematic Joints
4.2 Kinematic Motion
4.3 Simulation Results
5 Force Analysis in MSC ADAMS
5.1 Hinge Moment Calculations
5.2 Aerodynamic (Actuator) Load Calculations
5.3 ADAMS Force Model
5.4 Validation of Force Analysis Carried Out Using MSC ADAMS
5.5 Results of Force Simulation
6 Inertia Effects of Control Surfaces on Pilot Controls
7 Conclusions
References
Kinematic Synthesis and Optimization of a Double-Slotted Fowler Flap Mechanism
1 Introduction
2 Problem Definition
3 Methodology
3.1 Kinematic Synthesis of the Six Bar Mechanism
3.2 Parametric Analysis of the Design Parameters
3.3 Dimensional Synthesis Using MATLAB Program
3.4 Optimization of Link Lengths
3.5 Dimensional Synthesis of Aft-flap Mechanism
4 Verification of Synthesis Procedure with CAD Models
5 Fabrication of Prototype Model of the Mechanism
6 Conclusion
References
Intelligent Modeling of Dilution Percent in Laser Surface Alloying of AlxCu0.5FeNiTi High Entropy Alloy
1 Introduction
2 Experimental Method
3 Results and Discussions
3.1 Regression-Based Modeling
3.2 ANN-Based Modeling
3.3 Comparison of RSM and ANN Models
4 Conclusions
References
Spinal Needles Insertion and Traversal Based on Fiber Bragg Gratings—From Conceptual Approach to Prototype Development
1 Introduction
2 Device Design and Development
3 Experimental Procedure and Results
3.1 Device Calibration Methodology
3.2 Experimentation, Results, and Discussion
4 Conclusion
References
Kinematic and Dynamic Analysis of Sliding Door Operating Mechanism for Internal Weapon Bay
1 Introduction
2 Kinematic Analysis of Mechanism (Analytical)
2.1 Input Motion Profile of Connecting Link-A
2.2 Displacement Analysis: Point-B (Slider)
2.3 Velocity Analysis: Point-B (Slider)
2.4 Acceleration Analysis: Point-B (Slider)
3 Kinematic and Dynamic Analysis of Mechanism (Software)
3.1 Kinematic Analysis of Slider (Point-B)
3.2 Kinematic Analysis of Point-E
3.3 Dynamic Analysis
4 Conclusion
5 Further Scope of Works
References
A Method to Detect Isomorphism in Planar Kinematic Chains
1 Introduction
2 Basic Terminologies in Kinematic Chains
3 Method for Detection of Isomorphism
4 Analysis of Four-Bar Kinematic Chain
4.1 Comparison Between Watt and Stephenson Chains
4.2 Comparison Between a Pair of 8-links Single Degree of Freedom Kinematic Chains
4.3 Analysis of 8-link Single Degree of Freedom Kinematic Chains
4.4 Analysis of 9-link Double Degree of Freedom Kinematic Chains
5 Conclusion
References
Generation of Coupler Curves for Planar Kinematic Chains Using Link Joint Equations
1 Introduction
2 Method
3 Applications and Results
4 Conclusions
References
Tribological Behaviour of Polymer-Based Composite Reinforced with Molybdenum Disulphide
1 Introduction
2 Materials and Methodology
2.1 Materials and Fabrication Process
2.2 Analysis of Tribological Properties
2.3 Estimation of Volumetric Wear of the Pin
3 Result and Discussion
3.1 Sample Designation
3.2 Tribological Behaviour of Polymer Composite
4 Conclusion
References
Prediction of the Blood Flow Through Stenosis in AVF for Hemodialysis
1 Introduction
2 Geometrical Model
3 Numerical Modeling
4 Result
5 Conclusion
References
Tool Quality Monitoring in Friction Stir Welding Process
1 Introduction
2 Experimental Procedure
3 Result and Discussion
4 Conclusion
References
Condition Monitoring and Fault Diagnosis of Induction Motor in Electric Vehicle
1 Introduction
2 Faults in Induction Motor
3 Condition Monitoring Techniques
4 Proposed Methodology
5 Data Acquisition and Processing
6 On-Board Diagnosis
7 Conclusion
References
Mechanical Design Calculations of Flywheel Generator
1 Introduction
2 Design Schematic
3 Modelling and Design Calculation
3.1 NdFeB Magnet and Equivalent Layer
3.2 Stress Analysis of Rotor
3.3 Max Deflection of System
3.4 Anisotropic and Isotropic Properties
3.5 Critical Speed of the System
3.6 Windage Losses
4 Conclusion
References
Numerical Analysis and Reduction of Blade-Vortex Interaction (BVI) Noise in Helicopter Using Numerical Simulation
1 Introduction
2 Computational Methodology
3 Results and Discussions
4 Conclusion
References
Dynamics of Underwater Manipulator: A Recursive Lagrangian Formulation
1 Introduction
2 Mathematical Modeling of Object Moving Through Fluid
2.1 Dynamic Effect Due to Added Mass
2.2 Hydrodynamic Drag
2.3 Buoyancy
3 Recursive Lagrangian Formulation for the Serial Link Underwater Manipulator
3.1 The Lagrangian
3.2 Kinetic Energy
3.3 Potential Energy
3.4 Drag Force and Rayleigh Dissipation Function
3.5 Derivation for Recursive Formulation
4 Modeling and Simulation of Two-Link Underwater Manipulator
5 Experiments and Validation
6 Conclusion
References
Effect of Tool Geometry on Chip Morphology Formed During CNC Turning of a Round Shaped Product
1 Introduction
2 Workpiece and Tool Material
2.1 Workpiece Material
2.2 Tool Material and Geometry
3 Development of Mathematical Model
4 Experimental Verification
5 Discussion
6 Conclusion
References
Optimization of Equal Multi-square Cell Crash Box for Enhanced Energy Absorption
1 Introduction
2 Mathematical Model
3 Simulation Using LS-DYNA
4 Experimentation
5 Parametric Study
6 Overall Optimization Process
6.1 Optimization Algorithm
6.2 Optimization Results
7 Conclusions
Reference
Joint Stiffness Estimation Between Spindle-Tool Holder by Considering Clamping Forces
1 Introduction
2 Geometric Model of Spindle-Tool Holder Joint Interface
3 Contact Stiffness Model
3.1 Elastic Deformation of the Aspirates
3.2 Plastic Deformation of the Aspirates
4 BT-40 Spindle Shaft Case Study
4.1 Experimental Modal Analysis
4.2 Finite Element Analysis
5 Conclusion
References
Buckling Analysis of Nonlinear First-Order Shear Deformation Composite Plates
1 Introduction
2 Theoretical Formulations
2.1 Basic Assumptions
2.2 Kinematics
2.3 Equation of Motion
2.4 Constitutive Equations of a Laminate
3 Analytical Solutions
4 Numerical Results
5 Conclusions
References
Path Tracing and Object Avoidance Algorithm for Robotic Manipulators Incorporating Constrained Filters
1 Introduction
2 Representation of Rigid Bodies
3 Methodology
3.1 Workspace and C-Space
3.2 Collision Filter
3.3 Nearest State Filter
3.4 Smoothing Filter
3.5 Implementation of Approach
3.6 Results
4 Conclusion
References
Characterisation of Composites Made by Prepreg Waste
1 Introduction
2 Material Preparation
2.1 Chips Chopped from Prepreg Waste
2.2 Fabrication of PBDC Material
3 Experimentation
3.1 Aspect Ratio Size of Chopped Chips
3.2 Fibre Volume Fraction
3.3 Void Fraction
3.4 Tensile Test Setup for PBDC
4 Results and Discussions
4.1 Chips Size Results from Image J
4.2 Fibre and Void Volume Fraction
4.3 Tensile Testing Results
5 Conclusions
References
Experimental Investigation on the Effect of Process Parameters for CNC Turning of UNI Al 3055 Alloy Under MQCL Based Cooling Technique
1 Introduction
2 Experimental Procedure
2.1 Experimental Setup
2.2 Work Piece Materials
2.3 Design of Experiment
3 Results and Analysis
4 Discussion
5 Conclusions
References
Reciprocating Wear Behaviour of Al–SiC Composite Processed with MWS
1 Introduction
2 Experimental Details
2.1 Materials
2.2 Processing Details and Characterization Study
3 Results and Discussion
3.1 Microstructure Analysis of the Milled Powder Material
3.2 Hardness
3.3 Wear analysis
4 Conclusion
References
Dynamics and Control of a 6-DOF Biped Robot on MATLAB/SimMechanics
1 Introduction
2 Modeling of System in MATLAB/SimMechanics
2.1 Torso Modeling
2.2 Leg Modeling
2.3 PID Control Modeling
2.4 Ground Contact Modeling
3 Simulation and Results
4 Dynamic Stability
5 Conclusion
References
Modal Analysis of 3-RRR SPM Model
1 Introduction
2 Modeling of the 3-RRR SPM
3 Simulation of SPM
4 Experimentation of SPM
5 Comparison of Results
6 Conclusion
References
Mode Based Crack Identification of Rotor
1 Introduction
2 Finite Element Formulation for Cracked Beam Element
3 Procedure for Identifying the Crack and Location
4 Illustrative Examples
4.1 Case 1: Single Crack at 42nd Element
4.2 Case 2: Single Crack at 111th Element
4.3 Case 3: Multi-Crack at 45th and 161st Element
5 Conclusion
Annexure A
References
A Note on Implementation of Raghavan–Roth Solution for Wrist-Partitioned Robots
1 Introduction
2 Raghavan and Roth Algorithm Preview
3 Modified Architecture
3.1 Failure of Raghavan–Roth Solution
4 Numerical Example Case of KUKA KR 5 Robot
5 Discussion
6 Conclusion
References
Experimental Identification of Residual Unbalances for Two-Plane Balancing in a Rigid Rotor System Integrated with AMB
1 Introduction
2 Influence Coefficient Method
3 Working of AMB
4 Experimental Simulation
4.1 Physical and Electrical Connections
4.2 Testing of the Physical System
4.3 Dynamic Balancing
5 Conclusions
References
Determination of Steering Actuator Mounting Points of a Load Haul Dump Machine for Optimum Performance
1 Introduction
2 Geometric Modeling of the Articulated Steering Arrangement
2.1 Flowchart
2.2 Results Observed from MATLAB Programming
3 Force Analysis Using MSC ADAMS
4 Results and Discussion
5 Conclusion and Future Scope of Work
5.1 Conclusion
5.2 Future Scope of Work
References
Dynamic Analysis of Helicopter Boom with Different Payload Configurations
1 Introduction
2 Helicopter Boom and Its Arsenal
3 Configurations of Weapons Systems and Their Dynamic Behavior
4 Impact Hammer Test
5 Post Flight Data
6 Sensitivity Analysis
7 A Passive Damping Solution
8 Conclusion
References
Characterization of Composites Made with In-House Prepregs at Different Curing Cycle
1 Introduction
2 Material Making
2.1 Prepreg Making Process
2.2 B-Stage Curing
2.3 Laminate and Test Specimen
3 Experimentation
3.1 Fiber and Void Volume Fraction
3.2 Tensile Testing
4 Result and Discussion
4.1 Tensile results
4.2 Comparison of Strength and Modulus with Available Literature
4.3 Failure Analysis Using Microscope
5 Conclusions
References
A Parametric Approach to Detect Isomorphism and Inversion in the Planar Kinematic Chains
1 Introduction
2 Methodology
3 Isomorphism
4 Inversion
5 Results
6 Conclusion
Appendix
References
Feasibility of Tensegrity-Based Walking Robot
1 Introduction
2 Guiding Principle of Gait Generation in Tensegrity
2.1 Sensitive Analysis for Cable Actuation
2.2 Concept of Locomotion
2.3 Simulation of Gait of a Four-Bar Tensegrity Mechanism
2.4 Gate Generation of Other Tensegrity Mechanism
3 Conclusion
References
Design and Development of a Climb-Free Telescopic Mechanism for Harvesting from Tall Trees
1 Introduction
2 Design Concept
3 Selection of Appropriate Telescopic Mechanism
3.1 Types of Telescopic Mechanism
3.2 Comparison of Telescopic Mechanisms
3.3 Selection of Mechanism
4 Concept of Support System
5 Testing of Prototype
6 Conclusion
References
Simulation Modeling of 37 Degrees-of-Freedom ICF Coach
1 Introduction
2 Simulation Modeling
3 Equations of Motion
4 Natural Frequencies
5 Response of the System
6 Conclusion
References
Transmission Efficiency and Surface Damage of Polymer–Polymer Gear Pair Under Wet Lubrication
1 Introduction
2 Methodology
2.1 Experimental Methodology
2.2 Numerical Analysis
3 Results and Discussion
3.1 Temperature Measurement
3.2 Transmission Efficiency Measurement
3.3 Test Gear Surface Wear Morphology
4 Conclusion
References
Effect of Acceleration of Moving Object During Collision with Stationary Object
1 Introduction
2 Coefficient of Restitution (COR)
3 Effect of Acceleration on Momentum Transfer
4 Effect of Retardation on Momentum Transfer
5 Conclusion
References
Finite Element Modelling of the Human Lumbar Vertebrae for Dynamic Analysis
1 Introduction
2 Solid Modelling
3 Results and Discussion
References
Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR)
1 Introduction
2 The Steering System of SW-MTR
2.1 Differential Steering System
2.2 Worm-Sector Steering
2.3 Hybrid Steering
3 Dynamic Modeling of Hybrid Steering
3.1 Motion Analysis
3.2 Equation of Motion
4 Comparison of Simulation Results
4.1 Comparison of Steering Radius
4.2 Comparison of Contact Forces
4.3 Comparison of Motion Torque
4.4 Comparison of Power Consumption
5 Conclusion
References
Flexible Coupling—A Research Review
1 Background of Present Work and Scope of the Research
2 Reported Work Pertaining to Flexible Couplings
3 Summary and Conclusion
References
Design of Post-Curing Inflator Using Bistable Locking Mechanism
1 Introduction
2 Design Requirements
3 Bistable System
3.1 Bistable Beam
3.2 Pseudo-Bistable System
3.3 Retractable Mechanism
3.4 Compliant Mechanism
3.5 Four-Bar Linkage
4 Design Setup
4.1 Pressure Calculation
5 Rapid Prototyped Setup
6 Conclusion
References
Stability Analysis of a Dual-Rate Haptics Controller Using Discrete-Time Root-Locus Method
1 Introduction
2 Dual-Rate Haptics Controller
3 Stability Analysis Using DTRL Method
3.1 Root-Locus as a Function of B
3.2 Root-Locus as a Function of K
4 Discussions and Conclusions
5 Appendix
References
Design of Compliant Iris
1 Introduction
2 Iris Structure
3 Stiffness Analysis
3.1 Derivation of Stiffness Expression
4 Discussion
5 Conclusion
References
Investigation on the Effects of Nose Radius and Rake Surface of Cutting Tool for Machinability During Sustainable Turning of EN 31 Alloy Steel
1 Introduction
2 Experimentation
3 Results and Discussion
3.1 Analysis on MRR
3.2 Analysis for Cutting Temperature
4 Conclusions
References
Battery Performance Analysis of Static Temperature Variations for Medical Environment
1 Introduction
2 Experimentation
2.1 Implementation
2.2 Thermography
2.3 Scanning Electron Microscope
3 Results and Discussion
3.1 Thermography Image Analysis
3.2 Scanning Electron Microscope
4 Conclusion
References
Derivation of the Rotation Matrix for an Axis-Angle Rotation Based on an Intuitive Interpretation of the Rotation Matrix
1 Introduction
1.1 Problem Statement
2 Derivation of the Rotation Matrix
3 An Example
4 Conclusions
References
Resolving Hyper-Redundant Planar Serial Robots to Ensure Grasp
1 Introduction
2 Tractrices and Tractors
3 Methodology
4 Simulation Showing Caging and Grasp
4.1 Example
5 Conclusion
References
Boom Packaging with Yoshimura Pattern: Geometrical and Deformation Analysis
1 Introduction
2 Geometry of Yoshimura Pattern
2.1 Radius of Packaged Cylinder
2.2 Packaging Efficiency
2.3 Residual Volume Calculations
3 Packaging Behavior
4 Deployment of Single Story Yoshimura Cylinder
5 Summary
References
Multimodal Medical Image Fusion Based on Interval-Valued Intuitionistic Fuzzy Sets
1 Introduction
2 Literature Review
3 Proposed Method
4 Experimental Results
5 Conclusion
References
The Influence of Ultrasound for the Protection of Animals on Highways Through Electronic Circuits
1 Introduction
2 Literature Review
3 Proposed Method
3.1 Generation of Ultrasound Waves Using 555 Timer
4 Simulation Results
5 Conclusion
References
Workspace Analysis of a 5-Axis Parallel Kinematic Machine Tool
1 Introduction
2 Description of Parallel Kinematic Machine (PKM)
3 Position Analysis
3.1 Inverse Position Analysis (IK) and Forward Position Analysis (FK)
4 Workspace Analysis
5 Results
6 Conclusion and Future Scope
References
Reaction Solvability Analysis Using Natural Coordinates
1 Introduction
2 Equations of Motion and the RSA Algorithm
2.1 Equations of Motion of Multi-body Systems
2.2 The RSA Algorithm
3 The Optimal Coordinate Formulation for RSA
3.1 Joint Reactions from Natural Coordinates
3.2 Joint-Augmented Natural Coordinate Formulation
4 RSA of a Planar Mechanism Using Natural Coordinates
4.1 Assumptions
4.2 Modeling Using Natural Coordinates
4.3 Applying RSA Algorithm
4.4 Discussion
5 Conclusions
References
Design, Analysis and Development of Sweep Arm Scanner for Scanning Fast Breeder Reactor Core
1 Introduction
2 Sweep Arm Scanner (SAS)
2.1 General Aspects of SAS
3 Analysis of L-Motion
3.1 Kinematic Analysis
3.2 Dynamic Analysis
4 Analysis of R-Motion
4.1 R-Motion Using Bevel Gears
4.2 R-Motion Using Alternate Options
5 Summary
References
Kinematics of Three Segment Continuum Robot for Surgical Application
1 Introduction
2 Preliminary Design
2.1 Model and Design Specification
2.2 Kinematic Assumptions
3 Continuum Robot Kinematics
3.1 Forward Kinematics
3.2 Inverse Kinematics
4 Workspace and Dexterity Analysis
4.1 Workspace Analysis
4.2 Dexterity Analysis
4.3 Result and Discussion
5 Conclusion
References
Automatic Seed Cum Fertilizer Sowing Machine with Water Dripping on Seeds
1 Introduction
2 Literature Review
3 Proposed Method
4 Conclusion
References
Automatic Drip Irrigation Control System for Paddy Fields in Depleting Water Resource Areas
1 Introduction
2 Proposed System
3 Results and Discussion
References
Investigating the Ambient Thermal Loading Failure of Lead–Acid Battery Based on Thermal Analysis
1 Introduction
2 Experimentation
3 Result and Discussion
3.1 Differential Scanning Calorimetry Analysis
3.2 Thermogravimetric Analysis Result
4 Conclusion
References
Distance Operated Manipulator: A Case Study for Rose Plucking
1 Introduction
2 Literature Review
3 Novelty of the Device
4 General Specifications
5 Design Details and Methodology
5.1 Problem Identification
5.2 Possible Solutions
5.3 Design Considerations
5.4 Selection of Materials
5.5 Dimensions
5.6 Calculation of Forces
5.7 Anatomical Analysis
6 Conclusion
References
Topology Structure Design of Fish-Based Propulsive Mechanisms
1 Introduction
2 Creative Design Procedure
3 Procedure Followed
4 Equations
5 Results
6 Conclusion
References
Impact of SOC Estimation on Primary Frequency Regulation for Sustainable Grid Energy Storage System
1 Introduction
1.1 Introduction
2 State of Charge
2.1 Equivalent Circuit Model
3 State-of-Charge Estimation for PFR
4 Results and Discussion
5 Conclusion and Future Work
References
Kinematic Modelling of UR5 Cobot Using Dual Quaternion Approach
1 Introduction
2 Mathematical Preliminaries
3 Kinematic Modelling of Serial Manipulator
4 Application to UR5 Cobot
5 Results and Discussion
6 Conclusion
References
Design of XY Air Bearing Stage for Ultra-Precision
1 Introduction
2 Literature Review
3 XY Stage Design
3.1 Stiffness Analysis
3.2 Error Budget
3.3 Flexure Design
4 FEA
4.1 FE Model
4.2 Analysis and Results
5 Control
5.1 Results
6 Conclusion
References
Design and Fabrication of a Bio-inspired Soft Robotic Gripper
1 Introduction
2 Design of Gripper
3 Results and Discussion
4 Conclusion
References
Experimental and Simulation Study of Haptically Enabled Robotic Teleoperation for NOTES
1 Introduction
2 Method and Meterial
2.1 The Relation Between Joint Angle and Tip Position of Master Device Phantom Omni
2.2 The Relation Between Joint Angle and Tip Position of the Slave Miniature Robot
3 Experiement
3.1 Finding Joint Angles for the Tip Position of the Miniature Robot
3.2 Simulation Experiment for Haptic Feedback
3.3 Simulation in Haptic Environment to Get the Forces
4 Result and Discussion
4.1 Joint Angle for the Tip Position of Miniature Robot
4.2 Simulation Experiment for Haptic Feedback
5 Conclusion and Future Scope
References
Design of Robust Backstepping Controller for Four-Wheeled Mecanum Mobile Robot
1 Introduction
2 Kinematic Modeling
3 Dynamic Modeling
4 Controller Design
5 Simulation Results
6 Conclusion and Future Scope
References
Dynamic Analysis of a Magnetohydrodynamic Journal Bearing of Circular Cross Section in a Rotating Coordinate Frame
1 Introduction
2 Formulations
2.1 Assumptions
2.2 Non-Dimensionalization and Function Definitions
2.3 System Equations
2.4 Perturbed Equations and Solutions
3 Results and Discussion
References
Synergistic Effect of Pocket and Bionic Texture on the Performance Behaviours of Thrust Pad Bearing
1 Introduction
2 Governing Equations and Computational Procedure
2.1 Film Thickness Expressions
2.2 Reynolds Equation for Computation of Pressure in the Film
3 Results and Discussions
4 Conclusions
References
Analysis of a Soil-Moisture Sensor for Potential Failure Modes and Mass Manufacturing
1 Introduction
2 Reliability Analysis
3 Mass Manufacturing of the Sensor
3.1 Calculations for Feed Rate of the Copper Wire
3.2 Assembly of the Heater Probe
4 Closure
References
Evaluation and Validation of Weld Joint Fatigue in Vibration Using Notch Stress Approach
1 Introduction
2 Notch Stress Approach
3 Details of Analysis and Validation
3.1 Modeling
3.2 Simulation Results
3.3 Test Validation
4 Conclusions
References
Trajectory Control and Force Control of Biomimetic Fingers by Tendon-Based Actuation System Using Bond Graph
1 Introduction
2 Methodology
2.1 Prosthetic Fingers Mechanism
2.2 Lead-Screw and Slider-Nut Mechanism
2.3 String-Tube Mechanism
2.4 Experimental Setup
2.5 Force Sensor Design and Development
2.6 Bond Graph Modelling
2.7 Software Development
3 Results and Discussions
3.1 Results for Trajectory Control
3.2 Results for Force Control
4 Conclusion
References
Design of a Two Degrees of Freedom Actuator for Rehabilitation Robotic Applications
1 Introduction
2 A Single Degree of Freedom Actuator
3 A Two-Degrees-of-Freedom Actuator
4 Model of PAM in the Proposed Actuators
5 Mathematical Modelling of the Proposed Two-Degrees-of-Freedom Actuator
6 Conclusion
References
Kinematics/Dynamics Analysis with ADAMS/MATLAB Co-simulation of a SolidWorks Designed Spatial Robot Arm with Control and Validation of Results
1 Introduction
2 Kinematic Analysis, Dynamic Modelling and Control of the Manipulator
2.1 Kinematic Analysis
2.2 MATLAB/ADAMS Co-simulation
2.3 Dynamics and Control System of the Manipulator
3 Simulation and Results
4 Conclusions
References
Joining Aluminum Open Cell Sponge by Friction Stir Welding
1 Introduction
2 Experimental Procedure
3 Results and Discussion
4 Conclusions
References
Harmonic Response Analysis of Photovoltaic Module Using Finite Element Method
1 Introduction
2 Finite Element Analysis
3 Results
4 Conclusions
References
Development of a Micro-forming System for Micro-extrusion Process of Micro-pin in AZ80 Alloy
1 Introduction
2 Development of Micro-forming System and Micro-extrusion Process
3 Summary
References
Topological Analysis of Epicyclic Gear Trains—Symmetry and Redundancy
1 Introduction
2 Graph Representation of Kinematic Structure of EGTs
3 Determination of Symmetry Properties of EGTs
3.1 Application of Group Theory to Study Structure of EGT Graph
3.2 Topologically Similar Possibilities Owing to Symmetry—Examples
4 Identification of Redundant Links in an EGT Graph—Application of Connectivity Concept in Graph Theory
4.1 Background
4.2 Present Work
5 Conclusions
References
Condition Monitoring and Identification of Misalignment with Initial Unbalance of Flexible Rotor-Bearing System
1 Introduction
2 Mathematical Modelling
2.1 Disc Element
2.2 Shaft Misalignment
3 Result and Discussion
3.1 Modal Characteristics
3.2 Shaft Misalignment
4 Conclusion
References
Effect of Unbalance with Bearing Flexibility on Vibration Phenomenon of Geometrically Nonlinear Rotating Shaft with Ball Bearing
1 Introduction
2 Formulation of Mathematical Model
2.1 Hamilton’s Principle
3 Numerical and Experimental Investigation
3.1 Modal Characteristic
3.2 Unbalance Responses
4 Conclusion
References
Analysis of Parametric Influence on Control of a Two-Link Flexible Manipulator Incorporating a Payload
1 Introduction
2 Dynamic Modeling and Governing Equations
3 Closed Form Equations of Motion
4 Results and Discussions
5 Conclusions
References
An Assistive Chair Using a Series-Elastic Actuator
1 Introduction
2 Series-Elastic Actuator
2.1 Design and Retrofitting on Chair Frame
3 Motor Selection and Governing Equations
3.1 Kinematics
3.2 Dynamics
4 Control System
4.1 Error Calculation
4.2 Transfer Equation Form and System Block Diagram
5 Electronics and Integration
6 Multi-Body Dynamics Simulations
7 Device Operation
8 Closure
References
Natural Control of Virtual Models of Mechanisms Using Leap Motion for Interactive Learning
1 Introduction
2 Simulation of Mechanisms in Autodesk Inventor
2.1 Application Programming Interface (API) of Autodesk Inventor
3 Natural Control of Virtual Mechanism Using Leap Motion
3.1 Forward Kinematics of One DOF Mechanisms
3.2 Inverse Kinematics of Five-Bar Mechanism
4 Conclusions
References
Automated Calibration of Cervical Spine Motion Segment Finite Element Model for Physiological Kinematics
1 Introduction
2 Methods
2.1 Finite Element Model of Cervical Spine Motion Segment
2.2 Optimization Framework for Calibration
3 Results and Discussion
4 Conclusions
References
Identification of Inertial Parameters and Friction Coefficients for One-Link Manipulator
1 Introduction
2 Mathematical Modeling for Dynamic Identification
2.1 Linear-in-Parameter Form for One-Link Manipulator
2.2 Joint Trajectories for Dynamic Identification and Torque Reconstruction
3 Dynamic Identification of One-Link Manipulator
4 Discussion and Conclusions
References
A Task-Based Dimensional Synthesis of an Upper-Limb Exoskeleton: A Hybrid Configuration
1 Introduction and Background
2 Exercise-Based Data Collection
3 Planar Hybrid Configuration
3.1 Double Four-Bar Connected in Series
3.2 Kinematic Modelling and Jacobian Computation
4 Problem Formulation: An Optimization Approach
5 Results and Discussion: Task-Based Dimensional Synthesis
5.1 Case 1: Focus on Ergonomically and Aesthetically Compatible Configuration
5.2 Case 2: Focus on Avoiding Misalignment Issue
6 Conclusions
References
Topology Refinement from Design to Manufacturing Using Image Processing-Based Filtration Techniques
1 Introduction
2 Topology Optimization
2.1 Topology Optimization of Manipulator-Link
3 Methodology
3.1 Filtration Technique
4 Results and Discussion
5 Conclusion
References
Soft Robotic Gripper for Agricultural Harvesting
1 Introduction
2 Soft Actuator with Fiber-Reinforcement
3 Soft Actuator with PneuNets Methodology
4 Conclusions
References
Surface Profile Accuracy of Deployable Mesh Reflectors Based on Focal Offset
1 Introduction
2 Focus Offset Error
3 Closure
References
Design and Control of a Low-Cost EMG-Based Soft Robotic Ankle-Foot Orthosis for Foot Drop Rehabilitation
1 Introduction
2 Rehabilitation Techniques
2.1 Physiotherapy
2.2 Technology-Assisted
3 Existing Technologies for Rehabilitation
3.1 Rigid Actuation
3.2 Soft Actuation
4 Construction and Working
4.1 Design of the Prototype
4.2 Control of the Orthosis
5 Results and Statistical Data
5.1 Pneumatic Artificial Muscles Tests
5.2 EMG Control Tests
6 Conclusions
References
Comparison of PPC and LQR Controller for Stabilization of Cart Pendulum System: Simulation and Real-Time Study
1 Introduction
2 Experimental Setup
3 Modeling of Cart Pendulum System
4 Controller Design
5 Results and Discussions
6 Conclusions
References
Effective Education Using a 2-DOF Five-Bar Mechanism
1 Introduction
2 Kinematics of Five-Bar and Iris Mechanism
3 Prototype I
3.1 Effective Education Using Proposed Mechanism
4 Prototype II
5 Conclusions
References
Renewable Energy System Using Thermoelectric Generator (RESTEC)
1 Introduction
2 Previous works
2.1 CVT Heating
2.2 Thermoelectric Generator
3 RESTEC
3.1 Design
3.2 Implementation
4 Observation
5 Conclusion
References
Bending and Free Vibration Analysis of Exponential Graded FG Plate Using Closed-Form Solution
1 Introduction
2 Mathematical Formulation
2.1 Displacement Field
2.2 Energy Equations
2.3 Constitutive Equations
2.4 Analytical Solution
3 Results and Discussion
4 Conclusions
References
Structural Responses of Geometrically Imperfect Functionally Graded Plates with Microstructural Defects Under Hygrothermal Environment
1 Introduction
2 Mathematical Formulation
2.1 Displacement Field
2.2 Strain–Displacement Relation
2.3 Stress–Strain Relations
3 Finite Element Formulation
3.1 Expression for Characteristics Equation
4 Results and Discussion
4.1 Comparison Study
4.2 Parametric Study
5 Conclusion
References
Simplified Aerodynamic Modeling of a Bird Robot Using the DeNOC Matrices
1 Introduction
2 Mathematical Modeling for Simplified Bird Model
2.1 Kinematic Modeling of Robotic Bird
2.2 Dynamic Modeling of Robotic Bird
2.3 Reduced-Order Equations of Motion
3 Dynamic Simulation of Robotic Bird
4 Discussion and Conclusions
References
Parallel Mechanism-Based Master–Slave Manipulation
1 Introduction
2 Parallel Mechanism-Based Master–Slave Manipulation
2.1 Mechanical Master–Slave Parallel Manipulators
2.2 Forward Kinematics of Parallel Mechanisms
3 Communication Channel
4 Slave Manipulator
5 Conclusion
References
Comparative Stiffness and Damping Analysis for Various Flow Controlling Devices of Hole Entry Worn Hybrid Conical Journal Bearing Under the Variation of Speed
1 Introduction
2 Mathematical Formulations
2.1 Finite Element Formation
2.2 Fluid Film Thickness
2.3 Worn Zone Model
2.4 Restrictor Flow Equation
3 Solution Procedure
4 Results and Discussion
4.1 Variation of Stiffness overline2S11 and overline2S22 Coefficient with Speed
4.2 Variation of Damping Coefficient overline2C11 and overline2C22 with Speed Parameter Ω
5 Conclusion
References
A Comparative Study of Three Methods for the Computation of Determinants of Univariate Polynomial Matrices
1 Introduction
2 Mathematical Preliminaries
2.1 Method 1: Direct Expansion of the Determinant Inside a CAS
2.2 Method 2: Using Newton’s Identities
2.3 Method 3: Evaluation-Interpolation Method with FFT and IFT
3 Numerical Experiments
3.1 Setup of the Numerical Experiments
3.2 Comparative Analysis of the Results, for Accuracy
3.3 Comparative Analysis of the Results, for Computational Speed
4 An Application to Kinematics: FKP of the General SPM
5 Conclusions
References
A Comparative Study of Different Numerical Scanning Strategies for Finding the Safe Working Zone of a 3-DoF Parallel Manipulator
1 Introduction
2 Numerical Scanning Techniques
2.1 Cartesian Scanner
2.2 Polar Scanner
3 Case Study: Application to a Spatial 3-underlineRRS Manipulator
3.1 Setup of the Numerical Experiments
3.2 Details of the Boundary Functions
3.3 Numerical Results
3.4 Comparison of Computation Times
4 Conclusions
References
Motion Control of a Phalange Using Tendon-Based Actuation System: A Bond Graph Approach
1 Introduction
2 Proposed Model
2.1 Muscle Tendon Units and Their Modeling
2.2 Multibond Graph Model
3 Simulation and Results
4 Conclusion
References
Taguchi Optimization for Wear Behaviour of Drum Brake Shoe Interface
1 Introduction
2 Experimental Design
2.1 Materials
2.2 Selection of Process Variables and Design Levels
2.3 Experimental Setup
3 Results and Discussion
4 Conclusions
References
Analysis of a Hydrodynamic Journal Bearing of Circular Cross Section Lubricated by a Magnetomicropolar Fluid
1 Introduction
2 Formulations
2.1 Assumptions
2.2 Non-dimensionalization
2.3 System Equations
2.4 Perturbed Equations and Solutions
3 Results and Discussion
References
Twin-Plate Turbine Using Parallel Four-Bar Mechanisms
1 Introduction
2 Twin-Plate Turbine
3 Mathematical Model
4 Physical Prototype
5 Conclusion
Reference
Intuitive Manipulation of Delta Robot Using Leap Motion
1 Introduction
2 Kinematics of Delta Robot
3 Physical Prototype
4 Integration with Leap Motion
5 Conclusions
References
A Computation Model of Contact Interaction Between the Scaphoid and Its Neighboring Bones Using Bond Graph Approach
1 Introduction
2 Proposed Model
2.1 Bone Geometry
2.2 Cartilage Layer
2.3 Ligament
3 Bond Graph Model
4 Result and Discussion
5 Conclusion
References
Mathematical Model of SMA Spring Actuator in a Miniature Flexible Tube Robot
1 Introduction
2 Mathematical Model
3 Results
4 Conclusion
References
Analysis of Inner Block in a Roller Chain Using Glass Reinforced Composites
1 Introduction
2 Parts of a Chain and the Materials Usually Used
3 Fiberglass
4 Working Conditions
5 Results
5.1 Total Deformation
5.2 Strain
5.3 Stress
6 Conclusion
References
Study on the Effect of Process Parameters on Machinability Performance of AA7050/B4C Metal Matrix Composite on Wire Cut EDM
1 Introduction
2 Experimental Setup
2.1 Design of Experiments
3 Results and Discussions
3.1 Taguchi Methods
3.2 Confirmation Test and Prediction
4 Conclusions
References
Biomechanical Response of Seated Human Body Subjected to Vertical Vibrations Using Coupled Matrix Model
1 Introduction
2 Biomechanical Modelling
3 Parameters Optimization
4 Vehicle Human Integration
5 Conclusion
References
Categorization of the Indian Males’ Foot Data for Age 18–25 Years Based on Plantar Footprints
1 Introduction
2 Method
2.1 Experimental Setup
2.2 Subjects
3 Result
3.1 Foot Types Based on Chippaux-Simark Index (CSI)
3.2 Foot Types Based on Hallux-Valgus Angle (HVA)
4 Conclusion
References
Mechanization of Peppermint Oil Extraction Plant of Rural India
1 Introduction
1.1 Existing Traditional Method for Peppermint Oil Extraction Post-harvesting
1.2 Problems Associated with Existing System of Peppermint Oil Extraction Method
2 Literature Survey
3 Proposed Mechanized Models for Extraction of Peppermint Oil
4 Conclusions
References
Imperfection Sensitivity of Skewed FG Flat Plates Under Dynamic Loading
1 Introduction
2 Theoretical Formulation
2.1 Plate Kinematics
2.2 Finite Element Method
2.3 Skew Boundary Transformation
2.4 Governing Equation
3 Results and Discussion
3.1 Convergence and Validation
3.2 Parametric Study
4 Conclusion
References
A Bond Graph Model for the Estimation of Torque Requirements at the Knee Joint During Sit-to-Stand and Stand-to-Sit Motions
1 Introduction
2 Physical System
3 Bond Graph of the Physical System
3.1 Conditional Rotational Coupling
4 Controller Between Virtual and Actual Sub-models
5 Results
6 Conclusion
References
Detecting Cancerous Cells Using Data Augmentation In Deep Cascaded Networks
1 Introduction
2 Proposed Methodology
2.1 Basic Image Classification
2.2 Data Augmentation and Performing Image Classification
2.3 Dataset
2.4 Convolutional Neural Networks (CNNs)
2.5 Different Layers Of CNNs
3 Experiment
4 Results
5 Future Work
References
Free Vibration Analysis of the Sandwich Curved Panels with the Gradient Metallic Cellular Core
1 Introduction
2 Theoretical Formulations
2.1 Displacement Fields
2.2 Modeling of Cellular Core
3 Results and Discussions
4 Conclusions
References
A Study on Clean Coal Technology in the Indian Context
1 Introduction
2 Clean Coal Technology
2.1 Blast-Free Mining Technology
2.2 Case Study for Comparison of Energy Consumption
2.3 Case Study for Cost Savings
3 Clean Processing Technology
4 Silo
5 Conclusion
References
Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing with Partial and Fully Textured Surface
1 Introduction
2 Methodology
2.1 Fluid Film Thickness
2.2 FEM Formulation
3 Results and Discussion
3.1 Effect of Textures on Load Carrying Capacity and Coefficient of Friction:
3.2 Performance Enhancement Ratio (PER)
References
Big Turbo-Generator Shaft Vibrations Control Using Magnetorheological Fluid Damper
1 Introduction
2 Mathematical Formulation
2.1 Generator Modeling
2.2 Fault Modeling of Loaded Generator
2.3 Dynamic Modeling
2.4 Rotor with MR Fluid Damper Modeling
3 Validation
4 Results and Discussion
5 Conclusions
References
Antagonistic Actuation of Pneumatic Artificial Muscle (PAM) with Chain-Sprocket Mechanism
1 Introduction
2 Design of the Setup
2.1 Features of the Setup
3 Conclusion
References
Deep Neural Network Approach for the Prediction of Journal Bearing Static Performance Characteristics
1 Introduction
2 FEM Analysis of Journal Bearing
3 FEM Solution Procedure
4 Neural Network Model Development
5 Results and Discussion
6 Conclusion
Appendix
References
Kinematics and Foldability Analysis of Bennett Mechanisms and Its Networks
1 Introduction
2 Kinematics of Type 1 Bennett 4R Mechanism
3 Kinematics of Type 2 Bennett 4R Mechanism
4 Coupler Curves
5 Bennett Network
6 Replacement of R (Revolute) by S (Spherical)
7 Results
8 Different Cases of Twist Angles
9 Conclusion
References
Nanofibers for Sustainable Filtration: A Waste to Energy Approach
1 Introduction
2 Methodology
2.1 Materials
2.2 Electrospinning and Membrane Preparation
2.3 Characterization and Testing
2.4 Carbonization
2.5 Electrochemistry
3 Results
3.1 Fiber Morphology
3.2 Filtration Properties
3.3 Reusability Analysis
4 Conclusion
References
Effect of Heat Treatment on Wear Behaviour of Austenitic Stainless Steel
1 Introduction
2 Material
3 Experiment
3.1 Specimen Preparation
3.2 Heat Treatment
3.3 Pin on Disc Wear Test
4 Results and Discussion
4.1 Microstructural Examination:
4.2 Microhardness Measurement
4.3 Dry Sliding Wear Test
4.4 Wear Calculation on the Basis of Wear Scar Diameter
4.5 Wear Mechanism
5 Conclusion
References
Design and Development of Intelligent Moving Machine Using LabVIEW
1 Introduction
1.1 Intelligent Moving Machine
1.2 Motivation
1.3 Steps for Design and Development of Intelligent Machine
1.4 Proposed Work
2 Objective of Paper
2.1 Methodology
2.2 Block Diagram Panel for IMM
3 Results
4 Conclusion
5 Experimental Setup of Intelligent Moving Machine
References
Effect of Poling Orientation in Performance of Piezoelectric Materials
1 Introduction
2 Materials and Methodology
3 Results and Discussion
4 Conclusion
References
Parametric Analysis of Vertical Contact Mode Triboelectric Energy Harvester
1 Introduction
2 Mathematical Modeling
3 Result and Discussion
4 Parametric Study
4.1 Triboelectric Material
4.2 Effect of Dielectric Thickness on the Output Power
4.3 Investigation of Optimum Load Resistance
4.4 Effect of Motion Parameter (Frequency) on Output Power
4.5 Effect of Dielectric Surface Area on Output Power
4.6 Effect of Surface Charge Density on Output Power
5 Conclusion
References
Mathematical Model of Sliding Mode Triboelectric Energy Harvester
1 Introduction
2 Basic Mechanism of Working of Sliding Mode TENG
2.1 Initial Position
2.2 Sliding (X
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Lecture Notes in Mechanical Engineering
Rajeev Kumar Vishal S. Chauhan Mohammad Talha Himanshu Pathak Editors
Machines, Mechanism and Robotics Proceedings of iNaCoMM 2019
Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini, Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany
Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. Topics in the series include: • • • • • • • • • • • • • • • • •
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Rajeev Kumar · Vishal S. Chauhan · Mohammad Talha · Himanshu Pathak Editors
Machines, Mechanism and Robotics Proceedings of iNaCoMM 2019
Editors Rajeev Kumar School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India
Vishal S. Chauhan School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India
Mohammad Talha School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India
Himanshu Pathak School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India
ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-16-0549-9 ISBN 978-981-16-0550-5 (eBook) https://doi.org/10.1007/978-981-16-0550-5 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
Organization
Conference Proceedings 4th International and 19th National Conference on Machines and Mechanisms (5–7 December 2019) Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India.
Chief Guests Dr. (Ms.) Tessy Thomas, Distinguished Scientist and Director General—Aeronautical Systems, DRDO Shri. Sudheer Kumar N., Associate Director, CBPO, ISRO HQ Shri. Sanjay Gulati, Executive Director, Bharat Heavy Electricals Limited, Haridwar
Guests of Honour Shri. Jayant Patil, Whole Time Director (Defence and L&T-NxT), Member of the Board, Larsen & Toubro Limited Dr. S. Guruprasad, Distinguished Scientist and Director General—Production Coordination and Services Interaction (PC and SI) Commodore Mukesh Bhargava (Retd.), Vice President, Member of the Board, L&T Defence Shri. K. B. Batra, General Manager (Engineering and PCRI), BHEL, Haridwar
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Organization
Plenary Speakers and Keynote Speakers Prof. C. Amarnath, IIT Bombay Dr. Guirong Liu, University of Cincinnati Prof. Sambit Bhattacharya, Fayetteville State University Prof. G. K. Anathasuresh, IISc Bangalore Prof. Tarun Kant, IIT Bombay Prof. Subir Kumar Saha, IIT Delhi Prof. Alam Md. Mahbub, Shenzhen Graduate School of Harbin Institute of Technology, China Dr. Madhusudan A. Padmanabhan, Aeronautical Development Agency, Bangalore Dr. Lalit Singh, NPCIL-BARC
iNaCoMM 2019 Committees Patron Prof. Timothy Gonsalves (Director, IIT Mandi)
Advisory Committee C. Amarnath, IIT Bombay Suneel K. Agarwal, University of Delaware, Newark G. K. Ananthasuresh, IISc Bangalore, India S. Bandyopadhyay, IIT Madras Nilotpal Banerjee, NIT Durgapur Ranjan Bhattacharya, IIT Kharagpur Marco Ceccarelli, University of Cassino, Italy Javier Cuadrado, Universidad de La Coruña, Spain I-Ming Chen, NTU, Singapore Anirvan Dasgupta, IIT Kharagpur Santosha K. Dwivedy, IIT Guwahati Peter Eberhard, University of Stuttgart, Germany Ashitava Ghosal, IISc Bangalore, India Amitabha Ghosh, BESU, Shibpur India S. Guruprasad, R&D Estt., DRDO Pune Andres Kecskemethy, University of Duisburg-Essen, Germany Ashok Mallik, BESU, Shibpur, India J-P. Merlet, INRIA, France Rochdi Merzouki, Ecole Polytechnique, Universitaire de Lille, France
Organization
Arun Mishra, McGill University, Canada B. K. Mishra, IIT Roorkee Y. Nakamura, University of Tokyo, President, IFToMM Doina Pisla, Technical University of Cluj-Napoca Bernard Roth, Stanford University, USA Debanik Roy, BRNS, DAE, Mumbai Subir Kumar Saha, IIT Delhi Arun K. Samantaray, IIT Kharagpur Wolfgang Seeman, KIT Germany N. Shimizu, Iwaki Meisei University, Japan Pushparaj Mani Pathak, IIT Roorkee Joseph Anand VAZ, NIT Jalandhar N. S. Vyas, IIT Kanpur Indra Vir Singh, IIT Roorkee Anil Chand Mathur, ISRO Ahmedabad J. K. Agarwal, Bureau of Indian Standards, New Delhi Venkatakrishnan, HQ ISRO, Bangalore
Organizing Committee Chairman: Prof. Satish Chandra Jain, IIT Mandi Convener: Dr. Rahul Vaish, IIT Mandi Co-convener: Dr. Viswanath Balakrishnan, IIT Mandi Organizing Secretary: Dr. Rajeev Kumar, IIT Mandi Joint Organizing Secretary: Dr. Mohammad Talha, IIT Mandi Joint Organizing Secretary: Dr. Himanshu Pathak, IIT Mandi Treasurer: Dr. Vishal S. Chauhan, IIT Mandi Mr. Ahmed Raza, Research Scholar, IIT Mandi Mr. Ajay Kumar, Research Scholar, IIT Mandi Mr. Ashok Kumar Shivratri, Research Scholar, IIT Mandi Mr. Diwakar Singh, Research Scholar, IIT Mandi Mr. Gaurav Arora, Research Scholar, IIT Mandi Mr. Gokul Krishna, Research Scholar, IIT Mandi Mr. Jitendra Adhikari, Research Scholar, IIT Mandi Mr. Kamalpreet Singh, Research Scholar, IIT Mandi Ms. Margi Gajjar, Research Scholar, IIT Mandi Mr. Mohammad Amir, Research Scholar, IIT Mandi Mr. Mohammed Shakir, Research Scholar, IIT Mandi Mr. Nayan Pundhir, Research Scholar, IIT Mandi Mr. Nishant Verma, Research Scholar, IIT Mandi Mr. Prakash Poudel, Research Scholar, IIT Mandi Mr. Rishikant, Research Scholar, IIT Mandi Mr. Satish Kumar, Research Scholar, IIT Mandi
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Mr. Saurav Sharma, Research Scholar, IIT Mandi Ms. Shreya Lanjewar, Student, IIT Mandi Mr. Sumeet K. Sharma, Research Scholar, IIT Mandi Mr. Vikram Singh Chandel, Research Scholar, IIT Mandi
Organizing Institute
Organization
Preface
It is a great privilege for us to present the proceedings of the 4th International and 19th National in the series of biennial conferences on Machines and Mechanisms (iNaCoMM 2019) organized under the aegis of Association for Machines and Mechanisms (AMM) and International Federation for the Promotion of Mechanism and Machine Science (IFToMM) to be held at Indian Institute of technology Mandi, India, during 5–7 December 2019. iNaCoMM 2019 aims to provide a platform for discussing the issues, challenges, opportunities and findings of various aspects of design and analysis of machines and mechanisms by bringing together academicians, researchers, industry experts and students who are working in these domains. This conference has been a good opportunity for participants coming from different parts of the country to present and discuss the practical challenges encountered and the solutions adopted in their respective research areas. The responses to the call-for-papers had been overwhelming. Unfortunately, many papers from reputed institutions could not be accepted due to the reviewing outcomes. We would like to express our acknowledgment and appreciation to the reviewers who helped us in maintaining the high quality of manuscripts by making valuable suggestions for the authors. We would also like to extend our thanks to the members of the organizing team for their hard work. The deliberations in the conference consist of plenary sessions and keynote lectures by the leading experts and oral and poster presentations by the delegates on a wide range of topics in the field of machines and mechanisms. The exuberant programme will result in extensive exchange of information and networking among
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participants on the new technologies, as well as thorough discussions of the trends and driving forces in the field. Mandi, India
Assoc. Prof. Rajeev Kumar [email protected] Assoc. Prof. Vishal S. Chauhan [email protected] Asst. Prof. Mohammad Talha [email protected] Asst. Prof. Himanshu Pathak [email protected]
Contents
A Mechanical Contrivance for Acoustic Levitation and Mixing of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saurabh Yadav and Arpan Gupta
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Design and Validation of Flexure-Based Hinges for Space Deployable Antenna Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hemant Arora, B. S. Munjal, and Sudipto Mukherjee
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Effect of Implant Materials on Bone Remodelling Around Cemented Acetabular Cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ajay Kumar, Rajesh Ghosh, and Rajeev Kumar
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Influence of Ageing and High BMI on Lower Back Pain . . . . . . . . . . . . . . P. Praveen, M. S. Mallikarjunaswamy, and S. Chandrashekhara Design and Analysis of a Robotic Lizard Using Five-Bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. S. Rajashekhar, C. K. Dinakar Raj, S. Vishwesh, E. Selva Perumal, and M. Nirmal Kumar
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Development of an Automated Material Handling System Inside a Nuclear Containment Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anupam Saraswat and P. S. Somayajulu
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Nonlinear Modeling and Stability Analysis of Piezoelectric Energy Harvesting Mechanism Under Aeroelastic Vibration . . . . . . . . . . Rakesha Chandra Dash, Dipak Kumar Maiti, and Bhrigu Nath Singh
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Optimization of Surface Roughness of Laser Trepanned Hole in ZTA Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surendra K. Saini, Avinash K. Dubey, and B. N. Upadhyay
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PI Control-Based Modelling of Segway Using Bond Graph . . . . . . . . . . . A. Kumar, R. Singh, T. K. Bera, and Ashish Singla
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Strategic Coordination and Navigation of Multiple Wheeled Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Buddhadeb Pradhan, Nirmal Baran Hui, and Diptendu Sinha Roy Spur Gear Mechanism for Accurate Angular Indexing and Locking of Angular Position by Using Additive Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arivazhagan Pugalendhi, Rajesh Ranganathan, and C. Vivek
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Fabrication of Solid Lubricant Coating and Its Optimization Using Response Surface Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Tyagi, S. Kumar, A. K. Das, and A. Mandal
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Computing and Verification of IPMC Parameters Through Equivalent Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ankur Gupta and Sujoy Mukherjee
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Characterization of Mechanical Properties of Different Agro-derived Reinforcements Reinforced in Aluminium Alloy (AA6061) Matrix Composite: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arjun J. Deshmukh, Sanjaykumar S. Gawade, and Abhijeet B. Pawar Shoe-Based Energy Harvesting Using Ionic Polymer Metal Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satya Narayan Patel and Sujoy Mukherjee
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IoT-Based Health Monitoring System (IHMS) . . . . . . . . . . . . . . . . . . . . . . . P. Ramya and L. Padmalatha
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Search and Reconnaissance Robot for Disaster Management . . . . . . . . . Sarthak Narayan, Mohd Aquif, Abdur Rahman Kalim, Dharun Chagarlamudi, and M. Harshith Vignesh
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Experimental Validation of Various Existing Impedance Models for Acoustic Liners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ashutosh Tripathi, N. K. Jha, and R. N. Hota Design and Modeling of Pipeline Inspection Robot (PIR) for Underground Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ravi Kant Jain, Abhijit Das, A. Mukherjee, Santosha Goudar, Ankita Mistri, A. Mandal, and Pratap Karmakar
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Mechanism of Material Removal in Magneto Abrasive Flow Machining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Palwinder Singh, Lakhvir Singh, and Sehijpal Singh
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Optimization of Cutting Parameters of EN9 Steel with Plain Carbide Tool Using Response Surface Methodology . . . . . . . . . . . . . . . . . . Sachin Chauhan and Rajeev Kumar
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Design and Development of a Sit-to-Stand Assistive Device . . . . . . . . . . . Shoudho Das, Satyajit Halder, Sourabh Kumar Sahu, Sujatha Srinivasan, and Sourav Rakshit Effect of Structural Characteristics on Kinematics of Planar Kinematic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ashwini Biradar and Shubhashis Sanyal A Combined Experimental/Finite Element Model Analysis on Compressive Behavior of Tamarind Pod Shell Filler Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Santosha Goudar, Ravi Kant Jain, and Debashis Das Workspace Evaluation of Robotino-XT Under Reconfiguration . . . . . . . Vipin Pachouri, Pushparaj Mani Pathak, Mrunal K. Mishra, Arun K. Samantharay, Rochdi Merzouki, and B. O. Bouamama
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Assessment of Surface Water Quality Using Principal Component Analysis in the Yamuna River: A Case Study . . . . . . . . . . . . Bhuri Singh, Shahla Khan, and Shahjaha
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Performance Analysis of Gripper Assembly of an In-Vessel Fuel Handling Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anu Krishnan, R. Vijayashree, Sanjeev Kumar, and S. Raghupathy
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Design and Development of a Remote Racking Mechanism for Switchgear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alex Sherjy Syriac and Shital S. Chiddarwar
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Iwan Model for Bolted Joint with Residual Macroslip Stiffness and Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prabhat Ranjan and Ashok Kumar Pandey
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Kinematics Model of Bionic Manipulator by Using Elliptic Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mrunal K. Mishra, Arun K. Samantaray, Goutam Chakraborty, Vipin Pachouri, Pushparaj Mani Pathak, and Rochdi Merzouki A Review on the Effect of Biomechanical Aspects and the Type of Stability Fixation on the Bone Fracture Healing Process . . . . . . . . . . . Sandeep Rathor and Rashmi Uddanwadiker Mechanical Behaviour of Special Type Seals Used in the FBR Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nihal Kaushal, Sudheer Patri, R. Suresh Kumar, C. Meikandamurthy, B. K. Sreedhar, S. Murugan, and P. Selvaraj
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Realization of a Simple Mechanism to Simulate Core Subassembly Growth of FBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sudheer Patri, Muhammad Sabih, S. Krishnakumar, C. Meikandamurthy, S. Chandramouli, B. Babu, B. K. Sreedhar, B. K. Nashine, S. Murugan, and P. Selvaraj Investigation of Multiple Stable States of Tensegrity Structure . . . . . . . . P. K. Malik, C. Agrawal, Anirban Guha, and P. Seshu Design and Development of a Short-Wave Electric Infrared Heater of 215 kW Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Narendar and T. Srinivasa Kumar
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Kinematic and Dynamic Modeling of a Quadruped Robot . . . . . . . . . . . . Priyaranjan Biswal and Prases K. Mohanty
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Nonlinear Joint Stiffness Parameter Identification . . . . . . . . . . . . . . . . . . . Sanjay B. Ingole and Sanjay W. Rajurkar
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Design and Development of Reaper for Harvesting Maize . . . . . . . . . . . . Manjunath M. Ullegaddi, B. U. Balappa, and N. C. Mahendra Babu
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Transpose Jacobian Control of Flexible Joint Upper Limb Exoskeleton System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jyotindra Narayan, Mohamed Abbas, and Santosha K. Dwivedy
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Defect-Free Synthesis, Analysis and Optimization of Planar Lower Limb Assistive Device for Gait Rehabilitation . . . . . . . . . . . . . . . . . Ramanpreet Singh, Vimal K. Pathak, and Himanshu Chaudhary
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Theoretical and Experimental Investigation of Friction in Hydraulic Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jaidev Vyas, Aruna Rengasamy, L. Surya Narayanan, and Balamurugan Gopalsamy
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Support Vector Classifier-Based Broken Rotor Bar Detection in Squirrel Cage Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prashant Kumar and Ananda Shankar Hati
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Kinematic and Dynamic Analysis of Primary FCS Circuits of Typical 25 Seater Transport Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Manjunath, K Vinod Kumar, M Kiran, and B Rammohan
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Kinematic Synthesis and Optimization of a Double-Slotted Fowler Flap Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Vinod Kumar, Balamurugan Gopalsamy, H. K. Rangavittal, and C. Manjunath Intelligent Modeling of Dilution Percent in Laser Surface Alloying of Alx Cu0.5 FeNiTi High Entropy Alloy . . . . . . . . . . . . . . . . . . . . . A. A. Siddiqui and Avinash K. Dubey
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Spinal Needles Insertion and Traversal Based on Fiber Bragg Gratings—From Conceptual Approach to Prototype Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shikha Ambastha, Sharath Umesh, and Sundarrajan Asokan Kinematic and Dynamic Analysis of Sliding Door Operating Mechanism for Internal Weapon Bay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chandrakant K. Waghmare, Prasad S. Shivdas, Ravi Tiwari, and Sunil V. Nimje A Method to Detect Isomorphism in Planar Kinematic Chains . . . . . . . . Ankur Dwivedi, Anirudha Bhattacharjee, and Jai Narayan Yadav
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Generation of Coupler Curves for Planar Kinematic Chains Using Link Joint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harishankar Singh Yadav and Shubhashis Sanyal
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Tribological Behaviour of Polymer-Based Composite Reinforced with Molybdenum Disulphide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soumya Ranjan Guru, Prabhat Kumar, and Mihir Sarangi
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Prediction of the Blood Flow Through Stenosis in AVF for Hemodialysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suraj Shembekar, D. B. Zodpe, and Pramod M. Padole
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Tool Quality Monitoring in Friction Stir Welding Process . . . . . . . . . . . . Debasish Mishra, Rohan Basu Roy, Surjya K. Pal, and Debashish Chakravarty Condition Monitoring and Fault Diagnosis of Induction Motor in Electric Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Swapnil K. Gundewar and Prasad V. Kane Mechanical Design Calculations of Flywheel Generator . . . . . . . . . . . . . . Md Zafar Anwar, Nilanjan Sen, Jitendra Prasad Khatait, and Sudipto Mukherjee
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Numerical Analysis and Reduction of Blade-Vortex Interaction (BVI) Noise in Helicopter Using Numerical Simulation . . . . . . . . . . . . . . . John Sherjy Syriac and Narayanan Vinod
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Dynamics of Underwater Manipulator: A Recursive Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amarendra Kumar, Virendra Kumar, and Soumen Sen
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Effect of Tool Geometry on Chip Morphology Formed During CNC Turning of a Round Shaped Product . . . . . . . . . . . . . . . . . . . . . . . . . . Koustov Mondol, Goutam Paul, and Asim Gopal Barman
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Optimization of Equal Multi-square Cell Crash Box for Enhanced Energy Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ganesh Prasad, Issac Paul, M. V. Jaathaveda, K. S. Sridhar, and S. Harshitha
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Joint Stiffness Estimation Between Spindle-Tool Holder by Considering Clamping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ramesh H. Aralaguppi, K. B. Siddesh, and Ashok N. Bade
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Buckling Analysis of Nonlinear First-Order Shear Deformation Composite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ashes Maji and Prashanta Kr. Mahato
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Path Tracing and Object Avoidance Algorithm for Robotic Manipulators Incorporating Constrained Filters . . . . . . . . . . . . . . . . . . . . Vipul Garg and Vikas Rastogi
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Characterisation of Composites Made by Prepreg Waste . . . . . . . . . . . . . P. R. Krishna Mohan, Piyush, P. M. Mohite, Kunj Modi, and Dhwani Sharma Experimental Investigation on the Effect of Process Parameters for CNC Turning of UNI Al 3055 Alloy Under MQCL Based Cooling Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subrata Mondal, Goutam Paul, and S. C. Mondal
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Reciprocating Wear Behaviour of Al–SiC Composite Processed with MWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neeraj Kumar Bhoi, Harpreet Singh, and Saurabh Pratap
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Dynamics and Control of a 6-DOF Biped Robot on MATLAB/SimMechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Durbadal Kundu, Alinjar Dan, and Nirmal Baran Hui
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Modal Analysis of 3-RRR SPM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vijaykumar Kulkarni, C. V. Chandrashekara, and D. Sethuram
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Mode Based Crack Identification of Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . Ridha Ali, T. Pooja Priya, V. Rashmi, C. V. Chandrashekara, and Suneel Motru
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A Note on Implementation of Raghavan–Roth Solution for Wrist-Partitioned Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rajesh Kumar, Alinjar Dan, and K. Rama Krishna Experimental Identification of Residual Unbalances for Two-Plane Balancing in a Rigid Rotor System Integrated with AMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gyan Ranjan, Rajiv Tiwari, and Harshal B. Nemade
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Determination of Steering Actuator Mounting Points of a Load Haul Dump Machine for Optimum Performance . . . . . . . . . . . . . . . . . . . . SreeHarsha Rowduru, N. Kumar, and Vinay Partap Singh
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Dynamic Analysis of Helicopter Boom with Different Payload Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Gopikrishna, K. Kishore Kumar, and Y. R. Janarthanan
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Characterization of Composites Made with In-House Prepregs at Different Curing Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piyush Sute, P. R. Krishna Mohan, M. Anil Kumar, P. M. Mohite, and Mahesh A Parametric Approach to Detect Isomorphism and Inversion in the Planar Kinematic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kunal Dewangan and Arvind Kumar Shukla Feasibility of Tensegrity-Based Walking Robot . . . . . . . . . . . . . . . . . . . . . . P. K. Malik, Keshab Patra, and Anirban Guha Design and Development of a Climb-Free Telescopic Mechanism for Harvesting from Tall Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhupinder Singh, Sekar Anup Chander, and Vhatkar Dattatraya Shivling Simulation Modeling of 37 Degrees-of-Freedom ICF Coach . . . . . . . . . . Bharath B. Mahadikar, Charanpreet Singh, Akarsh K. S., and C. V. Chandrashekara
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Transmission Efficiency and Surface Damage of Polymer– Polymer Gear Pair Under Wet Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . Sarita Bharti and Selvaraj Senthilvelan
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Effect of Acceleration of Moving Object During Collision with Stationary Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pranav V. Deosant, H. T. Thorat, and Rupesh N. Tatte
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Finite Element Modelling of the Human Lumbar Vertebrae for Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raj Arjun S. I., Parth Goplani, Pavan Suswaram, and C. V. Chandrashekara Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ankit Nakoriya and Vijay Kumar Gupta Flexible Coupling—A Research Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harshanand P. Ramteke and Girish D. Mehta Design of Post-Curing Inflator Using Bistable Locking Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Md Zafar Anwar, Jitendra Prasad Khatait, and Sudipto Mukherjee
865
871 887
893
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Stability Analysis of a Dual-Rate Haptics Controller Using Discrete-Time Root-Locus Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suhail Ganiny, Majid H. Koul, and Babar Ahmad Design of Compliant Iris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vinay Arora, Prakhar Kumar, Rajesh Kumar, and Jitendra Prasad Khatait Investigation on the Effects of Nose Radius and Rake Surface of Cutting Tool for Machinability During Sustainable Turning of EN 31 Alloy Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sutanu Misra, Goutam Paul, and Asim Gopal Barman
901 911
919
Battery Performance Analysis of Static Temperature Variations for Medical Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Banuselvasaraswathy and R. Vimalathithan
929
Derivation of the Rotation Matrix for an Axis-Angle Rotation Based on an Intuitive Interpretation of the Rotation Matrix . . . . . . . . . . Roshan Kumar Hota and Cheruvu Siva Kumar
939
Resolving Hyper-Redundant Planar Serial Robots to Ensure Grasp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rajesh Kumar and Sudipto Mukherjee
947
Boom Packaging with Yoshimura Pattern: Geometrical and Deformation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hemant Sharma, Omkar Raj, and S. H. Upadhyay
955
Multimodal Medical Image Fusion Based on Interval-Valued Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Tirupal, B. Chandra Mohan, and S. Srinivas Kumar
965
The Influence of Ultrasound for the Protection of Animals on Highways Through Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . T. Tirupal and S. Fowzia Sultana
973
Workspace Analysis of a 5-Axis Parallel Kinematic Machine Tool . . . . . Anshul Jain and H. P. Jawale
981
Reaction Solvability Analysis Using Natural Coordinates . . . . . . . . . . . . . Shivam Sharma and Ashitava Ghosal
991
Design, Analysis and Development of Sweep Arm Scanner for Scanning Fast Breeder Reactor Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 Ashish Kumar, Y. V. Nagaraja Bhat, B. K. Sreedhar, S. I. Sundar Raj, S. Murugan, and P. Selvaraj Kinematics of Three Segment Continuum Robot for Surgical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 Shailesh Bamoriya and Cheruvu Siva Kumar
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Automatic Seed Cum Fertilizer Sowing Machine with Water Dripping on Seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 T. Tirupal and D. Rajasekhar Automatic Drip Irrigation Control System for Paddy Fields in Depleting Water Resource Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 K. Saravanakumar, M. Karthigai Pandian, T. Chinnadurai, and J. Dhanaselvam Investigating the Ambient Thermal Loading Failure of Lead– Acid Battery Based on Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 T. Chinnadurai, B. Banuselvasaraswathy, M. Karthigai Pandian, S. Saravanan, and K. Saravanakumar Distance Operated Manipulator: A Case Study for Rose Plucking . . . . . 1047 Utkarsha K. Mehta, Srushti R. Hippargi, Bhagyesh B. Deshmukh, and Roohshad Mistry Topology Structure Design of Fish-Based Propulsive Mechanisms . . . . . 1055 Gaikwad Pankaj Manik and Pankaj Dorlikar Impact of SOC Estimation on Primary Frequency Regulation for Sustainable Grid Energy Storage System . . . . . . . . . . . . . . . . . . . . . . . . 1067 J. Dhanaselvam and V. Rukkumani Kinematic Modelling of UR5 Cobot Using Dual Quaternion Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Mohsin Dalvi, Shital S. Chiddarwar, M. R. Rahul, and Saumya Ranjan Sahoo Design of XY Air Bearing Stage for Ultra-Precision . . . . . . . . . . . . . . . . . . 1087 Rajesh Kumar and Jitendra Prasad Khatait Design and Fabrication of a Bio-inspired Soft Robotic Gripper . . . . . . . 1105 Ayush Agarwal, Ankit Baranwal, G. Stephen Sugun, and Prabhat K. Agnihotri Experimental and Simulation Study of Haptically Enabled Robotic Teleoperation for NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 Sarvesh Saini and Pushparaj Mani Pathak Design of Robust Backstepping Controller for Four-Wheeled Mecanum Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Zeeshan Ul Islam, Shital S. Chiddarwar, and Saumya Ranjan Sahoo Dynamic Analysis of a Magnetohydrodynamic Journal Bearing of Circular Cross Section in a Rotating Coordinate Frame . . . . . . . . . . . 1135 Debasish Tripathy and Kingshook Bhattacharyya
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Synergistic Effect of Pocket and Bionic Texture on the Performance Behaviours of Thrust Pad Bearing . . . . . . . . . . . . . . . 1143 J. C. Atwal and R. K. Pandey Analysis of a Soil-Moisture Sensor for Potential Failure Modes and Mass Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 Mihir Mogra, Rajesh Aouti, N. S. Rakesh, Alishan Ahmed, R. Ashwin, Jose Joseph, and G. K. Ananthasuresh Evaluation and Validation of Weld Joint Fatigue in Vibration Using Notch Stress Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169 Ashish Patil and Swapnil Bhende Trajectory Control and Force Control of Biomimetic Fingers by Tendon-Based Actuation System Using Bond Graph . . . . . . . . . . . . . . 1177 Vijay Saini, Simran Pal Singh, Neeraj Mishra, and Anand Vaz Design of a Two Degrees of Freedom Actuator for Rehabilitation Robotic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189 Saurav Kumar Dutta, B. Sandeep Reddy, and Santosha Kumar Dwivedy Kinematics/Dynamics Analysis with ADAMS/MATLAB Co-simulation of a SolidWorks Designed Spatial Robot Arm with Control and Validation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 Vikash Kumar Joining Aluminum Open Cell Sponge by Friction Stir Welding . . . . . . . 1211 A. Chandru and S. V. Satish Harmonic Response Analysis of Photovoltaic Module Using Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219 Chaitanya V. Bhore, Atul B. Andhare, Pramod M. Padole, Chinmay R. Chavan, Vishal S. Gawande, V. Surya Prashanth, and R. Balagopal Chary Development of a Micro-forming System for Micro-extrusion Process of Micro-pin in AZ80 Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 D. Rajenthirakumar, N. Srinivasan, and R. Sridhar Topological Analysis of Epicyclic Gear Trains—Symmetry and Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 V. R. Shanmukhasundaram, Y. V. D. Rao, and S. P. Regalla Condition Monitoring and Identification of Misalignment with Initial Unbalance of Flexible Rotor-Bearing System . . . . . . . . . . . . . 1243 Sankalp Singh, Hanmant P. Phadatare, and Barun Pratiher
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Effect of Unbalance with Bearing Flexibility on Vibration Phenomenon of Geometrically Nonlinear Rotating Shaft with Ball Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261 Hanmant P. Phadatare, Sankalp Singh, and Barun Pratiher Analysis of Parametric Influence on Control of a Two-Link Flexible Manipulator Incorporating a Payload . . . . . . . . . . . . . . . . . . . . . . 1277 Pravesh Kumar and Barun Pratiher An Assistive Chair Using a Series-Elastic Actuator . . . . . . . . . . . . . . . . . . 1289 Ankur Kushwaha, Yash Agrawal, Sandeep Khandai, K. V. S. Hari, and G. K. Ananthasuresh Natural Control of Virtual Models of Mechanisms Using Leap Motion for Interactive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303 Sachin Pullil and Rajeevlochana G. Chittawadigi Automated Calibration of Cervical Spine Motion Segment Finite Element Model for Physiological Kinematics . . . . . . . . . . . . . . . . . . 1311 Dhinesh Natarajan, Jobin D. John, and Gurunathan Saravana Kumar Identification of Inertial Parameters and Friction Coefficients for One-Link Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1321 Anil K. Sharma, S. K. Saha, Virendra Kumar, and Soumen Sen A Task-Based Dimensional Synthesis of an Upper-Limb Exoskeleton: A Hybrid Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 Sakshi Gupta, Sameer Gupta, Anupam Agrawal, and Ekta Singla Topology Refinement from Design to Manufacturing Using Image Processing-Based Filtration Techniques . . . . . . . . . . . . . . . . . . . . . . 1337 G. Lakshmi Srinivas and Arshad Javed Soft Robotic Gripper for Agricultural Harvesting . . . . . . . . . . . . . . . . . . . 1347 S. M. G. Vidwath, P. Rohith, R. Dikshithaa, N. Nrusimha Suraj, Rajeevlochana G. Chittawadigi, and Manohar Sambandham Surface Profile Accuracy of Deployable Mesh Reflectors Based on Focal Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355 Shenoy S. Siddesh, R. Harisankar, and G. K. Ananthasuresh Design and Control of a Low-Cost EMG-Based Soft Robotic Ankle-Foot Orthosis for Foot Drop Rehabilitation . . . . . . . . . . . . . . . . . . . 1367 Nitish Gudapati, Koushik Kumaran, S. V. Deepak, R. Mukesh Kanna, R. Jinesh, and Himadri Poddar Comparison of PPC and LQR Controller for Stabilization of Cart Pendulum System: Simulation and Real-Time Study . . . . . . . . . . 1383 Gurminder Singh and Ashish Singla
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Effective Education Using a 2-DOF Five-Bar Mechanism . . . . . . . . . . . . . 1393 Shourie S. Grama, Prithvi Bharadwaj Mellacheruvu, S. Prasanth, V. Prathosh Kumar, S. Vignesh, and Rajeevlochana G. Chittawadigi Renewable Energy System Using Thermoelectric Generator (RESTEC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 Ritwik Dhar, Param Shah, Parth Kansara, and Niti Doshi Bending and Free Vibration Analysis of Exponential Graded FG Plate Using Closed-Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 Dheer Singh, Yogesh Kumar, and Ankit Gupta Structural Responses of Geometrically Imperfect Functionally Graded Plates with Microstructural Defects Under Hygrothermal Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 Ankit Gupta and Mohammad Talha Simplified Aerodynamic Modeling of a Bird Robot Using the DeNOC Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437 Anil K. Sharma, Sasanka S. Sinha, Rajesh Kumar, and S. K. Saha Parallel Mechanism-Based Master–Slave Manipulation . . . . . . . . . . . . . . 1447 Ravinder Kumar, S. K. Sinha, T A. Dwarakanath, and Gaurav Bhutani Comparative Stiffness and Damping Analysis for Various Flow Controlling Devices of Hole Entry Worn Hybrid Conical Journal Bearing Under the Variation of Speed . . . . . . . . . . . . . . . . . . . . . . 1455 Vikas M. Phalle and Sanjay R. Pawar A Comparative Study of Three Methods for the Computation of Determinants of Univariate Polynomial Matrices . . . . . . . . . . . . . . . . . . 1463 V. Safar, Anirban Nag, Bibekananda Patra, and Sandipan Bandyopadhyay A Comparative Study of Different Numerical Scanning Strategies for Finding the Safe Working Zone of a 3-DoF Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1471 Bibekananda Patra, V. Safar, and Sandipan Bandyopadhyay Motion Control of a Phalange Using Tendon-Based Actuation System: A Bond Graph Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479 Sandeep Kumar Uppal and Anand Vaz Taguchi Optimization for Wear Behaviour of Drum Brake Shoe Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 Vaibhav A. Kalhapure and H. P. Khairnar Analysis of a Hydrodynamic Journal Bearing of Circular Cross Section Lubricated by a Magnetomicropolar Fluid . . . . . . . . . . . . . . . . . . 1495 Debasish Tripathy and Kingshook Bhattacharyya
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Twin-Plate Turbine Using Parallel Four-Bar Mechanisms . . . . . . . . . . . . 1503 Bhanu Vardhan Chennoju, Sai Vikas Coca, and Rajeevlochana G. Chittawadigi Intuitive Manipulation of Delta Robot Using Leap Motion . . . . . . . . . . . . 1511 P. Giridharan, Rajeevlochana G. Chittawadigi, and Ganesha Udupa A Computation Model of Contact Interaction Between the Scaphoid and Its Neighboring Bones Using Bond Graph Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519 Arvind Kumar Pathak and Anand Vaz Mathematical Model of SMA Spring Actuator in a Miniature Flexible Tube Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529 Nisha Bhatt, Sanjeev Soni, and Ashish Singla Analysis of Inner Block in a Roller Chain Using Glass Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 V. Gyaneshwar and V. Sivasankaran Study on the Effect of Process Parameters on Machinability Performance of AA7050/B4C Metal Matrix Composite on Wire Cut EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545 Arvind Kumar, Subhan Pandey, Virendra Singh, and Ram Naresh Rai Biomechanical Response of Seated Human Body Subjected to Vertical Vibrations Using Coupled Matrix Model . . . . . . . . . . . . . . . . . 1555 Raj Desai, Anirban Guha, and P. Seshu Categorization of the Indian Males’ Foot Data for Age 18–25 Years Based on Plantar Footprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565 Pawan Mishra, Sachin Kumar Singh, Vinayak Ranjan, Sonu Singh, and Sabyasachi Souguny Mechanization of Peppermint Oil Extraction Plant of Rural India . . . . 1575 Mohd Anas and Abusad Imperfection Sensitivity of Skewed FG Flat Plates Under Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585 Mohammed Shakir and Mohammad Talha A Bond Graph Model for the Estimation of Torque Requirements at the Knee Joint During Sit-to-Stand and Stand-to-Sit Motions . . . . . . 1595 Vivek Soni and Anand Vaz Detecting Cancerous Cells Using Data Augmentation In Deep Cascaded Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605 Akshay Jain, Pallavi Chaturvedi, and Lalita Gupta
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Free Vibration Analysis of the Sandwich Curved Panels with the Gradient Metallic Cellular Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615 Mohammad Amir and Mohammad Talha A Study on Clean Coal Technology in the Indian Context . . . . . . . . . . . . 1623 Swayam Sampurna Panigrahi and Purna Chandra Panigrahi Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing with Partial and Fully Textured Surface . . . . . . 1635 Sanjay Sharma Big Turbo-Generator Shaft Vibrations Control Using Magnetorheological Fluid Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645 Tarun Kumar, Rajeev Kumar, and Satish Chandra Jain Antagonistic Actuation of Pneumatic Artificial Muscle (PAM) with Chain-Sprocket Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659 Bhaben Kalita, Arunjyoti Borgohain, and Santosha K. Dwivedy Deep Neural Network Approach for the Prediction of Journal Bearing Static Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 1669 Sunil Kumar, Vijay Kumar, and Anoop Kumar Singh Kinematics and Foldability Analysis of Bennett Mechanisms and Its Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683 Tony Punnoose Valayil Nanofibers for Sustainable Filtration: A Waste to Energy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693 Prakash Giri, Ashish Kakoria, Sahil Verma, and Sumit Sinha-Ray Effect of Heat Treatment on Wear Behaviour of Austenitic Stainless Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703 Waris Nawaz Khan, Furkan, and Rahul Chhibber Design and Development of Intelligent Moving Machine Using LabVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713 Amit Yadav, Ajeet Gaur, D. K. Chaturvedi, A. K. Saxena, and Dharvendra P. Yadav Effect of Poling Orientation in Performance of Piezoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1721 Jitendra Adhikari, Rajeev Kumar, Vikas Narain, and Satish Chandra Jain Parametric Analysis of Vertical Contact Mode Triboelectric Energy Harvester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733 Satish Kumar, Rajeev Kumar, Vikas Narain, and Satish Chandra Jain Mathematical Model of Sliding Mode Triboelectric Energy Harvester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745 Tarun Pratap Singh, Satish Kumar, and Rajeev Kumar
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Free and Forced Flexural Vibration Responses of the Laminated Composite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755 Niyaz Ahmad, Arshad Hussain Khan, and Mohammad Amir Shape Control of Piezolaminated Structure Using Poling Tuned Piezoelectric Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765 Saurav Sharma, Rajeev Kumar, Mohammad Talha, and Vikas Narain Power Optimization of a Wind Turbine Using Genetic Algorithm . . . . . 1777 Prakash Poudel, Rajeev Kumar, Vikas Narain, and Satish Chandra Jain Automated Design for Cam Profile Using CATIA V5 and Its Fatigue Life Assessment Using ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1791 Krishnajith Theril, M. R. Jithin, Bobby Xavier, Haris Naduthodi, P. A. Abdul Samad, and C. Arun Vibration Control of Smart Cantilever Beam Using Fuzzy Logic Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801 Kamalpreet Singh, Rajeev Kumar, Mohammad Talha, and Vikas Narain Multi-body Analysis for a Four-Bar Mechanism Using RecurDyn and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1813 Naman Chaudhary and Arpan Gupta Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825
Editors and Contributors
About the Editors Dr. Rajeev Kumar is currently Associate Professor in the School of Engineering, Indian Institute of Technology (IIT) Mandi, India. He received his Ph.D. in Machine Design from IIT Roorkee, India, in 2008. Prior to joining IIT Mandi, he served at the General Electric John F. Welch Technology Centre, Bengaluru, as a Technologist. His major research interests include smart material/structure, piezoelectric/triboelectric energy harvesting, finite element method, modelling, control, and optimization (genetic algorithm). He has published more than sixty research articles in international repute journals and more than 40 conference proceedings. Dr. Kumar has successfully guided 5 Ph.D. scholars and 6 M.S. thesis. He has received project funding from SERB-DST, Aeronautical Research Board, Naval Research Board, BHEL Haridwar, Indo Farm Equipment Ltd., and Baddi, Society for Technology and Development, Mandi. He has received National Doctoral Fellowship (NDF) Award-2004 by AICTE, Green Belt Certificate Award-2009 by General Electric (GE), IIT Mandi Foundation Award-2013 for teaching many courses and setting up mechanical workshop, IIT Mandi Foundation Award-2016 for developing a new course “Design Practicum”, and IIT Mandi Foundation Award-2019 for efficient management of the school activities as Chairperson of the School of Engineering and for playing key role in launching new academic programs. Dr. Vishal S. Chauhan is an Associate Professor in the School of Engineering at Indian Institute of Technology (IIT) Mandi, India. His research areas include deformation induced emissions for structural health monitoring, applications of ceramics and composites for deformation monitoring, thermal sensing and energy harvesting, glass ceramics for water treatment. Dr. Chauhan has published 55 papers in international journals, has received 4 externally funded projects, and has supervised 4 Ph.D. and 6 M.S. students. He has given invited talks in several workshops and conferences, and teaches courses related to Graphics for Design, Mechanics of Rigid Bodies, Theory of Machines, Design of Machines Elements, Deformation Behaviour of Materials. xxvii
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Editors and Contributors
Dr. Mohammad Talha is currently an Associate Professor in the School of Engineering, Indian Institute of Technology (IIT) Mandi, India. He received his Ph.D. in Aerospace Engineering from IIT, Kharagpur in 2012, and undergraduate and postgraduate degrees in Mechanical Engineering from Aligarh Muslim University, India. Dr. Talha has received the prestigious National doctoral fellowship from the Government of India for his doctoral degree. He has a passion for research in engineering and applied sciences, which includes computational solid mechanics, mechanics and composites structures, uncertainty quantification in aircraft analysis and design, imperfection sensitivity in composites, experimental and computational biomechanics. Dr. Talha has received project funding from SERB-DST, AR&DB, TBRL, DRDO and SEED grant from IIT Mandi. He has published more than 45 research articles in international journals, and more than 30 conference proceedings in India and abroad. Dr. Talha has successfully guided two Ph.D. scholars and 2 M.S. thesis at IIT Mandi. Dr. Himanshu Pathak is currently an Assistant Professor at Indian Institute of Technology (IIT) Mandi, India. Dr. Pathak has expertise on mesh independent computational methodology (like XFEM and meshfree methods), multi-scale modelling, solid mechanics, fracture and fatigue analyses of composite materials, etc. He has published more than 50 research articles in journals and conference proceedings of national and international repute. He has supervised 4 Ph.D. and 3 M.S. students. Dr. Pathak has given invited talks at several international workshops, conferences, colloquiums, etc., and teaches courses related to mechanical design and robotics.
Contributors Mohamed Abbas Indian Institute of Technology Guwahati, Guwahati, Assam, India P. A. Abdul Samad Production Engineering, Government Engineering College Thrissur, Kerala, India Abusad Integral University, Lucknow, India Jitendra Adhikari School of Engineering, Indian Institute of Technology Mandi, Mandi, India Ayush Agarwal MAdMatLab, Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, India Prabhat K. Agnihotri MAdMatLab, Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, India Anupam Agrawal Indian Institute of Technology Ropar, Rupnagar, Punjab, India
Editors and Contributors
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C. Agrawal Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India Yash Agrawal Department of Mechanical Engineering, IISc Bengaluru, Bengaluru, Karnataka, India Babar Ahmad Mechanical Engineering Department, NIT Srinagar, Hazratbal Srinagar, J&K, India Niyaz Ahmad Mechanical Engineering Department, AMU, Aligarh, Uttar Pradesh, India Alishan Ahmed Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Ridha Ali Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India Shikha Ambastha CSIR-Central Mechanical Engineering Research Institute (CMERI), Durgapur, India; Indian Institute of Science, Bengaluru, India Mohammad Amir School of Engineering, Indian Institute of Technology Mandi, Suran, Himachal Pradesh, India G. K. Ananthasuresh Department of Mechanical Engineering, Indian Institute of Science, Bengaluru, Karnataka, India Mohd Anas Integral University, Lucknow, India Atul B. Andhare Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India M. Anil Kumar Indian Institue of Technology, Kanpur, UP, India Md Zafar Anwar Indian Institute of Technology Delhi, New Delhi, India Rajesh Aouti Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Mohd Aquif National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India Ramesh H. Aralaguppi Bharat Fritz Werner Ltd., Bengaluru, India Hemant Arora Space Applications Centre, ISRO, Ahmedabad, India Vinay Arora Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India C. Arun Production Engineering, Government Engineering College Thrissur, Kerala, India R. Ashwin Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Sundarrajan Asokan Indian Institute of Science, Bengaluru, India
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Editors and Contributors
J. C. Atwal Department of Mechanical Engineering, I.I.T. Delhi, New Delhi, India B. Babu Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India N. C. Mahendra Babu Department of Mechanical Manufacturing Engineering, Ramaiah University of Applied Sciences, Bengaluru, India Ashok N. Bade Bharat Fritz Werner Ltd., Bengaluru, India B. U. Balappa Department of Mechanical Manufacturing Engineering, Ramaiah University of Applied Sciences, Bengaluru, India Shailesh Bamoriya Indian Institute of Technology Kharagpur, Kharagpur, India Sandipan Bandyopadhyay Indian Institute of Technology Madras, Chennai, Tamil Nadu, India B. Banuselvasaraswathy Department of Electronics and Communications Engineering, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India Ankit Baranwal MAdMatLab, Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, India Asim Gopal Barman Department of Mechanical Engineering, National Institute of Technology Patna, Patna, Bihar, India T. K. Bera Thapar Institute of Engineering and Technology, Patiala Punjab, India Sarita Bharti Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India Nisha Bhatt Thapar Institute of Engineering and Technology, Patiala, India Anirudha Bhattacharjee Smart Materials, Structures and Systems Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh, India Kingshook Bhattacharyya Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India Swapnil Bhende Eaton Technologies Pvt. Ltd., Kharadi, Pune, India Neeraj Kumar Bhoi Department of Mechanical Engineering, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India Chaitanya V. Bhore Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India Gaurav Bhutani Division of Remote Handling and Robotics, Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra, India Ashwini Biradar Department of Mechanical Engineering, NIT Raipur, Chhattisgarh, India
Editors and Contributors
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Priyaranjan Biswal Department of Mechanical Engineering, National Institute of Technology, Yupia, Arunachal Pradesh, India Arunjyoti Borgohain Yantrabot Technologies Pvt. Ltd, Guwahati, India B. O. Bouamama Université de Lille, Lille, France Dharun Chagarlamudi National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India Goutam Chakraborty Indian Institute of Technology Kharagpur, Kharagpur, India Debashish Chakravarty Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India Sekar Anup Chander CSIR-CSIO, Chandigarh, India B. Chandra Mohan Department of ECE, BEC, Bapatla, Andhra Pradesh, India S. Chandramouli Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India C. V. Chandrashekara Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India S. Chandrashekhara ChanRe Rheumatology & Immunology Center and Research, Bengaluru, India A. Chandru Department of Mechanical, PESIT-Bangalore South Campus Affiliated to Visvesvaraya Technological University, Belagavi, Bangalore, Karnataka, India R. Balagopal Chary Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India D. K. Chaturvedi Dayalbagh Educational Institute, Dayalbagh, Agra, India; ADRDE-DRDO, Agra, India Pallavi Chaturvedi Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh, India Himanshu Chaudhary Department of Mechanical National Institute of Technology Jaipur, Jaipur, India
Engineering, Malaviya
Naman Chaudhary Indian Institute of Technology Mandi, Mandi, India Sachin Chauhan Department of Mechanical Engineering, SIRDA Institute of Engineering and Technology, Sundernagar, Himachal Pradesh, India Chinmay R. Chavan Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India Bhanu Vardhan Chennoju Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
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Editors and Contributors
Rahul Chhibber Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, India Shital S. Chiddarwar Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India T. Chinnadurai Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India Rajeevlochana G. Chittawadigi Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India Sai Vikas Coca Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India Mohsin Dalvi Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India Alinjar Dan Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India A. K. Das Indian Institute of Technology (ISM), Dhanbad, India Abhijit Das CSIR-CMERI, Durgapur, West Bengal, India Debashis Das CSIR-Central Mechanical Engineering Research Institute (CMERI), Durgapur, West Bengal (WB), India Shoudho Das Indian Institute of Technology, Madras, India Rakesha Chandra Dash Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India S. V. Deepak National Institute of Technology, Tiruchirappalli, Tamil Nadu, India Pranav V. Deosant Visvesvaraya National Institute of Technology, Nagpur, India Raj Desai Department of Mechanical Engineering, IIT Bombay, Mumbai, India Arjun J. Deshmukh Rajarambapu Institute of Technology, Islampur, Maharashtra, India Bhagyesh B. Deshmukh Walchand Institute of Technology, Solapur, India Kunal Dewangan Kalinga University, Naya Raipur, Chhattisgarh, India J. Dhanaselvam Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India Ritwik Dhar Electronics and Telecommunication Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai, India R. Dikshithaa Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India
Editors and Contributors
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C. K. Dinakar Raj Department of Mechanical Engineering, Adhiparasakthi Engineering College, Melmaruvathur, Kanchipuram, Tamil Nadu, India Pankaj Dorlikar Department of Mechanical Engineering, Army Institute of Technology, Pune, India Niti Doshi Electronics and Telecommunication Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai, India Avanish K. Dubey Department of Mechanical Engineering, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, India T A. Dwarakanath Division of Remote Handling and Robotics, Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra, India Ankur Dwivedi Smart Materials, Structures and Systems Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh, India Santosha K. Dwivedy Indian Institute of Technology Guwahati, Guwahati, Assam, India S. Fowzia Sultana Department of ECE, GPCET, Andhra Pradesh, India Furkan Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, India Suhail Ganiny Mechanical Engineering Department, NIT Srinagar, Hazratbal Srinagar, J&K, India Vipul Garg Department of Mechanical Engineering, Delhi Technological University, Delhi, India Ajeet Gaur Dayalbagh Educational Institute, Dayalbagh, Agra, India; ADRDE-DRDO, Agra, India Sanjaykumar S. Gawade Rajarambapu Institute of Technology, Islampur, Maharashtra, India Vishal S. Gawande Department of Mechanical National Institute of Technology, Nagpur, India
Engineering,
Visvesvaraya
Ashitava Ghosal Indian Institute of Science, Bangalore, India Rajesh Ghosh School of Engineering, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh, India Prakash Giri School of Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India P. Giridharan Department of Vidyapeetham, Clappana, India
Mechanical
Engineering,
Amrita
Vishwa
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Editors and Contributors
Balamurugan Gopalsamy Structural Technologies Division, CSIR-National Aerospace Laboratories, Bengaluru, Karnataka, India R. Gopikrishna Defence Research and Development Laboratory, Kanchanbagh, Hyderabad, Telangana, India Parth Goplani Department Bengaluru, India
of
Mechanical
Engineering,
PES
University,
Santosha Goudar CSIR-Central Mechanical Engineering Research Institute (CMERI), Durgapur, West Bengal (WB), India; Academy of Scientific and Innovative Research (AcSIR), Ghaziabad, UP, India Shourie S. Grama Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India Nitish Gudapati National Institute of Technology, Tiruchirappalli, Tamil Nadu, India Anirban Guha Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India Swapnil K. Gundewar Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India Ankit Gupta School of Engineering, Shiv Nadar University, Greater Noida, Uttar Pradesh, India; Mechanical Engineering Department, Shiv Nadar University, Chennai, India Ankur Gupta PDPM IIITDM Jabalpur, Jabalpur, MP, India Arpan Gupta Indian Institute of Technology Mandi, Suran, Himachal Pradesh, India Lalita Gupta Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh, India Sakshi Gupta Indian Institute of Technology Ropar, Rupnagar, Punjab, India Sameer Gupta Indian Institute of Technology Ropar, Rupnagar, Punjab, India Vijay Kumar Gupta Professor, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India Soumya Ranjan Guru Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India V. Gyaneshwar Product Management, TIDC India, Chennai, India Satyajit Halder Indian Institute of Technology, Madras, India K. V. S. Hari Department of Electrical and Communication Engineering, IISc Bengaluru, Bengaluru, Karnataka, India
Editors and Contributors
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R. Harisankar Mechanical Engineering, Indian Institute of Science, Bengaluru, Karnataka, India M. Harshith Vignesh National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India S. Harshitha PES University, Bengaluru, Bengaluru, India Ananda Shankar Hati Indian Institute of Technology (Indian School of Mines), Dhanbad, India Srushti R. Hippargi Walchand Institute of Technology, Solapur, India R. N. Hota Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India Roshan Kumar Hota Indian Institute of Technology, Kharagpur, West Bengal, India Nirmal Baran Hui Department of Mechanical Engineering, National Institute of Technology Durgapur, Durgapur, West Bengal, India Sanjay B. Ingole Government College of Engineering, Chandrapur, Maharashtra, India Zeeshan Ul Islam Visvesvaraya National Institute of Technology, Nagpur, India M. V. Jaathaveda PES University, Bengaluru, Bengaluru, India Akshay Jain Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh, India Anshul Jain Visvesvaraya National Institute of Technology, Nagpur, India Ravi Kant Jain Academy of Scientific and Innovative Research (AcSIR), Ghaziabad, UP, India; CSIR-Central Mechanical Engineering Research Institute (CMERI), Durgapur, West Bengal (WB), India Satish Chandra Jain School of Engineering, Indian Institute of Technology Mandi, Kamand Campus, VPO Kamand, Mandi, HP, India Y. R. Janarthanan Defence Research and Development Laboratory, Kanchanbagh, Hyderabad, Telangana, India Arshad Javed Department of Mechanical Engineering, Birla Institute of Technology and Science-Hyderabad Campus, Secunderabad, Telangana, India H. P. Jawale Visvesvaraya National Institute of Technology, Nagpur, India N. K. Jha Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India R. Jinesh National Institute of Technology, Tiruchirappalli, Tamil Nadu, India
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Editors and Contributors
M. R. Jithin Production Engineering, Government Engineering College Thrissur, Kerala, India Jobin D. John Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India Jose Joseph Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Akarsh K. S. Department of Mechanical Engineering, PES University, Bengaluru, India Ashish Kakoria School of Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India Vaibhav A. Kalhapure Veermata Jijabai Technological Institute, Mumbai, India Abdur Rahman Kalim National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India Bhaben Kalita Indian Institute of Technology Guwahati, Guwahati, India Prasad V. Kane Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India Parth Kansara Information Technology Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai, India Pratap Karmakar CSIR-CMERI, Durgapur, West Bengal, India M. Karthigai Pandian Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India Nihal Kaushal Homi Bhabha National Institute, Mumbai, India H. P. Khairnar Veermata Jijabai Technological Institute, Mumbai, India Arshad Hussain Khan Mechanical Engineering Department, AMU, Aligarh, Uttar Pradesh, India Shahla Khan Department of Chemistry, Al-Falah University Dhauj, Faridabad, Haryana, India Waris Nawaz Khan Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, India Sandeep Khandai Department of Electrical and Communication Engineering, IISc Bengaluru, Bengaluru, Karnataka, India Jitendra Prasad Khatait Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India M Kiran Department of PG Studies, PES University, Bengaluru, Karnataka, India K. Kishore Kumar Defence Research and Development Laboratory, Kanchanbagh, Hyderabad, Telangana, India
Editors and Contributors
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Majid H. Koul Mechanical Engineering Department, NIT Srinagar, Hazratbal Srinagar, J&K, India P. R. Krishna Mohan Department of Aeronautical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, Kanpur, UP, India S. Krishnakumar Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Anu Krishnan Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Vijaykumar Kulkarni Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India A. Kumar Thapar Institute of Engineering and Technology, Patiala Punjab, India Ajay Kumar School of Engineering, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh, India Amarendra Kumar Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India Arvind Kumar NIT Agartala, Tripura, India Ashish Kumar Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Cheruvu Siva Kumar Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India N. Kumar Mining Machinery Department, IIT(ISM) Dhanbad, Dhanbad, Jharkhand, India Prabhat Kumar Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India Prakhar Kumar Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India Prashant Kumar Indian Institute of Technology (Indian School of Mines), Dhanbad, India Pravesh Kumar Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan, India Rajeev Kumar School of Engineering, Indian Institute of Technology Mandi, Kamand Campus, VPO Kamand, Suran, Mandi, Himachal Pradesh, India; Department of Mechanical Engineering, SIRDA Institute of Engineering and Technology, Sundernagar, Himachal Pradesh, India Rajesh Kumar Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India
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Editors and Contributors
Ravinder Kumar Division of Remote Handling and Robotics, Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra, India S. Kumar Indian Institute of Technology (ISM), Dhanbad, India Sanjeev Kumar Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Satish Kumar School of Engineering, Department of Mechanical Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India Sunil Kumar Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India Tarun Kumar Indian Institute of Technology Mandi, Kamand Campus, VPO Kamand, Mandi, HP, India Vijay Kumar Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India Vikash Kumar Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, Punjab, India Virendra Kumar CSIR-Central Mechanical Engineering Research Institute, Durgapur, West Bengal, India Yogesh Kumar School of Engineering, Shiv Nadar University, Greater Noida, Uttar Pradesh, India Koushik Kumaran National Institute of Technology, Tiruchirappalli, Tamil Nadu, India Durbadal Kundu Ashok Leyland, Kanpur, Uttar Pradesh, India Ankur Kushwaha Department of Electrical and Communication Engineering, IISc Bengaluru, Bengaluru, Karnataka, India G. Lakshmi Srinivas Department of Mechanical Engineering, Birla Institute of Technology and Science-Hyderabad Campus, Secunderabad, Telangana, India Bharath B. Mahadikar Department of Mechanical Engineering, PES University, Bengaluru, India Prashanta Kr. Mahato Indian Institute of Technology (ISM), Dhanbad, India Mahesh Punjab Technical University, Chandigarh, India Dipak Kumar Maiti Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India Ashes Maji Asansol Engineering College, Asansol, India P. K. Malik Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India
Editors and Contributors
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M. S. Mallikarjunaswamy Sri Jayachamarajendra College of Engineering, JSS Science & Technology University, Mysuru, India A. Mandal Indian Institute of Technology (ISM), Dhanbad, India; CSIR-CMERI, Durgapur, West Bengal, India Gaikwad Pankaj Manik Department of Mechanical Engineering, Army Institute of Technology, Pune, India C. Manjunath Department of PG Studies, PES University, Bengaluru, Karnataka, India Girish D. Mehta Priyadarshini College of Engineering, Hingna Road MIDC, Nagpur, India Utkarsha K. Mehta Walchand Institute of Technology, Solapur, India C. Meikandamurthy Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Prithvi Bharadwaj Mellacheruvu Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India Rochdi Merzouki Université de Lille, Lille, France Debasish Mishra Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India Mrunal K. Mishra Indian Institute of Technology Kharagpur, Kharagpur, India Neeraj Mishra Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab, India Pawan Mishra Indian Institute of Technology (IIT-ISM), Dhanbad, India Sutanu Misra Department of Mechanical Engineering, National Institute of Technology Patna, Patna, India; Department of Mechanical Engineering, University of Engineering and Management Kolkata, Kolkata, India Ankita Mistri CSIR-CMERI, Durgapur, West Bengal, India Roohshad Mistry Walchand Institute of Technology, Solapur, India Kunj Modi Department of Aerospace Engineering, Chandigarh University, Punjab, India Mihir Mogra Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Prases K. Mohanty Department of Mechanical Engineering, National Institute of Technology, Yupia, Arunachal Pradesh, India P. M. Mohite Department of Aeronautical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India
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Editors and Contributors
S. C. Mondal Indian Institute of Engineering Science and Technology (IIEST), Shibpur, Howrah, India Subrata Mondal University of Engineering and Management Kolkata, Kolkata, India Koustov Mondol Department of Mechanical Engineering, National Institute of Technology Patna, Patna, Bihar, India; Department of Mechanical Engineering, University of Engineering & Management Kolkata, Kolkata, West Bengal, India Suneel Motru Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India R. Mukesh Kanna National Institute of Technology, Tiruchirappalli, Tamil Nadu, India A. Mukherjee AcSIR, CSIR-CMERI, Durgapur, West Bengal, India Sudipto Mukherjee Department of Mechanical Engineering, Indian Institute of Technology Delhi, Delhi, New Delhi, India Sujoy Mukherjee Design and Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur, India B. S. Munjal Space Applications Centre, ISRO, Ahmedabad, India S. Murugan Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Haris Naduthodi Production Engineering, Government Engineering College Thrissur, Kerala, India Anirban Nag Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Y. V. Nagaraja Bhat Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Ankit Nakoriya Mechatronics Laboratory, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India Vikas Narain Shri Bhawani Niketan Institute of Technology and Management, Jaipur, India Jyotindra Narayan Indian Institute of Technology Guwahati, Guwahati, Assam, India Sarthak Narayan National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India L. Surya Narayanan CSIR-National Karnataka, India
Aerospace
Laboratories,
Bengaluru,
Editors and Contributors
xli
S. Narendar Defence Research and Development Laboratory, Hyderabad, Telangana State, India B. K. Nashine Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Dhinesh Natarajan Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India Harshal B. Nemade Department of Electronics and Electrical Engineering, IIT Guwahati, Guwahati, Assam, India Sunil V. Nimje DIAT (DU), Pune, Maharashtra, India M. Nirmal Kumar Department of Mechanical Engineering, Adhiparasakthi Engineering College, Melmaruvathur, Kanchipuram, Tamil Nadu, India N. Nrusimha Suraj Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India Vipin Pachouri Indian Institute of Technology Roorkee, Roorkee, India L. Padmalatha Gudlavalleru Engineering College, Gudlavalleru, India Pramod M. Padole Department of Mechanical National Institute of Technology, Nagpur, India
Engineering,
Visvesvaraya
Surjya K. Pal Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India Ashok Kumar Pandey Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, India R. K. Pandey Department of Mechanical Engineering, I.I.T. Delhi, New Delhi, India Subhan Pandey NIT Agartala, Tripura, India Purna Chandra Panigrahi Mahanadi Coalfields Limited, A Subsidiary of Coal India Limited, Sambalpur, India Swayam Sampurna Panigrahi Operations Management, International Management Institute, Bhubaneswar, India Vinay Partap Singh Mining Dhanbad, Jharkhand, India
Machinery
Department,
IIT(ISM)
Dhanbad,
Satya Narayan Patel Design and Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur, India Arvind Kumar Pathak Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology, Jalandhar, Punjab, India
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Editors and Contributors
Pushparaj Mani Pathak Mechanical and Industrial Engineering Department, Indian Institute of Technology, Roorkee, Uttarakhand, India Vimal K. Pathak Department of Mechanical Engineering, Manipal University Jaipur, Jaipur, India Ashish Patil Eaton Technologies Pvt. Ltd., Kharadi, Pune, India Bibekananda Patra Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Keshab Patra Mechanical Engineering Department, Indian Institute of Technology Bombay, Bombay, India Sudheer Patri Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Goutam Paul Department of Mechanical Engineering, University of Engineering and Management Kolkata, Kolkata, West Bengal, India Issac Paul PES University, Bengaluru, Bengaluru, India Abhijeet B. Pawar Rajarambapu Institute of Technology, Islampur, Maharashtra, India Sanjay R. Pawar VJTI, Mumbai, India Hanmant P. Phadatare Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur, India Vikas M. Phalle VJTI, Mumbai, India Piyush Department of Aeronautical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, Kanpur, India Himadri Poddar National Institute of Technology, Tiruchirappalli, Tamil Nadu, India T. Pooja Priya Department Bengaluru, Karnataka, India
of
Mechanical
Engineering,
PES
University,
Prakash Poudel School of Engineering, Indian Institute of Technology Mandi, Suran, India Buddhadeb Pradhan National Institute of Technology Durgapur, Durgapur, India Ganesh Prasad PES University, Bengaluru, Bengaluru, India S. Prasanth Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India V. Surya Prashanth Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India
Editors and Contributors
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Saurabh Pratap Department of Mechanical Engineering, Indian Institute of Technology Varanasi (IIT-BHU), Varanasi, UP, India V. Prathosh Kumar Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India Barun Pratiher Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan, India P. Praveen Dept. of E&IE, JSS Academy of Technical Education, Bengaluru, India; Sri Jayachamarajendra College of Engineering, JSS Science & Technology University, Mysuru, India Arivazhagan Pugalendhi Department of Mechanical Engineering, Coimbatore Institute of Technology, Coimbatore, TN, India Sachin Pullil Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India Tony Punnoose Valayil Coimbatore Institute of Technology, Coimbatore, India S. Raghupathy Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India M. R. Rahul Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India Ram Naresh Rai NIT Agartala, Tripura, India Omkar Raj Smart Materials and Structures Laboratory, MIED, IIT Roorkee, Roorkee, Uttarakhand, India D. Rajasekhar Department of ECE, GPCET, Kurnool, Andhra Pradesh, India V. S. Rajashekhar Tentacles Robotic Foundation, Kanchipuram, Tamil Nadu, India D. Rajenthirakumar Department of Mechanical Engineering, PSG College of Technology, Coimbatore, Tamil Nadu, India Sanjay W. Rajurkar Government College of Engineering, Chandrapur, Maharashtra, India N. S. Rakesh Indian Institute of Science, Bengaluru, Bengaluru, Karnataka, India Sourav Rakshit Indian Institute of Technology, Madras, India K. Rama Krishna Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India B Rammohan Department of PG Studies, PES University, Bengaluru, Karnataka, India Harshanand P. Ramteke Priyadarshini College of Engineering, Hingna Road MIDC, Nagpur, India
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Editors and Contributors
P. Ramya Gudlavalleru Engineering College, Gudlavalleru, India Rajesh Ranganathan Department of Mechanical Engineering, Coimbatore Institute of Technology, Coimbatore, TN, India H. K. Rangavittal Department of Mechanical Engineering, BMS College of Engineering, Bengaluru, India Gyan Ranjan Department of Mechanical Engineering, IIT Guwahati, Guwahati, Assam, India Prabhat Ranjan Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, India Vinayak Ranjan Bennett University, Greater Noida, India Y. V. D. Rao Department of Mechanical Engineering, BITS PILANI, Hyderabad Campus, Secunderabad, Telangana, India V. Rashmi Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India Vikas Rastogi Department of Mechanical Engineering, Delhi Technological University, Delhi, India Sandeep Rathor Visvesvaraya National Institute of Technology, Nagpur, India S. P. Regalla Department of Mechanical Engineering, BITS PILANI, Hyderabad Campus, Secunderabad, Telangana, India Aruna Rengasamy CSIR-National Karnataka, India
Aerospace
Laboratories,
Bengaluru,
P. Rohith Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India SreeHarsha Rowduru Mining Machinery Department, IIT(ISM) Dhanbad, Dhanbad, Jharkhand, India Diptendu Sinha Roy National Institute of Technology Meghalaya, Shillong, India Rohan Basu Roy Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India V. Rukkumani Sri Ramakrishna Engineering College, Coimbatore, Tamil Nadu, India Raj Arjun S. I. Department of Mechanical Engineering, PES University, Bengaluru, India Muhammad Sabih Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India V. Safar Indian Institute of Technology Madras, Chennai, Tamil Nadu, India
Editors and Contributors
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S. K. Saha Indian Institute of Technology Delhi, New Delhi, Delhi, India Saumya Ranjan Sahoo Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India Sourabh Kumar Sahu Indian Institute of Technology, Madras, India Sarvesh Saini Mechanical and Industrial Engineering Department, Indian Institute of Technology, Roorkee, Uttarakhand, India Surendra K. Saini Department of Mechanical Engineering, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, India Vijay Saini Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab, India Arun K. Samantharay Indian Institute of Technology Kharagpur, Kharagpur, India Manohar Sambandham Green Robot Machinery Private Limited, Bengaluru, India B. Sandeep Reddy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India Santosha Kumar Dwivedy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India Shubhashis Sanyal Department of Mechanical Engineering, NIT Raipur, Chhattisgarh, India Mihir Sarangi Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India Anupam Saraswat Bhabha Atomic Research Centre, Mumbai, Maharashtra, India Gurunathan Saravana Kumar Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India K. Saravanakumar Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India S. Saravanan Department of EEE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India S. V. Satish Department of Mechanical, PESIT-Bangalore South Campus Affiliated to Visvesvaraya Technological University, Belagavi, Bangalore, Karnataka, India Saurav Kumar Dutta Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India A. K. Saxena Dayalbagh Educational Institute, Dayalbagh, Agra, India; ADRDE-DRDO, Agra, India
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Editors and Contributors
E. Selva Perumal Department of Mechanical Engineering, Adhiparasakthi Engineering College, Melmaruvathur, Kanchipuram, Tamil Nadu, India P. Selvaraj Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Nilanjan Sen Indian Institute of Technology Delhi, New Delhi, India Soumen Sen CSIR-Central Mechanical Engineering Research Institute, Durgapur, West Bengal, India Selvaraj Senthilvelan Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India P. Seshu Department of Mechanical Engineering, Indian Institute of Technology Dharwad, Dharwad, India D. Sethuram Department of Mechanical Engineering, PES University, Bengaluru, Karnataka, India Param Shah Mechanical Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai, India Shahjaha Department of Chemistry, Al-Falah University Dhauj, Faridabad, Haryana, India Mohammed Shakir Indian Institute of Technology, Mandi, India V. R. Shanmukhasundaram Madanapalle Institute of Technology & Science, Madanapalle, Andhra Pradesh, India Anil K. Sharma Indian Institute of Technology Delhi, New Delhi, Delhi, India Dhwani Sharma Department of Aerospace Engineering, Chandigarh University, Punjab, India Hemant Sharma Smart Materials and Structures Laboratory, MIED, IIT Roorkee, Roorkee, Uttarakhand, India Sanjay Sharma School of Mechanical Engineering, Shri Mata Vasihno Devi University, Katra, India Saurav Sharma Indian Institute of Technology, Mandi, Suran, India Shivam Sharma Indian Institute of Science, Bangalore, India Suraj Shembekar VNIT, Nagpur, Maharashtra, India Prasad S. Shivdas DRDO R&DE (E), Pune, Maharashtra, India Vhatkar Dattatraya Shivling CSIR-CSIO, Chandigarh, India Arvind Kumar Shukla Lakshmi Narain College of Technology, Bhopal, Madhya Pradesh, India
Editors and Contributors
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K. B. Siddesh Bharat Fritz Werner Ltd., Bengaluru, India Shenoy S. Siddesh Mechanical Bengaluru, Karnataka, India
Engineering,
Indian
Institute
of
Science,
A. A. Siddiqui Mechanical Engineering Department, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, India Anoop Kumar Singh Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India Bhrigu Nath Singh Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India Bhupinder Singh CSIR-CSIO, Chandigarh, India Bhuri Singh Department of Chemistry, Al-Falah University Dhauj, Faridabad, Haryana, India Charanpreet Singh Department of Mechanical Engineering, PES University, Bengaluru, India Dheer Singh School of Engineering, Shiv Nadar University, Greater Noida, Uttar Pradesh, India Gurminder Singh School of Mechanical and Materials Engineering, University College Dublin, Dublin, Ireland Harpreet Singh Department of Mechanical Engineering, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India Kamalpreet Singh Indian Institute of Technology Mandi, Suran, Himachal Pradesh, India Lakhvir Singh BSB Engineering College, Fatehgarh Sahib, Punjab, India Palwinder Singh IKGPTU, Jalandhar, Punjab, India; BSB Engineering College, Fatehgarh Sahib, Punjab, India R. Singh Thapar Institute of Engineering and Technology, Patiala Punjab, India Ramanpreet Singh Department of Mechanical Engineering, Manipal University Jaipur, Jaipur, India Sachin Kumar Singh Indian Institute of Technology (IIT-ISM), Dhanbad, India Sankalp Singh Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur, India Sehijpal Singh GND Engineering College, Ludhiana, Punjab, India Simran Pal Singh Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab, India Sonu Singh Army Hospital, R&R, New Delhi, India
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Editors and Contributors
Tarun Pratap Singh Department of Mechanical Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India Virendra Singh 4I Lab, IIT Kanpur, Kanpur, India Ashish Singla Mechanical Engineering Department, Thapar Institute of Engineering and Technology, Patiala, Punjab, India Ekta Singla Indian Institute of Technology Ropar, Rupnagar, Punjab, India Sumit Sinha-Ray School of Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India S. K. Sinha Division of Remote Handling and Robotics, Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra, India Sasanka S. Sinha Indian Institute of Technology Delhi, New Delhi, Delhi, India V. Sivasankaran Product Management, TIDC India, Chennai, India P. S. Somayajulu Bhabha Atomic Research Centre, Mumbai, Maharashtra, India Sanjeev Soni Central Scientific Instruments Organization (CSIR-CSIO), Chandigarh, India Vivek Soni Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab, India Sabyasachi Souguny Bennett University, Greater Noida, India B. K. Sreedhar Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India K. S. Sridhar PES University, Bengaluru, Bengaluru, India R. Sridhar Department of Mechanical Engineering, PSG College of Technology, Coimbatore, Tamil Nadu, India S. Srinivas Kumar Department of ECE, JNTUA, Ananthapuramu, Andhra Pradesh, India T. Srinivasa Kumar Defence Research and Development Laboratory, Hyderabad, Telangana State, India N. Srinivasan Department of Mechanical Engineering, PSG College of Technology, Coimbatore, Tamil Nadu, India Sujatha Srinivasan Indian Institute of Technology, Madras, India G. Stephen Sugun MAdMatLab, Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, India S. I. Sundar Raj Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India
Editors and Contributors
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R. Suresh Kumar Homi Bhabha National Institute, Mumbai, India; Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Pavan Suswaram Department of Mechanical Engineering, PES University, Bengaluru, India Piyush Sute Indian Institue of Technology, Kanpur, UP, India Alex Sherjy Syriac Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India John Sherjy Syriac Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India Mohammad Talha School of Engineering, Indian Institute of Technology Mandi, Suran, Himachal Pradesh, India Rupesh N. Tatte Visvesvaraya National Institute of Technology, Nagpur, India Krishnajith Theril Production Engineering, Government Engineering College Thrissur, Kerala, India H. T. Thorat Visvesvaraya National Institute of Technology, Nagpur, India T. Tirupal Department of ECE, GPCET, Kurnool, Andhra Pradesh, India Rajiv Tiwari Department of Mechanical Engineering, IIT Guwahati, Guwahati, Assam, India Ravi Tiwari DRDO R&DE (E), Pune, Maharashtra, India Ashutosh Tripathi Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India Debasish Tripathy Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India R. Tyagi Indian Institute of Technology (ISM), Dhanbad, India Rashmi Uddanwadiker Visvesvaraya National Institute of Technology, Nagpur, India Ganesha Udupa Department of Mechanical Engineering, Amrita Vishwa Vidyapeetham, Clappana, India Manjunath M. Ullegaddi Department of Mechanical Manufacturing Engineering, Ramaiah University of Applied Sciences, Bengaluru, India Sharath Umesh ISRO-Laboratory for Electro-Optics Systems (LEOS), Bengaluru, India; Indian Institute of Science, Bengaluru, India B. N. Upadhyay Laser Development and Industrial Applications Division Lab, Raja Ramanna Centre for Advanced Technology, Indore, Madhya Pradesh, India
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Editors and Contributors
S. H. Upadhyay Smart Materials and Structures Laboratory, MIED, IIT Roorkee, Roorkee, Uttarakhand, India Sandeep Kumar Uppal Dr. B. R. Ambedkar NIT, Jalandhar, Punjab, India Anand Vaz Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab, India Sahil Verma School of Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India S. M. G. Vidwath Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India S. Vignesh Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India R. Vijayashree Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India R. Vimalathithan Department of Electronics and Communications Engineering, Karpagam College of Engineering, Coimbatore, India K. Vinod Kumar Structural Technologies Division, CSIR-NAL, Bengaluru, Karnataka, India Narayanan Vinod Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India S. Vishwesh Department of Mechanical Engineering, Adhiparasakthi Engineering College, Melmaruvathur, Kanchipuram, Tamil Nadu, India C. Vivek Department of Mechanical Engineering, Coimbatore Institute of Technology, Coimbatore, TN, India Jaidev Vyas CSIR-National Aerospace Laboratories, Bengaluru, Karnataka, India Chandrakant K. Waghmare DRDO R&DE (E), Pune, Maharashtra, India Bobby Xavier Production Engineering, Government Engineering College Thrissur, Kerala, India Amit Yadav Dayalbagh Educational Institute, Dayalbagh, Agra, India; ADRDE-DRDO, Agra, India Dharvendra P. Yadav Dayalbagh Educational Institute, Dayalbagh, Agra, India; ADRDE-DRDO, Agra, India; ISTRAC, ISRO, Bengaluru, India Harishankar Singh Yadav Department of Mechanical Engineering, NIT, Raipur, India Jai Narayan Yadav National Institute of Technology, Jamshedpur, Jharkhand, India
Editors and Contributors
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Saurabh Yadav Indian Institute of Technology Mandi, Suran, Himachal Pradesh, India D. B. Zodpe VNIT, Nagpur, Maharashtra, India
A Mechanical Contrivance for Acoustic Levitation and Mixing of Particles Saurabh Yadav
and Arpan Gupta
Abstract Standing wave acoustic levitation is an interesting technique to levitate small objects using sound. One of the common ways is to create a pressure node, or focal point, using standing waves, at which particles can levitate. In this study, the standing wave acoustic levitation phenomenon is studied numerically and validated with the experiments. Further, a mechanical design with five transducers having revolute joints is demonstrated numerically to create a focal point and hence levitate objects. By providing rotation to these transducers, two standing waves are created which are moved to merge two pressure nodes and hence mix two particles. Keywords Acoustic levitation · Finite element method · Mixing of particles
1 Introduction Acoustic levitation is a technique in which particles/objects can be levitated freely in the medium without any physical contact. It uses the sound waves to levitate the objects. Standing wave acoustic levitation consists of a driver and a reflector. There is a standing wave formed between the driver and the reflector. Particles can be levitated stably at the pressure nodes of the standing wave. Apart from levitating the particles, acoustic levitation technique can also be used to calculate the speed of sound, density and other properties of different materials in liquid state at normal temperature as well supercooled condition [1–5]. Standing wave acoustic levitation system has been used to study the melting and solidification of commercial grade succirsonitrile while levitating in the air [6]. Standing wave acoustic levitation method has been used to successfully levitate the materials in solid state as well as in liquid state having highest Supported by SERB (Science and Engineering Research Board) Through the Project YSS/2015/001245 S. Yadav (B) · A. Gupta Indian Institute of Technology Mandi, Suran, Himachal Pradesh 175005, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_1
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density on the earth [7]. Different studies have been performed to maximize the levitating force of the standing wave acoustic levitation [8, 9]. Polystyrene particles have been translated in the water medium along the axial direction of the standing wave acoustic levitation system [10]. Different studies have been performed in which the particles have been manipulated by controlling the phase of the ultrasonic sound sources [11, 12]. In this study, the levitation of polystyrene balls near the pressure nodes of the standing wave is demonstrated. Further, a one-dimensional array of five transducers connected through the revolute joints is used to create the focal point. Focal point of the array can be varied by providing the rotation to the transducers. By gradually varying the focal point of the transducer array, the mixing of the particles has been studied numerically.
2 Numerical Modeling 2.1 Geometry of Standing Wave Acoustic Levitation System For the numerical study of the standing wave acoustic levitation, 2D model of COMSOL Multiphysics software is used. Standing wave acoustic levitation system consists of a driver and a reflector. Geometry of the simple acoustic levitation system with one ultrasonic tweezer/driver is shown in Fig. 1a. The diameter of the tweezer casing (outer sides highlighted in blue) is 16 mm and height is 12 mm. Inside the tweezer casing, there is a vibrating part, of diameter 10 mm and height 7 mm, which produces the sound. The material of the driver is taken as aluminum. The medium around the tweezer is air (highlighted in gray) of width 20 mm. Further, the standing wave acoustic levitation system with one-dimensional array of five acoustic transducers is studied numerically. Figure 1b shows the geometry of the levitation system with five acoustic transducers arranged on a flat surface or having focal point at infinity.
Fig. 1 Geometry of standing wave acoustic levitation system with a One acoustic tweezer and b One-dimensional array of five acoustic transducers
A Mechanical Contrivance for Acoustic Levitation and Mixing of Particles?
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2.2 Procedure In standing wave acoustic levitation system, the distance between the driver and the reflector should satisfy the condition of resonance. In Fig. 1a, the upper boundary of the air domain is provided with the sound hard boundary condition and hence works as a reflector. Plane wave radiation condition is provided at the side boundaries of the air domain. Sound hard boundary condition and plane wave radiation condition can be represented by the following expressions respectively 1 − n − (∇ pt − q) = 0. ρc
(1)
1 k i − n − (∇ pt − q) + i p + T p = Q i . ρc ρc 2kρc
(2)
where k is the wave number, ρc is the density of the medium, q is the dipole source, Q i is the monopole source of incoming pressure field and T at a given point on the boundary denotes the Laplace operator in the tangent plane at that particular point. For the excitation frequency of 40 kHz, distance between the driver and the reflector for third resonance mode is calculated numerically. Numerical result is validated with the experimental result. Further, another numerical model with one-dimensional array of five acoustic transducers is developed. The intersection point of axes of all the transducers is considered as the focal point. The focal point of the transducer array is varied by providing the rotation to the transducers. The effect of varying the focal point of the transducers array is studied.
2.3 Finite Element Method and Convergence Study In this study, finite element approach is used to simulate the problem. Equation (3) represents the general wave equation ∂2 p − c2 ∇ 2 p = 0. ∂t 2
(3)
To obtain solution using finite element method, Eq. (3) is converted into the weak formulation ∂2 p Wi 2 d A − c2 Wi ∇ 2 pd A = 0. (4) ∂t D
D
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From the Green’s first identity (integration by parts)
Wi pd A + D
∇Wi .∇ pd A =
Wi (p.n) d S.
(5)
∂S
D
where = ∇ 2 and n is the normal of the surface ∂ S. Using the Green’s first identity and simplifying the terms, Eq. (3) can be represented as ∂2 p 2 Wi 2 d A + c ∇Wi .∇ pd A = 0. (6) ∂t D
D
where Wi is the trial function. Using the polynomial for approximate solution and applying the boundary conditions, p can be written asp=
N
Ci (t)Ni
(7)
i=1
where Ci (t) are the time dependent coefficients and Ni are basic functions. From Eqs. (6) and (7), the weak formulation can be obtained and can be written as N ∂ 2 Ci (t) i=1
∂t 2
D
Ni Wi d A + c
2
N i=1
∇ Ni .∇Wi d A = 0.
Ci
(8)
D
From Eq. (8), Ci can be obtained. Putting the values of Ci in the Eq. (7) gives the final expression of the pressure. Finite element-based software (COMSOL Multiphysics) is used for the numerical study. 2D, six nodded free triangular elements are selected to mesh the geometry. To find out the optimum size of the element, convergence study has been performed for single acoustic tweezer model as well as the one-dimensional array of five transducers model. Plot of absolute pressure at a point with respect to the mesh size (number of elements per wavelength) for both the models is shown in Fig. 2a, b respectively. As shown in Fig. 2, the solution is almost close to converged solution with element size λ/15, where λ is the wavelength. Further reducing the element size will lead to the higher computational cost. So, the element size λ/15 is selected for the simulation to get satisfactory results for both models without increasing the unnecessary computational cost. Figure 3 shows the meshing of the model of standing wave acoustic levitation system with one acoustic tweezer.
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Fig. 2 Convergence plot for the numerical model with a Single acoustic tweezer and b Onedimensional array of five transducers Fig. 3 Meshing of the model of standing wave acoustic levitation system with one acoustic tweezer
3 Results and Discussion In standing wave acoustic levitation, particles can be levitated near the pressure nodes of the standing wave. When the distance between the driver and the reflector is right, the resonance phenomenon occurs and the standing wave is formed. For the levitation system with single acoustic tweezer, the distance between the driver surface and the reflector for third resonance mode is calculated numerically. The distance for third resonance mode is obtained as 13.56 mm as shown in Fig. 4a. Distance for the third
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Fig. 4 Resonance condition corresponding to the third resonance mode calculated a Numerically as 13.56 mm and b Experimentally as 13.50 mm
resonance mode is also calculated experimentally as 13.50 mm. For third resonance mode, two pressure nodes are obtained outside the casing of the tweezer. At each pressure node, a small polystyrene ball of 3 mm diameter is levitating as shown in Fig. 4b. Figure 4 shows that the particles can be trapped near the pressure nodes of the standing wave. Further, using the same approach, a one-dimensional array of the five transducers (shown in Fig. 1b) is studied numerically. The excitation frequency of the transducers is 40 kHz. For the resonance condition, the distance between the driver surface and the reflector is taken as 130 mm. Initially, the focal point of the transducer array is taken at a distance of 60 mm as shown in Fig. 5a. Two standing waves with separate pressure nodes/anti-nodes are constructed. As the focal point moves away from the transducers, the pressure nodes/anti-nodes of both the standing waves come closer as shown in Fig. 5b (focal point at 75 mm) and Fig. 5c (focal point at 90 mm). As the distance of the focal point increases further, both the standing waves combine into a single standing wave as shown in Fig. 5d (focal point at 105 mm). In standing wave acoustic levitation, particle gets trapped near the pressure node. As the pressure node moves, the particle also moves with it. Hence, two particles can be mixed together by gradually varying the focal point of the array of the transducers.
4 Summary In this study, standing wave acoustic levitation using ultrasonic acoustic tweezer is studied numerically and validated with the experiment. It is found that the particles can be levitated near the pressure nodes of the standing wave. Further, the similar approach is used to propose a mechanical design for the acoustic levitation using an one-dimensional array of five transducers. The effect of varying the focal point
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Fig. 5 Surface plot of total acoustic pressure field with focal point of the transducers array at a 60 mm, b 75 mm, c 90 mm and d 105 mm
of the transducer array is studied numerically. It is observed that the mixing of two particles can be achieved by gradually varying the focal point of the array of the transducers.
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References 1. Apfel RE (1976) Technique for measuring the adiabatic compressibility, density, and sound speed of submicroliter liquid samples. J Acoust Soc Am 59(2):339–343 2. Trinh EH, Apfel RE (1980) Sound velocity of supercooled water down to -33 C using acoustic levitation. J Chem Phys 72(12):6731–6735 3. Trinh EH, Hsu CJ (1986) Acoustic levitation methods for density measurements. J Acoust Soc Am 80(6):1757–1761 4. Trinh EH, Ohsaka K (1995) Measurement of density, sound velocity, surface tension, and viscosity of freely suspended supercooled liquids. Int J Thermophys 16(2):545–555 5. Tian Y, Holt RG, Apfel RE (1995) A new method for measuring liquid surface tension with acoustic levitation. Rev Sci Instrum 66(5):3349–3354 6. Ohsaka K, Trinh EH (1989) Melting and solidification of acoustically levitated drops. J Cryst Growth 96:973–978 7. Xie W-J, Cao CD, Lu YJ, Wei B-B (2002) Levitation of iridium and liquid mercury by ultrasound. Phys Rev Lett 89(10):104304 8. Xie WJ, Wei B (2001) Parametric study of single-axis acoustic levitation. Appl Phys Lett 79(6):881–883 9. Yadav S, Gupta A (2019) Maximization of acoustic levitating force for a single axis acoustic levitation system using the finite element method. Chin Phys Lett 36(3):034302 10. Whitworth G, Grundy MA, Coakley WT (1991) Transport and harvesting particles using modulated of suspended ultrasound. Ultrasonics 29:439–444 11. Matsui T, Ohdaira E, Masuzawa N, Ide M (1995) Translation of an Object Using PhaseControlled Sound Sources in Acoustic Levitation. Japanese Journal of Applied Physics 34(5B):2771–2773 12. Marzo A, Seah SA, Drinkwater BW, Sahoo DR, Long B, Subramanian S (2015) Holographic acoustic elements for manipulation of levitated objects. Nat Commun 6:1–7
Design and Validation of Flexure-Based Hinges for Space Deployable Antenna Reflector Hemant Arora , B. S. Munjal , and Sudipto Mukherjee
Abstract Space exploration arises the demand for launching large-diameter antenna reflectors to satisfy the need of high-bandwidth telecommunication, earth observation and deep space interplanetary missions. Launching of monolithic LDR antennas of sizes 3 m or more is not feasible due to limited launch fairing space of state-of-the-art launch vehicles. Therefore, the development of innovative deployment mechanisms is the present need of the hour. Many researchers have proposed various mechanism options to fold large size antenna reflectors in compact size and deploy in space to full configuration. Deployment process of antenna is process of transition from mechanism to structure which is one of the unreliable stages due to availability of many conventional rotary joints which causes loss of energy due to backlash, friction and misalignment. This paper proposes a solution of replacement of conventional hinges with flexure hinges in state-of-the-art space deployable configuration of large size reflectors, reforming it as compliant configuration which eliminates the factors causing loss of energy. Tape flexures are explored as a suitable candidate for compliant deployable configuration. The proposed configuration consists of two tape flexures mounted in such a way that concave curve of each tape will face to each other. Options with single tape flexure and double tape flexures are explored and compared. Experimental evaluation of a proposed joint configuration is carried out for the development of a deployable antenna reflector. Keywords Deployable antenna reflector · Space compliant mechanism · Flexure hinges · Tape flexures
H. Arora (B) · B. S. Munjal Space Applications Centre, ISRO, Ahmedabad, India e-mail: [email protected] S. Mukherjee Department of Mechanical Engineering, IIT Delhi, Delhi, New Delhi, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_2
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1 Introduction Deployable structures are used for launching large size space antenna reflectors of the order of more than 3 m. The main requirement is to fold the antenna reflectors to compact size so that it can be easily accommodated in launch vehicle fairing space during launching and later deployed to full configuration in space after launch. Deployable structures are useful to overcome this type of challenge. Deployable mesh antenna is the subject of research last two decades, due to its simple configuration, high thermal stability, repeatability, easy in folding to compact size. Deployment of this deployable mechanism is most vulnerable stage for the antenna which is intended to happen in space, if any failure in full deployment, the complete mission will may be be ruined. This type of failure will be considered as single-point failure. Deployment of an unfurlable antenna reflector is categorized in three types, viz. selfactuated initial deployment, motorized deployment and final deployment to deployed lock position. During all these stages of deployment, actuation power needs to overcome the energy losses due to friction, backlash and misalignment. To eliminate this problem, a solution of flexure-based joints is proposed and presented in this paper. Flexure-based joints consist of flexible hinges which are folded elastically in stowed configuration. The flexible hinges can self-deploy to certain angle by releasing stored strain energy which helps in initial deployment when released from stowed configuration. There are many advantages of flexible hinges like easy to manufacture, easy to assemble, low mass to deployed stiffness ratio, low cost and self-latching capability at fully deployed configuration. There is a lot of demand of these flexible hinges which is being widely used in space deployable structures such as synthetic aperture radars (SAR), solar arrays and deployable antenna booms. Design of flexure hinges using tape springs is presented in this paper as an objective to form a compliant-based deployable reflector configuration.
2 Design Configuration of Deployable Mesh Reflector 2.1 Description Simplest type of ring structure used for mesh antenna is known as Astromesh [1] ring structure, in which four-bar mechanism is used as a basic entity with one diagonal telescopic member and a cable passing thru the diagonal telescopic member which helps in deployment of Astromesh configuration. In order to provide a synchronous deployment, a gear pair is incorporated as represented by schematic view of single bay of Astromesh deployable mechanism (see Fig. 1).
Design and Validation of Flexure-Based Hinges for Space …
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Fig. 1 Exploded view of deployable mesh reflector with description of single bay
2.2 Kinematics of Deployable Mesh Reflector The bay with diagonal telescopic member and joints at each corner is repeated number of times to get closed-loop ring structure. Deployment of this configuration is done by contracting the telescopic member with help of cable winding mechanism. Each entity is tied with top mesh layer and rear net which are further connected with each other at multiple points thru unidirectional cable ties. When deployed fully, the connection with entities pull the mesh and cable ties prestress the mesh layer to take parabolic shape. Kinematic formulation of deployment path is plotted the different intermediate position of ring structure (see Fig. 2) which are considered to be vulnerable in full deployment of antenna mesh reflectors. Red color members are upper circumferential members, and blue color members represent lower circumferential members. Vertical members are shown in black color, and telescopic members are described in green color.
3 Design of Joints with Flexure Hinges Each bay of deployable configuration consists of two types of joints, viz. 3J and 5J joint configuration. 3J joint configuration comprises a gear pairs for simultaneous motion of circumferential links, and 5J joint consists of conventional hinges connecting circumferential and diagonal links. An attempt has been made to convert circumferential conventional hinges of 5J joint configuration into flexure hinges using tape flexures. These flexible hinges will be in fully bend position of tape flexures maximum up to 90° in stowed configuration during launch and will be allowed to relax to reach at straight position for full deployment in orbit (see Fig. 3).
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Fig. 2 Deployment sequence of deployable antenna reflector
Conventional 5J joint
Compliant 5J joint Fig. 3 Stowed and deployed configuration of deployable ring structure with CAD illustrations of conventional and compliant 5J joints
3.1 Geometry of Tape Flexures Tape flexures [2] have been explored for making 5J joint as frictionless compliant joint because of its geometry simplicity. It consists of simple metallic strips which have a slight curvature along transverse direction (see Fig. 4). Best example of these
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Fig. 4 CAD model of tape flexure
tape flexures is steel tapes used for measurement. Material of strips could be of low modulus of elasticity for providing flexibility during folding sequence and hardened to gain flexure strength to produce bending greater than 90° without having permanent deformation. The curve portion provides the stiffness in straight configuration which will be useful in final deployed configuration. Tension in mesh to form a parabolic profile will certainly generate forces in transverse direction on tape flexures. Therefore, stiffness in transverse direction is achieved by taking two tapes with opposite curved faces, and stiffness can further be enhanced with multiple numbers of layers of tape flexures.
3.2 Kinematics of Tape Flexures Buckling behavior of tape flexure [3] exists differently when bending moment applied on concave and convex curvature (see Fig. 5). When a tape flexure subjects to bending moment applied on convex curvature side, it shows the higher rigidity initially, then its stiffness suddenly breaks down with a snap and form an elastic bend up to critical moment M+ max , and further, bending moment drops down and stabilizes at M+ * .
Fig. 5 Bending behavior of tape flexure
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Fig. 6 Finite element structural analysis of single-layered tape flexures
Table 1 Von Mises stresses (MPa) for various tape flexure configurations Material
Hardened steel (carpenter steel tape)
CFRP laminate
Be–Cu alloy
Single-layer tape flexures
403.5
82.6
196.5
Double-layer tape flexures
387.6
75.6
165.1
Similarly, when tape flexure subjects to bending moment applied on concave curvature side, it shows lower rigidity and perform elastic bend with lower small bending moment M-*. Therefore, combination of tape spring with opposite curvature will result into smooth bending and provide adequate stiffness.
3.3 Structural Analysis of Tape Flexures A set of single-layer tape flexures facing with opposite curvature and double-layered tape flexures facing with opposite curvature are structurally analyzed [4] with rotational moments applied at ends, and tangential contacts boundary conditions [5] have been applied (see Fig. 6). This analysis is a kind of nonlinear large deflection analysis which is solved using Altairs Radioss explicit dynamic simulation tool. Deflection and stresses are compared with various materials and flexure configuration (see Table 1). Each case is analyzed with 0.12 mm thickness of tape, and rotational moment of 85° is applied at each end.
3.4 Implementation of Compliant 5J Joint in Deployable Reflector 5J joint of a deployable reflector is modified to accommodate double-layer tape flexures in place of conventional hinge. Carpenter steel tape strips are evaluated for experimentation (see Fig. 7). It was observed that the configuration with single-layer
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Fig. 7 Deployment stages of reflector with compliant joint
flexures are deflecting with the tension loads produced by RF mesh, whereas doublelayer flexures are well capable to accommodate mesh stretching tension loads and behaving as a stable joint in fully deployed configuration which further eliminates the additional locking requirement. The configuration in folding state is achieved fully by bending of flexures near to 90°, and flexure rotational stiffness helped in initial deployment of 10°–15°.
4 Advantages of Tape Flexures in Deployable Configuration Conventional joints have stiffness in both radial and tangential axis of deployed antenna reflector, whereas in case of compliant joints, equal stiffness can be varied by combination of tape spring configuration and adequate layers. Compliant joints are frictionless joints as compared to friction between mating parts of conventional joints. There are no chances of loss of energy in compliant configuration due to rotation. Therefore, chances of obstruction of deployment are eliminated. Compliant joints are easy to manufacture and assemble as compared to multi-parts configuration of conventional joints. Auto locking/latching is achieved in compliant joints by its geometry as an added advantage, whereas additional locking mechanism is required in case of configuration with conventional joints. Torsion springs are required in conventional hinges to generate initial torsional moment, whereas initial bending configuration of tape flexures will itself behave like a spring to produce required torsional moment which eliminates the requirement of additional torsional springs.
5 Conclusion This paper has addressed the criticality of deployment path for a mesh reflector to deploy fully in space. To solve the problem, a solution is proposed to adapt compliant configuration of deployable reflector by replacing conventional hinges into flexure hinges. Tape flexures are studied as a prominent option for flexible hinges. Buckling behavior of curved tapes is studied with a conclusion of using a combination of two tapes positioned with opposite curve faces. The combinations with single and
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double layer of tapes are structurally analyzed with various material options and found that flexible tapes made of CFRP material are more suitable with benign stresses under maximum rotational moment. Experimental validation of the concept is demonstrated with modified design of joints incorporating tape flexures. Tape flexures are ascertained to be a suitable option for deployable configuration of large size space antenna reflectors.
References 1. Thomson MW (1999) The astromesh deployable reflector. IEEE Antenna Propogation Soc 1516–1519 2. Seffen KA (2001) On the behavior of folded tape-springs. J Appl Mech ASME 68:369 3. Zuo Y, Jin G, Xie P (2018) Calculative and experimental study of the CFRP tape spring. J Mech Sci Technol 32(8):3603–3609 (Springer) 4. Yang H, Liu R, Wang Y, Deng Z (2015) Experiment and multiobjective optimization design of tape-spring hinges. Struct Multidisc Optim 51:1373–1384 (Springer) 5. Guan F, Wu Y, Wang X, Y (2010) The mechanical behavior of the double piece of tape spring. In: international conference on intelligent computing (ICIC) 2010. Communications in computer and information science book series (CCIS), vol 93, Springer, Berlin Heidelberg, pp 102–110
Effect of Implant Materials on Bone Remodelling Around Cemented Acetabular Cup Ajay Kumar , Rajesh Ghosh , and Rajeev Kumar
Abstract Bone resorption around cemented acetabular cup caused failure in the long tenure. This study aims to investigate the effect of different implant materials on bone remodelling around the cemented acetabular cup. Finite element (FE) models of intact and implanted pelvic models were established using CT-scan data sets. Three combinations (acetabular cup-femoral head) of implant materials, UHMWPE-CoCrMo, CoCrMo-CoCrMo, and alumina-alumina ceramic, were considered to predict the effect of implant materials on bone remodelling around cemented acetabular cup. Less bone resorption was observed for UHMWPE acetabular cup. Bone remodelling was nearly similar for the ceramic and metallic cup. Considering the bone remodelling scenario, the results of the current study stated that ceramic could be a good alternative to metallic implant in the cemented acetabular cup since ceramic has better wear resistance property than metal. Keywords Acetabular cup · Implant material · Bone remodelling
1 Introduction Total hip replacement (THR) is an invasive restoration process that has gained much popularity in the past few years. In THA, the diseased part of the acetabulum is replaced by an artificial cup commonly known as an acetabular cup. The reports, as suggested by NJR for England and Wales indicated, around 11% of hip replacement surgery had to go through revision during the period from 1 April 2003 to 31 December 2015 [1]. NJR provided that 38,310 cases out of 78,130 first hip rerevisions were associated with aseptic loosening of the acetabular cup [1]. The dominant cause for the failure of the cemented acetabular cup is aseptic loosening [1]. Excessive bone resorption is one of the causes of late loosening of the cemented acetabular cup and that may promote earlier with the improper selection of implant. Decrease in cup bearing diameter would reduce volumetric wear and subsequently A. Kumar (B) · R. Ghosh · R. Kumar School of Engineering, Indian Institute of Technology Mandi, Kamand 175005, Himachal Pradesh, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_3
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increase bone resorption. Selection of implant material also shows a decisive role in implant-induced adaptive bone remodelling. A stiff implant can promote earlier loosening of the cup due to bone resorption. Elevated cement mantle stress and high contact stress also increase the risk of failure. Earlier studies suggested that cemented acetabular cup performs better as compared to uncemented one for elderly patients [2]. Clinical interest is focused on the more durable acetabular cup that may reduce loosening. Polyethylene acetabular cup has problem-related to excessive wear generation and that promote late loosening due to osteolysis. The ceramic implant would reduce the wear-related problem; however, bone remodelling is the leading causes of concerns. Owing to less computational cost and economic, finite element (FE) analysis is generally considered for pre-clinical studies plus the assessment of the orthopaedic implants. Past research on cemented acetabular cups was focused on the immediate post-operative state [3–13]. The research was constrained to the influence of stress/strain shielding [4–6] or stress in cement mantle owing to changes in the thickness of the cement mantle [3, 10–14] or enhancing the capability of cement–bone interface or fatigue (repeated variations of stress) failure analysis of cement mantle [9–11, 15–17]. Long-term analysis, including bone remodelling, around of cemented acetabular cups are few [18, 19]. A clinical study by Shetty et al. [18] used only one type of acetabular material (polyethylene) to investigate the bone remodelling around the cemented acetabular cup. A numerical analysis of bone remodelling around the cemented acetabular cup specifies concluding bone density decrement around the acetabulum [19]. However, to date, the investigation of different implant material on bone remodelling around the cemented acetabular cup is still not well reported. Enhancement in cement mantle thickness would mitigate tensile stress in the cement mantle [8]. However, in that study, the only immediate post-operative state was considered, and the effect of implant material properties was not considered. Regardless of failure due to osteolysis, it is foreseen that implant material might affect bone remodelling around the acetabulum. Elevated tensile stress in the cement mantle can initiate a crack, which can further propagate due to loading conditions and eventual loosening of the implant [20]. The immediate post-operative state might not be able to predict the loosening of the implant accurately. Considering this hypothesis and background, the objectives of this present study is to determine the effect of different implant materials on bone remodelling around the cemented acetabular cup.
2 Materials and Method Considering the success of the cemented acetabular cup for elderly patients, a right hemipelvis of 62 years, and the female patient was considered for the current study. Three-dimensional (3D) FE model of this right hemipelvis was made from CTscan data like previous evaluations [21–23]. The developing technique of the 3D FE model of the pelvic bone was corroborated and tested [24, 25]. Ten-noded tetrahedral
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elements were considered for the meshing the geometry of implanted hemipelvis. Mesh convergence study was made (established on stress value generated in the implanted pelvic bone) in such a way that the outcome does not govern by the element size. Coarse, medium and fine element size was taken for meshing. The coarse, medium and fine mesh were consisting of 180,217, 312,352 and 376,930 number of elements. So, the second FE model containing 312,352 number of elements (element edge length varied from 0.5–3 mm) was considered for the present analysis to reduce the computational cost. The intact and implanted pelvic bone model was produced to comprehend bone remodelling and fatigue in the cement mantle. The solid model and FE sub-model of the implanted pelvic bone, along with all cups, are represented in Fig. 1. Three millimetres (3 mm) uniform or constant cement mantle thickness polymethylmethacrylate (PMMA) was considered for cemented fixation of the acetabular cup, which already has been reported as the safe value of cement thickness for cemented acetabular cups [3, 17]. The acetabular cup was situated and tilting at an angle of 45-degree inclination angle and 15-degree anteversion angle [26]. Ceramic (alumina), cobalt-chromium molybdenum (Co–Cr–Mo) and ultrahigh molecular weight polyethylene (UHMWPE) were taken for the acetabular cup. The dimensions of each acetabular cup kept the same, where the bearing diameter is 28 mm, and the outer diameter is 48 mm. A spherical femoral head was considered for the presentation of the hip joint reaction force (Fig. 1). Fig. 1 FE model of the implanted hemipelvis
Fixed constraints
Hip Joint Force
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2.1 Material Properties of Cortical Bone, Cancellous Bone, Cement Mantle and Implant The construction of the pelvic bone is close to the sandwich structure. Modelling technique of the FE model of the pelvic bone was similar to an earlier paper [21]. An elastic, isotropic and heterogeneous material property was taken for the cancellous (spongy) part of the bone. Materials property of spongy bone was taken as heterogeneous and distributed using CT-scan data as same as an earlier published study [21]. A linear relationship was formed among density and HU and expressed in the below equation to get the heterogeneous bone density of cancellous part of the bone. ρ = 0.022 + 0.001038 × HU
(1)
In this equation, 0 HU relates to the bone density of 0.022 g cm−3 , and 1646 corresponds to the highest cortical bone density of 1.73 g cm−3 [21]. After obtaining the heterogeneous cancellous bone density, the following empirical relation was taken to determine Young’s modulus (E in MPa) of cancellous part of the bone. Poisson’s ratio was taken as 0.2. E = 2017.3ρ 2.46
(2)
Direct one-to-one validation was not likely since CT data were based on in vivo. The current pelvis FE model was corroborated in earlier experimental analysis accomplished on a composite hemipelvis [24]. The cortical bone was considered to have, linear, elastic, isotropic and homogeneous material properties. Cortical bone was considered to have an elastic modulus of 17 GPa, Poisson’s ratio of 0.3 and density 1.73 gm cm−3 . The elastic modulus of the UHMWPE (polymer), metallic (Co–Cr–Mo) and ceramic (alumina) acetabular cup is taken as 1.174 GPa, 210 GPa and 350 GPa. The Poisson’s ratio for subsequent implant material is taken as 0.4, 0.3 and 0.26. The elastic modulus and Poisson’s ratio for cement mantle were taken as 2 GPa and 0.33, respectively [11, 23].
2.2 Loading Boundary Conditions and Bone Remodelling Models In this investigation, the loading cycle was considered as a normal walking cycle. Eight load cases (static) were considered in this whole period of the normal walking cycle [27]. 21 muscle forces, along with the hip joint reaction force, were taken into account in applied loading boundary conditions for each static load case [21, 27]. The fixed constraint was set at the pubis and sacroiliac joint [14, 21]. In this analysis, a sub-modelling technique was used to analyse the effect of different implant materials on bone remodelling [26]. Bone remodelling process used in this analysis
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was similar to earlier published studies [26, 28]. Though, eight static load cases of a normal walking cycle were taken in place of two load cases. Months or years were represented as a result of bone remodelling, and this time was calculated by using the bone adaptation rate, as explained in an earlier study [29]. Center position of the spherical femoral head was set as the application point of hip joint reaction force.
2.3 Interfacial Conditions The acetabular cup is fixed to the bone and bone cement (PMMA) for cemented fixation. Fully bonded implant-cement and cement-bone interfacial condition were taken in this analysis [17]. Therefore, frictionless contact was analysed at the interface of the acetabular cup and femoral head [17] to characterise the good lubricating articulating surface. A 0.5 mm redial allowance was given between the femoral head and acetabular cup. Contact analysis was done using the asymmetric surface to surface contact elements. Contact stiffness 100 N/mm2 and penetration tolerance 0.1 (factor) were suited in this analysis for convergence [ANSYS Manual]. A sensitivity analysis was completed with the disparity of contact factors to realise the consequence of these factors on contact results. These contact parameters were selected in a system with a particular procedure that these factors do not have any impact on the simulations results as well as computational time. In this study, all the simulation and solutions were done using ANSYS v18 (ANSYS Inc. PA, USA).
3 Results The results of bone remodelling around cemented acetabular cup were presented in terms of months, and the adaptation rate of bone was used to calculate the time (months) as shown in Fig. 2. The implant-induced variations in bone density due to different implant materials in different locations are presented in Fig. 2. It has been seen that implant material affects the bone density distribution because of bone remodelling around the acetabular cup (Fig. 2). Bone resorption, fall in bone density, was seen in cancellous bone from immediate post-operative condition to after 6 months of bone remodelling in the cemented acetabular cup (Fig. 2). Bone density decrement is not the same in all the locations (dependent on the location or sections) as shown in Fig. 2. There is less bone density variation or reduction among immediate post-operative and after 6 months of bone remodelling considering polyethylene acetabular cup as compared to metallic and ceramic acetabular cup. However, there are very fewer changes (or almost similar) in bone density distribution among metallic and ceramic acetabular cup (Fig. 2). Comparison of average bone density distribution for immediate post-operative and after 6 months of bone remodelling with different implant materials is represented in Fig. 3. The average bone density reduced from 0.446 to
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Fig. 2 Bone density distribution after 6 months of bone remodelling: equilibrium in bone remodelling for different implant materials (UHMWPE, Co–Cr–Mo and ceramic)
0.5 0.45
Average Bone Density
Fig. 3 Comparison of average bone density distribution (gm cm−3 ) for immediate post-operative and after 6 months of bone remodelling with different implant materials
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.4 after the 6 months of bone remodelling for the polyethylene (UHMWPE) cup. However, the reduction in bone density was more for metallic and ceramic acetabular cup as compared to polyethylene acetabular cup. The main concern associated with the polyethylene was the polyethylene wear particles, which was responsible for osteolysis and loosening in cemented acetabular cup. The average bone density reduced from 0.446 to 0.36 after the 6 months of bone remodelling is for the metallic
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(Co–Cr–Mo) acetabular cup. The average bone density reduced from 0.446 to 0.356 after the 6 months of bone remodelling is for the ceramic (alumina) acetabular cup (Fig. 3).
4 Discussion Subject-dedicated FE model of implanted pelvic bone was created based on the CTscan data. Three different implant materials (UHMWPE, metallic and ceramic) were considered for this study. The dimensions of each acetabular cup are kept the same. This investigation aims to predict the outcome of different implant material on bone remodelling around the cemented acetabular cup. The result of the present study showed that bone resorption and apposition were found around the acetabular cup due to bone remodelling. Results of our study also indicated that implant materials affect the bone density distribution around the cemented acetabular cup. Our study highlighted the decrement in the bone density distribution in cemented acetabular cup owing to different implant materials. Less bone density reduction was observed for UHMWPE acetabular cup as compared to metallic and ceramic acetabular cup. However, considering wear factor ceramic acetabular cup would perform better as compared to the metallic acetabular cup. The bone density distribution around the acetabular cup was found to be almost similar in case of the ceramic and metallic cup. However, after achieving the equilibrium, a little higher bone densities of 2–7% were detected in the central part of the acetabulum in the superior region for the ceramic cup as likened to the metallic cup as shown in Fig. 2. Average bone density reduction is less in polyethylene (UHMWPE) acetabular cup as compared to the metallic and ceramic acetabular cup (Fig. 3). Result of our study showed that there is approximately 10% reduction in the bone density among the immediate post-operative and polyethylene acetabular cup. However, both metallic and ceramic acetabular cup shows the approximately 19.6% reduction in the bone density (Fig. 3). There is some limitation associated with this study. An adaptation rate of 129.6 gm mm−2 months was considered for the bone remodelling analysis, which was similar to the femur; however, the actual adaptation rate for the pelvis is unknown. The adaptation rate appears to be dependent on a particular location as well as might also change from patient to patient. Bone remodelling is well known to be reliant on loading boundary conditions. However, another limitation of our study is that we have considered the eight static load cases of a normal walking cycle. Cancellous bone was taken to be isotropic; however, pelvic trabecular bone was found to be not very much anisotropic. In this investigation, we have taken only three implant materials (UHMWPE, metallic and ceramic) to analyse the bone remodelling around cemented acetabular cup. However, so many studies in the past literature have been concentrated on the advancement of new biomaterial in the arena of synthetic hip joints considering mechanical strength, biocompatibility, wear resistance and mechanical dependability as well as to reduce wear and osteolysis [30]. Therefore, a separate
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study would be beneficial by considering the stainless steel, zirconia, titanium alloys, silicon nitride, high crosslinked UHMWPE, zirconia-toughened alumina, metals with specified coatings and alumina–zirconia composites as the new biomaterials for different combination of implant materials to better understand the bone remodelling analysis of cemented acetabular cup owing to different implant materials. The ceramic as an implant material has several advantages over metallic and polyethylene implants; these are as follows: • The heads of implant prepared from ceramics have high hardness as well as more scratch counteraction than metals. • The hard and ultra-smooth surface of ceramic implants also helps to diminish the wear rate as compared to metallic and polyethylene implants. • As per the earlier investigations, the wear rate of ceramic implants could be reduced by 95% less than the metallic and other implants. • Along with that the liners used in ceramic implants could also further reduce the wear rate by a very significant quantity.
5 Conclusion 3D FE models of implanted pelvic bone have been really helpful to understand the bone remodelling around cemented acetabular cup due to different implant materials (UHMWPE, Co-Cr–Mo and ceramic). Results of this study indicated that ceramic could be an excellent alternative to metallic implant in the cemented acetabular cup since ceramic has better wear resistance property than metal. Ceramic also has less surface fatigue, less wear rate, less osteolysis and harmless wear particle due to direct solid contact and has long-term results as well as higher survival rate. Ceramic also provide better clinical outcomes. Likewise, the implants made of ceramic have a smaller coefficient of friction as compared to metallic implant materials. Ceramic also has necessary qualities as suitable biocompatibility and simplicity of fabrication. Acknowledgements We acknowledge the Department of Science and Technology (DST), India (Grant No: ECR/2016/000023) for supporting this study.
References 1. National Joint Registry (2016) National Joint Registry for England and Wales, 13th Annual Report, NJR 2. Fang M, Noiseux N, Linson E, Cram P (2015) The Effect of advancing age on total joint replacement outcomes. Geriatr Orthop Surg Rehabil 6(3):173–179 3. Lamvohee JMS, Ingle P, Cheah K, Dowell J, Mootanah R (2014) Total hip replacement: tensile stress in bone cement is influenced by cement mantle thickness, acetabular size, bone quality, and body mass index. J Comput Sci Syst Biol 7:72–78
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4. Carter DR, Vasu R, Harris WH (1982) Stress distributions in the acetabular region II: effects of cement thickness and metal backing of the total hip acetabular component. J Biomech 17(3):165–170 5. Vasu R, Carter DR, Harris WH (1982) Stress distributions in the acetabular region-I. Before and after total hip replacement. J Biomech 15(3):155–164 6. Crowninshield RD, Brand RA, Pedersen DRA (1983) Stress analysis of acetabular reconstruction in protrusio acetabuli. J Bone Joint Surg Am 65(4):495–499 7. Oh I, Sander TW, Treharne RW (1985) Total hip acetabular cup flange design and its effect on cement fixation. Clin Orthop Relat Res 195:304–309 8. Revie I, Wallace M, Orr J (1994) The effect of PMMA thickness on thermal bone necrosis around acetabular sockets. Proc Inst Mech Eng Part H J Eng Med 208:45–51 9. Mootanah R, Ingle P, Dowell J, Cheah K, Shelton JC (2000) Fixation of the acetabular cup in cemented total hip replacement: improving the anchorage hole profile using finite element method. Technol Health Care 8:343–355 10. Mootanah R, Ingle P, Cheah K, Dowell JK, Shelton JC (2004) Total hip replacement: results of a postal survey of current practice on the cement fixation of the acetabular cup in the UK. Hip Int 14:155–162 11. Mootanah R, Dowell JK, Cheah K, Ingle P, Shelton JC (2007) Configuration of anchorage holes affects fixation of the acetabular component in cemented total hip replacement—a finite element study. Comput Methods Biomech Biomed Eng 10:439–445 12. Kumar, Y.S., Pant, B., Darunkumar, Singh K.: Thickness effects on maximum von-Mises Stress of a cement mantle in total hip replacement—a finite element study. J Appl Biomater Biomech 7:111–115 13. Coultrup OJ, Hunt C, Wroblewski BM, Taylor M (2010) Computational assessment of the effect of polyethylene wear rate, mantle thickness, and porosity on the mechanical failure of the acetabular cement mantle. J Orthop Res 28:565–570 14. Kumar A, Sanjay D, Mondal S, Ghosh R, Kumar R (2020) Influence of interface crack and non-uniform cement thickness on mixed-mode stress intensity factor and prediction of interface failure of cemented acetabular cup. Theor Appl Fract Mech 107:102524. https://doi.org/10. 1016/j.tafmec.2020.102524 15. Tong J, Zant NP, Wang JY, Heaton P, Hussell JG (2008) Fatigue in cemented acetabular replacements. Int J Fatigue 8:1366–1375 16. Heaton-Adegbile P, Zant NP, Tong J (2006) In vitro fatigue behaviour of a cemented acetabular reconstruction. J Biomech 39:2882–2886 17. Zant NP, Wong CK, Tong J (2007) fatigue failure in the cement mantle of a simplified acetabular replacement model. Int J Fatigue 29:1245–1252 18. Shetty NR, Hamer AJ, Kerry RM, Stockley I, Eastell R, Wilkinson JM (2006) Bone remodelling around a cemented polyethylene cup: a longitudinal densitometry study. J Bone Joint Surg Br 88(4):455–459 19. Ghosh R (2016) Assessment of failure of cemented polyethylene acetabular component due to bone remodeling: a finite element study. J Orthop 13:140–147 20. Mann KA, Gupta S, Race A, Miller MA, Cleary RJ, Ayers DC (2004) Cement microcracks in thin mantle regions after in vitro fatigue loading. J Arthrop 19:605–612 21. Ghosh R, Pal B, Ghosh D, Gupta S (2015) Finite element analysis of a hemi-pelvis: the effect of inclusion of cartilage layer on acetabular stresses and strain. Comput Methods Biomech Biomed Eng 18(7):697–710 22. Ghosh R, Mukherjee K, Gupta S (2013) Bone remodelling around uncemented acetabular prostheses. Bone Joint J Br 95:184 23. Kumar A, Ghosh R, Kumar R (2020) Effects of interfacial crack and implant material on mixed-mode stress intensity factor and prediction of interface failure of cemented acetabular cup. J Biomed Mater Res Part B Appl Biomater 108:1844–1856. https://doi.org/10.1002/jbm. b.34526 24. Ghosh R, Gupta S, Dickinson A, Browne M (2012) Experimental validation of finite element models of intact and implanted composite hemi-pelvises using digital image correlation. Trans ASME J Biomech Eng 134(8):081003 (1–9)
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25. Ghosh R, Gupta S, Dickinson A, Browne M (2013) Experimental validation of numerically predicted strain and micromotion in intact and implanted composite Hemi-Pelvises. Proc Inst Mech Eng H J Eng Med 227(2):162–174 26. Ghosh R, Mukherjee K, Gupta S (2013) Bone remodelling around uncemented metallic and ceramic acetabular components. Proc Inst Mech Eng H J Eng Med 227(5):490–502 27. Dalstra M, Huiskes R (1995) Load transfer across the pelvis bone. J Biomech 28(6):715–724 28. Ghosh R, Gupta S (2014) Bone remodelling around cementless composite acetabular components: the effects of implant geometry and implant-bone interfacial conditions. J Mech Behaviour Biomed Mater 32:257–269 29. Weinans H, Huiskes R, van Reitbergen B, Sumner DR, Turner TM, Galante JO (1993) Adaptive bone remodelling around bonded noncemented total hip arthroplasty: a comparison between animal experiments and computer simulation. J Orthop Res 11:500–513 30. Hu CY, Yoon TR (2018) Recent updates for biomaterials used in total hip arthroplasty. Biomater Res 22–33
Influence of Ageing and High BMI on Lower Back Pain P. Praveen, M. S. Mallikarjunaswamy, and S. Chandrashekhara
Abstract Lower back pain (LBP) is the common problem prevailing in most of the individuals. The individuals suffering from LBP would be of any age group. It is found common in subjects with higher body mass index (BMI), older age, inferior posture and occupation, etc. This study analyses the influence of ageing and high BMI on LBP by applying image processing techniques on X-ray images of individuals of various age groups. In this work, the vital features corresponding to degenerative discs are measured and analysed. The results indicated that the effects of ageing and high BMI are influencing LBP. Keywords Low back pain · Degenerative disc · Medical image processing · Body mass index
1 Introduction Lower back pain (LBP) is the common problem all along with the world. The lumbar spine, or low back, consists of a complex network of interconnecting bones, joints, nerves, tendons, ligaments, and muscles. Injury to any of these body parts can cause LBP. It is affecting the people of all ages from children to elders. The pain in such cases can be acute, sub-acute or chronic. There are few risk factors associated to LBP, namely age, obesity, body posture, occupation, and BMI, etc. [1]. The various imaging modalities are being used for identification of musculoskeletal pain. The imaging modalities include conventional X-ray, digital radiology, ultrasonography
P. Praveen (B) Dept. of E&IE, JSS Academy of Technical Education, Bengaluru, India P. Praveen · M. S. Mallikarjunaswamy Sri Jayachamarajendra College of Engineering, JSS Science & Technology University, Mysuru, India e-mail: [email protected] S. Chandrashekhara ChanRe Rheumatology & Immunology Center and Research, Bengaluru, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_4
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(US), conventional scintigraphy, computed tomography (CT) and magnetic resonance imaging (MRI). Grassi et al. [2] discussed the various modalities used in analysis of musculoskeletal pain and identified techniques used to find origin of musculoskeletal pain. Few of the modalities are not suitable for the diagnosis of LBP, and they may lead to adverse effects and serious complications. Ben Darlow et al. in their illustrated the harms due to inappropriate imaging of LBP suffered individuals [3]. David G. Borenstein [4] classified the disorders as mechanical namely, osteoarthritis and lumbar spinal stenosis; and non-mechanical disorders like infectious, neoplastic, rheumatologic, endocrinologic, vascular, and gynecologic. Damian Hoy [5], narrated LBP as a global issue and its influence and prevalence period were analyzed. Luoma Katariina [6] carried out a study on 164 subjects with various occupations and showed that there is an increased risk of LBP in relation to all signs of disc degeneration. Podichetty VK [7] shown that the frequency of degeneration, especially lumbar degeneration increases sharply with age and is regarded as a major cause of discogenic low back pain. Nilanjan Dey [8] discussed the use of thermal images for disease diagnosis. In this work the objectives was set to extract features indicating the LBP using image processing techniques and analyze to understand the influence of age and body mass index on LBP.
2 Methodology The segmental inter-vertebral disc position is essential to understand the LBP. The images of individuals suffering with LBP were obtained from hospital and processed. The dataset includes candidates of age 25–85 years which includes both the genders. The study was reviewed by institutional ethical committee of ChanRe Rheumatology and Immunology Centre and Research, Bengaluru. The study was informed, and consent was obtained from participants. The image processing steps which involve segmentation, feature extraction and analysis are depicted in Fig. 1. There is a necessity of eliminating noise before the processing of such images without blurring the edges of inter-vertebral discs. Therefore, the preprocessing operations were performed to eliminate the noise induced in such X-ray images. The images were operated using median filter to eliminate noise and also preserve edges of the inter-vertebral discs. The window size of median filter was 9 × 9, and in the next step of processing, images were segmented. The images were further processed to segment inter-vertebral discs. The acquired images of dataset were preprocessed which involves region localization and enhancement techniques. Then, the images were segmented for feature extraction of diagnostically pertinent region of interests, i.e. the herniated disc or degenerative disc which is vital in respective cases. The images were further processed for the extraction of feature, i.e. inter-vertebral distance. These extracted features were analysed in subjects of different age groups and subjects with various BMI. Further, the statistical analysis was done. Finally, the influence of ageing and BMI on LBP was analysed.
Influence of Ageing and High BMI on Lower Back Pain Fig. 1 Image processing steps in analysis of LBP
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Image Acquisition
Preprocessing
Segmentation
Feature Extraction
Analysis
3 Results X-ray of LBP suffered individuals exhibited the degeneration of the disc. The degenerative disc was evident as the inter-vertebral disc distance in LBP suffered individuals seemed to be less compared to the inter-vertebral distance in the healthy individuals. Figure 2a shows the X-ray of the lower back of LBP suffered individual. The preprocessed images and the segmented images of lower back are shown in Fig. 2b, c, respectively. The feature extraction in lower back with disc degeneration is shown in Fig. 3. The inter-vertebral disc (foramen) gets deteriorated by making the inter-vertebral distance smaller due to the ageing, drying over time, prolong wear and tear. Hence, this shows the symptoms of degenerative disc. Based on the statistical analysis, obesity and higher BMI are found to be the massive causes for the herniated disc as the weight of the upper portion of the body directly impacts on the lower back. The cases of herniated disc were mostly found in obese individuals compared to the normal or lower BMI individuals. The measurements are made in region of lateral posterior, lateral middle and lateral anterior for each of the inter-vertebral disc. The mean of measurement indicating the features of LBP in typical five cases is tabulated in Table 1. The results are grouped and correlated based on the individual age group and BMI. The two groups of individuals were made on the basis of age, namely less than 50 years and equal to and greater than 50 years of age. Similarly, the subjects were
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(a)
(b)
(c)
Fig. 2 a X-ray of the lower back, b preprocessed X-ray image of lower back and c segmented image of X-ray of lower back Fig. 3 Feature extracted image showing degenerative disc
Degenerative disc
Table 1 Mean inter-vertebral distance in different cases Cases
Age years
BMI
L1–L2 mm
L2–L3 mm
L3–L4 mm
L4–L5 mm
Case 1
26
34.0
11.46
12.69
16.15
17.09
Case 2
38
29.3
11.49
12.74
16.12
16.19
Case 3
71
32.6
11.11
12.31
15.08
16.15
Case 4
53
34.9
11.47
12.98
15.25
16.88
Case 5
49
24.7
11.49
12.94
15.18
16.97
Influence of Ageing and High BMI on Lower Back Pain
31
grouped on the basis of BMI, namely overweight whose BMI is less than 30 and obese who’s BMI is equal or greater than 30. The results showed that the subjects with the age group more than 50 years and the subjects of obese class were the major victims of the LBP due to degenerative disc. This study infers that the influence of the ageing and the BMI plays a vital role in causing the lower back pain.
References 1. Taimela S, Kujala UM, Salminen JJ, Viljanen T (1997) The prevalence of low back pain among children and adolescents: a nationwide, cohort-based questionnaire survey in Finland. Spine 22:1132–1136 2. Grassi W, Filippucci E, Carotti M, Salaffi F (2003) Imaging modalities for identifying the origin of regional musculoskeletal pain. Best Pract Res Clin Rheumatol 17(1):17–32 3. Darlow S, Forster BB, O’Sullivan K, O’Sullivan P (2017) It is time to stop causing harm within appropriate imaging for low back pain. Br J Sports Med 5:14–415 4. Borenstein DG, Md (1996) Chronic low back pain. Rheum Dis Clin N Am 22:439–456 5. Hoy D, Bain C (2012) A systematic review of the global prevalence of low back pain. Am Coll Rheumatol 64(6):2028–2037 6. Luoma K, MD, Riihimäki H (2000) Low back pain in relation to lumbar disc degeneration. Spine 25(4):487–492 7. Podichetty VK (2007) The aging spine: the role of inflammatory mediators in intervertebral disc degeneration. Cell Mol Biol (Noisy-le-grand) 53(5): 4–18 8. Dey N, Ashour AS, Althoupety AS (2017) Thermal imaging in medical science. Recent advances in applied thermal imaging for industrial applications, IGI Global, 87–117
Design and Analysis of a Robotic Lizard Using Five-Bar Mechanisms V. S. Rajashekhar, C. K. Dinakar Raj, S. Vishwesh, E. Selva Perumal, and M. Nirmal Kumar
Abstract Legged robots are being used to explore rough terrains as they are capable of traversing gaps and obstacles. In this paper, a new mechanism is designed to replicate a robotic lizard using integrated five-bar mechanisms. There are two fivebar mechanisms from which two more are formed by connecting the links in a particular order. The legs are attached to the links of the five-bar mechanism such that, when the mechanism is actuated, they move the robot forward. Position analysis using vector loop approach has been done for the mechanism. A prototype has been built and controlled using servomotors to verify the robotic lizard mechanism. Keywords Robotic lizard · Topological design · Position analysis · Five-bar mechanism
1 Introduction Mimicking animals for the purpose of solving human centered problems plays an important role in the field of robotics. Among these, four legged creatures are gaining attention in the past few decades [1]. Among them, MEMS-based robots are gaining importance [2]. Miniaturized robots can be used for surveillance and locomotion in cramped spaces. A lizard is a commonly seen reptile whose mechanics have been studied in detail in the following literature [3]. It was found that the walking and trotting gaits [4] exhibited by the lizards were similar to that of the mammals. The biomechanics and kinematics of the sprawling pattern exhibited by the lizards were studied extensively
Supported by Tentacles Robotic Foundation. V. S. Rajashekhar (B) Tentacles Robotic Foundation, Kanchipuram, Tamil Nadu 603202, India C. K. Dinakar Raj · S. Vishwesh · E. Selva Perumal · M. Nirmal Kumar Department of Mechanical Engineering, Adhiparasakthi Engineering College, Melmaruvathur, Kanchipuram, Tamil Nadu 603319, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_5
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in [5]. The sprawling gait has be experimentally shown on a dynamic model in [6]. It is observed that the compliant and flexible trunk of the lizards help in reducing the peak power [7]. The lizard-based bio-inspired robots have been made in the past, such as a water running robot [8], a leg mechanism [9], and a quadrupeds [10, 11]. Motion analyis [12], motion planning [13], and gait planning [14] based on kinematics of the Gecko have been done on robotic lizards. A seven degrees of freedom robotic Gecko has been developed for the experimental verification of the kinematic analysis [15]. Although there are works that exist in the literature that try to replicate a real lizard, they have not been able to mimic them closely. The novelty of this paper lies in the core mechanism design and its abilty to replicate the gaits of a real lizard. The real lizard that served as a source of inspiration is shown in Fig. 1a, and the fabricated robotic lizard is shown in Fig. 1b. Topological design has been done using the method mentioned in [16]. The position analysis has been done using the vector loop approach. Finally, the prototype of the robotic lizard has been shown exhibiting the walking gait.
Fig. 1 a The real lizard that served as an inspiration. b The bio-inspired robotic lizard introduced in this work
Design and Analysis of a Robotic Lizard …
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2 Topological Design of the Robotic Lizard Mechanism The design of robotic lizards using mechanisms like Watt-I planar linkage mechanism [17] and CPG-driven modular robots [18] have been done in the past. In this work, the development of the robotic lizard mechanism using two active and two passive five-bar mechanisms is done. The degree of freedom analysis is done to find out the number of degrees of freedom of the mechanism and also to find out whether the right driving links have been chosen for the mechanism. This is a eight step process, and it is done using the method given in [16]. F=
m
fi −
i=1 v
v
ξLj
(1)
j=1
j
ξLj = dim.((∩i=1 Mbi ) ∪ Mbj+1 )
(2)
j=1
where, F—Degree of freedom of parallel manipulator (PM). fi —Degree of freedom of the ith joint. m—Total number of joints in the parallel manipulator. v—Total number of independent loops in the mechanism, where v = m − n + 1. n—Total number of links in the mechanism. ξLj —Total number of independent equations of the jth loop. j ∩i=1 Mbi —Position and orientation characteristic (POC) set generated by the sub-PM formed by the former j branches. Mbj+1 —POC set generated by the end link of j + 1 sub-chains. Calculating the number of independent loops putting n = 13 and m = 16, we get v = 4. The eight steps involved in the calculation of the degrees of freedom of the mechanism are as follows. 1. The topological structure is mentioned symbolically here. Branches: SOC1 (−R1 ||R2 ||R3 −); SOC2 (−R4 ||R5 −); SOC3 (−R1 ||R7 ||R8 −); SOC4 (−R11 ||R10 −); SOC5 (−R5 ||R7 ||R9 −); SOC6 (−R12 ||R13 −); SOC7 (−R11 ||R14 ||R16 −); SOC8 (−R12 ||R15 −) Platforms: Fixed platform: R1 , R5 , R11 , R12 Moving platform: The rest are all moving platforms 2. An arbitrary point o is chosen on the moving platform. 3. Determining the POC set of branches 2 t (⊥ Ri ) i = 1, 2, 3 Mb1 = r 1 ||(Ri )
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Mb2 =
2 t (⊥ Rj ) j = 4, 5 r 1 ||(Rj )
4. Finding the total number of independent displacement equations. Topological structure of the independent loop t2 t2 ∪ ξL1 = dim(Mb1 ∪ Mb2 ) = dim r 1 (||R1 ) r 1 (||R4 ) t2 = dim =3 1 r (|| (R1 , R4 )) Similarly, it can be done for the other three loops, and the value of the number of independent displacement equations is found to be the same in each case. 5. Calculating the DOF of the mechanism m v fi − ξLj = 16 − (3 + 3 + 3 + 3) = 4 F= i=1
j=1
6. Finding the inactive pairs. Based on the calculations done in [16], similar steps were followed. It was found that there is no inactive pair in the mechanism except for the tail which is regarded as passive linkages. 7. Determining the position and orientation characteristic set of the robotic lizard mechanism. Based on the formula given in [16], MPa = Mb1 ∩ Mb2 2 t2 t2 t t2 ∩ ∩ ... ∩ = 1 r 1 (||R1 ) r 1 (||R4 ) r 1 (||R12 ) r
MPa =
The degree of freedom of the mechanism is four, and the dimension of the above POC set is three. Hence, the module has two translational and one rotational degree of freedom. The discrepancy in the values is due to the fact that there is one more rotational degree of freedom on the same plane which the position and orientation characteristic analysis has ignored. This is because both the rotational degrees of freedom are the same. 8. Select driving pairs The joints R1 , R5 , R11 and R12 are chosen to be the driving pairs, and hence, they are fixed. Calculating the degrees of freedom of the new mechanism, m v fi − ξLj = 12 − (3 + 3 + 3 + 3) = 0 F∗ = i=1
j=1
Since F ∗ = 0, the joints R1 , R5 , R11 and R12 can be used as driving pairs simultaneously. Thus, the topological design of the mechanism has been done.
Design and Analysis of a Robotic Lizard …
37
3 Position Analysis of the Robotic Lizard Mechanism The robotic lizard mechanism is as shown in Fig. 2c. The vector loop method adopted from [19] is used for the position analysis of the robotic lizard. There are five vector loop equations framed, one for the head, two for the body, one for the trunk, and one for the tail. These equations are solved by keeping the known angles on one side and the unknown angles on the other side. The position of each part of the body is known by substituting the known four servomotor angles. These equations will be presented in detail in this section. The vector loop equation of the head which forms a five-bar mechanism in the robotic lizard is written in Eq. 3. It is shown in Fig. 2b. R2 + R3 − R4 − R5 − R1 = 0
(3)
Collecting the sin and cos terms together, Eqs. 4 and 5 are obtained. L2 sin(θ2 ) + L3 sin(θ3 ) − L4 sin(θ4 ) − L5 sin(θ5 ) − L1 sin(θ1 ) = 0
(4)
L2 cos(θ2 ) + L3 cos(θ3 ) − L4 cos(θ4 ) − L5 cos(θ5 ) − L1 cos(θ1 ) = 0
(5)
Solving for the θ3 and θ4 by rearranging Eqs. 4 and 5, the following are obtained. √ B2 − 4AC θ4 = 2 arctan 2A √ −E + E 2 + 4DF θ3 = 2 arctan 2D −B −
(6)
(7)
Fig. 2 a The topological design of the robotic lizard mechanism. b The four five-bar mechanisms that form the robotic lizard. It is used in position analysis. c The prototype of the robotic lizard. d The graphical user interface developed for the robotic lizard
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where, K1 = L5 sin(θ5 ) − L2 sin(θ2 ); K2 = L5 cos(θ5 ) − L2 cos(θ2 ) + L1 K3 = −L5 sin(θ5 ) + L2 sin(θ2 ); K4 = −L5 cos(θ5 ) + L2 cos(θ2 ) + L1 2 2 2 2 3 +K2 +K1 A = L4 −2 K2 L4−L ; 2 B = 2 K1 L4 ; 2 2 2 2 3 +K2 +K1 C = L4 +2 K2 L4 −L 2 2 2 2 2 D = L4 −L3 +2 K24 L3 −K4 −K3 ; E = 2 K3 L3 ; 2 2 2 2 F = L4 −L3 −2 K24 L3 −K4 −K3 Similarly, for the rest of the three five-bar mechanisms, the vector loop equations are as follows. For the body side 1, body side 2, and tail, R6 + R7 − R8 − R9 − R10 = 0
(8)
R12 + R13 − R14 − R15 − R11 = 0
(9)
R17 + R19 − R16 − R18 − R20 = 0
(10)
From the above equations, the values of θ can be found that are similar to Eqs. 6 and 7. Thus, the position analysis has been done for the mechanism.
3.1 Coordiates of Linkages in the Mechanism The four joints are actuated in a particular order so that the robotic lizard mechanism exhibits the forward walking gait. The equations governing the coordinates of the linkages of the head are as follows. x1 = 0 y1 = 0 x2 = L2 cos(θ2 ) y2 = L2 sin(θ2 ) x3 = x2 + L3 cos(θ2 + θ3 ) y3 = y2 + L3 sin(θ2 + θ3 ) x5 = L5 cos(θ5 ) y5 = L5 sin(θ5 ) x4 = x5 + L4 cos(θ5 + θ4 ) y4 = y5 + L4 sin(θ5 + θ4 )
(11)
In the similar way, it can be derived for the left and right side of the body and the tail of the mechanism.
Design and Analysis of a Robotic Lizard …
39
3.2 Workspace of Robot Parts The workspace of the robot is plotted by writing a Python Code. The algorithm used for plotting the workspace of the robot is as follows. Algorithm 1 Plotting the workspace using position analysis Declare the range for θ2 and θ5 Declare the length of linkages For θ5 = 135, θ5 −−, while θ5 < 0 For θ2 = 45, θ2 ++, while θ2 < 160 Calculate Equation 11 plot([x1 ,x2 ],[x1 ,y2 ]) plot([x2 ,x3 ],[y2 ,y3 ]) plot([x3 ,x4 ],[y3 ,y4 ]) plot([x4 ,x5 ],[y4 ,y5 ]) EndFor EndFor
Fig. 3 The workspace of the a head, b tail, c body left, d body right
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Fig. 4 The robotic lizard exhibiting the walking gait
Figure 3a–d shows the reachable workspace of the head, tail, body left, and body right of the robotic lizard mechanism.
4 Prototype of the Robotic Lizard Mechanism The links of the mechanism were fabricated using the balsa wood in a CNC router. The mechanism was actuated using the four servomotors in a particular order. The control used here was Arduino UNO, and processing IDE was used as a graphical user interface (GUI) to control the robot. The GUI is as shown in Fig. 2d. The robotic lizard was able to move on a flat surface and exhibit the walking gait as shown in Fig. 4.
5 Conclusions In this work, a new robotic lizard mechanism was developed. The topological design of the mechanism was done. Then, the position analysis of the mechanism was done to find the various angles between the links. Finally, a prototype was made and controlled using servomotors. A graphical user interface was used to operate the robotic lizard. The future work would be to implement deep reinforcement learning to obtain a stable gait in unknown environments.
Design and Analysis of a Robotic Lizard …
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References 1. Siegwart R, Nourbakhsh IR, Scaramuzza D (2011) Introduction to autonomous mobile robots. MIT press, Cambridge 2. Doshi N, Jayaram K, Goldberg B, Manchester Z, Wood RJ, Kuindersma S (2019) Contactimplicit optimization of locomotion trajectories for a quadrupedal microrobot. arXiv preprint arXiv:1901.09065 3. Farley CT, Ko TC (1997) Mechanics of locomotion in lizards. J Exp Biol 200(16):2177–2188 4. Kim C-H, Shin H-C, Lee H-H (2013) Trotting gait analysis of a lizard using motion capture. In: 2013 13th International conference on control, automation and systems (ICCAS 2013), pp 1247–1251. IEEE 5. Russell A, Bels V (2001) Biomechanics and kinematics of limb-based locomotion in lizards: review, synthesis and prospectus. Comp Biochem Physiol Part A Mol Integr Physiol 131(1):89– 112 6. Kim C, Shin H, Jeong K (2014) Trot gait simulation of four legged robot based on a sprawled gait. In: 2014 14th International conference on control, automation and systems (ICCAS 2014). IEEE, pp 1031–1036 7. Gu X, Guo Z, Peng Y, Chen G, Yu H (2015) Effects of compliant and flexible trunks on peak-power of a lizard-inspired robot. In: 2015 IEEE International conference on robotics and biomimetics (ROBIO). IEEE, pp 493–498 8. Floyd S, Keegan T, Palmisano J, Sitti M (2006) A novel water running robot inspired by basilisk lizards. In:2006 IEEE/RSJ international conference on intelligent robots and systems. IEEE, pp 5430–5436 9. Dai Z, Zhang H, Li H (2009) Biomimetics of gecko locomotion: from biology to engineering. In: 2009 ASME/IFToMM international conference on reconfigurable mechanisms and robots. IEEE, pp 464–468 10. Park HS, Floyd S, Sitti M (2008) Dynamic modeling of a basilisk lizard inspired quadruped robot running on water. In: 2008 IEEE/RSJ international conference on intelligent robots and systems. IEEE, pp 3101–3107 11. Park HS, Floyd S, Sitti M (2009) Dynamic modeling and analysis of pitch motion of a basilisk lizard inspired quadruped robot running on water. In: 2009 IEEE International conference on robotics and automation. IEEE, pp 2655–2660 12. Kim C-H, Shin H-C, Jeong T-W (2012) Motion analysis of lizard locomotion using motion capture. In: 2012 12th International conference on control, automation and systems. IEEE, pp 2143–2147 13. Ratliff N, Zucker M, Bagnell JA, Srinivasa S (2009) Chomp: gradient optimization techniques for efficient motion planning 14. Son D, Jeon D, Nam WC, Chang D, Seo T, Kim J (2010) Gait planning based on kinematics for a quadruped gecko model with redundancy. Robot Auton Syst 58(5):648–656 15. Nam W, Seo T, Kim B, Jeon D, Cho K-J, Kim J (2009) Kinematic analysis and experimental verification on the locomotion of gecko. J Bionic Eng 6(3):246–254 16. Yang T-L, Liu A, Shen H, Hang L, Luo Y, Jin Q (2018) Topology design of robot mechanisms. Springer, Berlin 17. Xu L, Mei T, Wei X, Cao K, Luo M (2013) A bio-inspired biped water running robot incorporating the watt-i planar linkage mechanism. J Bionic Eng 10(4):415–422 18. Vonásek V, Saska M, Winkler L, Pˇreuˇcil L (2015) High-level motion planning for cpg-driven modular robots. Robot Auton Syst 68:116–128 19. Norton RL et al (2004) Design of machinery: an introduction to the synthesis and analysis of mechanisms and machines. McGraw-Hill Higher Education, Boston
Development of an Automated Material Handling System Inside a Nuclear Containment Structure Anupam Saraswat and P. S. Somayajulu
Abstract This paper presents design and development of an automated material handling and transfer system inside a containment structure of a nuclear facility. In the nuclear fuel cycle facilities, since, the feed material being handled is radiotoxic in nature all the process operations are carried out inside a mechanical containment structure known as a Glove Box. Main challenges for design are constraints arising due to the presence of highly radioactive material, confined space, limited access for maintenance and operation, maintaining vacuum inside the glove box and remote handling. To address the above-mentioned challenges, a conveyor system is designed for material transfer inside the shielded glove boxes. The system has been developed considering prime design objectives of modularity, ease of remote maintenance, standardization, provision for manual override, interchangeability and the absence of pneumatic devices inside the glove boxes. It is proposed to use the customized roller conveyor system. All the rollers are designed for the in situ maintenance. Three different types of power transmission systems shall be tested, viz. belt mechanism, gear mechanism and chain mechanism. The best out of three fulfilling the abovementioned design objectives will be finally selected for the installation. It is proposed to initially develop a prototype and then, later on, expand it further to complete the entire fabrication line of glove boxes. Keywords Material handling system · Automation · Remote handling · Glove box
1 Introduction In a nuclear fuel fabrication facility, various process operations are carried out in specially designed glove boxes as shown in Fig. 1. These glove boxes are dynamic containment system which acts as a barrier between operator and the radiological material inside it. Glove boxes are always maintained under dynamic negative pressure to prevent outward leakage of radioactivity. They are a leak-tight system with A. Saraswat (B) · P. S. Somayajulu Bhabha Atomic Research Centre, Mumbai, Maharashtra 400085, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_6
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A. Saraswat and P. S. Somayajulu
Fig. 1 Fabrication line showing series of interconnected glove boxes
a permissible leak rate of 0.05% GB volume per hour as given by relevant standards [1–3]and are designed as class I component as per the ASME section III NB guidelines [4]. Process equipment is kept inside the glove boxes to carry out various fabrication steps. Normally, material transfer between one glove box to another in the fabrication line is carried out manually by the operators. In case of reprocessed feed material coming out of fast reactors, radiological doses to the operator are expected to be higher. Hence, complete automation of material transfer is the need of the hour. Initially, a prototype system has been planned to connect few sample glove boxes using developed material handling system. The next section shall highlight design philosophy involved in the development of automatic material transfer system.
2 Design Considerations and Constraints To develop any system for functioning inside a glove box is a challenging task. There are various technical considerations required before starting the design. Following are the important considerations for the design of automatic material transfer system for glove box applications: 1.
Modular Design—In case of breakdown of any component, user of the system shall be able to take out the component, after decontamination, from the bag out port of the glove box and replace it with a new component. Size of each
Development of an Automated Material Handling System Inside …
2.
3.
4.
5.
45
individual component shall be such that it can be bagged out (removed) from the glove box using standard Ø 0.3 m bag-out port. Hence, all the component modules shall be designed based on this size constraint. Standardization—All the important components which may require frequent maintenance or replacement shall be standardized. For example, rollers in a roller conveyor, drive motors, fasteners, and supporting channels, etc. This helps in reducing maintenance downtime of the system. Remote In Situ Maintenance—A system inside a glove box has a very limited access from the outside. Hence, provision of remote maintenance is required. Remote maintenance tools shall be developed to fulfil the requirement. In situ maintenance of conveyor system is possible by using fast changing mounting fixtures for the rollers. Similarly, quick changing couplers for motors and drive mechanisms shall be preferred. Quick changing tools for the conveyor mechanism shall be available during maintenance. Radiation Resistance Components—Environment inside a glove box will have high radiation field. This requires two prong design philosophy. First design objective shall be to keep electronic components outside the shielded glove box as far as possible. If it is not possible, then use radiation-hardened cables and electronic components inside the glove box. Fail-safe Design—The system design shall be fail safe to avoid any radiological consequences. The system shall be simple and rugged in construction.
Considering the above-mentioned design objectives, a prototype material handling system is being designed. The system is being developed considering connection of two glove boxes with a material transfer tunnel. Once material transfer system is successfully demonstrated in the prototype; the same module would be scaled up to cover the entire fabrication line. Next section describes the details of the proposed system.
3 System Details In the proposed system, roller type of conveyors is planned to be used for the material transfer in a tray, as shown in the conceptual drawings from Figs. 2, 3 and 4. As can be seen in the drawings two bigger glove boxes (Type-VI) are connected with a smaller glove box (Type-I) with the help of material transfer tunnels. The conveyor passing through transfer tunnel has been designed to be modular in construction, so that in case of a radiological emergency, this can be retracted back and tunnel can be closed using a bunk. Hence, for a module of three interconnected glove boxes, five parts of conveyor will be connected together to make a complete conveyor system. Smooth transition of material placed in a carrier tray from one conveyor segment to the other has to be ensured. In case of power failure, it is expected that operator shall be able to move the material to a predetermined location. A provision to switch to manual handling mode using a clutch mechanism has been provided.
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Fig. 2 Conceptual scheme of the conveyor system for material transfer inside glove boxes
Fig. 3 Front view of the conveyor system
Development of an Automated Material Handling System Inside …
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Roller Conveyor
Carrier Tray Type VI Glove boxes
Type I Glove box
Material transfer tunnels
Omni directional transfer units
Fig. 4 Top view of the rollers type conveyor system
Details of various components of the conveyor system are given below: • Roller Conveyors: A number of independent roller types of conveyor modules are interconnected to form a continuous chain. Length of each segment has been selected considering isolation of the glove boxes (Type VI and Type I) from each other. A collapsible type of mounting support for the conveyors has been provided, so that in case of disassembly, it can be easily folded and taken out. It shall be noted that conveyors are required to work in a dusty environment with a very limited access for maintenance. Hence, provision for dust collection/removal and cleaning has been designed. In case of any breakdown, roller replacement inside a glove box shall be possible. It is proposed to use three different types of transmission mechanisms for power transmission from the master drive roller (MDR) to the slave rollers. These are as follows: (1) belt mechanism, (2) gear mechanism and (3) chain mechanism. Bidirectional drive with a speed control has been provided. For quick assembly/disassembly of the rollers from the top, clips have been provided. • Omnidirectional Transfer units: A number of transfer units are required to impart perpendicular/omnidirectional motion to the carriage based on the process sequence. These are required to be placed in appropriate location in the conveyor segment passing through Type VI GB as shown in Figs. 2, 3 and 4. Roller transfer units have been provided to carry out change in direction of motion using rotary action of embedded rollers. These units act as an Omni drive unit which can change
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A. Saraswat and P. S. Somayajulu
motion to the pallet/carrier tray in varied directions making angle of 0°–180° from the line of motion. • Carrier Trays: A number of carrier trays have been designed to carry material containers of different shapes. Total payload to be carried is 20 kg. These trays shall be provided with pre-defined grooves to position the material containers accurately inside them as shown in Fig. 4. These trays have been standardized. • Drive system: All the independent roller conveyor modules have been powered with the motors. The system requires precise movements and positioning of the material located in a transfer tray, and hence, motors with VFD have been provided. Independent drives have been provided to the omnidirectional transfer units and each individual conveyor module. It shall be noted that in case of any power failure or manual mode of operation, provision for automatic disengagement of the drive unit form the conveyor has been designed. It can be possible by using motors with the clutch arrangement. Bidirectional motion of the conveyor shall be possible with the drive unit. • Controller: All the drives and actuators required in the system shall be controlled through a PLC. It shall be possible to generate random and simultaneous sequence of material/tray from one glove box to the other. Provision for auto and manual mode of operation has been provided. An HMI has been provided to control the conveyor system remotely.
4 Conclusions A material handling system for the transfer of feed materials inside the glove box system has been developed. It has been designed considering all the possible constraints available for the operation and maintenance inside a containment system. The developed system shall be initially tested in a normal environment. Once the design features of automation and remote maintenance are demonstrated, then the system shall be installed in a radioactive environment.
References 1. 10648-1 (1997) ISO, Containment enclosures—Part 1: design principles. International Organisation for Standardisation 2. I. 10648-2 (1992) Containment enclosures—Part 2: classification according to leak tightness and associated checking methods. International Organization for Standardization 3. C852-09, ASTM standard -. Standard guide for design criteria for plutonium glove boxes. 2009 4. American Society of Mechanical Engineers (ASME) (2010) Boiler and pressure vessel code, Section-III, Divison 1-Subsection NB. ASME, New York
Nonlinear Modeling and Stability Analysis of Piezoelectric Energy Harvesting Mechanism Under Aeroelastic Vibration Rakesha Chandra Dash, Dipak Kumar Maiti, and Bhrigu Nath Singh
Abstract The present research work is focused on the mechanism of harvesting electrical energy from the oscillation of the bluff body placed in airflow. A rectangular bluff body is attached to the tip of a cantilever beam which has a portion embedded with piezoelectric layers. A geometrically nonlinear distributed parameter model is derived using extended Hamilton’s principle for both parallel and series connections of piezoelectric patches and solved using Newmark-beta method. Polynomial representation of aerodynamic force is done using quasi-steady hypothesis. It is found that nonlinear damping coefficients play a significant role in determining the stability of the system. System bifurcates (Hopf-bifurcation) and goes into a limit cycle after a particular wind speed. The radius of the limit cycle increases with wind speed. Approximately, 0.36 mW electric power can be generated at a wind speed of 7 m/s. Keywords Energy harvesting · Bifurcation · Galloping · Piezoelectric · Limit cycle oscillation
1 Introduction Flow-induced vibrations can be found in structures like airplanes and power transmission lines. A special kind of flow-induced vibration is transverse galloping. It is induced in prismatic cylinders. A certain threshold value of wind speed causes galloping to start, and it is called the onset speed of galloping. During the past few years, many studies have been done to convert surrounding vibrations to electrical energy by using piezoelectric material. This phenomenon is first studied and explained by Hartog [1]. Aerodynamic forces are described using quasi-steady hypothesis in his study. This approximation is valid when characteristic time scale of the airflow is much lower than the characteristic timescale of the structure oscillation. The effect of changing the geometry of the cross section of bluff body on R. C. Dash (B) · D. K. Maiti · B. N. Singh Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_7
49
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the amplitude of the structure is studied by Novak [2]. A. Barrero-Gil et al. used theoretical calculation to extract energy from transverse galloping [3]. They used a lumped-parameter model to describe the mechanical behavior of the system. A polynomial representation is used based on quasi-steady approximation. Abdelkefi et al. showed that for relatively high Reynolds number, galloping force can be approximately represented by a cubic polynomial function of the angle of attack [4]. Effect of D-shaped cross section geometry on energy harvesting is theoretically and experimentally investigated by Sirohi et al. [5]. A linear and nonlinear analysis is performed by Yan et al. to determine the effect of the electrical load resistance on the onset speed of galloping and level of the harvested power [6]. The nonlinear characterization of energy harvester system applied with combination of vibratory base excitations and aerodynamic loading is studied by Yan et al. [7]. Other than energy harvesting, Moutlana MK et al. (2015) used piezoelectric patches to control cantilever beam with end mass [8]. In this work, the effect of linear and nonlinear representation of Euler–Bernoulli’s beam theory on amplitude of vibration and voltage developed is discussed. A nonlinear distributed parameter model is derived for bimorph configuration of piezoelectric patches following Bibo et al. [9]. Quasi-steady theory is considered to model aerodynamic forces. Effects of coefficients in representing the aerodynamic force on energy harvesting are discussed.
2 Theoretical Background See Fig. 1. The dynamics of a beam can be completely described by using the longitudinal displacement, u(s, t), and transversal displacement, w(s, t), following Euler’s theory as depicted in Fig. 2. Nonlinearity in strain is considered for axial direction Fig. 1 Cross section of energy harvester in airflow
Nonlinear Modeling and Stability Analysis of Piezoelectric …
51
Fig. 2 Differential beam element deformation
only. So strain can be given by: ∈x = ∈0 + ∈r = u +
w2 − zw 2
(1)
Two Cartesian coordinate systems are used in this expression to describe a beam element at different instant of deformation. Global coordinate system is (x, y, z) , and local system is (x, y, z). Following the linear constitutive relationships, axial σx stress and axial strain ∈x can be given by σxb = Y b ∈bx
(2)
σxp = Y p ∈xp −d31 E 3
(3)
where b and p superscripts stand for structural and piezoelectric layers, respectively. Ohms law can be used to relate voltage and current via the relation V (t) = R Q˙ R (t), where Q˙ R (t) is current passing through the resistive load R.
3 Mathematical Formulation 3.1 Nonlinear Distributed Parameter Model Following the Hamilton’s variational principle
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t2 δL + δwext dt = 0
(4)
t1
where t1 to t2 is interval of time taken arbitrarily, L = T − U is the Lagrangian, δ is the variational operator, and Wext is the external non-conservative work. Expression for the kinetic energy T can be written as ⎧⎡ ⎫ ⎤2 ⎨ ∂ s 1 ⎬ 2 1 w2 ds ⎦ + w˙ 2 ds + It w˙ L b M(s) ⎣ ⎩ ∂t ⎭ 2 2 0 0 ⎡ ⎛ ⎞2 ⎤ 2 L b ∂ 1 1 ⎢ Dt Dt ⎥ w˙ L b + w| w ds − w L b w˙ L b ⎠ ⎦ + Mt ⎣ ˙ Lb + ⎝ 2 2 ∂t 2 2
1 T = 2
L b
(5)
0
where Dt is width of the bluff body, and M(s) is given by M(s) = m b + m p H (s) − H s − L p , m b = ρb Wb tb m p = ρ p W p t p
(6)
The total potential energy of the system, U, can be represented by the expression 1 U= 2
b b 1 σx ∈x +σxp ∈xp dv − 2 v
E 3 D3 dv
(7)
v
where v is total domain, and D3 is the electric displacement and can be expressed by the following linear piezoelectric relation D3 = d31 Y p ∈xp −e33 E 3
(8)
where e33 is the permittivity of the piezoelectric element at constant strain. Putting all relations in Eq. (7) and integrating over the cross section area of each layer, we obtain 1 U= 2
L b
2
2
YI w w +w
2
2 1 1 2 2 2 ˙ + θ w + w w R Q R ds + C p R Q˙ R 2 2
0
(9) where
Nonlinear Modeling and Stability Analysis of Piezoelectric …
YI =
2 p bY 3
Cp =
hs hp + 2
3
−
hs 2
3
53
2b b h s 3 + Y 3 2
e33 W p L p d31 Y p b 2 h p + h p hs θ= tp 2h p
Non-conservative Forces There are three types of work done due to non-conservative forces considered in this system, namely lateral aerodynamic force, Fy , mechanical viscous damping, Ca , and due to resistive load, R. Non-conservative virtual work can be expressed as
δW
ext
= Fy δw| L b + Fy L c δw
L b
Lb
−
˙ − R Q˙ R (t)δ Q R (Ca wδw)ds
(10)
0
Quasi-steady assumption is used to model the aerodynamic force, Fy . Fy =
w| ˙ Lb 1 2 ρa C Fy (α)Dt L t U∞ C Fy = a1 α + a3 α 3 + a5 α 5 α = w L b + 2 U∞
where ρa , U∞ , C Fy , and L t are the air density, wind speed, coefficient of lift, and length of bluff body, respectively. Substituting Eqs. (5), (9), and (10) in (4), then taking virtual variations with respect to w and Q R , and setting the coefficients of δw and δ Q R to zero, the following equations and boundary conditions can be obtained: M(s)w¨ + ca w˙ + Y I (s)w + w Y I (s)w w ⎡ ⎡ s ⎤ ⎤ ! s 1 1 2 2 + θ (s) 1 + w V (t) + ⎣w M(s)⎣ w¨ ds ⎦ds ⎦ 2 2 0 Lb = Fy δ(s − L b ) − L c δ (s − L b ) ⎤ ⎡L b 1 ∂ 1 C p V˙ (t) + V (t) = ⎣ θ (s)w 1 + w2 ds ⎦ R ∂t 2
(11)
(12)
0
where δ(s) is Dirac delta function. The boundary conditions associated with the above expression are given by w|s=0 = 0 w s=0 = 0
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! Dt Dt2 w¨ . At s = L b Y I (s)w = −Mt w¨ − It + Mt 2 4
Reduced-Order Model We express w(s, t), representing the lateral vibrations of the beam, in the form of a convergent series given by w(s, t) =
∞ "
φi (s)qi (t)
(13)
i=1
where qi (t) is the generalized time-dependent coordinates, and φi (s) are the set of mass-normalized orthonormal eigenfunctions. These eigenfunction also represents the mode shapes of defined configuration of beam with bluff body attached at the end. Substitution of Eq. (13) into (11) is done, and then, integration is done over the length of the beam by multiplying φn (s) on both the sides. The orthonormality properties of the eigenfunctions are used, and the following linearly decoupled set of nonlinear ordinary differential equations is formed for value of n = 1, 2, 3, …. q¨n + 2ζn ωn q˙n + ωn2 qn + θn∗ V (t) +
∞ "
Ani j qi q j V (t)
i, j
+
∞ "
∞ " Bni jk qi q¨ j qk + 2q˙i q˙k + q j q¨k + Cni jk qi q j qk = Fn (q, q) ˙
i, j,k
(14)
i, j,k
where L b ωn2
=
Y
I φn ds
0
Ani j =
L b =
φθ ds
0
1 2
L b
φn θ φi φ j ds
0
⎛ s ⎞ ⎤ L b s L b φn ⎣ ⎝ ⎠ ⎦ = M φ j φk ds ds ds Cni jk = φn φi Y I φ j φk ds φi 2 ⎡
Bni jk
θn∗
0
L
0
0
1 Dt 2 φn (L b ) + a1 α ∗ + a3 α ∗3 + a5 α ∗5 Fn = φn (L b ) ρa Dt L t U∞ 2 2
Nonlinear Modeling and Stability Analysis of Piezoelectric … ∗
α =
∞ "
φi (L b )qi
i
55
∞ 1 " + φi (L b )q˙i U∞ i
Also substituting Eq. (13) into (12), we will get C p V˙ (t) +
∞
∞
" V (t) " ∗ θn q˙n + An,i, j q˙n qi q j + qn q˙i q j + qn qi q˙ j = R n ni j
(15)
Equations 14 and 15 are the coupled governing equations of the system. These two equations are solved by using fourth-order Runge–kutta method. Geometric and physical parameters used to calculate the results are given in Table 1. Value of R is taken to be 172 × 103 [10].
4 Results and Discussion Phase portrait for galloping-based energy harvester before and after onset speed is plotted in Fig. 5 and 6, respectively. Only linear term in the polynomial expansion of lateral force is used. Before onset speed given any initial condition, the velocity and amplitude of vibration go to zero, but after onset speed amplitude and velocity drastically increases. Initial conditions are taken in order of 10−7 . The amplitude of vibration remains constant for a particular speed in practice. So, linear representation of aerodynamic force gives erroneous result. Variation of tip deflection with wind speed for a linear system is shown in Fig. 4. Taking cubic term also in consideration variation of tip deflection with wind speed is represented by Fig. 3. System bifurcates at wind speed 4 m/s and goes into limit cycle oscillation. This type of bifurcation is called Hopf bifurcation. Figure 8 depicts the variation of amplitude of vibration of the system with wind speed for a fifth-order representation of aerodynamic force. The effect of linear and nonlinear representation of Euler–Bernoulli beam theory is also shown. There is a sudden jump in amplitude due to the present of subcritical Hopf bifurcation. There is a hysteresis present due to this instability. Variation of voltage with wind speed is given in Fig. 7. Electrical power can be calculated using the formula: P=
V2 R
Therefore, electrical power corresponding to wind speed 7 m/s can be calculated to be 0.36 mW.
E p (pa)
b p (m)
h p (m)
L p (m)
190
1.4 × 10−2
1.2 × 10−3
14.5 × 10−2
E s (pa)
h s (m)
L b (m)
ρ p (kg/m3 )
Parameters
bs (m)
9873
ρs (kg/m3 )
× 109
Values
Parameters
8.5 × 10−2
.4 × 10−3
7 × 10−3
62
× 109
7800
Values
Table 1 Geometric and physical parameters of the energy harvester [10]
Mt (kg)
L t (m)
Dt (m)
d31 (m/V)
e33 (F/m)
Parameters
15 × 10−3
10.2 × 10−2
2 × 10−2
−320
a5
a3
a1
ζ
ρa (kg/m3 )
27.3 × 10−9 × 10−12
Parameters
Values
−437.036
8.24
2.3
.005
1.24
Values
56 R. C. Dash et al.
Nonlinear Modeling and Stability Analysis of Piezoelectric … Fig. 3 Amplitude versus wind speed for a third-order aerodynamic damping system
57
0.05
Amplitudde (m)
0.04 0.03 0.02 0.01 0.00
0
1
2
3
4
5
6
U (m/s)
Amplitude (m)
Fig. 4 Amplitude versus wind speed for a linear aerodynamic damping system
8.0x10
41
6.0x10
41
4.0x10
41
2.0x10
41
0.0 0
1
2
3
4
5
U (m/s)
5 Conclusions In the present work, closed-form solution of galloping-based beam energy harvester is discussed. Governing differential equation for energy harvester has been derived and solved considering nonlinearity in both geometry and damping. Comparison of amplitude variation for both geometric linear and nonlinear model is done for different wind speed. Both harvesters have a bluff body of square cross section attached at the tip. Amplitude and voltage variation for different wind speed are calculated and plotted for a beam harvester. Using quasi-steady theory, lateral lifting force is represented by polynomials of odd order. Effect of the coefficient of different order polynomial on galloping behavior is observed. • Linear coefficient a1 has a significant impact of onset speed of galloping.
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Fig. 5 Phase portrait before galloping speed for a linear aerodynamic damping system
Fig. 6 Phase portrait after galloping speed for a linear aerodynamic damping system
• Nonlinear coefficients have a significant impact on the type of instability (bifurcation). It is found from the above study that dynamic behaviors of energy harvesting system are totally dependent on the representation of the galloping force. The performed study shows the cruciality of aerodynamic force representation in order to design an accurate model of galloping-based piezoelectric energy harvesters.
Nonlinear Modeling and Stability Analysis of Piezoelectric … Fig. 7 Voltage versus wind speed using both linear and nonlinear Euler–Bernoulli beam theory
59
Modarately large deflection theory Small deflection theory
8
Voltage (V)
6
4
2
0 0
1
2
3
4
5
6
7
8
U (m/s) 0.07
Fig. 8 Tip deflection versus wind speed using both linear and nonlinear Euler–Bernoulli beam theory
Modarately large deflection theory Small deflection theory
0.06
Amplitude (m)
0.05 0.04 0.03 0.02 0.01 0.00 0
1
2
3
4
5
6
7
U (m/s)
References 1. Hartog JPD (1984) Mechanical vibrations. McGraw-Hill, NewYork 2. Novak M (1969) Aeroelastic galloping of prismatic bodies. ASCE J Eng Mech Div 95(1):115– 142 3. Barrero-Gil A, Alonso G, Sanz-Andres A (2010) Energy harvesting from transverse galloping. J Sound Vib 329(14):2873–2883 4. Abdelkefi A, Hajj MR, Nayfeh AH (2012) Power harvesting from transverse galloping of square cylinder. Nonlinear Dyn 70(2):1355–1363 5. Sirohi J, Mahadik R (2012) harvesting wind energy using a galloping piezoelectric beam. J Vib Acoust 134(1):11009
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6. Yan Z, Abdelkefi A, Hajj MR (2014) Piezoelectric energy harvesting from hybrid vibrations. Smart Mater Struct 23(2) 7. Yan Z, Abdelkefi A (2014) Nonlinear characterization of concurrent energy harvesting from galloping and base excitations. Nonlinear Dyn 1171–1189 8. Adali S (2015) Vibration of a cantilever beam with extended tip mass and axial load subject to piezoelectric control. Res Dev J 31(1):60–65 9. Bibo A, Daqaq MF (2018) Modeling and characterization of a piezoelectric energy harvester under combined aerodynamic and base excitations. J Vib Acoust 137:1–12 10. Javed U, Abdelkefi A (2017) Impacts of the aerodynamic force representation on the stability and performance of a galloping-based energy harvester. J Sound Vib 400:213–226
Optimization of Surface Roughness of Laser Trepanned Hole in ZTA Plate Surendra K. Saini, Avanish K. Dubey, and B. N. Upadhyay
Abstract Zirconia toughened alumina (ZTA) is employed to make components for aerospace, chemical, biomedical and cutting tool industries due to its excellent hardness, fracture toughness and strength. Improved properties of ZTA make it difficult-to-cut ceramic composite. Achieving a better surface quality in machining of ceramic composites has been very challenging due to the presence of surface cracks. Recent researchers have revealed that laser beam machining can overcome the machining limitations of ceramic composites. The present study tries to optimize the surface roughness of laser cut holes in ZTA plate using artificial intelligence tool. The optimum result shows an improvement of 17.5% in surface finish as compared with surface finish obtained at non-optimal parameters levels. The predicted optimum results have been tested by confirmation experiments. Keywords ZTA · Laser trepan drilling · Surface roughness · Genetic algorithm
1 Introduction Unfavorable machining characteristics such as hardness, fracture toughness and brittleness of zirconia toughened alumina (ZTA) make it difficult-to-cut ceramic composite. ZTA is used to manufacture armor, nozzles and biomedical components, etc. [1, 2]. Traditional hole generating processes found insignificant for ceramic composites due to their own limitations [3, 4]. However, few advanced hole generating processes are used for ceramic-based composites but they also have their limitations [5]. Laser beam machining is a nonconventional machining process which used focused laser beam of high intensity to machine any type of materials [6, 7]. Laser trepan drilling (LTD) is a laser hole generating process employed for generally S. K. Saini (B) · A. K. Dubey Department of Mechanical Engineering, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh 211004, India B. N. Upadhyay Laser Development and Industrial Applications Division Lab, Raja Ramanna Centre for Advanced Technology, Indore, Madhya Pradesh 452013, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_8
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holes above 1.2 mm diameter in difficult-to-cut materials [2]. Study of different pulse duration-based laser drilling process for different structural ceramics has reviewed by Wang et al. [8]. Bharatish et al. [9] studied the effect of hole diameter, laser power, pulse frequency and scanning speed on hole taper (HT), hole circularity and heat-affected zone (HAZ). They found laser power and hole diameter as significant parameters on hole circularity, while laser frequency and laser power on HAZ and HT. Adelmann et al. [10] performed laser drilling on aluminum nitride and alumina ceramics. They reported that lower laser focus position yields lesser HT. Saini et al. [11] done laser drilling on YSZ ceramic. They determined optimum hole quality characteristics using artificial neural network. Saini et al. [12] studied recast layer, microhardness and microcrack width in laser trepanned hole in ZTA. They reported that trepanning speed and air gas pressure were significant factors for considered hole characteristics. Zhang et al. [13] have done laser trepan drilling on silicon nitride ceramic. They obtained high RLT at high assist gas pressure. Murray and Tyrer [14] have done laser drilling on partially stabilized tetragonal zirconia (PSTZ) ceramic. They investigated the effect of heating temperature on recast layer microcracking. They reported that heat treatment before and after drilling of PSTZ reduced the formation of microcracks. Preliminary literature review revealed that most of the researchers have studied hole diameter, circularity, recast layer, HAZ and HT of laser drilled hole. But study of laser trepanned hole surface quality seldom is found. The present research paper investigates the optimum set of control factors for surface roughness in laser trepan drilled hole of 6.0 mm thick ZTA. Regression model has been developed from experimental observations. Further, developed model has been used as objective function to elect the optimum set of control factors using artificial intelligence tool (i.e., genetic algorithm) within range of control factors. The confirmed optimum results have been compared with best experimental observation.
2 Experimentation Laser trepan drilling experiments have been conducted on ZTA (6.0 mm thick) using pulsed Nd:YAG laser. Table 1 shows the range and levels of selected control factors. Figure 1 shows the measurement procedure of drilled hole surface roughness. The observed experimental values are drawn by Fig. 2 [15]. Table 1 Control factors with ranges and levels Control factors
Ranges
Levels −2
−1
0
1
2
Pulse width (x 1 ) (ms)
4
12
4
6
8
10
12
Pulse frequency (x 2 ) (Hz)
8
14
8
9.5
11
12.5
14
Trepanning speed (x 3 ) (mm/min)
5
40
5
10
20
30
40
Assist gas pressure (x 4 ) (kg/cm2 )
8
10
8
8.5
9
9.5
10
Optimization of Surface Roughness of Laser Trepanned Hole …
63
Fig. 1 Representation of surface roughness measurement of laser trepanned hole
Surface Roughness (μm)
Surface Roughness (μm)
10 8 6 4 2 0 1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31 Number of experiment
Fig. 2 Laser trepanned hole surface roughness corresponding to number of experiment
3 Modeling and Optimization To realize the influence of LTD process on drilled hole surface quality characteristic, a second-order regression model is developed using response surface method, and it is represented by Eq. (1) [16]. Y K = β0 +
u i=1
βi X i +
u i=1
βii X i2 +
u u
βi j X i X j
(1)
i=1 j=u+1
where Y K is the hole characteristic, βs and β0 are regression coefficient and constant, and u is number of control factors, while X i = linear, X i2 = square, andX i X j = interaction of control factors. In the developed regression model, back elimination approach using p-value has used for eliminating the insignificant terms. The p-value of significant control factors
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S. K. Saini et al.
is considered less than 0.05 [16]. Equation (2) represents the developed regression models for surface roughness. YSR = 208.7 − 12.23x1 − 1.51x2 − 1.701x3 − 30.3x4 + 0.2868x22 + 0.00405x32 + 1.382x42 + 0.0334x1 x3 + 1.296x1 x4 − 0.572x2 x4 + 0.1417x3 x4
(2)
To validate the developed regression model, calculated F-ratio of analysis of variance (ANOVA) results has been compared with critical F-ratio that found less than calculated F-ratio value at the confidence level (95%). Source of regressions pvalues is less than 0.05. S-value, R-sq and R-sq (adj) developed regression models are 0.8095, 86.9% and 79.32%, respectively. From these values, it may be summarized that developed regression model is adequate and reliable at the 95% confidence level. Next, robust optimization technique, i.e., genetic algorithm (GA) has used to determine the optimum value of surface roughness and their corresponding value of control factors [17]. The inbuilt code of GA for single optimization in MATLAB® software is used for optimization. The developed regression model of response has employed as objective functions. The objective of present research work is to minimize the drilled hole surface roughness. Therefore, functions YSR (Eq. (2)) need to be minimized within boundary of control factors (as 4 ≤ X 1 ≤ 12; 8 ≤ X 2 ≤ 14; 5 ≤ X 3 ≤ 40; 8 ≤ X 4 ≤ 10). After defining the objective function in MATLAB file editor, next step is to select the GA optimization parameters such as crossover probability, population size, number of generation and mutation probability. In this study, crossover probability 0.95, population size 40, number of generation 100 and mutation probability 0.01 have been employed after distinct trials to get desired results. After choosing all above parameters, GA optimization process was started. The optimization process has been terminated after satisfaction of termination criteria (objective/fitness function is optimized when mean fitness curve converges with best fitness curve without any further improvement in the best fitness value). Figure 3 shows the fitness and optimum value for drilled hole surface roughness. The predicted optimum results compared with best experimental value of response. The comparison result is shown in Table 2 which shows percentage improvement 17.5%. To verify the optimized result obtained from GA, confirmation test has also been performed for surface roughness at predicted optimum parameters levels. The confirmation test results revealed that variation between optimum predicted and confirmed values for response surface roughness is 4.69% as shown in Table 2. The microscopic images of surface roughness of drilled hole at non-optimal and optimal parameter levels are shown in Fig. 4.
Optimization of Surface Roughness of Laser Trepanned Hole …
65
Fig. 3 Fitness value and optimum values of drilled hole surface roughness
Table 2 Optimum result for surface roughness Value of control factors
Response
X1
X2
X3
Best experimental value
4.000
12.539
34.050
9.866
1.857
Predicted optimum value
4.000
12.539
34.050
9.866
1.532
Confirmed optimum value
4.000
12.500
34.000
10.000
1.460
Improvement (%) Variation (%)
X4
Surface roughness (µm)
17.50 4.69
Fig. 4 SEM images of surface roughness at a non-optimal parameter level, b optimum parameter level
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4 Conclusions Trepan drilling experiments have been conducted on 6.0 mm thick ZTA using pulsed Nd:YAG laser, successfully. Developed second-order regression models found reliable at 95% confidence level. Drilled hole surface roughness has been reduced by 17.5% at pulse frequency = 12.5 Hz, pulse width = 4 ms, trepanning speed = 34 mm/min and assist gas pressure = 10 kg/cm2 . The observed SEM images of drilled hole surface at non-optimum and optimum control factor levels also confirm the above improvement. The confirmation test result shows errors 4.7%. Acknowledgements Financial support for present work was granted by Aeronautics R & D Board, New Delhi, Government of India through the sanction order no. AR&DB/01/2031751/M/I. The authors also thankfully acknowledge the financial assistance provided by TEQIP-III.
References 1. Wang J, Stevens R (1989) Review zirconia toughened alumina (ZTA) ceramics. J Mater Sci 24:3421–3440 2. Saini SK, Dubey AK, Upadhyay BN, Choubey A (2018) Study of hole characteristics in laser Trepan drilling of ZTA. Opt Laser Technol 103:330–339 3. Jain VK (2007) Advanced machining processes. Allied Publishers Private Limited Delhi 4. Tuersley IP, Jawaid A, Pashby IR (1994) Review: various methods of machining advanced ceramic materials. J Mater Process Technol 42:377–390 5. El-Hofy H (2005) Advanced machining processes. McGraw Hill Mechanical Engineering Series, New York 6. Samant AN, Dahotre NB (2009) Laser machining of structural ceramics—a review. J Eur Ceram Soc 29:969–993 7. Dubey AK, Yadava V (2008) Laser beam machining—a review. Int J Mach Tools Manuf 48:609–628 8. Wang H, Lin HT, Wang C, Zheng L, Hu X (2017) Laser drilling of structural ceramics-a review. J Eur Ceram Soc 37:1157–1173 9. Bharatish A, Murthy HNN, Anand B, Madhusoodana CD, Praveena GS, Krishna M (2013) Characterization of hole circularity and heat affected zone in pulsed CO2 laser drilling of alumina ceramics. Opt Laser Technol 53:22–32 10. Adelmann B, Hellmann R (2015) Rapid micro hole laser drilling in ceramic substrates using single mode fiber laser. J Mater Process Technol 221:80–86 11. Saini SK, Dubey AK, Pant P, Upadhyay BN, Choubey A (2017) Study of laser drilled hole quality of Yttria Stabilized Zirconia. Lasers Manuf Mater Process 4:121–135 12. Saini SK, Dubey AK (2019) Study of material characteristics in laser trepan drilling of ZTA. J Manuf Process 44:349–358 13. Zhang J, Long Y, Liao S, Lin HT, Wang C (2017) Effect of laser scanning speed on geometrical features of Nd:YAG laser machined hole in thin silicon nitride sub-strate. Ceram Int 43:2938– 2942 14. Murray AJ, Tyrer JR (1999) Nd:YAG laser cutting and drilling of PSTZ-influence of substrate heating temperature on recast layer microcracking. J Laser Appl 1:128–135 15. Saini SK, Dubey AK, Upadhyay BN (2019) Study and optimization of recast layer thickness and surface quality in laser trepan drilled ZTA. Int J Adv Manuf Technol https://doi.org/10. 1007/s00170-019-03704-3
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16. Montgomery DC (2015) Design and analysis of experiments, Wiley 17. Deb K (2012) Optimization for engineering design. PHI Learning Private Limited, New Delhi
PI Control-Based Modelling of Segway Using Bond Graph A. Kumar, R. Singh, T. K. Bera, and Ashish Singla
Abstract A personal transporter vehicle, called segway, is based on the stabilization principle of inverted pendulum system. In this work, the bond graph model of segway with PI control is developed. The simulation results for the forward and backward motion of segway are presented as pitch angle, speed of the vehicle with respect to time. For the turning of the vehicle, controller based on Ackermann steering mechanism is adopted to modulate the voltages of two motors. The results of turning motion of the segway are also presented as yaw angle response of main vehicle body. The focus of this paper is to develop the dynamic model and simulate the response of segway using non-model-based control system design. The objective of the work is accomplished by developing the stabilization controller based on PI control scheme using bond graph approach. Also, the swing up controller is developed for nonlinear behaviour of the segway. Keywords Segway · PI controller · Bond graph · Stabilization control · Swing up control
1 Introduction The study on modelling and balancing of two-wheeled robots (segway) has gained momentum over the last decades worldwide in robotics domains. This is due to the inherent unstable dynamics and nonlinearity of such systems. The principle of inverted pendulum system has been employed into segway, which is a self-balancing human transporter and looks like a high technological scooter (see Fig. 1a). The configuration of segway consists of two wheels and an inverted pendulum as a handle attached to the vehicle body [1–3]. To move forward or backward, the rider just has to lean the pendulum forward or backward. The segway consists of a control system, series of sensors and driving system. Segway is equipped with two sensors, i.e. a A. Kumar · R. Singh (B) · T. K. Bera · A. Singla Thapar Institute of Engineering and Technology, Patiala Punjab 147001, India A. Singla e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_9
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(a)
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Fig. 1 a Concept of balancing a broomstick on index finger applied to segway and b schema of a segway
gyroscopic sensor and an accelerometer for measuring the pitch angle and pitch angular velocity. The system is stabilized with the help of a feedback gain and delay [1]. The system is controlled by feedback linearization technique using LQR controller [2]. Two types of control techniques are used to control the two-wheeled robotic machine [2]. Gyroscopic sensor with Kalman filter is used to measure the pitch angle of the vehicle. The dynamics of the model are derived with the help of Lagrangian mechanics [3]. Control algorithms and signal processing are distributed among the three microprocessors, one for each drive and one is responsible for the stabilizing control [4]. One is the proportional derivative control and other one is the fuzzy logic control to balance the vehicle [5]. By using the liberalized models, various control laws have been applied to the wheeled inverted pendulum [6, 7]. The two-wheeled inverted pendulum system is implemented and designed with the compensation of the friction. The stabilization concept of inverted pendulum system has been employed in this paper to form a segway. The simulation results for straight line motion as well as turning motion are also presented. The segway moves forward if the handle bar is tilted in the front and moves backward if the handle bar is tilted in the rear. For the left turn, the bar is turned towards left and for the right turn, and the bar is tilted towards right. The motivation of this paper is the validation through modelling and simulation of a non-model-based control system design, i.e. energy-based control technique is used. Modelling and simulation play a vital role in the area of engineering design process. The accurate mathematical description of any system provides an easy way to understand the system in very quick and accurate manner. The data and information about any system can be obtained from modelling and simulation before implementing it in real-life situation. Bond graph (BG) model is an efficient tool for modelling engineering systems. Bond graph modelling technique is a way to describe the dynamic behaviour of a physical system graphically. The technique is very similar to the block diagram and signal flow graph, but the major difference is that the bonds in the bond graph represent bi-directional energy exchange [8, 9]. The
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segway (vehicle) is composed of an inverted pendulum attached to the vehicle body as shown in Fig. 1b, which is connected to the two wheels through an axle. There is wheel bushing system located between the wheel axle and mounting point. The bushing system is modelled as a parallel spring damper system. The structure of the paper is as follows. Initially, bond graph modelling of segway is discussed and developed in Sect. 2. The word bond graph model of the segway is presented first and then final bond graph model of segway is developed. The modelling of various components used such as vehicle body, handle of segway, wheel, wheel bushing, electric motor with PI controller and steering mechanism is also done. In Sect. 3, the control scheme of segway is discussed. The control approach is applied on single wheel of the segway. In Sect. 4, simulation results of segway are discussed. The two cases are further taken into consideration for validation of the proposed model of the segway. At last, conclusions are drawn in Sect. 5.
2 Bond Graph Modelling of Segway 2.1 Word Bond Graph of Segway The modelling of segway is done by the combination of five sub-models, i.e. vehicle body, wheel, handle, electric motor and wheel bushing system. Figure 2 represents the whole system, segway in word bond graph form where the bonds with two parallel lines denote multi-bonds between two modules. The bushing system is used to connect the left and right wheels to the main body of the vehicle. The electric motors are connected to the main body of the vehicle and the two wheels with the help of scalar bonds. Likewise, the inverted pendulum (handle) is also connected to the main vehicle body via scalar bonds.
Fig. 2 Word bond model of segway
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Fig. 3 a Sketch for illustration of velocities in the moving frame and b bond graph modelling of vehicle body and attached components
Vehicle Body. The model of vehicle body is shown in Fig. 3a. The vehicle is assumed to be symmetrical about the longitudinal axis. The bushing is taken as a simple linear spring damper system. The main vehicle body’s motion can be expressed as the three linear displacements about the three body fixed coordinate axis, and rotational motion of the body is given as yaw, pitch and roll motion. The Newton–Euler (NE) equations are used to model the main body of the vehicle [12]. These equations contain attached body fixed axis which are aligned to the inertia principal axis. The Newton–Euler equations are given in [10–12]. Now, further step in the modelling is to compute the velocities at different points on the rigid body and transforms them to the inertial coordinate system and physical constraint implementations at chosen points. For three linear velocities of left bushing reference point (Fig. 3a) in the moving system of axes, the equations can be written as x˙1 = x˙c + z 1 θ˙cy − y1 θ˙cz
(1)
y˙1 = y˙c + x1 θ˙cz − z 1 θ˙cx
(2)
z˙ 1 = z˙ c + y1 θ˙cx − x1 θ˙cy
(3)
Same type of expressions can be written for the right-side bushing. The transformation of velocity from the moving frame to the inertial frame is given in detail in the appendix and Euler angle rates from the body fixed angular velocities are given in [10]. Figure 3b represents the models of main vehicle body and transformation of velocities (linear to angular) to the bushing reference points. Three moments and forces (three sets each) act upon the body. The bond graph coordinate transformations block (CTF) for coordinate transformation from inertial to moving frame and from moving frame to inertial frame are given in [10] and also explained in the
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Fig. 4 Bond graph model of inverted pendulum attached with segway
appendix. As the weight of the vehicle body and the aerodynamic forces act in the inertial frame, these forces are to be converted to body fixed forces through coordinate transformation before being applied to the vehicle body. Here, in this simulation, the aerodynamic forces are neglected as the speed of the segway is not too high. Modelling of Handle (Inverted pendulum) for Segway. The bond graph modelling for inverted pendulum system, which is attached to the vehicle main body is shown in Fig. 4. The inverted pendulum system is connected to the main body of the vehicle through the two coordinate transformation blocks. The angle θ shown in the bond graph of inverted pendulum in Fig. 4 represents the pitch angle, which is to be controlled by the PI controller. Model of Wheel. The schematic diagram of wheel is shown in Fig. 5a. The bond graph model for the wheel is shown in Fig. 5b. It is modelled as a rigid body having five degrees of freedoms and the rolling about x-axis is neglected here. There are six input ports in the bond graph modelling of wheel. Three effort inputs are coming for the bushing and torque is supplied from the differential. The contact point between the wheel and road is not a fixed point. It changes as per the wheel rotation about the axle. The gravity force and the normal contact force between the wheel and road always act in the inertial direction. The weight of the wheel is acting in inertial zdirection, which can be represented by source of effort (Se: wm g). The mass of the wheel is represented as (I : wm ) and rw is the radius of the wheel. The tractive force, which is used to generate the motion between the wheel and road surface, is given by Se : Ft . The mass moments of inertias are represented by Iwy and Iwz in y- and z-directions, respectively. Model of Wheel Bushing. The schematic diagram of wheel bushing is shown in Fig. 6a. The bond graph model of wheel bushing that is located between the wheel axle and mounting point is shown in Fig. 6b. The bushing is modelled as a parallel spring damper system. In the bond graph model, the stiffness (C: K S ), damping
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Fig. 5 a Schematic diagram and b bond graph model of wheel
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Fig. 6 a Schematic diagram and b bond graph model of wheel bushing
(R : R S ) in x-, y-directions and stiffness (C: K z ) and damping (R : Rz ) in inertial z-direction are connected to 0-junction between the flow input from the vehicle body (mounting point) and effort output to the wheel. Model of Electric Motor with PI Control. The bond graph modelling for electrical motor with PI controller is shown in Fig. 7a. Two separate electrical DC motors are used for the motion of the segway. The PI controller is used a stabilization controller for balancing the pitch angle θ (as shown in Fig. 7) of inverted pendulum attached to the main vehicle body of segway. In case of PI controller, the desired value (θd ) is the inverted position of the pendulum, and feedback value (θ ) is the value of the pitch angle measured by the sensor. The error between these flows is converted to the voltage error signal with the help of gain V v . Now according to the voltage error signal, a control signal is generated by the PI controller to modulate the voltage applied to the motor. In this block diagram representation, G P is the proportional gain and G I is the integral gain. In the bond graph modelling of motors (Fig. 7a), there are two output ports from 0-junction providing the equal torque to both the left/right wheel and vehicle body.
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Fig. 7 Bond graph model of a motors and b steering controller
Model of Steering Controller. The steering is provided for the turning motion of the vehicle. The bond graph modelling for steering controller is given in Fig. 7b. Steering is basically a moment which has applied to the axles of the two wheels about z-axis causing the change in the yaw angle of the wheel (also the relative torque on the body of the vehicle). The steering is provided by changing the rotational speed of the motors. The speeds of the motors are modulated by and where and are the angular position of the wheel about z-axis. These angular positions are determined by steering controller. The modulus of two transformers and as shown in Fig. 7b are determined by the Ackermann’s formulae [10] in the controller domain and these are given as a cos2 θ1 + c tan θ1 cos2 θ1 δ˙ a cos2 θst − c tan θst cos2 θst a cos2 θr − c tan θr cos2 θr ˙δo = δ˙ a cos2 θst + c tan θst cos2 θst δ˙i =
(4)
(5)
3 Bond Graph Modelling of Segway 3.1 Word Bond Graph of Segway Initially, single wheel model of segway without controller (Fig. 8) is developed. M2, M6, M35, M16, M20, M23 and M30 are inductance of motor, flywheel along with wheel mass moment of inertia, mass of wheel, mass of vehicle (segway), pendulum mass, mass moment of inertia of pendulum rod and pendulum mass, respectively. The signal flow graph of the single wheel model is shown in Fig. 9.
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Fig. 8 Single wheel model of segway without controller
Fig. 9 Signal flow graph of single wheel model of segway without controller
The transfer function between the angular velocity of segway arm and the voltage applied to the motor, when linearized about the vertical position is
(6) From the open-loop transfer function of the system as given in Eq. (6), it is found that there is a double pole at the origin, which makes the system unstable. Rest all the poles are either at imaginary axis or in the left-half plane. Further, the controllability of the open-loop system is calculated and it is found that all the 11 states are controllable. The bond graph model of single wheel model of segway with PI controller is shown in Fig. 10. The transfer function between the angular velocity of segway arm and the voltage applied to the motor with PI controller is
(7) After implementing the PI controller, all the closed-loop poles are lying in the left-half plane, which ensure the stability of the system. When PI controller is added in the forward loop, there is a new pole that lies at the negative real axis
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Fig. 10 Single wheel model of segway with PI controller
(−0.51 + 0. (i)) due to the use of integral term and a new zero is also located at (−0.64 + 0. (i)) on the same axis.
4 Bond Graph Simulation Result of Segway The parameter values for simulating segway are given in Table 1. The bond graph simulation results for the segway are given here by simulating the various conditions. Case 1: Response of the segway with a stabilization controller The bond graph simulation results for the pitch angle of handle bar/pendulum and speed response of segway are presented in Fig. 11, when a constant force is provided to the handle in the forward direction. A constant force of 11 N is provided in the forward direction to the handle of segway. As the force is provided to the handle in the forward direction, the pitch angle is displaced for its reference position (inverted position). For the vehicle, the control is designed in such a manner that the vehicle will not move in the forward or backward direction to balance the force on the vehicle due to change in the pitch angle until the pitch angle is displaced through 15° from its reference position. Up to the time equals to 0.8 s, the vehicle will not move and the linear and angular velocities of the vehicle will be zero. When the pitch angle is displaced through 15° in the forward direction from the reference position, the stabilization control part (PI controller) will be activated. The voltage signal is applied to the motor to minimize the change in the pitch angle. As a result of this, torque is applied to the two wheels and vehicle starts moving in the forward direction. As the force will be continued in the forward direction, the vehicle will continue to move in the forward direction. Now, at 2 s, the force from the handle is removed and
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Table 1 Simulation parameters for segway Nomenclature
Description
Values
a
Distance of the axle form the vehicle cg
0.05 m
Gp
Proportional gain
1.0
Gi
Integral gain
1.21
h
Height of the vehicle cg from bushing reference point
0.05 m
J cx , J cy , J cz
Moment of inertia of vehicle about x-, y- and z-directions
26, 111, 137 kg m2
J
Mass moment of inertia of pendulum about z-direction
1 kg m2
Jm
Mass moment of inertia of flywheel
0.1 kg m2
J wy
Mass moment of inertia of wheel about y-direction
2 kg m2
J wz
Mass moment of inertia of wheel about z-direction
2 kg m2
K
Stiffness
105 N/m
K sx , K sy
Bushing stiffness in x- and y-direction
105 N/m
K sz
Bushing stiffness in z-direction
109 N/m
Kw
wheel stiffness in z-direction
106 N/m
L
Length of pendulum
0.5 m
L
Inductance of motor
1.0 H
M
Mass of the pendulum
5 kg
mc
Mass of the vehicle body
160 kg
rw
Radius of wheel
0.3 m
Rd
Viscous damping of motor
0.03 Ns/m
Rm
Resistance of motor
2.5
Rsx , Rsy
Bushing damping in x- and y-direction
1.6 × 105 Ns/m
Rsz
Bushing damping in z-direction
2 × 105 Ns/m
Rw
Damping in z-direction
103 Ns/m
Vv
Voltage to angular displacement conversion gain
118 V/rad
wm
Mass of wheel
15 kg
Θ st
Steering input
0.1 rad/s
θ max
Maximum angle rotation
0.25 rad
M
Coefficient of friction between tyre and road
0.98
μm
Motor torque constant
1 Nm/A
PI controller will try to take the handle to the reference position (inverted position). But due to inertia, the pitch angle is displaced in backward direction due to forward motion of segway. And then due to this change in the pitch angle, segway moves in the backward direction to minimize the change in pitch angle, and finally, vehicle comes to the rest after 4 s and handle will come to its reference position after a little bit oscillations about the reference position in 7 s. The plot for pitch angle with time (Fig. 11a), plot for linear speed of the vehicle with time (Fig. 11b) and plot for angular speed of the wheel with time (Fig. 11c) are shown.
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Fig. 11 Plot for a pitch angle, b linear speed of the vehicle, c angular speed of the wheel for positive force applied to the handle and d pitch angle for backward force
Case 2: Yaw response of segway The simulation results are shown here in which a constant positive voltage of 68 V is provided to the motors, torque is given to the wheels of the vehicle, and vehicle moves in the forward direction with a linear speed of about 5 m/s. Then, after 25 s, the voltage to the left motor of the wheel is reduced to 62 V unless yaw angle of the wheel reaches to 0.25 rad about the z-axis of the wheel. The maximum angle of rotation is limited to the 0.25 rad. In first plot, Fig. 12a shows the linear speed of the vehicle with time. This plot shows that after 25 s, there is decrease in the vehicle speed as the turning motion starts and also there is change in the yaw angle of the wheel (Fig. 12b). As the yaw angle of the wheel reaches to the 0.25 rad, wheels stop turning further and the vehicle moves with a constant speed of 4.8 m/s. After 50 s, the voltage applied to the left motor is increased from 62 to 68 V for the returned back of the segway. As the steer is returned back, the speed of the vehicle increased up to the earlier speed of 5 m/s. The yaw angle of the wheel also comes to the initial position before steering. The variation of the yaw angle of the vehicle is shown in Fig. 12c. There are very small changes in the pitch angle during the steering motion as shown
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in Fig. 12d. The angular speeds of left and right wheels are shown in Fig. 12e, f, respectively. The speed of the left wheel is lesser than the speed of the right wheel during turning of the segway.
Fig. 12 Plot for a linear speed of vehicle, b yaw angle at the centre, c yaw angle of the vehicle body, d pitch angle of vehicle, e angular speed of left wheel and f angular speed of right wheel during steering
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5 Conclusions Bond graph modelling technique is very efficient tool for modelling any physical and mechanical system, and simulating various kinds of conditions for the same. As a segway is an under-actuated, nonlinear and unstable system, designing controller for such a system is a very complicated task. In this paper, an effort was made to develop the stabilization and swing up controller for segway using bond graph approach. Also, the bond graph modelling of the segway and its various parts such as vehicle body, handle, wheel, wheel bushing and electric motor with PI control were developed. The proportional integral (PI) control was well demonstrated to be capable of measuring the change in the pitch angle from the reference position and generates a control input signal for forward and backward motion of the segway to minimize that change.
References 1. Martynenko G, Formal AM (2005) The theory of the control of a monocycle. J Appl Math Mech 69:516–528 2. Nawawi A, Ahmad MN, Osman JH (2008) Real-time control of a two-wheeled inverted pendulum mobile robot. Int J Comput Inf Eng 79:316–356 3. Dai F, Gao X, Jiang S, Guo W, Liu Y (2015) A two-wheeled inverted pendulum robot with friction compensation. Mechatronics 30:116–125 4. Pathak K, Franch J, Agrawal S (2005) Velocity and position control of a wheeled inverted pendulum by partial feedback linearization. IEEE Trans Rob 21:16–28 5. Goher KM, Tokhi MO, Siddique NH (2011) Dynamic modelling and control of a two wheeled robotic vehicle with a virtual payload. ARPN J Eng Appl Sci 9:131–150 6. Grepl R (2009) Balancing wheeled robot: effective modelling, sensory processing and simplified control. Eng Mech 16:141–154 7. Delgado S, Kotyczka P (2015) Energy shaping for the robust stabilization of the wheeled inverted pendulum, vol 48. IFAC, Elsevier Ltd., pp 093–098 8. Mukherjee A, Karmakar R, Samantaray AK (2006) Bond graph in modelling, simulation and fault identification. CRC Press, Florida 9. Karnopp DC, Margolis DL, Rosenberg RC (2000) System dynamics, modelling and simulation of mechatronic system. Wiley, New York 10. Bera TK, Bhattacharya K, Samantaray AK (2011) Evaluation of antilock braking system with an integrated model of full vehicle dynamics. Simul Model Pract Theory 19:2131–2150 11. Bera TK, Merzouki R, Ould Bouamama B, Samantaray AK (2012) Design and validation of the reconfiguration strategy for a redundantly actuated intelligent autonomous vehicle. Proc Inst Mech Eng Part I J Syst Control Eng 226(8):1060–1076 12. Bera TK, Bhattacharya K, Samantaray AK (2011) Bond graph model based evaluation of a sliding mode controller for combined regenerative and antilock braking system. Proc Inst Mech Eng Part I J Syst Control Eng 225(7):918–934
Strategic Coordination and Navigation of Multiple Wheeled Robots Buddhadeb Pradhan , Nirmal Baran Hui , and Diptendu Sinha Roy
Abstract Multi-robot navigation and coordination are addressed in this paper. All the robots are subjected to their own kinematic and dynamic constraints. Genetic algorithm tuned fuzzy logic-based motion planner is compared with the potential field-based motion planner. To avoid the conflicts during the navigation, two different coordination schemes, namely strategic and heuristic, are implemented. Results are compared through computer simulation. Simulation experiments were started with eight number of robots initially, and the number of robots has been increased up to 17 later on due to the need of coordination scheme for the maximum number of robots. Strategic coordination scheme along with the genetic fuzzy-based motion planner is found to perform better than the other combinations concerning the quality of solutions and time taken to reach the goal positions. Computational complexity of different methods has also been compared and presented. Keywords Multi-agent system (MAS) · Potential field method (PFM) · Fuzzy logic controller (FLC) · Genetic algorithm (GA) · Game theory · Coordination
1 Introduction Multiple robots negotiating in a dynamic workspace may confront collision among each other. Several approaches have been implemented, and different researchers solved the problem in a diverse way. But the prominent approach is still underdeveloped. Strategy-based coordination techniques are developed by most of the researchers, but in case of real robot experiments, it fails to show its efficacy. Heuristic coordination has also been attempted in a static environment. This paper proposes a game theory-based strategic coordination for solving multi-agent robotics problem. B. Pradhan · N. B. Hui (B) National Institute of Technology Durgapur, Durgapur 713209, India e-mail: [email protected] D. S. Roy National Institute of Technology Meghalaya, Shillong 793003, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_10
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Lots of researchers have put their effort to solve the mobile robot navigation problem. Meng et al. [14] studied a leader—follower coordinated tracking problem for multiple heterogeneous Lagrange systems to control a group of followers to track a time-varying global objective function (often denoted as a leader) by using only local information interactions. Peng et al. [6] solved the navigation problem considering it as a mixed-integer nonlinear programming problem. Here, the path of each agent is decomposed into collision-prone and collision-free segments. But, this approach is not at all suitable for real-time systems, as the environment is highly dynamic [6]. Morinaga et al. [15] discussed a motion planning problem for a spherical rolling robot actuated by two internal rotors that are placed on orthogonal axes. A motion planning strategy composed of two trivial and one nontrivial manoeuvre is devised. Neural network [8]-based motion planning as well as traditional methods like potential field method [17, 19] also have been implemented to solve this navigation problem. The artificial neural network (ANN) can learn the scenario of the environment, and a few researchers have successfully applied the neural network [8] to develop the motion planning algorithm of mobile robots. Motion planning as well as coordination techniques have also been developed simultaneously. Recently, Kia et al. [7] proposed a novel algorithm for networked robots which localize itself in a global coordinate frame by local dead reckoning and opportunistically corrects its pose estimate whenever it receives a relative measurement update message from a server. Load distribution strategy has been developed for cooperative manipulators, and kinematic constraints imposed for manipulator ensemble have also been done by Erhart and Hirche [3]. GA fuzzy [11] and neuro fuzzy [5] have been applied to solve such problems with dynamic obstacles. Obstacle avoidance problem [13] of mobile robots using FLC has also been studied by many researchers for years. In such a case, primary fact is to develop a high-efficiency rule set. The performance of FLC is influenced by its knowledge base (rule set) and the membership functions. Therefore, it is essential to adjust the rules of fuzzy controller and the membership functions of input and output variables to get better performance. Genetic algorithm has been applied to tune the input and output membership functions of the fuzzy logic controller [20]. Three different algorithms have been developed with different effects. Very few researchers as like [20] have also used the GA optimized FLC in this research problem. Most of them experimented on the regular (circle shape) obstacles for simulation. But these approaches might not work efficiently in the presence of irregular obstacles. Here, we have done simulation experiment based on the navigation problem of a wheel mobile robot in the unknown environment. FLC has been used to sense the data from knowledge base though it is manually constructed by authors. Then, a GA has been applied to optimize the membership functions as well as the rules set of the FLC. We have studied some extensive literature survey based on different research direction. Percentage calculation of different research direction has been presented via Fig. 1. Here, it is evident that researchers are focusing on coordination characteristics but very few researchers have successfully implemented strategic coordination on navigation problem [1, 10, 23]. In the present research, different approaches have been tried to solve the motion planning problem as well as coordination of multi-agent systems in the dynamic
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Fig. 1 Summary of studied literature review
environment. For the motion planning purposes, three different approaches have been developed, and some strategies are proposed to evolve the coordination’s among the agents. The remaining part of the paper is structured as follows: in Sect. 2, developed cooperative multi-agent systems have been studied. Results and discussion are presented and discussed in Sect. 3. Comparison with previous research works has been depicted in Sect. 4. Finally, some concluding remarks and the scopes for future work are discussed in Sect. 5.
2 Developed Cooperative Multi-agent Systems Multi-agent system (MAS) can be developed by multiple mobile robots participating in a dynamic environment. In such a system, all the robots will have a single objective to find their collision-free paths during navigation among themselves. In a competitive situation, a robot should search for not only collision-free path but also a time-optimal one satisfying all the kinematic and dynamic constraints [9] of the twowheeled robots [5]. The motion planning of all the robots can be made sequentially. For example, the motion of the first robot will be planned first, then the second robot and so on. When the motion of the first robot is planned, it is considered as planning robot, and others all become obstructing robot. If the Euclidean distance between the planning robot and any of the obstructing robots becomes less than the maximum distance (dmax ), the robot will have to slow down its maximum speed (vmax ). Not only that was depending upon the location of the obstructing robots concerning the goal direction; the robot may need to take a turn either in the right or left of goal direction to avoid collision with the obstructing robots. There may be some obstructing robots to a planning robot. However, to reduce the complication of the problem,
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the obstructing robot which is closest to the planning robot is treated as the critical robot, and motion of the planning robot is made based on the critical robot. Here, we consider two inputs (distance, di j and angle, θi for the ith planning robot treating jth robot as critical one) and two outputs (deviation, φi and acceleration αi for the ith planning robot) for the motion planner. All the inputs and outputs are restricted to some of their lower and upper limits. In this manner, the motion of all the robots is made. After that, positions and orientations of the robots are updated [5], and time step is updated to next one. This process is repeated for all the robots until or unless all of them reach the goal. Finally, total travelling time of each robot is calculated by multiplying with the number of time steps (say, n i for the ith robot) a robot has taken to reach the goal with the T (time step) value. However, last time step may not be the full-time step, and it will be a fraction of T say (ki ∈ 0–1) and depend on the distance between the planning robots to its goal (called as goal distance, dgoal ). Moreover, if goal distance (dgoal ) is less than the maximum distance (dmax ), motionplanning scheme is not activated, and no collision with the other robots is presumed. Total Rtravelling time taken by the robots to achieve the task for a particular scenario (n i + ki )T , where R denotes the number of robots. The primary requireis i=1 ment of this study is to minimize the total travelling time. Overall, motion planning scheme is explained via activity diagram in Fig. 2. In a conflicting situation, one robot might be treated as most critical to other robots. In such case, according to the process explained above, a motion planner will be providing reduced velocities and deviations for both the robots. It is a conflicting situation, and reduction of motion of both the robots is unnecessary. One might move along the goal direction with a maximum velocity of it, and the other may compromise its motion. But, the question is who will compromise and whom to be given the upper hand in a competitive environment, where the performance of each one of them is measured. Also, a robot will get this motivation only when they get some rewards for this. It demands a cooperation scheme. Proper strategies to be developed to decide who will cooperate and how much? We have developed two different motion planning approaches along with two cooperative schemes to solve this kind of problems, and they are explained below in brief.
3 Developed Motion Planning Approaches 3.1 Approach 1: Potential Field-Based Motion Planner This traditional method is very popular and effective. Due to the generation of attractive and repulsive forces, this method has shown the directions to calculate the deviation angle and other necessary things. The concept of this method has been taken from the literature [5, 19]. For more details, interested readers may go through [9].
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Fig. 2 Activity diagram of motion planning scheme
3.2 Approach 2: Manually Constructed Mamdani-Type Fuzzy Logic-Based Motion Planner The fuzzy theory can deal with uncertain and approximate information. In the field of mobile robot navigation, this technique is generated additional significance amongst researchers. In this approach, both the database and rule base of the fuzzy logic-based motion planner are extracted from the published literature [5]. Four each linguistic terms for deviation and acceleration and five each linguistic terms for angle and
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deviation is considered. Therefore, a rule base will have twenty antecedents (input combinations), and for each antecedent, there could be twenty different consequents. So, four hundred possible rules might exist in the rule base and out of which twenty rules must be there. Detail of this is available in the book [22].
3.3 Automatically Evolved Genetic-Fuzzy Motion Planner Due to the non-optimal nature of FLC, it is required to tune the FLC rule base as well as database. In this paper, knowledge base of FL-based motion planner has been designed manually. Thus, a binary-coded genetic algorithm (GA) has been used to tune the two input and two output-based knowledge base of FLC. Working principles of the genetic fuzzy system may be extracted from the book of Pratihar [22]. The information regarding GA strings used here has been extracted from [18, 21]. The essential purpose of GA is to optimize the travelling time taken by the robot and is mentioned in Eq. 1. Minimize total travelling time (T ) =
N 100
(n i j + ki j )T
(1)
j=1 i=1
where n i j and ki j can be understood from Sect. 2. A robot will take the least time to reach the goal if it does not deviate from its goal line and move with the maximum possible velocity. Therefore, above function can be minimized indirectly by minimizing the error due to deviation and acceleration of the robots. Therefore, fitness function of GA is considered as the half mean squared error (HMSE) of all the 100 training scenarios and expressed as HMSE =
N M 100 2 1 amax − ai jm + φi jm 2 200N × M j=1 i=1 m=1
(2)
where ai jm and φi jm are the acceleration and deviation of the ith robot in jth situation following mth time step. The above function is to be minimized satisfying all the constraints of the robots.
4 Implementation of Strategic Coordination During multi-robot navigation, following questions may be arise • Why is coordination needed? • When is it required? • How to implement?
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Let us discuss the answers to these queries. To survive daily life, human beings need to show coordination with either static things or dynamic elements. The attributes or elements might be living or non-living. As robots are designed/developed like human beings, so these characteristics are also to be incorporated. Otherwise, they may suffer at the time of navigation and cannot handle the conflicting situations. Different parameters are involved in developing coordination between robots [2, 12, 16]. Suppose, in a multi-agent system, there are four robots where robots 1 and 2 are treating each other to be critical. In such a situation, motion planner will ask both the robots to compromise their motion to avoid the collision. However, this is not required for both the robots, one may negotiate and other may not. The robot which compromises is called cooperative robot, and other is a selfish robot. Question is how to segregate them and is there any reward for becoming cooperative or penalty for showing non-cooperative. There could be at least three possible strategies for a robot. • No Coordination (NC) If the robot does not want to sacrifice its motion generated by the motion planner. • Full Coordination (FC) The robot is ready to sacrifice both the outputs which might take different deviation and acceleration other than the generated by the motion planner. • Partial Coordination (PC) The robot is ready to sacrifice only one output which might take different deviation but considers the same acceleration as generated by the motion planner. The robot is ready to sacrifice only one output which might take different deviation but consider the same acceleration as generated by the motion planner. Partial coordination is better than the full coordination; however, wheeled robots are limited by its turning speed and cannot take any amount of turn. Thus, we will have to identify the best option for a situation. The robot which will cooperate will be awarded a reward, and the non-cooperative robot will be given a penalty. Reward and penalty values are presented in Table 1.
Table 1 Reward and penalty of cooperative robots NC FC PC ⎡ φ − φ ⎤ φ−φm Reward (R) 0 m φmax T + ⎢ φmax ⎥ ⎢ ⎥ ⎣ a − am ⎦ T a max ⎡ φ ⎡ φ − φ ⎤ ⎤ φ−φm Penalty (P) m + φ T ⎢ φmax ⎢ ⎥ T ⎥ ⎢ ⎢ φ ⎥ ⎥ ⎣ amax − a ⎦ T ⎣ a − am ⎦ 2 + a a max Here, φ and φm are the deviations generated by the motion planner and revised deviation required to avoid the collision. a and am are the acceleration generated by the motion planner and revised deviation due to showing coordination
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Table 2 Pay-off matrix of Robot i
Robot i
NC
FC
PC
Robot j NC i i − PNC RNC − j j R − PNC NC i i RFC − − PFC j j R − PNC NC i i RPC − − PPC j j RNC − PNC
FC i i − PNC RNC − j j R − PFC FC i i RFC − − PFC j j R − PFC FC i i RPC − − PPC j j RFC − PFC
PC i i − PNC RNC − j j R − PPC PC i i RFC − − PFC j j R − PPC PC i i RPC − − PPC j j RPC − PPC
Now, the question is who will follow which strategy in a two robot facing the conflicting situation. Heuristic coordination can solve the said problem [19]. However, it lacks adaptability and not optimal in any sense. Thus, we are proposing a strategic coordination scheme based on game theory in this paper. Here, both the robots have three alternatives and are competing players. Gains or losses called pay-off and profit of one robot will be the loss of the other robot. Pay-off matrix for different strategies adopted by the robots is placed in Table 2. It suggests that multiple strategies can be developed. But in between nine developed strategies, the best plan will be selected. The concept of two-person zero-sum pure strategy games [4] is used to identify the best strategy, and accordingly, the revised motion of the robots is calculated. Finally, three motion planning approaches can be clubbed with the two coordination schemes (Heuristic or Strategic) to solve the navigation problems of multiple mobile robots and tested through computer simulations.
5 Results and Discussion The efficacy of the designed approach has been experienced through computer simulation to solve different cases of car-like robots. Three different cases have been considered with eight, twelve and seventeen robots navigating in a grid of 40 × 40 m2 . Some of the important parameters of the robots are mentioned below. • Dimension: Two-wheeled mobile robot (70 mm × 70 mm × 80 mm; wheel diameter: 40 mm) • Maximum Velocity: 4 m/s • Acceleration: 0.005–0.05 m/s2 • Time interval(dt): 2 s.
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Performances of the approaches have been quantified through computer simulations against one hundred random scenarios. A scenario is different from other regarding starting and goal positions of the robots.
5.1 GA parametric Study and Identification of Best Rule Base of FLC The optimal GA parameters are achieved from the GA parametric study and in Table 3 for three different cases. Optimal rule base for three different cases is also obtained and presented in Table 4. It can be observed that the robots are slowing down the acceleration and taking more deviation with the increase in the number of robots present in the environment.
5.2 The Need for Coordination Schemes Once the tuning of Approach 3 is over, all the three approaches along with the two coordination schemes are compared to solve the navigation problem of multiple mobile robots. Let us first try to identify the need of different coordination schemes. A robot “ j” can take n i j number of time steps (out of which last time step may be fractional) to reach the goal in a particular scenario “i”. Therefore, a robot “ j” can run 100 n number of time steps, and all the robots as a whole can traverse for a total i=1 8/12/17 100 i j i=1 n i j many time steps. It is sure that the motion planning schemes are j=1 not activated for all the time steps.
Table 3 Optimal parameters of GA for the three different cases for Approach 3 Cases Crossover Mutation Population size Maximum probability probability number of generations Case 1 (involves 8 robots) Case 2 (involves 12 robots) Case 3 (involves 17 robots)
0.5
0.009
90
90
0.011
70
100
0.011
70
130
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Table 4 Optimized rule base for FLC using Approach 3 Rule No. Inputs Case 1 Case 2 Outputs Outputs 1 2 1 2 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
VNR VNR VNR VNR VNR NR NR NR NR NR FR FR FR FR FR VFR VFR VFR VFR VFR
LT AHL A AHR RT LT AHL A AHR RT LT AHL A AHR RT LT AL A AHR RT
A AHR A AHL LT A A AHL LT LT AHR A AHL A A A AHR AHL A A
H H H VH H VH VH H H L VH VH VL VH VH L H H H VH
A LT A A AHR AHL A AHR LT LT AHR A LT A LT A AHR AHR AHL LT
VL H L VH L VL VL VH VH VL VH VH H H L VH VL L H VH
Case 3 Outputs 1
2
A LT LT AHL A AHL AHR AHR A A A AHR AHL AHR LT A A AHR AHL AHL
VH H H VH H VL VH L L H VH H VL H L VH L VL VL H
Table 7 presents the data relevant to the number of time steps that the cooperation schemes have been fired. It is also for sure that not all the robots will be cooperative for an equal number of times. Therefore, we have tried to identify the robot that has cooperated maximum times. Although it depends on the position and orientations of the robots, the maximum velocity for all the robots is same. Therefore, the robot which has shown more cooperativeness is a more autonomous robot compared to the other. In that perspective, it can be observed that the Approach 3 in a combination of strategic coordination scheme (SC2) has provided the best result and generated a number of cooperative scenarios. The eighth robot has become more cooperative for most of the cases and therefore said to be most adaptive robot amongst all.
83.33
115.25 115.36 116.23 116.48 116.07 116
82.96
R7
R8
85.43
84.26
84.68
84.09
84.29
85.38
21.36
21.59
21.54
21.56
21.46
20.5
20.6
22.16
22.17
22.18
21.98
70.97
18.02
21.38
70.91
S.D.
71.52
73.16
71.21
58.57
62.37
48.62
79.24
52.81
61.15
43.32
85.22 59.8
Avg.
52.79
61.2
42.91
83.1 54.18
86.03
53.25
62.1
43.48
83.54 54.14
R17
53.04
61.62
43.2
84.39 54.89
64.68 73.21
83.95 54.59
86.38
57.4
61.64
49.31
59.71
R16 73.04
59.59
64.97
73.33
63.46
56.69
64.51
70.88
61.79
57.05
64.91
71.28
62.22
SC2
Approach 2 SC1
85.68
84.63
85.17
56.92
65.21
72.15
61.16
84.44
85.08
105.27 106.1
56.5
64.39
70.41
61.77
SC2
Approach 3 SC1
18.03
73.24
86.07
64.77
86.48
61.7
62.57
58.07
62.35
49.66
60.14
85.52
19.56
71.06
83.87
63.61
84.57
56.77
59.35
53.77
62.08
43.39
55.37
84.24
19.55
71.48
84.04
63.95
84.82
57.17
59.85
54.12
62.95
43.68
55.94
84.8
19.51
70.83
83.6
63.47
84.19
56.76
59.25
53.44
61.88
43.26
54.94
83.94
19.3
71.01
83.62
63.22
84.43
57.26
59.05
53.24
62.06
43.47
56.29
82.72
115.24 115.46 116.86 117.37 116.65 115.29
85.72
R15
80.14
59.97
65.14
73.03
63.73
SC2
Approach 1 SC1
61.66
79.88
56.11
64.54
70.85
60.63
SC2
R14
80.27
55.94
64.01
69.79
61.31
SC1
62.36
79.98
56.44
64.19
70.6
61.93
SC2
17-Robots case
R13
57.48
R12
85.33
61.38
79.35
56.16
64.06
70.31
61.53
SC1
Approach 3
105.18 105.32 105.56 104.97 105.74 105.19 105.31 105.64 105.9
59.03
64.5
72.67
62.89
SC2
Approach 2
115.18 115.36 116.49 116.76 116.33 115
85.61
48.49
83.71
83.93
R11
83.55
83.95
R10
84.15
84.37
59.14
59.81
83.68
84.07
56.34
R9
83.21
83.27
55.7
64.23
72.54
R6
56.13
64.17
70.35
103.85 104.12 104.99 105.29 104.87 105.37 104.7
55.88
63.84
69.8
R5
55.47
63.99
70.1
55.5
63.82
69.89
62.55
R4
63.4
69.34
61.25
SC1
63.26
61.28
SC2
69.24
61.64
SC1
R3
61.28
SC2
R2
60.64
SC1
60.55
SC2
Approach 1
SC1
12-Robots case
Approach 3
Approach 1
Approach 2
8-Robots case
R1
Robots
Table 5 Average travel time in seconds of the robots for different approaches
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31, 46, 94
5, 8, 66
2
3
12
5, 20, 71
1, 8, 27
1, 19, 31
19, 39, 71
1
2
3
SC1
1
8
1, 49, 52
2, 20, 31
1, 3, 32
1, 7, 49
1, 20, 31
3, 5, 32
SC2
Three best scenarios out of hundred scenarios
Approaches
No. of robots
Table 6 Best scenario out of 100 scenarios
52
31
3
7
31
5
Best scenario
[65.16, 36.93, 87.86, 24.12, 120.85,83.96, 120.06, 94.06, 83.93, 32.27, 60.23, 33.52]
[18.94, 33.45, 25.99, 87.00, 123.56, 87.31, 129.58, 80.57, 33.09, 44.66, 47.44, 67.68]
[119.37, 118.89, 119.03, 119.55, 121.92, 118.50, 119.14, 118.99, 118.49, 20.23, 118.87, 118.73]
[69.71, 70.58, 95.52, 73.53, 113.45, 96.73, 126.96, 82.29]
[18.94, 33.45, 25.99, 87.00, 121.54, 87.56, 128.11, 82.65]
[115.57, 79.73, 129.03, 87.12, 116.96, 96.74, 125.88, 91.09]
SC1
[65.16, 36.93, 86.17, 24.12, 132.65, 83.96, 119.56, 95.75, 81.39, 32.27, 63.77, 37.96]
[18.94, 34.55, 25.99, 48.98, 124.81, 97.58, 124.24, 80.57, 34.91, 44.17, 47.44, 64.32]
[121.94, 89.11, 100.38, 18.27, 124.80, 82.50, 119.14, 93.01, 73.51, 21.52, 98.87, 65.27]
[69.71, 74.32, 84.15, 73.53, 113.45, 96.73, 126.96, 88.68]
[18.94, 34.55, 25.99, 52.98, 124.81, 87.89, 124.24, 82.09]
[117.28, 82.00, 129.03, 87.12, 118.01, 97.08, 125.88, 95.07]
SC2
Travelling time in seconds of different robots for the best scenario
70.25
64.94
111
91.1
73.16
105.3
SC1
71.64
62.21
84.03
90.94
68.94
106.4
SC2
Average travel time in seconds
(continued)
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1, 27, 35
5, 16, 19
1, 27, 52
2
3
SC1
1
17
5, 27, 52
27, 46, 72
3, 5, 71
SC2
Three best scenarios out of hundred scenarios
Approaches
No. of robots
Table 6 (continued)
52
46
71
Best scenario
[76.01, 79.80, 100.54, 44.74, 89.34, 76.1, 102.6, 98.33, 16.14, 54.31, 44.67, 45.43, 13.88, 44.31, 53.49, 68.94, 52.09]
[46.10, 103.26, 68.66, 55.23, 109.4, 93.47, 100.42, 69.72, 41.6, 33.74, 15.19, 46.66, 35.3, 58.13, 87.28, 46.73, 97.96]
[81.87, 110.44, 95.38, 48.44, 87.52, 73.51, 79.95, 95.43, 61.75, 55.81, 46.41, 43.71, 47.89, 126.03, 73.37, 4.56, 75.51]
SC1
[88.39, 79.80, 103.81, 46.67, 79.51, 76.1, 102.6, 97.02, 16.14, 54.31, 31.33, 34.57, 13.88, 31.70, 53.49, 68.94, 52.09]
[45.90, 100.23, 68.66, 52.77, 114.45, 93.47, 104.12, 69.72, 41.6, 33.74, 23.19, 46.66, 35.30, 61.87, 95.48, 46.73, 104.8]
[71.43, 119.61, 114.38, 51.56, 87.52, 73.51, 84.91, 80.14, 63.62, 55.81, 45.59, 30.00, 44.11, 126.03, 73.37, 4.53, 81.01]
SC2
Travelling time in seconds of different robots for the best scenario
62.4
65.23
71.03
SC1
60.61
66.98
71.01
SC2
Average travel time in seconds
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Table 7 Estimation of cooperativeness of the robots Approaches Case 1 Case 2 No. times cooperation schemes are fired SC 1 SC2 SC 1 SC 2 App. 1 App. 2 App. 3 App. 1 App. 2 App. 3
Case 3 SC 1
65 129 121 219 210 75 142 133 203 240 118 143 171 218 280 Best cooperative robot and a total number of cooperation Robot 8 Robot 8 Robot 8 17 25 19 27 22 Robot 8 Robot 5 Robot 5 Robot 5 Robot 8 20 26 27 28 27 Robot 5 Robot 8 Robot 5 Robot 8 Robot 5 24 28 25 30 24
SC 2 373 353 362
30 Robot 5 30 Robot 8 31
5.3 Comparison of Traveling Time Taken by the Robots Average travelling time for all the one hundred scenarios is presented in Table 5. It has been observed that heuristic coordination scheme (SC1) has resulted in less average travelling time for the robots in compared to the strategic coordination scheme (SC2). Increase in the number of cooperative scenarios may be the reason for the increase in travelling time. However, the difference is not very high. For eight robots case, performance regarding the travelling time of Approach 1 is the best. But, with the increase in robots, Approach 3 has superseded the others. It indicates that the Approach 3 is more adaptive and robust. We have also identified the three best scenarios regarding a higher number of cooperative situations and noted the best scenarios in which highest number of cooperative schemes is fired (refer to Table 6). It has been observed that the strategic cooperation scheme has generated the best result in all such cases. Movement of eight, twelve and seventeen robots for the outstanding scenario of different methods has been envisioned in Figs. 3, 4, 5, 6, 7 and 8, respectively.
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Fig. 3 Navigation of eight, twelve, sixteen and twenty robots using PFM (heuristic coordination), respectively
6 Concluding Remarks Traditional as well as soft computing-based approaches are found to solve the coordination problems of multiple mobile robots working in a dynamic environment. Heuristic-based coordination schemes have been considered in our earlier study, but it lacks in adaptability and robustness. Therefore, a game theory-based strategic coordination scheme is presented in this paper. Two different types of motion planning schemes are coined with two coordination strategies to evolve the best-suited navigation algorithm. Following observations are noted.
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Fig. 4 Navigation of eight, twelve, sixteen and twenty robots using fuzzy (heuristic coordination), respectively
• Performance of genetic fuzzy-based motion planner is the best. However, tuning of its knowledge requires a considerable amount of time. • The presence of coordination schemes within the motion planning schemes improves the quality of solutions. • Strategic coordination shows better solutions compared to the heuristic-based coordination schemes.
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Fig. 5 Navigation of eight, twelve, sixteen and twenty robots using GA fuzzy (heuristic coordination), respectively
We could not test the performance of the developed approaches with real robots. We are presently doing this. Also, we are thinking of collecting data from the real market scenario. A big data-based approach can be integrated to enhance the complexity of the existing problem.
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Fig. 6 Navigation of eight, twelve, sixteen and twenty robots using PFM (strategic coordination), respectively
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Fig. 7 Navigation of eight, twelve, sixteen and twenty robots using fuzzy (strategic coordination), respectively
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Fig. 8 Navigation of eight, twelve, sixteen and twenty robots using GA fuzzy (strategic coordination), respectively
References 1. Cui R, Gao B, Guo J (2012) Pareto-optimal coordination of multiple robots with safety guarantees. Auton Robots 32(3):189–205. https://doi.org/10.1007/s10514-011-9265-9 2. Cui R, Guo J, Gao B (2013) Game theory-based negotiation for multiple robots task allocation. Robotica 31(6):923–934. https://doi.org/10.1017/S0263574713000192 3. Erhart S, Hirche S (2015) Internal force analysis and load distribution for cooperative multirobot manipulation. IEEE Trans Robot 31(5):1238–1243. https://doi.org/10.1109/TRO.2015. 2459412 4. Hillier F, Lieberman G (2010) Introduction to operations research. McGraw-Hill, New York, NY. https://books.google.co.in/books?id=NvE5PgAACAAJ 5. Hui NB, Mahendar V, Pratihar DK (2006) Time-optimal, collision-free navigation of a car-like mobile robot using neuro-fuzzy approaches. Fuzzy Sets Syst 157(16):2171–2204. https://doi. org/10.1016/j.fss.2006.04.004
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Spur Gear Mechanism for Accurate Angular Indexing and Locking of Angular Position by Using Additive Manufacturing Arivazhagan Pugalendhi, Rajesh Ranganathan, and C. Vivek
Abstract Ratchet and pawl mechanism is used neither in many applications to provide precise locking of components either in linear nor in rotational directions. The number of locking positions depends on the total number of teeth on the ratchet. The objective is to increase the number of locking positions and thus minimizing the interval angle. For this objective to be fulfilled, spur gear is treated as ratchet, and meshing spur gear teeth is treated as pawl (named as lock). Due to the geometry of the spur gear teeth, the apparatus facilitates bidirectional locking. Spur gear permits 18 teeth in angular positions (P) achieved by a traditional ratchet mechanism with interval angles (A) of 20°. The locking of desired angle is impossible except for 18 positions of 20°, 40°, and 60° up to 360°. This study gives a better solution for increasing the number of locking position as well as decreasing the interval angles by introducing an innovative mechanism. Here, rearranging the position of the lock results in increasing the number of angular positions without increasing the number of teeth. Outcome of the paper is developed design which provides four times increase and reduction of angular positions. The locking position is derived by a mathematical formula, and simulation of the assembly was performed by using CAD software. Finally, the concept was proven by physical prototype fabricated through additive manufacturing technology and compared with traditional ratchet mechanism. Keywords Ratchet and pawl mechanism · Spur gear · Angular positions · Indexing rate · Additive manufacturing
1 Introduction Recent advancements in equipments/apparatus depend upon the mechanism that is needed to prevent the apparatus from failure [1]. Ratchet and pawl mechanism can
A. Pugalendhi (B) · R. Ranganathan · C. Vivek Department of Mechanical Engineering, Coimbatore Institute of Technology, Civil Aerodrome Post, Coimbatore, TN 641014, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_11
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replace the conventional spring elements that meet the design specifications for selflocking purposes. Ratchet is gear tooth mechanism with a pawl that engages the teeth [2]. The main components of a ratchet and pawl mechanism are gear (uniform asymmetric teeth), pawl and base [3]. When ratchet rotates in forward, pawl slides over the teeth, and if ratchet rotates in the opposite direction, pawl gets inserted into the teeth and gets locked restricting backward motions. Ratchet mechanism provides uni-directional transmission both in linear and rotational direction against heavy loading conditions [4]. The purpose the paper is to develop a compact novel mechanism, which is capable of locking in bi-direction with more number of locking positions and thereby decreasing the interval angles without fasteners.
2 Literature Review The indexing mechanism converts rotatory motion into a series of step movements of the output link. Common types of indexing mechanisms are ratchet and pawl, rack and pinion, Geneva mechanism and cam drive [5]. Ratchet and pawl mechanism used for large wheels that rotate in one direction along the specified axis [6]. The most common examples of the ratchet and pawl mechanism are clocks, jacks and hoists [5]. One of the experimental studies conducted by Jalili (2003), piezoelectric ceramic-based ratchet actuation device is utilized to control the engine valves. In the study, it revealed that the opening and closing of engine valves are computer controlled by cam timings [7]. The usage of electromechanical actuator can be used to control the ratchet and pawl mechanism [8]. Need for bearings and springs is eliminated by a new design of the ratchet mechanism [2]. Under that study, the micro-wire EDM process is identified as the potential manufacturing process for making the prototype model of the mechanism and inferred that it was a potentially working prototype model. Ratchet mechanism utilized for preventing the vehicle from descending down [9]. A microgripper with a ratchet self-locking mechanism manufactured using silicon-on-insulator wafer with 30 µm layer and with a locking interval as 10 µm. With the application of ratchet locking, damage of the microgripper has been reduced [10]. During locking, pawl moves in the reverse direction; as a result, it may lead to damage of the pawl due to the force acting on the ratchet [11]. Thereby, an alternative mechanism is needed to reduce the damage of the pawl and permits the bi-direction movement. Gears are basically a toothed profile that transmits power and motion by the gear tooth interlocked in mesh [12]. Parameters that are considered for the gear tooth profile are pitch circle diameter, tooth thickness, pressure angle, number of teeth, the rotational angle of the gear and rotational speed of the gear. Manufacture of gears requires high accuracy and specialized manufacturing process for production of gear tooth profile [13]. Spur gear is relatively simple in design and easy to produce for operating at low-speed applications. Comparing the bending and contact stress for both normal contact ratio (NCR) and for high contact ratio (HCR), spur gear using finite element analysis inferred that NRC gear set has less contact ratio than HCR
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gear set [14]. Most of the spur gear applications are with respect to 20° involute teeth that are manufactured by gear hobbing machine. However, if the involute profile is less than 17 teeth, it was identified that low involute profile thickness leads to crack initiated at the root fillet area of the profile geometry [15]. Contact stress and bending stress on the spur gear are affected by the shape of the gear tooth profile, and it is identified by nonlinear finite element approach [16]. By using the boundary element method, effect of circular root fillet over the standard trochoidal root fillet was studied. The study revealed that teeth having circular fillet gave high bending strength [17]. Fillet radius of nylon 6/6 gear produced through an injection moulding process affects the performance of the transmission [18]. The fatigue life of the gear is low when the number of teeth is less and has to optimize at the earlier stage of gear design [19]. Design iteration work was carried out to find the optimum tooth profile that reduces vibration and noises [20]. A numerical fault detection model of a dynamic transmission error (DTE) is used to examine the effects of tooth crack length for symmetric and asymmetric tooth profiles [21]. The parameters of the spur gear are towards minimizing the transmission errors that are governed by nominal torque. The study also inferred that low contact ratio gears will result in more noise and vibrations when used for transmitting power. The mathematical model is governed by line equation which utilizes simple curve features for improving the tooth profile and its taking pressure angle as a parametric variable [22]. Mathematical model of spur gear is based on the curvature that engages with the tooth profiles. Study revealed that mathematical model developed to enhance the contact strength decreases the wear of the tooth surfaces [23]. Non-circular gear trains are used to create an indexing mechanism where the indexing period is different [24]. Conventional gear hobbing process and injection moulding process are involved in manufacturing of gear tooth profile [25]. Additive manufacturing (AM) deals with developing a wide range of products with complicated profiles from the CAD model. AM technology having the possibility to produce low volume and small quantities of customizable products that are in complicated shapes with low cost [26]. AM technology is suited for producing functional end products that can be used directly [27]. It finds potential applications in the area of healthcare such as implantable prosthesis, orthotics, bio-engineering and in surgical planning [28]. AM technology benefitted industries such as automotive, aerospace, consumer products and architectural modelling catering to new product development [29]. The application of AM technology allows the development of non-assembly mechanism and for robotic systems. The study revealed that evaluation of the mechanism was easy as it was able to visualize the joints clearly, able to visualize the internal mechanisms and achieve multiple degrees of freedom as in the case of robotic systems. Mavroidis et al. studied the various kinematic models using rapid prototyping technologies and found out that rapid prototype products developed are the exact replica of the physical models of the machines [30]. In a study conducted by Plooij et al., different locking was identified; in the study, Geneva mechanism was able to lock at 5 positions with 72° interval. Ratchet mechanism was capable of locking at 15 positions with 24° interval, while cam follower-based locking is
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capable of locking at 45° interval positions. Considering the principles of mechanical locking and friction locking, developed a locking mechanism for robotic system [31]. Giorgio et al. studied the Geneva mechanism and its pure-rolling cam-equivalent for 4, 6, 7 and 8 slots on the Geneva wheel [32]. Coaxial eccentric indexing cam mechanism gives the higher indexing rate and more squeezed construction than the same inputs of the conventional indexing mechanisms [33]. Locking mechanism actuates the multifunctional prosthetic hand with shape memory alloy actuators [34]. Geneva mechanism is utilized for positioning and locking thumb and wrist of dextrous robots [35]. Behrens et al. presented a prototype of trans femoral prosthesis for effective mobility and control of the prosthesis. A mechanism was developed for achieving different range of motions, with spur gear and other linkages. Angles achieved are 15° as in the case of ankle movements [36]. By using the AM technology, Saharan et al. developed a lightweight robotic prosthetic hand and locking mechanism to hold and grip the objects based on the requirements of individual [37]. Diameter of 18– 20 mm pipe robot with large traction utilizes the bilateral self-locking mechanism [38]. From the above-said literature, the functions, applications and limitations of ratchet mechanism are carried out. Spur gear design, working principles, factors influencing the performance, applications and manufacturing methods also clearly explored by the above literature. Advantage and applications of AM in a variety of fields are detailed from the literatures. In this regard, the main drawbacks of the locking mechanism are limited positions and its accuracy; more number of components are included as fasteners, size and weight. Thereby, an alternative approach is required for developing locking mechanism that overcomes the above issues and more complicated in design. The objective of this paper is to develop a mechanism for accurate angular indexing and locking of angular positions that is less in weight and complicated in design through AM technology.
3 Methodology 3.1 Machine, Materials and Software In this study, fused deposition modelling (FDM)-based uPrint AM machine produced by Stratasys is used. The nominal size of the build volume is 203 × 152 × 152 mm (8 × 8 × 6 in) in X, Y and Z planes, respectively. Layer thickness, which is the minimum height of material (measured in Z axis) that deposited in each run, is 0.254 mm (Source: Stratasys). Build or model material is a strong, accurate and white prototyping plastic material of ABS Plus-P430 in ivory. It is particularly used for determining form, fit and function in everything from ergonomics to manufacturing processes. SR-30 is used as a support material.
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Softwares used in this study are SolidWorks 2016 and CatalystEX. Modelling and simulation of the components are performed by SolidWorks. After the accepted simulation results, final parts to be printed are obtained from SolidWorks in the STL format. CatalystEX is used to interact with the machine such as tool path generation, part orientation, model and support material creation and its layer gap as well as printing time.
3.2 Design of Modified Ratchet Mechanism In conventional ratchet mechanism, unidirectional locking is accomplished by insertion of the pawl. Perimeter of the ratchet consists of uniform asymmetrical teeth with steep slope and gradual slope. Continuous contact of the pawl slides over the ratchet; while pawl struck into the steep slope of the ratchet, it restricts the reverse motion. Working principle of modified ratchet mechanism is similar to meshing of two spur gears as shown in Fig. 1. Here, asymmetrical tooth of ratchet is replaced by symmetrical tooth profile of spur gear. Similarly, portion of the spur gear is used for locking instead of pawl and named as a lock. Both gear and lock are working in inner side of the housing. Due to size and weight constraint 18 teeth spur gear with module 2 is chosen for this study. Generally, number of position (P) and accuracy or interval angles (A) are mainly depend upon the number of teeth (Z). In this regard, basic expressions of locking mechanism are as follows, P=Z
(1)
Equation 1 infers that the total number of positions is directly proportional to total number of teeth on gear. A = 360◦ × Z −1
(2)
Accuracy or interval of angle is determined by complete rotation of the gear which is divided by total number of teeth as per Eq. 2. From Eqs. 1 and 2, 18 teeth spur gear results that 18 positions with 20° of interval angles. While changing the lock position with in the 20°, it increases the number of positions with same 20° interval angles instead of without of increasing the number of teeth. Proposed mechanism is achieved by adjusting the lock positions, and it allows the bi-directions locking. So, altering the above equations with number of changing positions of the lock (L) is as follows, P = Z × L
(3)
A = 360◦ × Z −1 L −1
(4)
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Fig. 1 Schematic of performance of the proposed spur gear mechanism
LPx =
i x 360◦ × L −1 + A , n > 1, n ∈ Z + 0 if n = 1
(5)
where P = Total number of positions after increasing the position of the lock A = Minimized interval angle after the introduction of position of the lock LPx = Starting position/origin of the provision of the lock By using Eqs. 3 and 4, if the lock is able to change their positions in four locations means, it results 72 positions and interval angles of 5°, respectively. Figure 1 illustrates the performance of the proposed spur gear mechanism with four locking locations. Initially, desired position is achieved by insertion of lock into the gear as shown in Fig. 1a. Here, interval angle of 20° is possible in both directions. Red colour solid line represents the origin, and blue colour centre line represents the interval angle accomplished by modified ratchet mechanism. At this juncture, the apparatus facilitates the interval angles of 20°, 40° and 60° up to 360°. Adjustment of the 10°
Spur Gear Mechanism for Accurate Angular Indexing and Locking … Table 1 Specifications of the spur gear
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S. No.
Description
Symbols
Values
1
Module
m
2 mm
2
Number of teeth
z
18
3
Pressure angle
α
20°
4
Reference diameter
d
36 mm
5
Addendum
ha
2 mm
6
Dedendum
hf
2.5 mm
7
Tooth depth
h
4.5 mm
8
Tip diameter
da
40 mm
9
Root diameter
df
31 mm
10
Face width
b
25 mm
(for third quadrant = 5° + 5°) angular position of lock from starting position leads to double the total number of locking positions (36 No’s) and thereby reducing half of the interval angle (10°). Now, the apparatus facilitates the interval angles of 10°, 30° and 50° up to 350° as shown in Fig. 1c. Similarly, again 5° adjustment of the position of lock in second (5°) and fourth (5° + 5° + 5°) quadrants and its interval angles are shown in Fig. 1b and Fig. 1d, respectively. Thereby, the apparatus facilitates 5° interval angle and 72 angular positions. Due to space constraint, adjustment of lock position is performed on upcoming quadrants. Starting position or origin of the provision of lock is determined by Eq. 5, and the results of first, second, third and fourth lock position are 0°, 95°, 190° and 285°, respectively.
3.3 Design of Spur Gear Capabilities of design flexibility, inferior inertia, reduced friction and enhanced quality of shock and vibration absorption plastic gears are better than traditional metal gears. Compared to metal gears, plastic gears are lighter in weight and inexpensive. Symmetrical profile of teeth permits the forward and reverse rotation of the gear. Geometry of the spur gear is derived from standard design calculations of spur gear, and modelling of 3D gear model is done by SolidWorks. The view of lock is in the form of spur gear that has only three teeth, dimensionally modified to alignment and space constraint. Table 1 displays the specifications of the spur gear.
4 Analysis In this study, the stress analysis of spur gear mechanism was carried out using SolidWorks simulation. Both identical part materials are ABS Plus-P430 and its detailed
112 Table 2 Properties of ABS Plus-P430
A. Pugalendhi et al. S. No. Properties
Standard (ASTM) Values
1
Ultimate tensile strength
D638
2
Tensile modulus of D638 elasticity
2200 MPa
3
Impact strength
D256
106 J/m
4
Flexural strength
D790
99.6 MPa
5
Hardness (rockwell)
D785
109.5 HRD
33 MPa
Fig. 2 Von Mises stress plot (a) and displacement plot (b) of proposed spur gear mechanism
properties are shown in Table 2 (Stratasys, 2008). High-quality curvature-based solid mesh is applied on both gear and lock. Minimum and maximum element size is 0.567 mm and 2.839 mm, respectively. The total number of elements is 241,901, and total number of nodes is 365,246. Bottom flat surface of the lock is referred as a fixed geometry, and bore for shaft in the gear is referred as a fixed hinge. Both gear and lock are no penetration contact between the meshing teeth. The applied torque of 150 N and the results of von Mises stress and resultant displacement are shown in Fig. 2a and Fig. 2b, respectively. The obtained results of minimum and maximum von Mises stress are 0.000870995 MPa and 1750.58 MPa. Similarly, maximum displacement of the teeth is 2.7823 mm. From the analysis, it was found that the load carrying capacity of the assembled spur gear mechanism made of ABS Plus P-430 was good.
5 Development of Spur Gear Mechanism Following the design and simulation of the model, STL file format is converted by SolidWorks. This final STL file is ready for printing by uPrint machine through
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pre-processing software named as CatalystEX. Layer gap (gap between the adjacent layers), part orientation and placement of the printed parts are controlled by CatalystEx. 3D printed parts strength, surface finish, total printing time and materials consumptions of both model and support material are mainly affected by part orientation and its layer gap. Envelope temperature of the uPrint machine and model material temperature is 70 °C and 210 °C, respectively. The fabrication of the spur gear mechanism is divided into four parts such as gear, lock, shaft and housing. Before printing the whole model, the working and accuracy of the mechanism with the help of sample miniature prototype are checked. Here, shaft and gear coupled by single part. All parts are manufactured by same procedure and same fabrication conditions in single print. Solid-type layer gap for model material and smart-type layer gap for support material are chosen. Due to minimum Z axis height and less support material consumption, auto orientation is selected for all the three parts. All the three parts are printed in single run with desired locations on build tray. Finally, packed chromeleon backup archive (CMB) file is send to printer for fabrication. Once the printing was finished, printed parts are removed from build tray and removing the all support materials using wave wash technology. This fabrication process consumes 0.48 in3 of model material and 0.24 in3 of support material, and it took 0.38 h. 3D printed miniature model of spur gear mechanism is shown in Fig. 3. Individual parts of miniature model are shown in Fig. 3a, and assembly of all parts are shown in Fig. 3b. Gear with shaft is placed on bore of the housing, and it is rotated freely in both directions. Placing the lock into teeth of the spur gear results, arrest the rotation of gear. Movement of the lock is arrested by slots provided in the outer periphery of the housing. For easy understanding, pointer position and lock provisions are marked in both housing and gear. Despite, exact locking is ensured by angle readings on the gear. For example, 75° angle is achieved by lock engaged in fourth provision of housing as shown in Fig. 3b.
Fig. 3 Miniature model of spur gear mechanism a individual parts and b assembly
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Fig. 4 Final spur gear mechanism fabricated by additive manufacturing a individual parts and b assembly
After ensuring the performance and functionality of the proposed spur gear mechanism, actual prototype is manufactured similar to miniature prototype. 3D printed individual parts of final spur gear mechanism and assembly are shown in Fig. 4a and b. This fabrication process consumes 5.27 in3 of model material and 0.47 in3 of support material, and it took 7.22 h.
6 Results and Discussion The proposed spur gear mechanism is fabricated by AM technology, which results in good functionality. Identified the exact problem in the existing mechanism and designed the new mechanism without sacrificing the performance as well as functionality. After the final CAD modelling, computationally simulated the mechanical response of the spur gear mechanism during the meshing. Competency of the spur gear mechanism allows the clock-wise and anti-clock wise rotation of the gear. The difference between the traditional ratchet mechanism and proposed spur gear mechanism having same 18 teeth is shown in Table 3. Increasing the number positions (P) without increasing the number of teeth (Z) was proven. Because increasing the number of teeth leads to increase in the tip diameter (da) as well as weight of the system. Increasing the number of teeth without altering the diameter results, weak teeth profile. So, adjusting the position of lock (S) is overcome by the above issues. Initially, this mechanism starts with four positions of the pawl with manual locking. Symmetrical profile of the both gear and pawl permits the bi-directional rotation. Spur gear mechanism increases the number positions from 18 to 72 and reduces the interval angle from 20° to 5°. Quantitatively, spur gear mechanism increases the number of positions and interval angles by 75%. Elimination of the springs and fasteners, it reduces the number of parts and overall size of the mechanism. Exact position of the shaft is determined by readings of the gear. Optimal number of lock positions with self-locking mechanism fabricated by traditional manufacturing methods are used in wide range of applications.
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Table 3 Difference between traditional ratchet mechanism and proposed spur gear mechanism S. No.
Description
Ratchet mechanism
Spur gear mechanism
1
Rotary motion
Ratchet
Spur gear
2
Locking by
Pawl
Spur teeth
3
Contact between ratchet and pawl
Continuous
Discontinuous
4
Direction of rotation
Uni-directional
Bi-directional
5
Self-locking
Yes
No
6
Fasteners requirement
Yes
No
7
Total number of positions
18
72
8
Interval angle
20°
5°
9
% of increases in position
NA
75%
10
% of increases in accuracy
NA
75%
11
Mode of locking
Automatic
Manual/automatic
12
Measurement of angle
Partially
Exactly
13
Play between ratchet and pawl
Reverse direction of ratchet
Purely eliminated
14
Size
Larger
Compact
7 Conclusion This study initiates the successful implementation of AM technology that integrates the principles of traditional ratchet mechanism and spur gear mechanism. It aids to create an impression towards the application of AM in fabrication of machines and mechanisms. Ability to manufacture accurate, efficient and cost-effective mechanisms will discover the influences of FDM in new product development. The mechanism allows the bi-directional rotation and lock position by insertion of the lock. The main advantages of the spur gear mechanism in comparison to the traditional ratchet mechanism are as follows: (1) increases the number positions, (2) reduces the interval angle, (3) measurement of the exact positions can be achieved and (4) the capability to control the mechanical behaviour by tuning the geometry. This novel method finds a potential use in variety of fields, and this is specifically used in medical instruments used for post-operative treatments. Further work can be undergone for self-locking spur gear mechanism, and understanding the influences of material and geometrical parameters over the mechanical performance would be a future focus. Acknowledgements We gratefully acknowledge the financial support for establishing the Centre of Excellence in Manufacturing Sciences at Coimbatore Institute of Technology, India from Ministry of Human Resource Development (MHRD), Government of India where the R&D work is carried out.
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21. Dogan Oguz, Karpat Fatih (2019) Crack detection for spur gears with asymmetric teeth based on the dynamic transmission error. Mech Mach Theory 133:417–431. https://doi.org/10.1016/ j.mechmachtheory.2018.11.026 22. Zhang-Hua F, Ta-Wei C, Chieh-Wen T (2002) Mathematical model for parametric tooth profile of spur gear using line of action. Math Comput Model 36(4–5):603–614. pii:SO8957177102100185-l 23. Liu H, Mao K, Zhu C, Xu X, Liu M (2012) Parametric studies of spur gear lubrication performance considering dynamic loads. Proc Inst Mech Eng Part J: J Eng Tribol 226(9):731–737. https://doi.org/10.1177/1350650112445672 24. Fangyan Z, Lin H, Xinghui H, Li Bo, Dingfang C (2016) Synthesis of indexing mechanisms with non-circular gears. Mech Mach Theory 105:108–128. https://doi.org/10.1016/j.mechma chtheory.2016.06.019 25. Tang Y, Chang S, Wang Z, Zhang K (2011) Study on helical tooth profile modification of planetary gear transmission on the basis of gear transmission error. Appl Mech Mater 86:47–50. https://doi.org/10.4028/www.scientific.net/AMM.86.47 26. Ngo T, Kashani A, Imbalzano G, Nguyen K, Hui D (2018) Additive manufacturing (3D printing): a review of materials, methods, applications and challenges. Compos B Eng 143:172–196. https://doi.org/10.1016/j.compositesb.2018.02.012 27. Campbell I, Bourell D, Gibson I (2012) Additive manufacturing: rapid prototyping comes of age. Rapid Prototyp J 18(4):255–258. https://doi.org/10.1007/978-1-4419-1120-9 28. Webb P (2002) A review of rapid prototyping (RP) techniques in the medical and biomedical sector. J Med Eng Technol 24(4):149–153. https://doi.org/10.1080/03091900050163427 29. Edgar J, Tint S (2019) Additive manufacturing technologies: 3D printing, rapid prototyping, and direct digital manufacturing, 2nd edn. http://dx.doi.org/10.1595/205651315X688406 30. Mavroidis C, DeLaurentis K, Won J, Alam M (2001) Fabrication of non-assembly mechanisms and robotic systems using rapid prototyping. J Mech Des 123(4):516. https://doi.org/10.1115/ 1.1415034 31. Plooij M, Mathijssen G, Cherelle P (2015) Review of locking devices used in robotics. IEEE. https://doi.org/10.1109/MRA.2014.2381368 32. Giorgio F, Pierluigi R, Jorge A (2006) The pure-rolling cam-equivalent of the Geneva mechanism. Mech Mach Theory 41:1320–1335. https://doi.org/10.1016/j.mechmachtheory.2006. 01.002 33. Yuhu Y, Jianyong W, Shicai Z, Tian H (2019) Design of a novel coaxial eccentric indexing cam mechanism. Mech Mach Theory 132:1–12. https://doi.org/10.1016/j.mechmachtheory.2018. 10.012 34. Andrianesis K, Tzes A (2014) Development and control of a multifunctional prosthetic hand with shape memory alloy actuators. J Intell Rob Syst 78(2):257–289. https://doi.org/10.1007/ s10846-014-0061-6 35. Pons J, Rocon E, Ceres R, Reynaerts D, Saro B, Levin S (2004) The MANUS-HAND dextrous robotics upper limb prosthesis: mechanical and manipulation aspects. Auton Robots 16(2):143– 163. https://doi.org/10.1023/B:AURO.0000016862.38337.f1 36. Unal R, Behrens S, Carloni R, Hekman E, Stramigioli S, Koopman B (2018) Conceptual design of a fully passive transfemoral prosthesis to facilitate energy-efficient gait. IEEE Trans Neural Syst Rehabil Eng 26(12):2360–2366. https://doi.org/10.1109/IEMBS.2011.6090111 37. Saharan L, Tadesse Y (2016) Robotic hand with locking mechanism using TCP muscles for applications in prosthetic hand and humanoids. Bioinspiration Biomimetics Bioreplication 9797. https://doi.org/10.1117/12.2219535 38. Yang J, Xue Y, Shang J, Luo Z (2014) Research on a new bilateral self-locking mechanism for an inchworm micro in-pipe robot with large traction. Int J Adv Rob Syst 11(10):174. https:// doi.org/10.5772/59309
Fabrication of Solid Lubricant Coating and Its Optimization Using Response Surface Methodology R. Tyagi , S. Kumar , A. K. Das , and A. Mandal
Abstract The present paper elaborates process for production of solid-lubricating coating over the mild steel by electrical discharge technique through green compact electrode made of MoS2 + Cu powder. Electrical discharge coating which is based on EDM principle and performed with the help of powder compact electrode formed via hot mounting press. The transportation of material occurred through electrical discharges which helps in melting and simultaneous deposition of powder. The loose bonding between particles facilitates the material deposition over the substrate. The effect of input parameters variation on the output parameters, such as, tool wear rate, mass deposition rate, and coating layer thickness is investigated by response surface methodology (RSM). The experiments results achieved from this work show that experimental values are in good agreement with the predictive results. The morphology study showed minimum amount of defects at the top coated surface. EDS study confirmed the successful deposition of principal elements presence in the coating. Keywords Mild steel · EDC · RSM · Green compact electrode
1 Introduction Surface modification is generally applied to improve the tribological behavior of components, which are exposed to friction and wear in extreme working conditions, such as in a high vacuum and high-temperature environment. From the last few years, many researchers tried economical processes as alternatives, and among these, EDC process has emerged as one of the most popular among them. The advantages of surface modification by EDC includes (1) good adhesion between substrate and coating, (2) ability to produce thick coating, (3) simultaneous coating of multiple materials having different melting point and in situ inter-metallic compounds formation [1]. With the help of electro-discharge coating (EDC), these problems can be rectified by applying solid lubricant coating on mild steel. In EDC, the loose bonding R. Tyagi (B) · S. Kumar · A. K. Das · A. Mandal Indian Institute of Technology (ISM), Dhanbad 826004, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_12
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between particles was observed to be helpful in achieving higher deposition rate and thick layered coating. Beri et al. well defined that the tool electrode (green compact) fabricated by powder compaction process is called green compact [2]. The loose bonding between particles facilitates the material deposition over the substrate. EDC is capable to deposit the thick coating over the workpiece by alternating the various process parameters as implemented by the Patowari et al. [3]. Krishna and Patowari analyzed melting and vaporization takes place in EDM process due to the occurrence of multiple electrical discharges between tool electrode and the work surface in presence of hydrocarbon oil [4]. Beri et al. also discussed that the electrical discharge process can be applied in aerospace and automotive industry. At high temperature, the chemical reaction starts between the eroded tool and carbon present in dielectric oil and results in the formation of carbide [2]. Goto verified the phenomenon of thick layer formation when the carbide piled up over the work surface [5]. Solid lubricant was revealed to be the finest choice for components that require high temperature/vacuum for working [6]. EDC has been widely studied by the researchers but limited research work carried out on the formation of solid lubricant coating using EDC process through MoS2 + Cu powder green compact tool. In this paper, solid lubricant coating of powder mixture (MoS2 + Cu) in green compact form is done over a mild steel substrate. The green compact is prepared by compressing the powder at a specific pressure and temperature to provide loose bonding between powder particles. The effect of various experimental input process parameters (duty factor, peak current, and mixing ratio) on the output parameters (tool wear rate (TWR), mass deposition rate (MDR), and coating thickness) is also studied. For this purpose, response surface methodology (RSM) is applied to apply the mathematical relation among independent and dependent variables. The surface plots of all the output responses with change in input have been drawn. Many experiments carried out in this work to find out that if experimental values are in good agreement with those predictive results.
2 Experimentation Various trials were conducted to achieve a desirable mixture ratio of MoS2 and Cu powder for the green compact. Then for the final electrode, to be used in the main experiment, MoS2 powder was mixed with Cu powder in a ratio of 0.6:1, 1:1 and 1.5:1 by percentage weight. The powder compact tool was made with mixture of molybdenum disulfide (MoS2 ) and copper (Cu) powders in a hot mounting press. The compaction of prepared powder mixture was done in hot mounting press at a pressure 200 kg/cm2 , the temperature 130 °C, heating time 10 min, and cooling 5 min. A powder compact having diameter 15 mm and height 3 mm was formed and that size depends on the powder mixture amount used. A green compact electrode shown in Fig. 1a was then pasted with copper rod end using the conductive paste. At the end, the complete tool electrode was used to perform the coating process. The
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(b) Tool electrode
Workpiece
Clamping vice
Fig. 1 a Tool electrode, b setup arrangement for EDC process
EDC process was carried out using die sink EDM by connecting tool and workpiece to reverse polarity in presence of hydrocarbon oil as shown in Fig. 1b. The process started with a series of electrical discharges. A chemical reaction takes place between dielectric particles and the tool material, as a result of which a layer forms over the work piece. The tool melts faster than the workpiece and material deposition starts from the tool to the work surface due to loose bonding between particles. Few EDM parameters those kept constant were gap voltage: 40 V, time of coating: 4 min, pulse on time: 29 µs, for all the experiment and output parameters [layer thickness (LT), tool wear rate (TWR), mass deposition rate (MDR)] were evaluated as given in Table 1.
3 Results and Discussion 3.1 Response Surface Methodology A central composite design of experiment is used with 3 levels, 3 variables, and 15 runs. The influence of coating parameters upon tool wear rate (TWR), mass deposition rate (MDR)m and layer thickness (LT) is observed. Response surface methodology (RSM) was applied to apply the mathematical relation among independent and dependent variables. Three variable parameters and three responses are taken into account; therefore, a large range of experiments were performed. The application of RSM can deduct the total experiments up to some extent, and interactions of these are possible to be effectively studied. With the application of design of experiments (DOE) along with regression analysis, response of independent input parameters has been retrieved. Independent process parameters are represented quantitatively as given below: y = f (x1 , x2 , x3 , . . . , xn ) ± ε
(1)
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Table 1 Experimental parameters and results S. No.
Input process parameters Current (A)
Output process parameters
DF (%)
(MoS2 :Cu)
TWR (g/m)
MDR (g/m)
LT (mm) 0.553
1
7
30
1.50
0.20
0.031
2
4
90
1
0.06
0.015
0.300
3
10
60
1.50
0.07
0.004
0.495
4
10
90
1
0.11
0.024
0.388
5
7
90
1.50
0.03
0.009
0.400
6
7
30
1
0.08
0.040
0.363
7
4
60
1
0.23
0.056
0.546
8
7
60
0.66
0.06
0.001
0.207
9
10
30
0.66
0.13
0.082
0.322
10
7
30
1
0.32
0.057
0.558
11
7
60
1
0.23
0.040
0.546
12
7
60
1
0.23
0.080
0.546
13
7
90
0.66
0.06
0.005
0.283
14
10
60
0.66
0.21
0.075
0.522
15
4
60
1.50
0.21
0.027
0.513
Here, desired response is y, response function is f , independent input variables are x 1 , x 2 , …, x n , and ε is fitting error. After that, expected response is plotted to obtain requisite surface termed as response surface. The RSM suitability is defined by the approximate f value obtained by second order polynomial regression model, which also termed as quadratic model. The quadratic model is given below: f = a0 +
n i=1
ai xi +
n i=1
aii xi2 +
n
ai j xi x j + ε
i< j
A final regression equation becomes: MDR = −0.212 − 0.00037duty factor + 0.0484y + 0.251z − 0.000011x ∗ x − 0.001648y ∗ y − 0.0968z ∗ z − 0.000022x ∗ y + 0.001091x ∗ z − 0.01925y ∗ z TWR = −1.361 + 0.01188x + 0.1491y + 1.333z − 0.000067x ∗ x − 0.00306y ∗ y − 0.368z ∗ z − 0.000528x ∗ y − 0.00198x ∗ z − 0.0575y ∗ z
(2)
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LT = −1.5475 + 0.01532x + 0.1904y + 1.5471z − 0.000105x ∗ x − 0.005500y ∗ y − 0.3529z ∗ z − 0.000297x ∗ y − 0.002262x ∗ z − 0.06607y ∗ z − 0.000105x ∗ x − 0.005500y ∗ y − 0.3529z ∗ z − 0.000297x ∗ y − 0.002262x ∗ z − 0.06607y ∗ z Here, x = duty factor, y = peak current, z = composition. The experiments were conducted by varying the current, duty factor, and mixing ratio which showed the extensive effect on the change in output responses as presented in Table 1. The peak current was varied from 4 to 10 A because low current below 4 A decreases the material deposition rate and high current above 10 A results in unstable coating process. The mixing ratio value for MoS2 :Cu was taken in such a way that green compact tool exhibit sufficient strength that can provide easy transfer of material over the mild steel. After the trial experiments, the suitable ratio was MoS2 :Cu/1, 1.50, 0.66. The effect duty factor value was observed for 30 to 90% as increase and decrease in this value only tend toward arcing process. In this way, the process parameters were selected.
3.2 Analysis of Variance (ANOVA) The ANOVA of TWR, MDR, and LT is given in Tables 2, 3 and 4. The p-value indicates that probability of F-value > calculated owning to noise. F value is used to find the significance of values. When p-value goes beyond 0.05, the term becomes insignificant which lacks fit. It shows that non-significant term is discarded from the model to allow it to fits well. As the table shows that P values are significant as all the p values ac ) are undergoes elastic deformation. Therefore, it undergoes both elastic and plastic deformation [5]. For the plastic deformations of the contacted aspirates, the stiffness induces at the interface are almost equal to zero. For elastic–plastic deformation of the contact interface aspirates, the total force and the total real contact area consists of two parts, i.e. elastic and plastic deformation. Hence, the total elastic–plastic force and area are as shown in Eqs. (13) and (14) [5]. (3−2D) (3−D) D D K σ y (2−D) c1 D c1 D 2 2 al 2 + ac ac al 2 − F= 2− D 3 − 2D 3 − 2D
Ar =
(2−D) D D 3D al + ac 2 al 2 2(2 − D) 2(2 − D)
(13) (14)
The normal contact stiffness kn is originated by the elastic deformation of the aspirates is given in Eq. (15)
al Kn =
kn n(a)da ac
(15)
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Substitute Eq. (10) and (12) in Eq. (15) the final equation of the contact stiffness is obtained as. √ (2−D) D 1 2 2D(3 − D) Kn = √ (16) E ∗ ac 2 al 2 − al2 3 π (2 − D)(D − 1)
4 BT-40 Spindle Shaft Case Study The study and understanding of the behaviour of joint parameter of spindle and the tool holder interface is most significant task for the machine tool designers, because the stability of the machining component greatly depends on the stiffness between the spindle and the tool holder joint. The stiffness at that interface changes with the change in the critical area of aspirates contact and largest truncated area of contact aspirates and this behaviour depends on the tool holder clamping force. Hence, here we consider the fractal topology study to calculate the interface stiffness using roughness of the spindle and tool holder. Here, author considers BT-40 spindle shaft and the tool holder whose dimensional parameters are considered as per ISO standard and its taper angle is 8.29°. The yield strength for the softer surface σ y of the spindle and tool holder and young’s modulus E 1 and E 2 , respectively. Whose values are tabulated in Table 1. The spindle and the tool holder substructure are well grinding with better heat treatment process having the roughness value is around 0.2 μm, for this surface roughness parameter, the characteristics length G and the Fractal dimensions D are calculated using the power spectral density which was reported by Zhao [15]. These values are tabulated in Table 2. The spindle and the tool holders are connected by using the bolted mechanism to induce the different clamping forces. One end of the tool holder is connected to the long bolted screw passes through the hallow section of the spindle shaft, and then other ends of the bolts are used to vary the different clamping force by tightening the nut through flange (rear end of the spindle shaft). Here, author considers different condition of clamping force to understand how the fractal surface behaves Table 1 Material Property of tool holder and spindle shaft N E 2 (GPa) E 1 (GPa) σ y mm 2 1300
200
Table 2 Fractal parameters [15]
200
ϑ1
ϑ2
0.3
0.3
D (fractal dimension)
G (Characteristic length)
1.26
2.7 × 10−12 mm
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as change in the contact area of the truncated aspirates, this means to say that the largest truncated area of contact of the aspirates changes with the change in clamping forces. Hence, the projected normal contact force induced on the contacted interface for different clamping force can be calculated using the following Eq. (17) F = Fd sin α
(17)
From blue match technique, visual area of contact is around 90% of the apparent area of contact. But in reality, the real area of contact is extremely small. However, the actual contact area is Aa = 7.49×10−3 m2 from Eq. (1) and for 12 kN of drawbar force the real area of contact Ar = 1.02 × 10−6 m2 (from Eq. (14)). As per the above discussion of the bolted mechanism, author used the 16 mm diameter steel bolt, and its co-efficient of friction is 0.2. By using this parameter the clamping force can be altered for different torque and is tabulated in Table 3. Where F d is the clamping force, F is the normal projected force at the contact interface and P is the contact pressure. These forces influence the joint interface behaviour this means to say that, as the force changes the joint stiffness also varies. This is because of elastic–plastic deformation of the aspirate. Then the largest area of the truncated aspirates can be calculated by using Eq. (13) similarly the critical area of aspirates for the fractal parameters of roughness value 0.2 μm is ac = 7.37 × 10−8 m2 calculated by using Eq. (6). The largest truncated are for different condition of drawbar force is as tabulated in Table 4. From the above calculation, the largest area of the truncated aspirates is greater than the critical area of contact; this concludes that deformation of the aspirates at the spindle-tool holder interface is elastic–plastic in nature. Then the total real area of contact can be calculated by integrating the distribution function n(a) from smallest contact area to largest truncated contact aspirates. As per the fractal topology theory, the stiffness induced at the joint interfaces depends on the fractal parameters and different clamping forces, for that the overall Table 3 Force at different torque
Table 4 Largest truncated area for different clamping force
Torque (Nm)
Fd (kN)
F (kN)
P (MPa)
20
6.25
0.9
0.12
38.5
12
1.73
0.23
40
12.5
1.8
0.24
60
18.75
2.7
0.36
Fd (kN)
al m2
6.25
2.46 × 10−7
12
4.8 × 10−7
12.5
4.98 × 10−7
18.75
7.4 × 10−7
602 Table 5 Real area of contact and their stiffness
R. H. Aralaguppi et al. al m2
Ar m2
Kn
N
2.46 × 10−7
6.13 × 10−7
3.35 × 108
4.8 × 10−7
1.02 × 10−6
4.69 × 108
4.98 × 10−7
1.04 × 10−6
4.71 × 108
7.4 × 10−7
1.43 × 10−6
5.82 × 108
m
stiffness induced at the interface is calculated by using Eq. (16). Similarly, for different clamping forces, the overall stiffness induced is as tabulated in Table 5. The above calculation indicates that the ratio of apparent area of contact to real area of contact is extremely small (e.g. Ar /Aa = 0.013618% for 12 kN drawbar forces). In order to verify the efficiency of theoretically calculated contact stiffness value, a finite element model of Eigen frequency study is established in the COMSOL multiphysics software. Then obtained Eigen frequencies are correlated with the frequencies of experimental modal analysis.
4.1 Experimental Modal Analysis The dynamic inherent characteristics like natural frequencies and mode shapes of the spindle and tool holder can be experimentally calculated using modal analysis technique. These results will indicate the dynamic behaviour of spindle –tool holder subsystem and how the structures will response to the dynamic loading. To obtain the natural frequencies of spindle and tool holder substructure for different clamping forces (or contact stiffness) is need to be subjected for experimental modal analysis. The different clamping force can be induced by varying the torque of tighten nut, the arrangement of spindle with bolted mechanism is as shown in Fig. 4. In experimental modal analysis, the free-free boundary condition is achieved by hanging the spindle using optimum stiffen spring is as shown in Fig. 5.
Fig. 4 Spindle and tool holder substructure
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Fig. 5 Free-free boundary condition of spindle shaft
Brief explanation of experimental analysis procedure: 1.
2.
3.
As it is an Impact hammer test, initially mark the points on the spindle and the tool holder. Then select any one of the marked points for the excitation using impact hammer. Place an accelerometer at any one of the marked points and make an excitation at the selected excitation point. Then capture the vibration response. Similarly, repeat the same procedure for all the marked points in sequential manner. After collecting vibration data from all the points are subjected to curve fitting technique, then we get the frequency response function of spindle and tool holder, i.e. natural frequencies and mode shapes.
Case 1 In the first case, we consider the spindle shaft and the tool holder is held by projected normal contact force of 0.9 kN (i.e. for 20 Nm Torque and 6250 N Clamping force). Spindle shaft and tool holder consists of 52 marked points. The frequency response (FRF) obtained at all the marked point on the spindle shaft is as shown in Fig. 6. The intermediate mode shape at frequency around 2300 Hz, mostly indicates the maximum deformation at the spindle-tool holder joint section, and therefore, it can conclude by the joint stiffness (3.35 × 108 N/m). The mode shape at that frequency is as shown in Fig. 6 and the frequency response function is as shown in Fig. 7. Case 2 In second case, author considers the typical condition of 12.5 KN clamping force, by providing the torque rate around 40 Nm, whose joint stiffness, i.e. K n = 4.71 × 108 N/m for that the frequency response function is taken from all marked 52 points including both spindle and tool holder is as shown in Fig. 8. From this frequency
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Fig. 6 Experimental mode shapes at 2360 Hz
Fig. 7 Frequency response function for 6.25 kN clamping force
analysis, the mode shape at frequency 3360 Hz indicates the maximum deformation occurs at the spindle and tool holder interface, and therefore, it is mainly decided by the contact stiffness.
4.2 Finite Element Analysis The 3D CAD model of spindle shaft and tool holder is modelled, whose dimensions are considered as per the ISO standards, the CAD model is as shown in Figs. 9 and 10.
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Fig. 8 Frequency response function for 12.5 KN clamping force
Fig. 9 3D CAD model of spindle shaft and the tool holder
Fig. 10 Cross-sectional view of shaft and tool holder
The 3D CAD model is imported to COMSOL multiphysics and is subject to Eigen frequency analysis to get the dynamic inherent characteristics like natural frequency and mode shapes. In order to provide the joint stiffness, select the two surfaces of spindle and tool holder joint (flexible connection) and input the theoretically calculated stiffness value for 12.5 kN drawbar forces, i.e. K n = 4.71 × 108 N/m without considering any damping value. In free-free boundary conditions, the first six rigid mode shapes of frequencies are almost equal to zero, this is because of 3 translational motion along x, y, z axes and their correspond rotational motion, respectively. Then one of the frequencies around
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Fig. 11 Mode shape obtained for 12.5 Kn clamping force
3208 Hz indicates the maximum deformation occurs at the spindle and tool holder interface, and therefore, it can be decided by the joint stiffness at that interface, the mode shape of that frequency is as shown in Fig. 11. From the above analysis, the intimidated frequency obtained by experimental result, i.e. 3360 Hz is predominantly influenced the maximum deformation at the spindle and tool holder interface. Similarly, the numerical study indicates the maximum deformation occurs at spindle and tool holder to 3208.8 Hz (for K n = 4.71 × 108 N/m joint stiffness). Hence, the theoretically calculated stiffness values from fractal topology theory and their corresponding dynamic results from the numerical analysis are almost correlated with the dynamic analysis experimental result.
5 Conclusion In this paper, author considered the fractal topology theory to understand the effect of joint stiffness by the dynamic behaviour of spindle shaft and the tool holder substructure assembly. The spindle and the tool holder substructure assembly is the delicate joint in the machine tool, and hence, it is the most significant task to understand these joint parameters. Many researchers were worked on this to understand the joint parameters by using Reacceptance Coupling Substructure Analysis (RCSA) method and Surface Fractal Topology Theory. Here, author considers the surface fractal topology theory to understand the joint parameter. In this real contact area of the spindle shaft and the tool holder surfaces are less than the apparent contact areas.
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This contacted area is calculated by integrating the aspirates present on the surface, which is obtained based on the surface roughness parameters like fractal dimension and the characteristics length. Then it is necessary to understand the relationship between clamping force and joint stiffness. This can be derived based on the contact stiffness model developed by surface fractal topology theory. The obtained theoretical calculated joint stiffness value is used in Eigen frequency study to understand the influence of joint stiffness on the dynamic behaviour of the spindle and tool- holder substructure. Then obtained frequencies and mode shapes in FEA are correlated with the frequencies obtained in experimental modal analysis. This experimental result shows the efficiency of proposed fractal topology model in contact stiffness determination. Therefore, understanding the joint behaviour is most prominent for the machine tool designers to optimize the joints.
References 1. Zhang GP, Huang YM, Shi WH, Fu WP (2003) Predicting dynamic behavior of a whole machine tool structure based on computer aided engineering. Int J Mach Tool Manuf 43(7):699–706 2. Budka E, Erturk A, Ozguven HN (2006) A Modeling approach for analysis and improvement of spindle-holder-tool assembly dynamics. CIRP Ann 55(1):369–372 3. Schmitz T, Smith L, Scott SK (2009) Machining dynamics-frequency response to improved productivity. Springer. ISBN 978-0-387-09645-2 4. Chikate PP, Basu SK (1975) Contact stiffness of machine tool joints. Tribol Int 8(1):9–14 5. Zhao Y, Song X, Cai L, Liu Z Cheng Q (2014) Surface fractal topography-based contact stiffness determination of spindle-tool holder joint. J Mech Eng Sci 0(0):I–9 6. Wang JH, Horng SB (1994) Investigation of tool holder system with a taper angle. Int J Mach Tool Manuf 34(8):1163–1176 7. Namazi M, Altintas Y, Abe T, Rajapakse N (2006) Modelling and identification of tool-holder spindle interface dynamics. Int J Mach Tool Manuf 47(9):1333–1341 8. Jhon S (2005) Agapiou a methodology to measure joint stiffness parameter for tool holder spindle interface. J Manuf Syst 24(1):13–20 9. Greenwood JA, Williamson J (1996) Contact of nominally flat surface. Proc Royal Soc Mathe Phys Eng Sci 295(1442). Online ISSN 2053–9169 10. Majumdar A, Tien CL (1990) Fractal characterization and simulation of rough surface. Wear 136(2):313–327 11. Majumdar A, Bhushan B (1991) Fractal model of elastic-plastic contact between rough surface. American Soc Mech Eng J Tribol 113(1) 12. Johnson KL Contact mechanics. Cambridge University Press, Cambridge, UK, pp. 153– 155,175–176, 196–201,208–210 and 427–428. 13. Sackfield A, Hills DA Some useful results in the tangentially loaded hertzian contact problem. J Strain Anal 18:107–110 14. Wang S, Komvopoulos K (1994) A fractal theory of the interfacial temperature distribution in the slow sliding regime: Part II—multiple domains, elastoplastic contacts and applications. American Soc Mech Eng J Tribol 116(4) 15. Zhao Y, Song X, Cai L (2015) Surface fractal topology based contact stiffness determination of spindle holder joints 230(4)
Buckling Analysis of Nonlinear First-Order Shear Deformation Composite Plates Ashes Maji
and Prashanta Kr. Mahato
Abstract This paper developed the buckling analysis of nonlinear behavior of firstorder shear deformation laminated composite plates. The first-order displacement functions are used based on some simplifying assumptions. This article further simplifies the transverse displacement into bending and shear components at the midplane have reduced the number of variables and governing differential equations. The von Karman assumptions are used to derive the nonlinear strains based on the displacement functions. Hamilton’s principle is used to derive the equation of motion and boundary conditions. The analytical solutions are developed from the present theory and compared the results of the critical buckling loads with exact and three-dimensional solutions. Result shows that the present theory has closely associated with the exact and three-dimensional theory. Keywords Laminated composite plate · Shear deformation theory · Buckling
1 Introduction The increasing use of composite materials in plate construction the conventional methods of analysis are inadequate. Therefore, accurate analysis is more important than conventional. Nowadays, laminated composites are tremendously used in every fields of applications such as spacecraft, aircraft, automotive and marine structural applications because of different advantageous features such as high stiffness and strength to weight ratio and low cost of maintenance. The enormous increase in engineering applications the variation of laminated theories has been developed. Many reviews articles based on laminated plates and shells have been published in Refs [1–3]. The classical laminated plate theory (CLPT) is an extension of Love– Kirchhoff’s classical plate theory for isotropic plates applied for thin laminates. This theory omitted the transverse shear deformation and rotary inertia effects [4–6]. To A. Maji (B) Asansol Engineering College, Asansol 713305, India P. Kr. Mahato Indian Institute of Technology (ISM), Dhanbad 826001, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_60
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defeat the constraints of CLPT, Reissner [7] first developed the shear deformation theory with the effect of transverse shear deformation and rotary inertia. Mindlin [8] developed a plate theory based on first-order well known as Mindlin’s first-order plate theory [9–11] where he considers the transverse shear effects in conjunction with shear correction factors [12–15]. Many studies are presented based on the buckling of laminated composites plates using first-order shear deformation plate theory (FSDPT) can be found in Refs. [16–19]. This paper is to develop the buckling analysis of nonlinear behavior of orthotropic type laminated composite plates by using FSDPT method. This paper considers Von Karman nonlinear strain to develop the relations between strain and displacement. The conventional first-order shear deformation theory has five functions and these have been done by considered constant transverse normal strain and transverse shear strains at the top and bottom surface of the laminates are zero by using the theory of Lo et al. [20]. In this article, we simplifying transverse displacement w = wb + ws , where wb and ws are the bending and shear component of the midplane respectively initially proposed by Refs. [21–23] and also φx = −∂ Wb /∂X and φ y = −∂ Wb /∂y where φ is the rotation vector about x and y axis, respectively. Split of transverse displacement in the FSDPT method the number of unknown functions and governing differential equations has been reduced to four. This assumption makes in-plane rotation is constant within the laminate plane and has little effect on the stability of the orthotropic and composite laminate plates. Hamilton’s principle is used to derive the equation of motion and boundary condition. A closed-form solution has been developed and compared the results with the exact and three-dimensional theory.
2 Theoretical Formulations 2.1 Basic Assumptions Consider a laminated plate of thickness h and edge dimension a and b. The plate is assumed to a Cartesian coordinate system x-y-z, where x, y plane is the middle plane of the plate and z axis is normal to the middle surface of the plate. For laminated composite plates, the basic assumption is taken as. (Fig. 1). 1. 2.
The displacement is small in comparison with the plate thickness ‘h’ and therefore, consider plain stress problems. The transverse displacement ‘w’ dividing three components extension wa , bending wb and shear ws . These components are functions of coordinates x and y only. w(x, y) = wa (x, y) + wb (x, y) + ws (x, y)
3.
(1)
The In-Plain Normal Stress σx and σy is Large in Compression with Transverse Normal Stress σz Therefore, Neglected.
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Fig. 1 Coordinate system and layer number of typical laminate.
2.2 Kinematics The displacement in the longitudinal and transverse direction is given by [7] u(x, y, z) = u 0 (x, y) + zφx (x, y) v(x, y, z) = v0 (x, y) + zφ y (x, y) w(x, y, z) = w0 (x, y)
(2)
where u0 , v0 and w0 are the in-plane and transverse displacement functions of the corresponding coordinate points x and y on the reference surface and φ x and φ y are the rotations about y and x axes respectively of the normal to the mid-surface in the undeformed plates. The transverse displacement ‘w’ making further assumptions that the extension is very low as compare to bending and shear {i.e., w(x, y) = wb (x, y) + ws (x, y)}. Also, φx = −∂wb /∂x and φ y = −∂wb /∂y, the displacement field of the FSDPT can be rewritten as [21]: ∂wb ∂x ∂wb v = v0 − z ∂y w = wb (x, y) + ws (x, y) u = u0 − z
(3)
The Von Karman nonlinear strain associated with the displacement in Eq. (3) is given by ∂u 0 1 ∂wb ∂ws 2 ∂u ∂ 2 wb = + + −z ε1 = εx = ∂x ∂x 2 ∂x ∂x ∂x2 2 ∂v0 1 ∂wb ∂ws ∂v ∂ 2 wb = + + ε2 = ε y = −z ∂y ∂y 2 ∂y ∂y ∂ y2
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ε3 = εz = ε4 = 2ε yz ε5 = 2εzx ε6 = 2εx y
∂w =0 ∂z ∂ws = ∂y ∂ws = ∂x ∂v0 ∂wb ∂ws ∂wb ∂ws ∂ 2 wb ∂u 0 + + + + − 2z = ∂y ∂x ∂x ∂x ∂y ∂y ∂ x∂ y
The associated strains have the form ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ε10 ⎪ ε11 ⎪ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ ε2 ⎪ ⎨ ε21 ⎪ ⎬ ⎪ ⎬ ⎨ ε2 ⎪ ⎬ 0 1 ε6 = ε6 + z ε6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε4 ⎪ 0 ⎪ ⎪ 4⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ε0 ⎪ ⎩0 ⎪ ⎭ ⎪ ⎭ ⎩ε ⎪ ⎭ 5 5 ⎧
b 2 ∂u 0 ⎧ ⎫ ⎪ s + 21 ∂w + ∂w ⎪ 0 ∂ x ∂ x ∂ x ⎪ ⎪ ε1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ∂ws ∂v0 1 ∂wb ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ + + ε ⎨ ⎨ ⎬ ∂ y 2 ∂y ∂ y 2 0
∂wb ∂wb ∂ws ∂u 0 ∂v0 ε = ε60 = + + + + ⎪ ⎪ ⎪ ∂ y ∂ x ∂ x ∂ x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ε40 ⎪ ∂w ⎪ ⎪ ⎪ s ⎪ ⎪ ⎩ ε0 ⎪ ⎭ ⎪ ∂y ⎪ ⎩ 5 ∂ws ⎧ ∂ 2 wb ⎪ − ∂x2 ⎧ 1⎫ ⎪ ⎪ 2 ⎪ ⎪ − ∂∂ yw2b 1 ⎨ ε11 ⎬ ⎨ 2 ε = ε2 = −2 ∂ wb ∂ x∂ y ⎩ 1⎭ ⎪ ⎪ ⎪ ε6 ⎪ 0 ⎪ ⎩ 0
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
∂x
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4)
(5)
∂ws ∂y
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
(6)
2.3 Equation of Motion Hamilton’s principle is used to derive the equation of motion as: T 0=
(δU + δW − δ K )dt
(7)
0
where δU, δW and δK are the virtual strain energy, virtual work done by applied external forces and virtual kinetic energy, respectively.
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The variation of strain energy can be expressed by: δU =
a b +h/2
σ1 δε1 + σ2 δε2 + σ6 δε6 + σ4 δε4 + σ5 δε5 dx d y dz 0 0 −h/2
=
⎧ ⎪ h ⎪ a b ⎪ ⎨ 2 ⎪
⎪ ⎪ h 0 0 ⎩ −2 a b
=
N1 0 0
⎫ ⎪ ⎪ ⎬ ⎪ σ1 δε10 + zδε11 + σ2 δε20 + zδε21 + σ6 δε60 + zδε61 + σ4 δε40 + σ5 δε50 dz dx d y ⎪ ⎪ ⎪ ⎭
∂δu 0 ∂wb ∂ ∂ws ∂ 2 δwb + N1 + + M1 − 2 ∂x ∂x ∂x ∂x ∂x
∂δv0 ∂wb ∂ ∂ws ∂ 2 δwb + N2 + + M2 − 2 ∂y ∂y ∂y ∂y ∂y ∂ ∂δu 0 ∂wb ∂δv0 ∂ws + N6 + N6 + + ∂y ∂x ∂y ∂x ∂x ∂ ∂wb ∂ws + N6 + ∂x ∂y ∂y ∂δws ∂δws ∂ 2 δwb +M6 −2 dx d y + Q1 + Q2 ∂ x∂ y ∂y ∂x + N2
(8)
We used the strain from Eq. (4) and the following stress resultants N, M and Q is define by: h
2
σi (1, z)dz (i = 1, 2, 6)
Ni, Mi = − h2 h
2 (Q 1 , Q 2 ) =
(σ5 , σ4 )dz
(9)
− h2
The variation of work done by external force is calculated by a b qδ(wb + ws )dxdy
δW = − 0
0
where q is the external load of transverse directions. The Kinetic energy variation is calculated by:
δK =
a b +h/2 ρ(uδ ˙ u˙ + vδ ˙ v˙ + wδ ˙ w)dzdxdy ˙ 0 0 −h/2
(10)
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A. Maji and P. K. Mahato h
a b 2 =
ρ 0 0 −h 2
u˙ 0 − z
∂ w˙ b ∂x
∂δ w˙ b ∂ w˙ b ∂δ w˙ b δ u˙ 0 − z + v˙0 − z δ v˙0 − z ∂x ∂y ∂y
+(w˙ b + w˙ s )δ(w˙ b + w˙ s ) dzdxdy a b =
I0 u˙ 0 δ u˙ 0 + v˙0 δ v˙0 + (w˙ b + w˙ s )δ(w˙ b + w˙ s )
0 0
∂δ w˙ b ∂ w˙ b ∂δ w˙ b ∂ w˙ b u˙ 0 + δ u˙ 0 + v0 + δv0 ∂x ∂x ∂y ∂y ∂ w˙ b ∂δ w˙ b ∂ w˙ b ∂δ w˙ b +I2 + ∂x ∂x ∂y ∂y −I1
⎫ ⎪ ⎪ ⎬ dxdy ⎪ ⎪ ⎭
(11)
where dot-superscript indicates the differentiation with respect to time variable, the mass density is given by ρ and (I 0 ,I 1 ,I 2 ) are the mass moment of inertias is given by: +h/2
1, z, z 2 ρdz
I0 , I1 , I2 =
(12)
−h/2
Substituting the appropriate energy expressions δU,δW and δK into the virtual work statement in Eqs. (8), (10) and (11) and integrating by parts and collecting the variables of δu0 ,δv0 ,δwb and δws to zero. The following equation of motion as: ∂ N6 ∂ w¨ b ∂ N1 + = I0 u¨ 0 − I1 ∂x ∂y ∂x ∂ N2 ∂ w¨ b ∂ N6 + = I0 v¨0 − I1 δv0 : ∂x ∂y ∂y 2 ∂ 2 M2 ∂ M1 ∂ 2 M6 + q − N (w) + + 2 δwb : ∂x2 ∂ x∂ y ∂ y2 ∂ u¨ 0 ∂ v¨0 + − I2 ∇ 2 w¨ b = I0 (w¨ b + w¨ s ) + I1 ∂x ∂y ∂ Q2 ∂ Q1 + + q − N (w) = I0 (w¨ b + w¨ s ) ∂ws : ∂x ∂y ∂wb ∂wb ∂ws ∂ ∂ws ∂ N1 + + N6 + N (w) = ∂x ∂x ∂x ∂y ∂x ∂x ∂wb ∂wb ∂ws ∂ ∂ws ∂ N6 + + N2 + + ∂x ∂y ∂y ∂y ∂y ∂y δu 0 :
The natural boundary conditions are obtained by the coefficients as:
(13)
(14)
Buckling Analysis of NonLinear First-Order Shear Deformation …
δu 0 : N1 n x + N6 n y δv0 : N6 n x + N2 n y δwb :
∂ M1 + ∂∂My6 − I1 u 0 + I2 ∂∂w¨xb ∂x ∂ Mns b b Nn ∂w + ∂s + Nns ∂v ∂n ∂s
nx +
∂ M6 ∂x
+
∂ M2 ∂y
615
− I1 v¨0 + I2 ∂∂w¨yb n y
(15)
∂δwb : ∂n
Mn
s s δws : Q n + Nn ∂v + Nns ∂v ∂n ∂s where
Nns = (N2 − N1 )n x n y + N6 n 2x − n 2y Nn = N1 n 2x + 2N6 n x n y + N2 n 2y Qn = Q1nx + Q2n y ∂ ∂ ∂ ∂ ∂ ∂ = nx + ny, = − ny + nx ∂n ∂x ∂y ∂s ∂x ∂y
(16)
where nx and ny be the direction cosines of the outward unit normal to the mid-plane boundary in the x and y directions. M n and M ns are the same forms as N n and N ns .
2.4 Constitutive Equations of a Laminate Consider a composite laminate of n orthotropic layers of thickness h rest on the coordinate of x, y and z. Having assumption that each lamina have a plane of material symmetry parallel to the x-y plane, the constitutive equations of a lamina layer as: ⎧ ⎫ ⎡ ⎪ K 11 σ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢K ⎪ ⎪ ⎪ ⎪ σ 2⎪ ⎢ 12 ⎪ ⎪ ⎬ ⎢ ⎨ ⎪ σ3 ⎢K = ⎢ 13 ⎪ ⎢ 0 ⎪ σ6 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎣ 0 ⎪ σ4 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ σ5 0
K 12 K 22 K 23 0 0 0
K 13 K 23 K 33 0 0 0
0 0 0 K 66 0 0
0 0 0 0 K 44 0
⎤⎧ ⎫ ε1 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ε 2⎪ ⎪ ⎥⎪ ⎪ ⎬ ⎥⎨ ⎪ 0 ⎥ ε3 ⎥ 0 ⎥⎪ ε ⎪ ⎪ 6⎪ ⎥⎪ ⎪ ⎪ ⎪ 0 ⎦⎪ ε 4⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ε5 K 55
(17)
where K ij is defined as the engineering constant of the material in a lamina layer. Substituting Eq. (4) into Eq. (17) and the strain resultants of Eq. (9) the stress resultants are obtained related to the displacement gradients (u0 , v0 , wb , ws ) are: ⎧ ⎧ ⎫ ⎡ ⎤⎪ A11 A12 A16 ⎪ ⎨ ⎨ N1 ⎬ ⎣ N2 = A21 A22 A26 ⎦ ⎪ ⎩ ⎭ ⎩ ∂u 0 + N6 A16 A26 A66 ⎪ ∂y
b 2 ∂u 0 s + 21 ∂w + ∂w ∂x ∂x ∂x 2 ∂v0 b s + 21 ∂w + ∂w ∂ y ∂y ∂ y
b ∂wb ∂v0 s + ∂w + ∂w ∂x ∂x ∂x ∂y
⎫ ⎪ ⎪ ⎬ +
∂ws ∂y
⎪ ⎪ ⎭
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A. Maji and P. K. Mahato
⎤⎧ ∂ 2 wb ⎫ B11 B12 B16 ⎪ ⎨ ∂2x 2 ⎪ ⎬ ∂ wb ⎣ ⎦ − B12 B22 B26 2 ⎪ ∂ y2 ⎪ B16 B26 B66 ⎩ 2 ∂∂ x∂wyb ⎭ ⎧
b 2 ∂u 0 s ⎧ ⎫ ⎡ ⎤⎪ + 21 ∂w + ∂w ⎪ ∂ x ∂ x ∂ x B11 B12 B16 ⎨ 2 ⎨ M1 ⎬ ∂ws ∂v0 1 ∂wb + + M2 = ⎣ B12 B22 B26 ⎦ ∂ y 2 ∂y ∂ y ⎩ ⎪ ⎭ ⎩ ∂u 0 + ∂v0 + ∂wb + ∂ws ∂wb + M6 B16 B26 B66 ⎪ ∂y ∂x ∂x ∂x ∂y ⎡ ⎤⎧ ∂ 2 wb ⎫ D11 D12 D16 ⎪ ⎨ ∂2x 2 ⎪ ⎬ ∂ wb ⎣ ⎦ − D12 D22 D26 2 ⎪ ∂ y2 ⎪ D16 D26 D66 ⎩ 2 ∂∂ x∂wyb ⎭ # ∂ws " A44 A45 Q2 ∂y =k ∂ws Q1 A45 A55 ∂x ⎡
⎫ ⎪ ⎪ ⎬ ∂ws ∂y
⎪ ⎪ ⎭
(18)
where k be the shear correction factor and Aij , Bij, Dij are the stiffness defined as h
2
Ai j , Bi j , Di j =
Q i j 1, z, z 2 dz (i, j = 1, 2, 6)
(19)
− h2
3 Analytical Solutions We consider different boundary condition in the explicit form: Simply supported edge (cross − ply laminate) N1 = v0 = wb = ws = M1 = 0 at x = 0, a u 0 = N2 = wb = ws = M2 = 0 at y = 0, b Simply supported edge (angle − ply laminate) u 0 = N6 = wb = ws = M1 = 0 at x = 0, a N6 = v0 = wb = ws = M2 = 0 at y = 0, b
(20)
(21)
A plate is considered with all edges simply supported of length a and width b under external transverse load q that are based on Navier solution as: wb (x, y, t) =
∞ $ ∞ $ m=1 n=1
Cbmn sin
mπ x nπ y sin a b
Buckling Analysis of NonLinear First-Order Shear Deformation …
ws (x, y, t) =
∞ $ ∞ $
Csmn sin
nπ y mπ x sin a b
Au 0 mn cos
nπ y mπ x sin a b
Bv0 mn sin
nπ y mπ x cos a b
Au 0 mn sin
mπ x nπ y cos a b
Bv0 mn cos
nπ y mπ x sin a b
m=1 n=1
617
(22)
For antisymmetric cross-ply u 0 (x, y, t) = v0 (x, y, t) =
∞ $ ∞ $ m=1 n=1 ∞ $ ∞ $ m=1 n=1
(23)
For antisymmetric angle-ply u 0 (x, y, t) = v0 (x, y, t) =
∞ $ ∞ $ m=1 n=1 ∞ $ ∞ $ m=1 n=1
(24)
√ where i = −1, (Au0mn , Bv0mn , C bmn , C smn ) are coefficient and ω is the natural frequency of free vibration. The transverse load q(x, y) is expressed in terms of double Fourier sin series as: q(x, y) =
∞ $ ∞ $
Q mn sin
m=1 n=1
nπ y mπ x sin a b
(25)
The Fourier coefficient Qmn can be determined from the relationship
Q mn
4 = ab
a b q(x, y) sin 0
nπ y mπ x sin dxdy a b
(26)
0
Q mn = q0
for sinusoidal load
(27)
The constitutive boundary condition of Eqs. (22)–(24) put into the governing Eq. (13) are obtained the following equation as: [[K ] − λ[M]]{ } = {0}
(28)
Therefore, any fixed value of m and n in the element stiffness matrix is denoted by [K] and element mass matrix is denoted by [M]. The coefficient { }T = {Au0mn , Bv0mn , C bmn , C smn }T and also the coefficient λ denoted the buckling load parameter. The elastic constants for element stiffness matrices are used in this articles from
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Ref. [24], K 22 /K 11 = 0.543103, K 33 /K 11 = 0.530172, K 12 /K 11 = 0.23319, K 13 /K 11 = 0.010776, K 23 /K 11 = 0.098276, K 66 /K 11 = 0.262931, K 55 /K 11 = 0.159914 and K 44 /K 11 = 0.26681, where K ij is defined by σ i = K ij εj and i, j = 1, 2, … 0.6.
4 Numerical Results In this article, the critical buckling load of simply supported composite plates are analyzed and compared the result with exact solution of Reddy and Phan [25], threedimensional result of Srinivas and Rao [24] and the result of Senthilnathan et al. from Ref. [26]. The material constant used in this article are E 1 /E 2 = 40, G23 /E 2 = 0.5, G12 /E 2 = G13 /E 2 = 0.6 and ν = 0.25 where subscripts used 1, 2 and 3 are x, y, and z axes of the laminated plates. Table 1 compared the critical buckling load parameter λ developed by the present theory with three-dimensional theory and theory by Senthilnathan et al. for homogeneous orthotropic plates with h/b = 0.05, 0.1 and 0.2. The observation can be made from this table that the classical plate theory gives constant buckling load even for thick homogeneous orthotropic plates. The accuracy of results in the threedimensional solutions and present theory are almost equally associated and difference between the two results are minimal. Figures 2 and 3 contains critical buckling loads vs. modulus ratios for both the antisymmetric cross-ply and angle-ply laminated composite two-layer plates. The following discussion may be made from the figures: (1) the bending-extension coupling effects on the critical buckling load is greatest when degree of anisotropy is increased (from Fig. 2). (2) The transverse shear stress on critical buckling load directly depends on the modulus ratio of the materials. The ratio increases with the increase in transverse shear on critical buckling load (from Fig. 3). Table 2 and 3 discussed the critical buckling loads of antisymmetric cross-ply and angle-ply laminated composite plates. From the tables observed that the numerical results calculated from the present analysis and result obtained from the exact and three-dimensional theory are closely associated. Therefore, for all plate side to thickness ratio and aspect ratio are in excellent agreement with each other. Table. 1 The critical buckling load of square homogeneous orthotropic plates where λ = (12N 1 /π 2 E 1 )(b/h)2 , (k = 5/6) h/b
CPTa
Exact theorya
FSDPTb
Present
0.05
3.039
2.966
2.9774
2.967
0.10
3.039
2.770
2.8074
2.778
0.20
3.039
2.210
2.2910
2.221
a using
values in Ref. [24],
b using
values in Ref. [26]
Buckling Analysis of NonLinear First-Order Shear Deformation …
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Fig. 2 The material anisotropy on the critical buckling load for antisymmetric Cross-ply orthotropic laminates [0°/90°/…] (a/h = 10)
Fig. 3 The material anisotropy on the critical buckling load for antisymmetric Angle-ply orthotropic laminates [45°/−45°/…] (a/h = 10)
5 Conclusions First-order nonlinear shear deformation theory has been used to determine the critical buckling load of laminated composite plates. The laminated composite plates are used with simply supported boundary condition is considered. The present solutions are obtained by using Von Karman nonlinear strain equations and compared the results with the classical plate theory, exact solution and three-dimensional solutions. The present observations suggest that the effect of critical buckling load of composite plates have closely associated with other investigators and also have minimal effect on the accuracy.
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Table. 2 The critical buckling loads for antisymmetric cross-ply square plates where λ = (N 1 b2 /E 2 h3 ), (k = 5/6) [0°/90°]
[0°/90°/0°]
[0°/90°/90°/0°]
(12.628)a
(35.831)
(35.831) FSDPT Present FSDPTb
FSDPT Present
a/h
FSDPTb
FSDPTc
Present
FSDPTb
5
8.142
8.277
8.277
10.525
–
10.625
11.533
–
11.534
10
11.099
11.352
11.353
21.643
–
21.648
23.270
–
23.271
12.5 11.605
12.882
12.887
25.144
–
25.346
26.518
–
26.518
20
12.208
12.515
12.526
30.664
–
30.667
31.432
–
31.431
25
12.356
12.671
12.672
32.332
–
32.431
32.872
–
32.872
50
12.559
12.886
12.887
34.883
–
34.885
35.037
–
35.038
12.611
12.947
12.947
35.589
–
35.592
35.629
–
35.629
100 a CPT
values, b using values in Ref. [25], c using values in Ref. [26]
Table. 3 The critical buckling loads for antisymmetric angle-ply square plates [45°/−45°] where λ = (N 1 b2 /E 2 h3 ), (k = 5/6)
(21.708)a a/h
FSDPTb
FSDPTc
Present
5
11.024
11.148
11.149
10
17.453
17.552
17.553
12.5
18.846
18.852
18.852
20
20.469
20.496
20.495
25
20.916
20.916
20.916
50
21.507
21.507
21.507
100
21.667
21.666
21.667
a CPT values, b using values in Ref. [25], c using values in Ref. [26]
References 1. Liu D, Li X (1996) An overall view of laminate theories based on displacement hypothesis. J Comp Mat 30(14):1539–1561 2. Ghugal YM, Shimpi RP (2002) A review of refined shear deformation theories of isotropic and anisotropic laminated plates. J Renf Plast Comp 21(9):775–813 3. Khandan R, Noroozi S (2012) The development of laminated composite plate theories: a review. J Mat Sci 47:5901–5910 4. Ambartsumyan SA (1969) Theory of anisotropic plates (translated from Russian). Technomic Stanford, CT 5. Stavsky Y (1961) Bending and stretching of laminated isotropic plates. J Eng Mech ASCE 87(6):31–56 6. Whitney JM, Leissa AW (1969) Analysis of heterogeneous anisotropic plates. J Appl. Mech 36(2):261–266 7. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–77 8. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J Appl. Mech 18:31–38
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9. Reissner E (1950) On a variational theorem in elasticity. J Math Phy 29:90–95 10. Dobyns AL (1981) Analysis of simply supported orthotropic plates subjected to static and dynamic loads. AIAA J 19(5):642–650 11. Reissner E (1972) A consistent treatment of transverse shear deformations in laminated anisotropic plates. AIAA J 10(5):716–718 12. Whitney, J.M.: Shear Correction Factors for Orthotropic Laminates under Static Load. ASME, J Appl. Mech, 40 (1), 303–304. (1973). 13. Chatterjee SN, Kulkarni SV (1979) Shear correction factors for laminated plates. AIAA J 17:498–499 14. Ferreira AJM (2003) A formulation of the multi-quadric radial basis function method for the analysis of laminated composite plates. J Comp. Struct 59:385–392 15. Castellazzi G, Krysl P, Bartoli I (2013) A displacement-based finite element formulation for the analysis of laminated composite plates. J Comp. Struct 95:518–527 16. Whitney JM, Pagano NJ (1970) Shear deformation in heterogeneous anisotropic plates. J Appl Mech Tr ASME 37(4):1031–1036 17. Khdeir AA (1989) Comparison between shear deformable and Kirchhoff theories for bending, buckling and vibration of antisymmetric angle-ply laminated plates. J Comp Struct 13(3):159– 172 18. Ferreira AJM, Castro L, Bertoluzza, S (2009) A high order collocation method for the static and vibration analysis of composite plates using a first-order theory. J Comp Struct 89(3):424–32 19. Nelson RB, Lorch DR (1974) A refined theory for laminated orthotropic plates. J Appl. Mech 41:177–183 20. Lo KH, Christensen RM, Wu EM (1977) A higher-order theory of plate deformation, Part 1: homogeneous plates. J Appl. Mech 44:663–668 21. Huffington NJ (1963) Response of elastic columns to axial pulse loading. AIAA J 1(9):2099– 2104 22. Krishna Murty AV (1987) Flexure of composite plates. J Comp Struct 7(3):161–177 23. Shimpi RP (2002) Refined plate theory and its variants. AIAA J 40(1):137–146 24. Srinivas S, Rao AK (1970) Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Sol. And Struct 6:1463–1481 25. Reddy JN, Phan ND (1985) Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sou Vib 98(2):157–170 26. Senthilnathan NR, Lim SP, Lee KH, Chow ST (1987) Buckling of shear deformable plates. AIAA J (Tech notes) 25(9):1268–1270
Path Tracing and Object Avoidance Algorithm for Robotic Manipulators Incorporating Constrained Filters Vipul Garg and Vikas Rastogi
Abstract This paper presents an algorithm for object avoidance while sustaining the trajectory of the end effector of the robotic manipulators. Multiple filters are developed that are based on classified constraints and applied to configuration space to obtain a collision-free path. The recursive and iterative nature of the algorithm makes it possible to acquire results under the permissible zone of error. The identification of desired configurations from millions of choices is achieved to obtain smooth and least deviated movements. MATLAB software is used to design, implement, visualize, and test the proposed model. Keywords Robotic manipulator · Configuration space · Object avoidance · Trajectory · Constrained -filters
1 Introduction During the last decade, manipulators of diverse classes have appeared as an imperative part of the automated industries. The flexible and configurable nature of most of the manipulators makes it feasible to stretch their work domain. The very prevalent objective of them is to move the tool located on the manipulator to some goal in 3D (Three Dimensional) space. The task becomes susceptible to failure when it is implemented in a dynamic environment. In such environments, object avoidance and real-time path planning become the fundamental task of the manipulator. In some applications, it becomes crucial to trace the desired path [1] to reach the goal such as autonomous grout filling of cracks on the dam. The imperialist Competitive Algorithm was used in [2] to solve the path-planning problem. The generation of C-Space (Configuration Space) is very computationally expensive because of the exponential growth in the possible arrangements of serial chain manipulators. Arjang Hourtash and Mahmoud Tarokh who presented their work on path planning decomposition in [3] have solved this issue. The division of the serial chain into sub-chains had been done to lessen the planning time. A* algorithm [4] is used to find the shortest V. Garg (B) · V. Rastogi Department of Mechanical Engineering, Delhi Technological University, Delhi, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_61
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path between the pick and place locations of objects. Most of the approaches were converged to reach the goal through any path between the initial and final state of the manipulator. To solve the trajectory problem, different algorithms were discussed and tested on a robotic tracking control testbed [5]. A nonlinear control design has been proposed in [6] to incorporate the dynamics of the manipulator while performing the tasks which require a trajectory control. Non-linear behavior of PD (Proportional Derivative) controllers have been proposed in [7]. A robust and novel approach based on multiple constraint filters is proposed in this paper to reduce the complexity and incorporate the general behavior of robotic manipulators. Multiple constraints can be added according to the environment, manipulator’s task, and the manufacturing constraints. The organization of the paper is as follows. Section 2 discusses the criteria of the description of rigid bodies over 3-dimensional space. The proposed methodology, its implementation, and the corresponding results are presented in detail in Sect. 3. The approach is illustrated in the first four sub-sections that is followed by the implementation and the simulation results. All simulations have been performed on MATLAB and are supported by comparing the desired and actual trajectories. Finally, a conclusion is made in Sect. 4, which also addresses future work.
2 Representation of Rigid Bodies A 6-Revolute open-chain anthropomorphic manipulator with six DoF (degrees of freedom) has been scrutinized in this work. Three DoF corresponds to the position of the manipulator in 3D space and other three DoF control its orientation. Its mechanism is analogous to the PUMA 560 [8] (refer Fig. 1) in such a way that the first three links (L 1 , L 2 , and L 3 ) control the position of the wrist-point, i.e., the intersection of the last three joint axes. The rotary motion of manipulator’s links can be achieved using different actuators. These actuators can be purely revolute joint such as servo and stepper motors or they can be a part of a complex system. Such a complex mechanism can be found in Yasukawa Motoman L-3, which makes use of two linear actuators coupled to links 2 and 3 with four-bar linkages [9]. For the simplicity of the manipulator, the links are assumed to be actuated through servo motors with a resolution of 1°. The main challenge in the interfacing of servos to industrial manipulators is their high cost, as compared to non-servo motors, for the required load capacity and joint limits. Considering the complexity of designs of manipulators and objects, components of the environment, modeling of rigid bodies can be achieved either by using superquadrics [10] or considering them as a spherical shell [11]. These spherical shells are the point clouds with a constant distance from their respective geometric primitives. These primitives could be points, line segments, or rectangles. A point cloud is generated by forming a sphere of optimum radius around primitive geometry and superimposed. Objects will be described using these point clouds, which fit into it perfectly and with minimum unoccupied space i.e. concave shape. The
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Fig. 1 PUMA 560 dimensions [8]
corresponding geometric shapes of point clouds for points, line segments, and rectangles are the sphere, cylisphere, and box with rounded edges respectively. These simple shapes reduce the computation cost while determining the characteristics of a collision. A simple geometry shape is selected based on the shape of the object to be avoided. Similarly, the individual links of the manipulator are considered cylinders and superimposed to construct the simplified structure of the manipulator. MATLAB software has been used to simulate the movement of manipulator because of its versatility in representing objects as a surface or a mesh. Also, the compatibility of the ROS (Robot Operating System) framework in MATLAB can be further utilized to simulate the manipulator in more dedicated simulators like RViz and Gazebo. The primary state, goal state, and the path traced are plotted in 3D space in MATLAB to verify the algorithm. Object-Oriented Programming is employed to store and handle data more efficiently. It reduces the complexity of the model and makes it adaptive to all small constraints.
3 Methodology In this paper, a novel approach has been suggested to avoid the objects and disturbances of surroundings while sustaining the desired trajectory. Trajectory represents the path that is required to follow by the end effector. This approach is divided into 4 main steps. Step 1 deals with the generation of the C-Space and Workspace by applying the initial parameters and mechanical constraints. Collision checking and
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avoidance filters are introduced in step 2. Objects are tested for their interference with the manipulator. Step 3 and step 4 further put more constraints on the collision-free states to obtain a smooth and least deviated path.
3.1 Workspace and C-Space A trajectory can be represented through a set of points or an equivalent polynomial equation in its task space. Task space [12] can be defined as a space in which a robot’s task is expressed naturally. It should be a subset of the workspace otherwise it will not be possible to trace the trajectory. To find out the possible configurations of a manipulator for a specific point in its workspace, the most popular approach is inverse kinematics. It uses algebraic and numerical methods to solve the configuration problem but on the other side, predetermined C-space solves this problem in a much more efficient way. Knowledge of only D-H (Denavit-Hartenberg) parameters is not sufficient to solve the non-linear complex equations of Inverse Kinematics. This C-space can also be generated with the help of the forward kinematics by considering the possible states of joints. With a span of 180 degrees for each servo motor, the total number of possible configurations are spanDOF i.e. 1806 which requires a lot of computational memory to store but only 1803 configurations are responsible for the position of the wrist. A vector in (2) representing the position of the wrist is developed by using the D-H notation [13]. This notation uses 4 parameters to describe a robot kinematically by assigning the values for each link. The values of parameters for the 6 R robot used in this work are provided in Table 1. All the parameters except the link twist are given integer values to simplify the matrix calculations. The link length and link offset are left as dimensionless numbers. A 4 × 4 square link-transformation matrix as in (1) defines the frame {i} in relative to frame {i−1} and acts as a pillar for the kinematics equations. Table 1 Denavit-Hartenberg parameters for 6 DoF manipulator i
α i–1 (°)
1
0
0
0
30
2
−90
0
5
30
3
0
5
−2
30
4
−90
5
5
30
5
90
0
0
30
6
−90
0
0
30
ai–1
di
θ i (°)
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⎤ − sin θi 0 ai−1 cos θi ⎢ sin θi cos αi−1 cos θi cos αi−1 − sin αi−1 −di sin αi−1 ⎥ ⎥ =⎢ ⎣ sin θi sin αi−1 cos θi sin αi−1 cos αi−1 di cos αi−1 ⎦ 0 0 0 1
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⎡
Tii−1
(1)
In the above equation, Tii−1 is the transformation matrix for two successive links, θ i is the angle of ith joint, α i−1 is the twist of ith link with respect to (i−1)th link, ai−1 is the length of ith link, and d i is the offset between links i and i−1. ⎡
⎤ cos θ1 (5 cos θ2 + cos θ2+3 − 5 sin θ2+3 ) − 5 sin θ1 P = ⎣ cos θ1 (5 cos θ2 + cos θ2+3 − 5 sin θ2+3 ) − 5 cos θ1 ⎦ −5 cos θ2 − cos θ2+3 − 5 sin θ2+3
(2)
The position vector P in (2) is solved for the combination of configurations to obtain the Cartesian space occupied by the wrist-point of the robot. The C-space obtained from the above perimeters and constraints is passed through several filters to optimize the path traced.
3.2 Collision Filter Object detection [14] and localization [15–17] are the primary tasks prior to the collision check. Localization of objects and mapping them to the simulated environment is not required to discuss at this point. All the generated configurations are iterated to measure the minimum distance from the object’s primitive. The minimum distance is the perpendicular distance between any two geometric primitives. The configurations, which failed to satisfy the condition (3) will be excluded from the original C-space. d(i, o) ≥ r (i) + r (o) + th
(3)
where, d(i, o) is the perpendicular distance between the ith link and the object o, r(i) is the radius of the ith link, and r(o) is the radius of object and th is the minimum required separation between the object and the manipulator. This geometric image is simplistic and less computationally expensive in comparison to superquadratics representation but ends in the exclusion of additional states of the manipulator that are not in a collision. To circumvent this, another image of the object can also be generated by creating a mesh of irregular shape in 3D space and the constraint is modified to (4). min(d(i, mesh(o))) ≥ r (i) + th
(4)
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In the above equation, mesh(o) is the point cloud of object and d(i, mesh(o)) is the distance of ith link from the point cloud. After passing from the collision filter, the remaining configurations are collision-free and can be processed further.
3.3 Nearest State Filter The desired trajectory is presented as a set of points. The separation between consecutive points plays an important role in deciding the planning time of the algorithm and the error between the theoretical and calculated trajectories. The former is negatively correlated and the latter is positively correlated with the separation distance. The greater the separation, the lesser will be the total number of points, planning time, and more will be the error in tracing the trajectory. An optimum value of separation can be decided on the basis of the computational speed and the toleration zone for error. Figure 2 shows the relation of the planning time and the number of coordinates i.e. points in 3D space used to represent the trajectory line. It can be inferred from the figure that both of the parameters are directly related to each other. Once the set of points is generated, each point is tested against the collision-free states to obtain the configurations that take the wrist to the path. The configurations responsible for crossing the toleration zone of error will be excluded. These are further sorted to determine the least error state. The extracted set of configurations is finally passed through a smoothing filter.
Fig. 2 Variation of planning time with the number of points on the trajectory line
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3.4 Smoothing Filter The continuous motion of actuators is very important during the action of the manipulator. This continuity enhances the life cycle of an actuator and diminishes the impulses, which generally occur during jerks in motion. continuous motion can be obtained by collecting the consecutive configurations of a manipulator, which converges towards the task space. Initially, each state is given a common identity. This identity is obtained from Eq. (5) by calculating the maximum movement of each actuator to reach the goal state from the initial state.
IN = max θ f − [θi ]
(5)
In the equation, θ f is the joint vector of goal state and θ i is the joint vector of the initial state. This number is characterized by an integer value which lies between the joint limits. It can either be positive or negative depending upon the reference state i.e. initial state. IN (Identity Number) of the initial state is unique with a value of zero. A subset of configuration space is determined with IN ranging from IN of initial state and goal state. Joint states with consecutive IN are extracted, which satisfies the constraint (6) in which θ IN is the joint vector with IN value. The joint vector is a column vector of dimension DoFx1 consisting of the joint states.
min θIN+1 − [θIN ] ≤ 1
(6)
3.5 Implementation of Approach The generation of configuration space is achieved during preplanning. Memory required to store the dataset exponentially increases with the joint limits. 4,096,000 configurations have been obtained for a joint limit [−90°, + 90°]. The algorithm is performed iteratively and recursively on this dataset as shown in the flow chart presented in Fig. 3. Figure 4a shows the desired trajectory as a set of points on a line. A spherical object of unit radius is placed between the desired path (refer Fig. 4b). Initially, goal state is decided in such a way that it is collisionfree, nearest to the first point on the trajectory and takes less time to reach, i.e., the difference of IN of goal state with IN of the initial state is minimized. The error between the goal state and the trajectory point is minimized by using the nearest state filter. If the error crosses the tolerance zone, the new goal state is searched with the second least IN value. The process continues until a desirable goal state is found. Initial and primary goal state along with the collision-free C-space is received by the smoothing filter. Joint states which provide continuous motion between initial and goal state are obtained by enforcing the continuity constraint on those states with IN
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Fig. 3 Flow chart of the algorithm
values ranging from [INi , INf ]. If any type of discontinuity is detected then it returns back to the nearest filter to pass another goal state. Once the desired joint movements are achieved, the initial state and goal state are updated. The previous goal state is updated to the initial state of the next iteration and goal state is updated to the next point of the trajectory. Iterations are performed until all points of trajectory are covered.
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Fig. 4 a The desired trajectory of the end effector of the manipulator. b Manipulator under collision with a spherical object. c The path traced during 1/4th of the period. d The path traced during 1/2nd of the period. e The path traced during 3/4th of the period. f The collision-free path traced at the end period
3.6 Results The ideal solution is plotted to achieve the desired path (refer Fig. 4b) irrespective of environmental interference. The links of the manipulator are considered to be cylinders with an assumption that end effector doesn’t play any role in determining its trajectory. 16% of the path is obstructed because of the object, it is required to be avoided without disrupting the trajectory of the end effector. Multiple filters based iterative algorithm is implemented to find a least deviated, collision-free, and smooth path. Figure 4c–f visualize the motion of manipulator for four progressive quarters of the time period. The period is defined by the time required by the end effector to meet the entire trajectory. It is controlled by the characteristics of the actuators such as the joint velocity, acceleration, and resolution. The object is placed near to the reachable workspace of the manipulator and so no collision-free joint configurations are left for the manipulator to overlap the desired position in the obstructed region and hence, the manipulator started to deviate from its mean path and looked for other nearest collision-free configurations at the end of the first quarter (refer Fig. 4c). The other reason for such deviation is the lower resolution of actuators which is negatively correlated with it. High resolution and a wide the range for joint movements will give the best results at a cost of high computational power. In the second quarter, when the manipulator is crossing the object, a maximum deviation
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Fig. 5 Desired and achieved trajectory
can be observed (refer Fig. 4d) and results into a bump in achieved trajectory as shown in Fig. 5. The manipulator started to converge to the required trajectory at the end of the quarter and the object is completely avoided while maintaining the smoothing action. Figure 4e–f shows its continuous motion in the unrestricted environment until the trajectory is completely traced. The required and reached trajectories are compared in Fig. 5 to depict the deviation of the manipulator from the desired path.
4 Conclusion A novel approach of path tracing and object avoidance algorithm has been presented in this paper. A 6R open-chain manipulator is simulated on MATLAB to trace the trajectory when an object is placed mid-between its path. Forward kinematics is used to generate the configuration space. The manipulator is characterized by parameters of D-H Notation and represented as a superimposition of multiple cylinders to cover the rigid links. Similarly, nearby objects are also treated as a spherical shell. Multiple filters are used to pass only those joint states, which show a smooth, collision-free and minimum deviated path. These filters are robust to adapt to any environment. This algorithm is highly configurable for either the planning time or the deviation. A large number of simulations have been performed and results have shown the good reliability of the proposed approach. Future objectives of this work are to reduce the planning time such that the algorithm can be implemented in the dynamic environment.
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References 1. Tsai CS (2014) Online trajectory generation for robot manipulators in dynamic environment--an optimization-based approach. Doctoral dissertation, UC Berkeley 2. Tabealhojeh H, Ghanbarzadeh A (2014) Two steps optimization path planning algorithm for robot manipulators using imperialist competitive algorithm. In: 2014 second RSI/ISM international conference on robotics and mechatronics (ICRoM), Tehran, pp 801–806. 3. Hourtash A, Tarokh M (2001) Manipulator path planning by decomposition: algorithm and analysis. IEEE Trans Robot Autom 17(6):842–856. https://doi.org/10.1109/70.976006 4. e Silva JS, Costa P, Lima J (2013) Manipulator path planning for pick-and-place operations with obstacles avoidance: an A* algorithm approach. In: International workshop on robotics in smart manufacturing. Springer, Berlin, Heidelberg, pp 213–224 5. Nagy PV (1989) Trajectory tracking control for industrial robots. J Mech Working Technol 20:273–281 6. Oliver JPO, Domínguez-Ramirez OA, Quezada ESE (2008) Trajectory tracking control for robotics manipulators based on passivity. In: 2008 electronics, robotics and automotive mechanics conference (CERMA’08). IEEE, pp 484–489 7. Liu F, Er MJ Trajectory tracking of robot manipulators using linear and nonlinear PD-type controllers 8. Hendarto HA, Munadi M, Setiawan JD (2014) ANFIS application for calculating inverse kinematics of programmable universal machine for assembly (PUMA) robot. In: 2014 The 1st international conference on information technology, computer, and electrical engineering. Semarang, pp 35–40 9. Craig JJ (1989) Introduction to robotics: mechanics and control, 2nd edn. Addison-Wesley Longman Publishing Co., Inc, Boston, MA, USA 10. Silva ECE, Costa MF, Erlhagen W, Bicho E (2016) Superquadrics objects representation for robot manipulation. In: AIP conference proceedings, vol. 1738(1). AIP Publishing, p 300004 11. Bosscher P, Hedman D (2009) Real-time collision avoidance algorithm for robotic manipulators. In: 2009 IEEE international conference on technologies for practical robot applications. Woburn, MA, pp 113–122 12. Lynch KM, Park FC (2017) Modern robotics: mechanics, planning, and control, 1st edn. Cambridge University Press, New York, NY, USA 13. Denavit J, Hartenberg RS (1955) A kinematic notation for lower-pair mechanisms based on matrices. Trans ASME E J Appl Mech 22:215–221 14. Cho JM, Kim K (2017) Precise object detection using local feature for robot manipulator. In: 2017 14th international conference on ubiquitous robots and ambient intelligence (URAI), Jeju, pp 497–499 15. Kim K, Cho J, Pyo J, Kang S, Kim J (2017) Dynamic object recognition using precise location detection and ANN for robot manipulator. In: 2017 international conference on control, artificial intelligence, robotics & optimization (ICCAIRO). Prague, pp 237–241 16. Kuehnle J et al (2009) 6D object localization and obstacle detection for collision-free manipulation with a mobile service robot. In: 2009 international conference on advanced robotics. Munich, pp 1–6 17. Yang Y, Cao Q-X (2012) Monocular vision based 6D object localization for service robot’s intelligent grasping. Comput Math Appl 64:1235–1241. https://doi.org/10.1016/j.camwa.2012. 03.067
Characterisation of Composites Made by Prepreg Waste P. R. Krishna Mohan, Piyush, P. M. Mohite, Kunj Modi, and Dhwani Sharma
Abstract The current work deals with the utilisation of prepreg waste in making composites. The leftover and out-of-shelf prepregs were chopped into chips of different lengths to make the laminates. The fibre volume and void content tests were performed to evaluate the quality of the manufacturing process. The prepreg waste collected from the industries consists of bidirectional glass/epoxy material. The mechanical characterisation of the laminates, made by the prepreg waste, was performed. The effect of chip length on the tensile strength and modulus were analysed. Keywords Prepreg wastes · Chip length · Fibre volume · Void content · Strength and stiffness
1 Introduction Composites made from epoxy/carbon prepregs have broad applications in the field of the aerospace industry due to their high performance [1]. Industries utilising prepregs usually landfill the leftover prepreg. Composites made of this wastage can be utilised in high stiffness applications. Feraboli et al. [2], have used discontinuous composites made by chopped prepregs for making the window frame of an aircraft. Vishnu et al. [3], have studied the size effect of chips on tensile properties of prepreg-based discontinuous composites (PBDC) with chip lengths of 2, 4 and 8 mm with a constant width of 2 mm and 6 × 24 mm chip size. Krishna Mohan et al. [4], have predicted the stiffness of PBDC for chip aspect ratio of 2, 3, 4, 5 and 6 by developing the RVE and studied the size effect of the RVE for a chip volume fraction of 22%. Anil et al.
P. R. Krishna Mohan (B) · Piyush · P. M. Mohite Department of Aeronautical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, Kanpur, India e-mail: [email protected] K. Modi · D. Sharma Department of Aerospace Engineering, Chandigarh University, Punjab, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_62
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[5], have studied the compression properties of PBDC. The primary objective of this study is (1)
(2) (3)
To develop a composite manufacturing technology using different sizes of the chips (6 × 6, 6 × 12 and 6 × 18 mm) chopped from prepreg waste collected from industry. To find the effect of chip length of the above-prepared laminates under tensile load. To analyse the failure mechanism.
2 Material Preparation 2.1 Chips Chopped from Prepreg Waste The chips were derived from the leftover and out-of-shelf prepregs by manual chopping. The width of the chip was maintained constant with a value of 6 mm and the length of chips were varied from 6 to 18 mm in the steps of 6 mm to study the effect of chip length at the different aspect ratio of 1, 2 and 3. An epoxy resin system was used to make the resin -hardener mixture. A weight ratio of 100: 80 was used for this purpose [6]. This mixture of epoxy was added to the prepreg waste.
2.2 Fabrication of PBDC Material The chopped systems are compression moulded at 0.3 bar using a hot press. The curing cycle adopted for this process was 75 ± 5 °C for 40 ± 5 min followed by a post-curing cycle of 135 ± 5 °C for 60 ± 15 min. A typical composite panel made by this procedure is shown in Fig. 1. Fig. 1 Glass/epoxy PBDC Panel
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Table 1 Manually chopped chip size (average and standard deviation) S. No.
Chip size
Average ± SD Length
Width
A.R
1
6×6
5 ± 0.05
5.2 ± 0.04
0.98 ± 0.15
2
6 × 12
10.7 ± 0.03
5.5 ± 0.03
1.96 ± 0.14
3
6 × 18
17.9 ± 0.06
6 ± 0.05
3 ± 0.26
Fig. 2 A typical Image of 6 × 18 chopped chips used for analysis in Image-J software
3 Experimentation In this section, we have briefly described qualitative and quantitative tests.
3.1 Aspect Ratio Size of Chopped Chips As the chips were chopped manually, there is a possibility of deviation in the size of chopped chips. This manual error was calculated using Image-J software. The pixel values are calibrated with the known reference, and the length and width of the chopped chips were measured. The average and standard deviation of these chips are calculated and are reported in Table 1. A typical image consisting of 50 chips, taken for this analysis is shown in Fig. 2.
3.2 Fibre Volume Fraction Adopting the procedure proposed in the work of Yee and Stephens (1996) [7], the Thermo Gravimetric Analysis (TGA) test was performed using ‘SDT Q600 apparatus. The fibre volume fraction was calculated by using Eq. (1). The weight ratios
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are obtained from TGA test. By using Eq. (2), the theoretical density (ρ c ) was calculated. The density of the matrix (ρ m ) was obtained by the relative density test. In this work, the density and mass/weight percentage are denoted by ‘ρ’ and ‘w’ with subscripts ‘f ’ and ‘c’ representing the fibre and composite, respectively νf = wf ∗ ρc =
wf ρf
ρc ∗ 100 ρf
(1)
1 +
(2)
wm ρm
3.3 Void Fraction Adopting the ASTM D792 standard, the experimental density (ρ e ) of the composite was found. ASTM D2734 standard was used to calculate the percentage of the void content and is shown in Eq. (3). υv =
ρc − ρe ∗ 100 ρc
(3)
3.4 Tensile Test Setup for PBDC We have performed the uniaxial tensile tests of the samples obtained from the PBDC material which are made from the leftover prepregs collected from the industry. The samples of different chip lengths (6, 12 and 18) were made according to the ASTM 3039D standards. The experiments were carried using Instron 1195 Universal Testing Machine (UTM) with 50 kN load cell and the test results are reported in Table 3.
4 Results and Discussions 4.1 Chips Size Results from Image J The average and standard deviations of the chopped chips are reasonably good, and they are very close to the desired size of the chip. The standard deviations of the chip aspect ratio were between 0.14 and 0.26. This indicates that the chopping process adopted is very satisfactory.
Characterisation of Composites Made by Prepreg Waste Table 2 Fibre volume fraction
Table 3 Average strength and modulus
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S. No
Sample
Fibre content (vf %)
Void content (vv %)
1
6×6
63.77
0.81
2
6 × 12
61.99
0.85
3
6 × 18
65.24
0.85
S. No
PBDC (chip size)
Strength (Mpa)
Modulus (Gpa)
(Avg ± SD)
(Avg ± SD)
1
6×6
50.36 ± 5.85
14.6 ± 0.57
2
6 × 12
52.71 ± 4.98
15.13 ± 1.58
3
6 × 18
54.01 ± 7.8
15.24 ± 1.03
4.2 Fibre and Void Volume Fraction The fibre content of the laminates made from BD glass/epoxy prepreg waste was between 61.99 and 65.24%. The void volume fraction is between 0.81 and 0.85%. The fibre volume fraction and void content of all the laminates made with different aspect ratio are reported in Table 2.
4.3 Tensile Testing Results The tensile behaviour of a laminate obtained from the chopping bidirectional glass/epoxy prepreg waste for aspect ratio 1–3 is shown in Fig. 3, 4 and 5. From Fig. 6, it can be seen that as the length of the chip increases the strength also increases. Fig. 3 6 × 6 glass fibre/epoxy (industry)
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Fig. 5 6 × 18 glass fibre/epoxy (industry)
Fig. 6 Average ultimate stress versus aspect ratio
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Among the aspect ratio of the tested samples, the maximum average ultimate strength was obtained for 6 × 18-chip size. Further, the tensile modulus increases with the increase in the chip size; this behaviour can be seen from Fig. 7. The failure mechanism of the samples was observed using a Zeiss optical microscope. A typical failed sample under tensile loading is shown in Fig. 8. The failures are mainly due to debonding of the fibre and matrix. The crack initiation is observed by splitting of the chips and gradually it progress to the surface by the chip delamination. Fig. 7 Average tensile modulus versus aspect ratio
Fig. 8 Crack microscopic images of glass/epoxy material
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5 Conclusions In this work, a novel manufacturing method of fabricating the structural components by utilising the industrial prepreg waste is presented. The PBDC panels were obtained by chopping bidirectional glass prepreg waste collected from the industry. The chip aspect ratio from 1 to 3 was considered to understand the behaviour of the material under tensile loading. The following are the main observations. (1)
(2)
(3) (4)
The ultimate strength of the material increases with the increase in the chip length. The maximum values of 54.01 MPa were observed for BD glass/epoxy PBDC with chip size being 6 × 18 mm. The tensile modulus of the material increases with the increase in the chip length. The maximum values of 15.24 GPa was observed for BD glass/epoxy PBDC with chip size being 6 × 18 mm. The failure mechanism is essentially due to the fibre matrix debonding. The crack initiated with the splitting of the chip and it progressed to the other surface by chip delamination.
Acknowledgements The current work on “Characterisation of composites made by prepreg waste” was carried out at Indian Institute of Technology, Kanpur. We are thankful to Composite Division of Hindustan Aeronautics Limited, Bangalore, India to provide all the necessary permissions to collect the prepreg waste from them. Therefore, we express our gratitude for their support in our research.
References 1. Mohan PRK, Kumar A, Mohite PM (2019) Development of in-house unidirectional carbon/epoxy prepregs and its characterization for aerospace applications. Procedia Struct Integr 14:176–183. 2. Feraboli P, Peitso E, Cleveland T, Stickler PB (2009a) Characterization of prepreg-based discontinuous carbon fiber/epoxy systems. J Reinf Plast Composi 28:1191–1214. 3. Vishnu AR, Krishna Mohan PR, Mohite PM (2017) Characterization micromechanical analysis and prediction of effective properties of prepreg based discontinuous composite. In: Proceedings of ICTACEM 2017, IIT Kharagpur 4. Mohan PRK, Anil Kumar M, Mohite P (2020) Representative volume element generation and its size determination for discontinuous composites made from chopped prepregs. Compos Struct 252:112633 5. Anil Kumar M, Mohan PRK, Piyush, Mohite PM (2019) Characterization of prepreg based discontinuous composites under compression. In: Sixth international conference on recent advances in composite materials (ICRACM-2019). IIT-BHU. 6. Rahul R, Kitey R (2016) Effect of cross-linking on dynamic mechanical and fracture behavior of epoxy variants. Compos B Eng 85:336–342 7. Yee R, Stephens T (1996) A TGA technique for determining graphite fiber content in epoxy composites. Thermochim Acta 272:191–199
Experimental Investigation on the Effect of Process Parameters for CNC Turning of UNI Al 3055 Alloy Under MQCL Based Cooling Technique Subrata Mondal, Goutam Paul, and S. C. Mondal
Abstract In the present research a novel cooling setup that can help the turning process to be performed in various media, i.e., dry, liquid CO2 or compressed air, cooling lubricant, and the mixture of the liquid CO2 or compressed air mixed with the cooling lubricant. The conventional cooling oil is used to dissipate the heat and increasing life of a cutting tool. So, the surface finish of the workpiece improves leading to a better quality product. Customarily, the contaminated cutting fluid needs to be disposed of after its use that involves a good amount of money of total manufacturing cost. Primarily this work is a turning operation on Al 3055 alloy under three different cutting media namely dry, Liq. CO2 and Liq. CO2 assisted the minimum quantity of liquid in order to get better quality of the product. Keywords Turning · Liquid CO2 · Minimum Quantity Liquid · Al 3055 Alloy · RSM
1 Introduction The conventional cooling lubrication system involves a commensurable amount of total manufacturing cost. Replacement of the traditional cooling lubricants with the dry or minimum quantity lubrication (MQL) can reduce the machining cost. The dry or MQL can also help in environmentally friendly machining increasing the safety to the labor health resulting in increased job satisfaction. Nowadays MQCL, i.e., minimum quantity cooling lubrication is one of the solutions to successful dry machining. Coolants and lubricants obviously enhance the surface quality and tool life whenever at the same time the hazards and the cost of machining increases [1]. Cost of coolants is around 7–17% of the total machining cost. Here the sustainability issues, i.e., eco-friendliness and cost-effectiveness come into the picture. Dry
S. Mondal · G. Paul (B) University of Engineering and Management Kolkata, Kolkata 700160, India S. C. Mondal Indian Institute of Engineering Science and Technology (IIEST), Shibpur, Howrah 711103, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_63
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machining might dominate the wet method if only economic and ecological perspectives are concerned. But dry machining has some limitations, like during machining of sticky material e.g. pure Aluminium. The material easily gets stuck to the cutting tool face as the turning of the material is going on. Consequently the machining rate decreases as well as the tool wear rate and cutting temperature increases. Thus the tool life is hampered. The good surface finish of the product achieved by the application of MQCL and liq. CO2 conditions compared to dry machining condition whereas MRR is greater in dry machining environment [2]. It has also been observed that proper selection of process parameters such as feed rate, depth of cut, cutting speed, and tool nose radius will produce good quality of surface roughness under MQL with Nano coolant [3]. Good quality of surface roughness value can be achievable by MQL compared with dry machining conditions as well as flooded cooling conditions [4]. This is also noted that higher value of feed rate and cutting speed under the MQL conditions can damage the surface quality [5]. This innovative MQL technique can be used in drilling, turning, milling, and grinding process also [6]. By means of the analytical model, it is noted that specific cutting energy reduces when the flow rate and pressure of the MQL process increase [7]. The usage of MQL technique with a cryogenically treated tool inserts shows improved results in terms of minimum surface roughness [8]. In addition to the above literature, it becomes customary to prepare sustainable machining setup in order to provide a multi-type of cooling medium like dry, near dry, liq. CO2 and MQL/MQCL etc. The sustainable machining setup followed by experimental validation has been necessary in order to survive in the field of manufacturing.
2 Experimental Procedure 2.1 Experimental Setup The experiments were performed in a SINEWAVE CNC turning center at the workshop of UEMK. The schematic diagram of the machine set up has been shown in Fig. 1a. The whole control of the cutting fluid can be accomplished with the help of a single mixing chamber as shown in Fig. 1a. The flow of CO2 gas and mixed coolant spray is controlled by the mixing chamber by regulating the knob left side and right side respectively. Dry machining is performed by adjusting the knob at the mid position. When the value of the liquid CO2 is kept closed and the knob is at right position then only coolant lubrication is emerged out of the mixing chamber. The nozzle as shown in Fig. 1b supplies the cutting fluid at the junction of workpiece and tip of the cutting tool.
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Fig. 1 a Schematic diagram of the experimental setup and b its original photograph
Table 1 The composition of the work piece material Alloy
Elements Si
Fe
Cu
Mn
Mg
Ni
Zn
Ti
Al
UNI 3055
2
0.5
0.05
0.8
0.75
0.01
0.05
0.15
Rest
2.2 Work Piece Materials Aluminium alloy 3055 (UNI3055) material was chosen to perform the turning operation in CNC machine under three different cutting environment (Dry, Liquid CO2 , Liquid CO2 assisted minimum quantity cooling conditions). Workpiece material composition is shown in Table 1. The design of experiment (DOE) was set as per response surface methodology to carry out the experiments. Three input parameters, i.e., spindle speed in rpm, feed rate in mm/min, and depth of cut in mm were varied and two sets of each experiment were chosen.
2.3 Design of Experiment The experiment in the present research work has been set in such a way so as to analyze the effects of the input parameters on the turning operation under various cooling media. Input parameters are spindle speed (A) in rpm, depth of cut (B) in mm and feed rate (C) in mm/min. Output responses are material removal rate (MRR) in g/min, surface roughness values (Ra) in micron, Chip thickness (t) in mm in degree Celsius. The experiment is designed based on response surface method especially central composite design. The factors and levels for the current experiment are tabulated in Table 2.
646 Table 2 Factors and levels of the parameters for the CNC turning operation
S. Mondal et al. Coded RSM level
Factor A
B
C
−1.414
793
0.39
5.51
−1
1000
0.5
20
0
1500
0.75
55
1
2000
1
90
1.414
2207
1.10
104.49
Based on Response surface methodology, experiments were carried out for measuring responses for each of the cutting conditions viz. dry, liq. CO2 and mixed coolant. The experimental results were analyzed using MINITAB 16 software. Also, a mathematical model is developed to quantify the effect of process parameters on the responses using Response Surface Methodology (RSM). Response Surface Methodology (RSM) is used to obtain the quantitative equation and optimum combination of the input parameters are found for optimum response.
3 Results and Analysis In the research work, the prime intention is to develop a system to supply dry, Liq. CO2 and MQL machining whichever is required during machining. To validate the performance of the setup the experiment was carried out in the setup as per the RSM design of the experiment. The 20 number of experiments were carried out. The responses that were measured are MRR, Ra value. The regression equations as obtained from response surface methodology are shown in Table 3. In support of these ANOVA tables are provided (Table 4) which illustrate that the quadratic equations can correlate the experimental results entirely. The regression equations as depicted in Table 3 are used to predict the responses and the predicted values are compared with the actual values from experiments (as shown in Fig. 2).
4 Discussion It has been observed from Fig. 2a that MRR hardly deviates when medium changes from dry to CO2 and then MQL. Though in liq. CO2 it is little bit lower than dry medium but as soon as the MQL medium is used then the MRR becomes almost the same as that of dry medium. Figure 2b depicts that surface roughness is distinguishably lower in MQL medium but it is maximum in dry medium. Liq. CO2 medium can help to improve the surface quality as depicted in the same figure.
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Table 3 Regression equations with coefficient of correlation (R2 ) Media MRR
R2 (%) Ra
R2 (%)
Dry
0.041751 + 0.010128*A + 99.07 0.030281*B + 0.00438*C − 0.0105*A*A + 0.032509*B*B − 0.0098*C*C + 0.008875*A*B + 0.014875*A*C − 0.00163*B*C (1)
0.064538 − 0.002374*A + 98.57 0.007024*B + 0.010536*C − 0.000577*A*A − 0.000077*B*B − 0.003078*C*C + 0.000000*A*B + 0.009500*A*C − 0.002000*B*C (4)
CO2
0.042674 + 0.006774*A + 99.89 0.026639*B + 0.006607*C − 0.012101*A*A + 0.030912*B*B − 0.010600*C*C − 0.003375*A*B + 0.018375*A*C − 0.005625*B*C (2)
0.044673 − 0.001722*A + 95.25 0.006392*B + 0.008156*C − 0.000096*A*A + 0.001155*B*B − 0.004847*C*C − 0.001625*A*B + 0.009375*A*C − 0.001375*B*C (5)
MQL
0.044039 − 0.007955*A + 99.71 0.028881*B + 0.004110*C − 0.011832*A*A + 0.032932*B*B − 0.007831*C*C + 0.006750*A*B + 0.013000*A*C + 0.000750*B*C (3)
0.028077 + 0.000549*A + 90.12 0.002561*B + 0.003837*C − 0.002904*A*A-0.003404*B*B − 0.002904*C*C + 0.003000*B*C − 0.001250*C*A (6)
Fig. 2 Comparison of a MRR, b surface roughness obtained experimentally at various environments versus predicted value
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Fig. 3 Surface plot for MRR at different combination of input parametric levels in various medium a dry, b liquid CO2 , and c MQCL
It implies that the process can produce better surface finish steadily when the MQL medium is used. In comparison to MQL medium liq. CO2 have obtained the ‘runner up’ position. The surface plots for MRR (Fig. 3), and Surface roughness (Ra) (Fig. 4) at various media and level setting are depicted. The surface plots are used to explain the effect of the process parameters on different responses (Table 4). Here Fig. 3a depicts that in the dry medium the MRR increases at first with the spindle speed but after the 1500 rpm the MRR decease whereas it increases steeply with the depth of cut. The causes of such observation are that with an increase of spindle speed up to 1500 rpm along with the increase of depth of cut at first the chip of greater thickness are coming out from the job. But as soon as the spindle speed is increased then the finer chips are coming out of the machining zone. But it is clearly understood that with the increase of feed rate MRR increases impressively as the process goes to rough turning condition consequently chips of higher thickness are released. The same kinds of graphs are also obtained in liq. CO2 and MQL medium as depicted in Fig. 3b and c. So it is observed that cooling medium has a comparatively lower effect on MRR.
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Fig. 4 Surface plot for surface roughness (Ra) at different combination of input parametric levels in various medium a dry, b liquid CO2 and c MQCL
Table 4 ANOVA for all the responses Responses
SS
DF
MS
F
P
Model
Model
Model
MRRDry
0.024999
9
0.002778
117.89
0.0
MRRCO2
0.022398
9
0.002489
990.01
0.0
MRRMQCL
0.023113
9
0.002568
380.55
0.0
RaDry
0.002835
9
0.000315
76.77
0.0
RaCO2
0.002273
9
0.000253
22.29
0.0
RaMQCL
0.002156
9
0.000214
23.84
0.0
As the surface roughness is coming into the picture, it has been observed from Fig. 4a that with the increase of spindle speed at the dry condition the Ra value decreases steeply. The same steepness of increase in Ra value is observed with the increase of depth of cut and feed rate. So the dry medium is responsible for poorer surface finish of the product. In line to the above discussion and supported by Fig. 4b
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that Ra value is not much improved in liq. CO2 medium but if Fig. 4c is taken as a reference it is clearly been seen that nature of the surface plots is convex. The convex surface as shown in Fig. 4c explains that Ra value increases up to an optimum and then decreases.
5 Conclusions In the above-mentioned research work it has been illustrated that for the sustainability issues it is time to manufacture an unusual setup which can be able to supply coolant in any form like dry, liq. CO2 /compresses air and MQL/MQCL. The authenticity of experimental results was analyzed and correlated by ANOVA table which shows that experimental results are well in the agreement with the regression equations for material removal rate (MRR) and surface roughness (Ra value) that are the important responses for getting better surface quality of the product. The RSM-based regression equations confirm of experimental results and it ensures fewer chances of error. The surface plot clearly depicts the optimum region for Ra value can be obtainable from the level setting of the process parameters. Without the delve into the optimization process, it can be seen from the analysis that though the liq. CO2 and MQL media cannot improve the MRR much over dry medium but it can be concluded from the above research work that surface roughness is lower in Liq. CO2 and MQL conditions hence the Ra value decrease. So, to get a better quality product from the UNI 3055 Al alloy the MQL conditions can be used sustainably.
References 1. Sharma VS, Dogra M, Suri NM (2009) Cooling techniques for improved productivity in turning Int J Mach Tools Manuf 49:435–453 2. Mondal S, Paul G, Mondal SC (2019) Investigation into the application of liquid CO2 and MQL for CNC turning of Al Alloy 3055. Research into design for a connected World proceedings of ICoRD 2019, vol 1, pp 977–987 3. Patole PB, Kulkarni VV (2018) Optimization of Process parameters based on surface roughness and cutting force in MQL turning of AISI 4340 using nano fluid. Mater Today: Proc 5:104–112 4. Joshi KK, Kumar R, Anurag (2019) An experimental investigation in turning of incoloy 800 in dry MQL and flood cooling conditions. Procedia Manuf 20:350–357 5. Çakır A, Ya˘gmur S, Kavak N, Küçüktürk G, Seker ¸ U (2016) The effect of minimum quantity lubrication under different parameters in the turning of AA7075 and AA2024 aluminium alloys. Int J Adv Manuf Technol 84:2515–2521 6. Boswell B, Islam MN, Davies IJ, Ginting YR Ong AK (2017) A review identifying the effectiveness of minimum quantity lubrication (MQL) during conventional machining. Int J Adv Manuf Technol 92:321–340 7. Chetan, Ghosh S, Rao PV (2018) Specific cutting energy modelling for turning nickel-based Nimonic 90 alloy under MQL condition, Int J Mech Sci https://doi.org/10.1016/j.ijmecsci.2018. 07.033
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8. Deshpande YV, Andhare AB, Padole PM (2018) Experimental results on the performance of cryogenic treatment of tool and minimum quantity lubrication for machinability improvement in the turning of inconel 718. J Braz Soc Mech Sci Eng 40:6
Reciprocating Wear Behaviour of Al–SiC Composite Processed with MWS Neeraj Kumar Bhoi , Harpreet Singh , and Saurabh Pratap
Abstract The present study involves the fabrication and characterization of Al–SiC composites by means of varying content of SiC (i.e., 5, 10, 15 and 20 wt. %). The material was synthesized with developed hybrid microwave sintering (MWS) technique incorporating powder metallurgy (PM) process. Findings are in good agreement for the selection of reinforcement content in the matrix material for the material development to be used in a lightweight material with improved properties. A systematic approach was made to validate the importance of reinforcement in the matrix material for enhanced product response for various functional applications. The hardness of the developed composite material increases significantly with maximum up to 38.1% for 20 wt.% reinforcement in aluminum. The higher reinforcement percentage in the composite material reduces the wear loss up to 96% as observed in the Al + 20% wt.% SiC composites. Keywords Al/SiC composite · Microwave sintering · Hardness · Reciprocating wear behaviour · Microstructure
1 Introduction In the last three-decade innovation and research in the field of processing and design of new material grows exponentially. Sustainability of the system depends largely on the selection of material so the basic and important step is to select a material having low weight, low cost and environment-friendly. Development in the direction of lightweight material leads to the processing of material like aluminum metal N. K. Bhoi (B) · H. Singh Department of Mechanical Engineering, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India H. Singh e-mail: [email protected] S. Pratap Department of Mechanical Engineering, Indian Institute of Technology Varanasi (IIT-BHU), Varanasi, UP, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_64
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matrix composites (Al-MMCs) which are very light in weight as well as their functionality which perform better in the desired environment [1–3]. However, the lower wear resistance of Al-based material limits the applicability in the different functional surface. To adlib, the wear resistance properties numerous hard ceramics, oxide-based material is utilized [4]. However, the wear behaviour of the material depends upon the type of application undergoing during the operating environment. Present paper labels the reciprocating wear behaviour of the Al-based composite material reinforced with a different weight percentage of silicon carbide [5, 6]. The replacement of the conventional material demands lightweight and improved wear resistance for different tribological applications [7]. In a similar context Samal et al., studied the wear resistance behaviour of AA5052 MMCs with the addition of TiC during the stir casting process. They found that the addition of a hard carbide phase reduces the volumetric wear loss by 24% compared to the monolithic phase of the material [6]. The effect of heat treatment on fretting wear of A356 was evaluated by Chen and their colleagues. It was observed that the 15 vol.% silicon carbide (SiC) offers greater wear resistance at lower load. However, T6 treatment found to be a good agreement for better wear resistance at higher load and higher working cycles [8]. The use of fine reinforcement in the matrix leads to greater strength and wear resistance compared to the coarser particle. As the finer particles hold better interfacial strength and load-bearing capacity of the matrix material [5]. However, the choice of processing mode and reinforcement percentage holds a unique position in the material response under extreme survival conditions. The present study uses the innovative developed hybrid microwave sintering (MWS) approach for material development. The use of MWS helps in improvement of densification, improved mechanical and tribological response with shorter processing time, environmental friendly with minimal wastage [2, 9]. From the literature, it is evident that the lesser properties have been given to investigate the reciprocating wear behaviour of the material. The Al/SiC composite material is developed by incorporating powder metallurgy and hybrid MWS method. The present study aims to develop the correlation between the reinforcement percentage in the matrix material and wear resistance of the material under the reciprocating test condition. The reciprocating test was carried out under constant sliding velocity of 0.3 m/s for a different travelled distance under dry environment. The test results will help in the selection of the amount of reinforcement for the wear resistance application in the field of automobile, aerospace and numerous structural applications.
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2 Experimental Details 2.1 Materials The pure form of Al powder (Assay>99.5%) with particle size in the range of 44–200 micron and SiC with an average particle size of 44 microns used as starting material for the development and characterization of Al-MMCs. The material was supplied by Alpha Chemika Pvt. Ltd. Mumbai, India. The material was used as supplied with any further purification and test.
2.2 Processing Details and Characterization Study The powder metallurgy and microwave sintering approach used for the development of Al/SiC composite material. For the development of Al/SiC composite material, four different reinforcement percentage 5, 10, 15 and 20 wt.% of SiC were used in the matrix material. In this process, the initial powder (i.e. matrix and reinforcement) were blended by means of mechanical ball milling. The milling was carried out for 2 hours at a rotational speed of 150 rpm at normal atmospheric pressure and at room temperature. The blended powder material is cold compacted by the application of uniaxial compaction pressure of 580 MPa [10]. The cold compacted part is further sintered by innovative developed microwave sintering approach. Figure 1 depicts the schematic representation of the hybrid microwave sintering approach. The sintering setup consists of silicon carbide susceptor material and alumina insulation. The susceptor material absorbs the microwave energy and converts it into heat. The susceptor assisted heating in microwave sintering approach helps in heating the samples in bidirectional mode (i.e. susceptor heating from outside to inside and
Fig. 1 Schematic view of hybrid microwave sintering
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Fig. 2 Pictorial representation of reciprocating wear test rig
microwave heating from the core of the material to the surface) [11]. The compacted part is exposed to 11 minutes in the microwave radiation to reach the sintering temperature of 640 °C (near the melting point of aluminium) with no holding time in this temperature. The sample is allowed to cool inside the developed setup to room temperature with natural cooling mode. The sintered samples are further used as different characterization and end-use. The microstructural observation of mechanically mixed powder material and sintered specimens were done by the help of scanning electron microscope (SEM). The quanta FEI 250 was for the examination of the microstructural images. The hardness of the monolithic and composite sample was evaluated by applying 60 Kgf load as per the ASTM standard E18-02. The load was applied for a dwell time of 5 s in the sample. For the repeatability of the test sample, an average of 5 successive indentations were reported. The hardness measurement was done with the help of Rockwell hardness test scale B. For the reciprocating wear analysis, in-house fabricated reciprocating test setup was employed. The pictorial representation of the test setup is given in Fig. 2. The test setup consists of an AC motor and gearbox which operate at a different rotational speed. The test was carried out at constant reciprocating velocity of 0.3 m/s. The total stroke length was kept constant throughout the test and kept as 0.6 m. During the test, the weight loss method was adopted for the calculation of wear rate and moss loss of the Al/SiC composite material. The wear test was carried out with a normal applied load of 20 N throughout the test. Prior to the test sample was polished against fine abrasive paper and cleaned with the help of acetone. The counter surface is made of hardened steel with HRC>55. The sample weight was measured before and after the test with the help of a precision balance of 0.1mg. The specific wear rate (wear co-efficient) of the developed composite material was calculated by the use of Archard model: k=
m pl
(1)
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Fig. 3 Morphology of the mechanically milled Al+5% SiC powder after 2 h under normal atmospheric conditions
where k is the wear co-efficient, m is the loss in weight of the sample, p resembles the normal applied load at the test sample, l is the total travelled distance of the test sample.
3 Results and Discussion 3.1 Microstructure Analysis of the Milled Powder Material The Al powder material is in nearly spherical morphology and SiC is in the form of nearly flakes likes structure. The Fig. 3 shows the morphology of the milled powder material after 2 h of milling. The morphology revealed that the powder material is homogenously mixed and some part of the reinforcement is diffused into the matrix material [12]. There are no substantial changes in the powder morphology is observed due to the lower milling speed and time. However, uniform mixing of the matrix and reinforcement is clearly seen from the SEM morphology of the Al/SiC powders. Fig. 3 represent the milled powder sample of Al+5%SiC composite powder.
3.2 Hardness The variation of the hardness value of the developed material is given in Fig. 4. The higher content of the SiC was attributed to a higher hardness value. The hardness of the sintered sample is found to be increasing with the higher content of SiC percentage with 4.76%, 11.9%, 23.81% and 38.1% for reinforcement percentage of 5, 10, 15 and 20 wt.% in aluminum respectively. The formation of strong interfacial bonding between the reinforcement and matrix resists the plastic deformation during
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Fig. 4 Variation in the hardness value
60
Hardness (HRB)
50
40
30
20
10
0 Pure Al
Al+5%SiC Al+10%SiC Al+15%SiC Al+20%SiC
Specimen
the application of load. Similar trends were reported by the researcher by the consideration of several reinforcements such as TiC, TiB2 , B4 C, Y2 O3 in the Al-MMCs [4, 6, 12]. The improvement in the material hardness can be related to homogeneous particle distribution, uniform microwave heating of the samples and the restriction of the grain movement to a greater extent. The well-known Hall-Petch relation and Orowon strengthening well described the stated observation in a beautiful manner for the property enhancement [11–14]. On a similar exploration by Samal et al. [6] shows the highest achieved microhardness of 32% higher in caparison with base matrix material [6]. The formation of intermetallic compound in the composite material is sole responsible for the improved material response under similar operating environment.
3.3 Wear analysis The reciprocating wear behaviour of the Al/SiC composite was assessed by means of applying reciprocating wear conditions as stated in Fig. 2. For the assessment of wear behaviour, a fixed amount of deadweight is applied over the sintered sample during the reciprocating actions. The weight loss and specific wear rate of the specimen are illustrated in Fig. 5. The presence of a hard-ceramic phase in the composite material softens the contact zone by the generation of stable and hard phase in the material. The higher reinforcement percentage in the material reduces the tendency of crack propagation with strong interfacial bonding between the intermolecular of the material during rubbing and sliding action [6]. The wear loss of pure Al is higher compared to other reinforced material with similar operating conditions. The variation of the specific wear rate is plotted in Fig. 5. The specific wear rate of the material reduces with a higher content of SiC in the aluminum. This can be related
Reciprocating Wear Behaviour of Al–SiC Composite Processed … Fig. 5 Wear behaviour of the sample
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0.06
mass loss (gm)
0.000020 0.04 0.000015 0.03 0.000010
0.02 0.01
0.000005
0.00
0.000000 0
5
10
15
Specific wear rate (g/NM)
0.000025
mass loss specific wear rate
0.05
20
wt.% of SiC
to the hardening of the phase formed during the reinforcement in the material which limits the worn-out stage of the material. In case Al/SiC composite the wear, the loss is reducing with a minimum of 40% and a maximum of 96% in the case of 5 and 20 wt.% SiC respectively. Higher hardness is responsible for the reduction in the wear loss of the material which lowers the overall particle fracturing during sliding and rubbing action. On a similar contrary, the presence of TiC in the aerospacegrade Al material exhibits better and higher sliding wear resistance compared to unreinforced material [6]. Similarly, the observations were made for the nanostructured B4 C composite material under different loading condition with sliding conditions on a pin on disc configuration. It was noted that presence of hard carbide particles and formation of intermetallic phases in the material reduces the wear rate to much extent [15].
4 Conclusion Al/SiC composite was successfully synthesized by the incorporation of powder metallurgy and hybrid microwave sintering approach. Microstructure analysis of milled powder, hardness and reciprocating wear behaviour were investigated for the developing the co-relation between the matrix and reinforcement percentage in the composite. The following conclusion can be drawn from the present study: • Mechanical milling is a powerful and easy method for the mixing of the powder material. Homogenous mixing and effective diffusion are observed in case of mechanical milling of Al and SiC powder. • A maximum of 38.1% improvement in the hardness of the Al/SiC composite material with the addition of 20 wt.% SiC in the aluminum. The results are in good agreement and responsible for the better interfacial bonding between the matrix and reinforcement.
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• A maximum of 96% reduction in wear loss is observed in the case of Al/SiC composite with reinforcement percentage of 20 wt.% in the matrix material. Higher hardness and strong interfacial bonding are the primary factors responsible for improved wear resistance of the developed composite material.
References 1. Tabandeh-khorshid M, Omrani E, Menezes PL (2016) Tribological performance of selflubricating aluminum matrix nanocomposites: Role of graphene nanoplatelets. Eng Sci Technol an Int J 19:463–469. https://doi.org/10.1016/j.jestch.2015.09.005 2. Singh H, Jain PK, Bhoi N, Pratap S (2018) Experimental study pertaining to microwave sintering (MWS) of Al-Metal Matrix Composite—A review. Mater Sci Forum 928:150–155. https://doi.org/10.4028/www.scientific.net/MSF.928.150 3. Pattnaik SK, Bhoi NK, Padhi S, Sarangi SK (2018) Dry machining of aluminum for proper selection of cutting tool: tool performance and tool wear. Int J Adv Manuf Technol 98:55–65. https://doi.org/10.1007/s00170-017-0307-0 4. Bhoi NK, Singh H, Pratap S (2020) Synthesis and characterization of alumina nanoparticles: a case study. J Inst Eng India Ser C 101:407–413. https://doi.org/10.1007/s40032-019-00542-w 5. Kumar S, Pandey R, Panwar RS, Pandey OP (2013) Effect of particle size on wear of particulate reinforced aluminum alloy composites at elevated temperatures. J Mater Eng Perform 22:3550– 3560. https://doi.org/10.1007/s11665-013-0642-8 6. Samal PR, Vundavilli PR, Meher A, Mahapatra MM (2019) Influence of TiC on dry sliding wear and mechanical properties of in situ synthesized AA5052 metal matrix composites. J Compos Mater 53(28–30):4323–4336. https://doi.org/10.1177/0021998319857124 7. Bhoi NK, Singh H, Pratap S (2020) Developments in the aluminum metal matrix composites reinforced by micro/nano particles—A review. J Compos Mater 54(6):813–833. https://doi. org/10.1177/0021998319865307 8. Chen R, Iwabuchi A, Shimizu T (2000) The effect of a T6 heat treatment on the fretting wear of a SiC particle-reinforced A356 aluminum alloy matrix composite. Wear 238:110–119. https:// doi.org/10.1016/S0043-1648(99)00328-2 9. Bhoi NK, Singh H, Pratap S, Jain PK (2019) Microwave material processing : a clean , green, and sustainable approach. In: Kumar K, Divya Z, Paulo D (eds) Sustainable engineering products and manufacturing technologies, 1st edn. Academic Press Elsevier, pp 3–23. https://doi.org/ 10.1016/B978-0-12-816564-5.00001-3 10. Dasari BL, Morshed M, Nouri JM et al (2018) Mechanical properties of graphene oxide reinforced aluminium matrix composites. Compos Part B Eng 145:136–144. https://doi.org/10. 1016/j.compositesb.2018.03.022 11. Bhoi NK, Singh H, Pratap S (2019) A study on microwave susceptor material for hybrid heating. J Phys Conf Ser 1240(1):012097. https://doi.org/10.1088/1742-6596/1240/1/012097 12. Kumar A, Bhoi NK, Singh H (2020) Corrosion behavior of microwave clad material under different acidic environment. In: Sharma V, Dixit U, Sørby K, Bhardwaj A, Trehan R (eds) Manufacturing engineering. lecture notes on multidisciplinary industrial engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-4619-8_43 13. Niste VB, Ratoi M, Tanaka H et al (2017) Self-lubricating Al-WS2 composites for efficient and greener tribological parts. Sci Rep 7:1–14. https://doi.org/10.1038/s41598-017-15297-6 14. Ma X, Zhao YF, Tian WJ et al (2016) A novel Al matrix composite reinforced by nano-AlN p network. Sci Rep 6:1–8. https://doi.org/10.1038/srep34919 15. Alizadeh A, Taheri-Nassaj E (2012) Mechanical properties and wear behavior of Al-2 wt.% Cu alloy composites reinforced by B4 C nanoparticles and fabricated by mechanical milling and hot extrusion. Mater Charact 67:119–128. https://doi.org/10.1016/j.matchar.2012.02.006
Dynamics and Control of a 6-DOF Biped Robot on MATLAB/SimMechanics Durbadal Kundu, Alinjar Dan, and Nirmal Baran Hui
Abstract This paper plans to demonstrate a biped robot on MATLAB/ SimMechanics, which tackles dynamics problems with time-efficient numerical models. Biped robot model in this paper has seven links and all the joints connecting links are revolute in nature. Two identical legs have hip joints between upper leg and torso, knee joints between the lower leg and upper leg parts, ankle joint between the lower leg and foot. A rigid body forms the torso. Modeling of ground contact forces is done using inbuilt MATLAB contact library. A PID controller is used in order to simulate the dynamics of the system. Results obtained from the dynamic simulation are presented. Keywords Biped robot · SimMechanics · Dynamic modeling · Control
1 Introduction In the most recent couple of decades objective of copying human movement through strolling biped robots has gotten huge consideration among scientists. Biped robots can move in obscure landscapes, climb staircases which wheeled robots are unfit to perform. They can be employed in hazardous works such as rescue operations [5], disaster situation [1], or rehabilitation of disabled people, for example, dynamically controlled prosthetics [10] and made humanoid and biped, to rescue injured victims into safety; they can prevent human life from being put in danger. Due to the complexity of human walking, which increases with the increase of no. of links and degree of freedom (DOF), research progress in this area has been limited.
D. Kundu Ashok Leyland, Kanpur, Uttar Pradesh 208002, India A. Dan (B) Indian Institute of Technology Delhi, New Delhi 110016, India N. B. Hui Department of Mechanical Engineering, National Institute of Technology Durgapur, Durgapur, West Bengal 713209, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_65
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Few researchers contributed to modeling a biped robot in MATLAB/SimMechanics environment. A SimMechanics model represents physical model through blocks and converts it into time-efficient mathematical model [4]. Mester-G [7] applied the Euler–Lagrange method for dynamic modeling of a 20-degree of freedom (DOF) underactuated biped robot and verified his results by Robotics toolbox of MATLAB/SimMechanics. Velásquez-Lobo et al. [13] presented a methodology for modeling of a 5-link biped robot, which is connected through revolute joints, in MATLAB/SimMechanics. They have modeled ground contact as well. Mathworks Student development team [12] developed a SimScape model of walking robot implementing genetic algorithm to find optimal trajectory for walking. The main objective of this project is to make a model of a biped robot and simulate it in MATLAB/Simulink and improve its dynamic stability changing controller parameters. The model has seven links and six revolute joints. Finally, proportional– integral–derivative (PID) control on ankle joint is applied.
2 Modeling of System in MATLAB/SimMechanics Figure 1 shows kinematic structure of 6-DOF biped robot to be modeled. Different approaches for biped locomotion control existing in the literature are zero moment point (ZMP) detection [11] Passive walking method [6] and Walking primitive [2]. Walking primitive approach is followed for our biped locomotion in MATLAB/SimMechanics. In this approach, positions for all joints are set in prior and required torque for motion is calculated at each step. Biped robot is modeled using SimMechanics which accepts its system as a combination of block diagrams and performs dynamic simulation using the standard Newtonian dynamics of forces and torques. The model will have foot, lower leg, and upper leg and torso/trunk blocks. The foot is connected with the lower leg through a rotary joint. Lower and upper legs
Fig. 1 Kinematic structure of the biped robot
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Fig. 2 SimMechanics block diagram for torso
are connected via knee joints and the hip joint connects the upper leg with the torso. Therefore, one leg is having three joints and four links (torso is common to both the legs). Torso, leg, and foot are modelled specifying geometric parameters and ground contact is modeled using ‘Sphere to plane force’ block of SimMechanics [8] from MATLAB contact library.
2.1 Torso Modeling Torso is modelled as a rectangular block connected from right to left hip. Representation of torso in SimMechanics environment is shown in Fig. 2. Dimensions of the torso are represented in Table 1.
Table 1 Torso parameters Torso parameter Length Breadth Width Density
Value 10 cm 8 cm 5 cm 950 kg/m3
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2.2 Leg Modeling Legs are modelled as rigid cylinders. Each leg contains three links named as foot, lower leg, and upper leg (Simulink block diagram is shown in Fig. 3). They are connected by revolute joints which are named as ankle joint (between foot and lower leg), knee joint (between the lower leg and upper leg) and hip joint (between upper leg and torso). All joints are actuated by input motions. Required parameters for legs are represented in Table 2.
2.3 PID Control Modeling Proportional–Integral–derivative controls are applied at ankle joints of each leg (block diagram of it is shown in Fig. 4). An initial torque is applied at ankle joints by motor. The difference between the reference and actual angular position of ankle joint is taken as input to the PID controller which actuates necessary torques to that joint. Parameters of PID controller as found after PID tuning are given as K p (proportional gain) = 0.1, K i (Integral gain) = 0.57,K d (differential gain) = 0.1. The equation for calculating actuation torque at ankle is given in Eq. 1 τ = K p (q − qr ) + K d
d(q − qr ) + Ki dt
(q − qr )dt
(1)
Here q is the actual ankle joint angle after application of torque whereas qr is the reference ankle joint angle.
Fig. 3 SimMechanics block diagram for leg Table 2 Leg parameters Leg parameter Leg radius Lower leg length Upper leg length
Value (cm) 0.75 10 10
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Fig. 4 SimMechanics block diagram for PID control at ankle
2.4 Ground Contact Modeling The contact force between foot and ground is unilateral, i.e., only repulsive force exists between them when they come to the contact. To model this kind of contact, Sphere to plane force [8] SimMechanics block has been used. This block implements a contact force between a sphere and a plane. Feet are placed at the spherical end and the ground is at the plane end. Block diagram of ground contact in SimMechanics environment is shown in Fig. 5. All parameters of ground contact are represented in Table 3.
Table 3 Ground contact parameters Ground contact parameter Contact stiffness Contact damping Ground static friction coefficient Ground kinematic friction coefficient Ground plane width Ground plane height Ground plane length
Value 2500 N m/deg 100 N m/deg/s 0.6 0.8 3m 0.025 m 25 m
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Fig. 5 SimMechanics block diagram for Ground contact
3 Simulation and Results Input gait is provided in the form of joint trajectories. In a walking cycle, all six joint trajectories (two ankle, two knee, and two hip joints) are expressed as cubic polynomials of time. A cubic polynomial requires four points at four given time instants to specify all coefficients. To make joint trajectories position, velocity, and acceleration continuous, additional three conditions are required. To take that into consideration, three more joint angles at three given time instants are specified. So seven joint angles have been specified at given time instants for generating a trajectory for each joint. All simulations are based on the ODE 15s solver which is used for solving stiff problems, i.e., problems in which two or more solution components vary on drastically different time scales [4]. Performance of PID controller also depends on the selection of controller gains. Initially, simulation was performed with default gains K p (proportional gain) = 1, K i (Integral gain) = 0 and K d (differential gain) = 2 (to maintain condition for critical damping). Later on, autotuning was applied with the help of transfer function-based PID tuner app available in SimMechanics and optimal PID controller gains are noted as K p (proportional gain) = 0.05, K i (Integral gain) = 0.57 and K d (differential gain) = 0.05. Simulation results of walking due to given trajectories and applied PID control at ankles are shown below. Bipedal motion achieved with these control parameters shows that robot walking is continuous, smooth, and dynamically stable, i.e., it does not fall off during locomotion. It is observed that increasing K p and K d and reducing K i from optimal values make biped walking non-human like gait, unstable. Details about dynamic stability are discussed in Sect. 4.
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Fig. 6 Plot of torques at joints at left leg
Fig. 7 Plot of torques at joints at right leg
Figure 6 shows that required torque at ankle joint is maximum when it leaves the ground (changes from support phase to swing phase). Otherwise, torque requirement at the hip joint is greater compared to the other joints. It can be validated from the fact that in a serial chain system revolute joint that connects fixed base and first link requires more torque than others as first joint needs to move the whole system. Here, hip joint is serving the same purpose. Torque profile is similar in right leg and left leg as shown in Figs. 6 and 7. Angular speeds of three joints (ankle, knee, and hip) for both the legs are presented in Figs. 8 and 9.
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Fig. 8 Angular velocity at three joints at left leg
Fig. 9 Angular velocity at three joints at right leg
Maximum speed is observed for ankles and knee and hip joints are moving in similar manner. Maximum speed of all the joints is not seen at the same time and maximum torque is seen for hip joint when it moves with maximum hip speed. The 3D model of the robot during simulation is shown in Fig. 10.
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Fig. 10 Biped robot model in SimMechanics environment
4 Dynamic Stability There are few methodologies for available in the literature for analyzing stability for a biped robot model. Most used approaches are based on zero moment point (ZMP) [11], limit cycle theory [9], and Lyapunov theory [3]. In this paper, limit cycle theory is used to check and improve the dynamic stability of the biped model. For a biped locomotion to be stable, phase portrait of controlled variable should conform to a limit cycle as walking progress [9]. In biped robot system which is discussed in this paper, PID controllers are applied at the right ankles and the left ankles only. So phase portrait of associated joint angles and angular velocities for last two walking cycles are studied and presented in Figs. 11 and 12. If K p and K d are improved further, distance between last two walking cycles increases. In Fig. 12 controller parameters used are K p (proportional gain) = 1, K i (Integral gain) = 0 and K d (differential gain) = 2. In Fig. 11, distance between last two walking cycle is lesser than that in Fig. 12. Varying K p , K i , and K d it can be stated that distance between last two walking cycle is minimum for optimal controller gains as mentioned in Sect. 2.3. It refers to the fact that with optimal controller gain values, biped robot is dynamically most stable. It is also observed that there are two areas in Fig. 12 where significant angular velocity is changed with small change in angular displacement. This phenomenon is observed at the time of impact on the ground with a high velocity. For walking of a robot, this may damage the system. So this kind of situation should be avoided as well.
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Fig. 11 Phase portrait for right ankle at optimal condition
Fig. 12 Phase portrait for right ankle at non-optimal condition
5 Conclusion This paper presented a way to model 6-DOF biped robot system in MATLAB/ SimMechanics. Simulation with and without PID tuning parameters makes noticeable difference in walking pattern and dynamic stability of the system. With PIDtuned controller, movement of each joint occurs in a more organized way and more human like movement is obtained. A limitation of this system is that it produces flat feet motion. To overcome this problem, toe joints can be implemented at each foot. In turn it will incorporate more complexity to the system as more controllers will be required and simulation of the system will require more time as degree of freedom of the system will increase. Walking of model can be made more humanoid if contact forces are fed as feedback to movement of the system.
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References 1. Bouyarmane K, Vaillant J, Keith F, Kheddar A (2012) Exploring humanoid robots locomotion capabilities in virtual disaster response scenarios. In: 2012 12th IEEE-RAS international conference on humanoid robots (Humanoids 2012). IEEE, New York, pp 337–342 2. Denk J, Schmidt G (2003) Synthesis of walking primitive databases for biped robots in 3denvironments. In: 2003 IEEE international conference on robotics and automation (Cat. No. 03CH37422), vol 1. IEEE, New York, pp 1343–1349 3. Galloway K, Sreenath K, Ames AD, Grizzle JW (2015) Torque saturation in bipedal robotic walking through control Lyapunov function-based quadratic programs. IEEE Access 3:323– 332 4. Inc. TM. https://in.mathworks.com/help/matlab/release-notes-R2015a.html/. Accessed 9 Apr 2019 5. Kuswadi S, Jazidie A et al (2011) Reinforcement learning based of five legs robot for rescue operations. Acad Res Int 1(2):6 6. McGeer T et al (1990) Passive dynamic walking. Int J Rob Res 9(2):62–82 7. Mester G (2011) Modelling of the humanoid robot motion. IPSI J Trans Adv Res, TAR, New York, Frankfurt, Tokio, Belgrade 7(1):21–25 8. Miller S (2019) https://in.mathworks.com/matlabcentral/fileexchange/47417-simscapemultibody-contact-forces-library/. Accessed 9 Apr 2019 9. Ono K, Takahashi R, Shimada T (2001) Self-excited walking of a biped mechanism. Int J Robot Res 20(12):953–966 10. Sinnet RW, Zhao H, Ames AD (2011) Simulating prosthetic devices with human-inspired hybrid control. In: 2011 IEEE/RSJ international conference on intelligent robots and systems. IEEE, New York, pp 1723–1730 11. Takanishi A, Naito G, Ishida M, Kato I (1985) Realization of plane walking by the biped walking robot WL-10R. In: Theory and practice of robots and manipulators. Springer, Berlin, pp 383–393 12. Team MSC. https://in.mathworks.com/matlabcentral/fileexchange/64227-matlab-andsimulink-robotics-arena-walking-robot/. accessed 9 Apr 2019 13. Velásquez-Lobo MF, Ramirez-Cortés JM, de Jesus Rangel-Magdaleno J, Vázquez-González JL (2013) Modeling a biped robot on matlab/simmechanics. In: CONIELECOMP 2013, 23rd international conference on electronics, communications and computing. IEEE, New York, pp 203–206
Modal Analysis of 3-RRR SPM Model Vijaykumar Kulkarni, C. V. Chandrashekara, and D. Sethuram
Abstract Spherical parallel manipulator (SPM) finds in many engineering applications like aerodynamic simulators, medical devices, precision machine tools and telescopes. The dynamic characteristic of SPM plays an important role in determining the performance and stability of the manipulator. The present paper reports modal analysis of a 3-DOF SPM with full-circle twist model. A solid model of the SPM is developed using Unigraphics Nx and imported into Ansys for modal analysis. First, three fundamental natural frequencies and mode shapes are extracted. Natural frequencies are validated with the experimental results. Keywords Parallel manipulator · SPM · 3-RRR
1 Introduction Spherical parallel manipulator (SPM) finds in many engineering applications like aerodynamic simulators, medical devices, precision machine tools and telescopes. The dynamic characteristics play an important role in the performance and stability of the manipulator. Many researchers worked on the kinematics and dynamics of many parallel manipulators from last three decades. Patel and George presented survey on [1] parallel manipulators applications concluded that a continued and incessant effort by researchers will lead to an increased capability and effectiveness of various parallel robotics et al. [2] presented a novel approach of workspace modeling with Euler parameters is proposed for a special class of spherical parallel manipulators with symmetrical structure [3] presented a multi-objective optimization problem is formulated to optimize the structural and geometric parameters of the spherical parallel manipulator. Castri and Messina [4] described the freely vibrating dynamics of multi-link flexible manipulator adopting Timoshenko’s theory. Rong et al. [5] reported a stiffness and modal analysis of precision parallel manipulator with flexure V. Kulkarni (B) · C. V. Chandrashekara · D. Sethuram Department of Mechanical Engineering, PES University, Bengaluru, Karnataka 560080, India C. V. Chandrashekara e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_66
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hinges. Wu et al. [6] proposed a typical 3-RRR type spherical parallel manipulator. Author demonstrated a graphically different configuration for workspace size, dexterity and singularity of the wrist manipulator and concluded that the reconfigurable wrist manipulator can meet various task requirements due to its real-time performance enhancement. Literature surveys indicate numerous participations of researcher in designing manipulator for various engineering application, but a limited work in the area of vibration analysis of manipulators for dynamic performance. The present paper considers a 3-RRR SPM to demonstrate the finite element simulation modal analysis. Initially a 3D model is developed using Unigraphics Nx 10.0 and modal analysis is carried in the Ansys 18.1 environment. First three natural frequency of the top plate is obtained. An experimental model of the same size is fabricated and experimental modal analysis is carried out using LMS Test.Lab. Results obtained by finite element simulation are having a very good agreement with experimental results. The frequencies and mode shapes directly influence the performance and stability of the attachment on the top plate.
2 Modeling of the 3-RRR SPM The 3D SPM model is generated using Unigraphics Nx 10.0. Four basic elements, viz., bottom circular base, three-curved guides, three-curved links and one mobile top plate forms a simple SPM. A circular guide is considered to be the base. Three side units are attached to the circular guide and rotate along the guide. Each side unit is attached to a curved link through a revolute joint. The mobile platform is attached to the side units through three curved links. The side of the top plate considered according to the angular orientation of the curved link angle. Depending on the orientation of the curved link, the top plate dimensions vary. The guides are placed at 120° apart to each other on the base plate, supports the top plate in horizontal position. Change in the position of the guides, the top plate will get a different orientation other than the horizontal position. The angular orientation of the curved link is considered 45°. Figure 1 shows the developed SPM model.
3 Simulation of SPM Simulation model includes the circular guide, side units, curved links and top plate made of aluminum alloy. The boundary condition considered is the fixed-free. The base circular guide is considered as fixed support and the mobile platform is considered as free. The curved link acts as the stiffness and the top platform acts as the mass. The finite element modal analysis of the model is carried out in Ansys 18.1. Figures 2, 3 and 4 show the mode shapes of SPM model. The first mode shape exhibits the structural deformation in lateral direction. The second mode shape exhibits the structural deformation in transverse direction.
Modal Analysis of 3-RRR SPM Model Fig. 1 SPM model
Fig. 2 First mode shape for SPM model isometric and front view
Fig. 3 Second mode shape for SPM model isometric and front view
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Fig. 4. Third mode shape for SPM model isometric and front view
Table 1 Comparison of simulation and experimentation results of SPM model
Mode
Natural frequency (Hz) Simulation
Experimentation
Error (%)
1
335.64
361.85
7.24
2
394.77
427.18
7.58
3
395.16
428.50
7.70
The third mode shape exhibits the structural deformation in transverse direction. The obtained natural frequencies through simulation are reported in Table 1 along with the experimentation result in the Sect. 5. The first mode describes the lateral deformation of the system. The second mode and third mode describe the transverse deformation of the system. The second and third mode deform along the same plane so there is no large variation in the natural frequencies. The mode shapes clearly define the deformation in the three degrees of freedom.
4 Experimentation of SPM Experimentation is carried out in a manner to evaluate the physical system. The dynamic behavior of the system can be easily understood by experimentation setup. The setup gives the clear visualization and the mechanism involved in the process. Experimentation is carried out in LMS Test.Lab. Test.Lab provides a complete modal testing solution from measuring frequency responses to performing modal analysis. The model is fixed to the base and excited using impact hammer. The care is taken in boundary condition to avoid damping and obtaining accurate results. The accelerometer is fixed at one point on the SPM model and model is excited at different location. Figure 5 shows the experimentation setup with appropriate boundary condition to analyze the dynamic characteristics of the system.
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(a) Experimentation setup with LMS Test.Lab
(b) Isometsic view
(c) Top view
Fig. 5 Experimentation setup of SPM model
The extracted results from experimentation are reported in Table 1. Selection of material and machining is given utter most cares. The boundary condition established played a crucial role in obtaining the experimental results.
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5 Comparison of Results The natural frequencies are obtained through the simulation using Ansys, and experimental results are extracted using LMS setup. The simulation and experimental results are compared with each other and listed in Table 1. The natural frequencies values obtained from the simulation and experimentation are in a good correlation and with a maximum error of 7.70% in the third mode.
6 Conclusion In this paper, 3D SPM simulation model is developed. Modal analysis of the developed SPM is carried out using Ansys 18.1. The natural frequencies and mode shapes of the system up to sixth mode is extracted and up to third mode is reported. The mode shapes of the system clearly indicate the deformation of link and top plate at each mode. Production drawing is developed and system get it fabricated using commercially available aluminum. Fabricated SPM model is used to conduct the experimental modal analysis. Proper boundary conditions are established. Using LMS Test.Lab, experimental natural frequencies are extracted. Experimental results obtained using LMS Test.Lab are in close agreement with the simulation results.
References 1. Patel D, George PM (2012) Parallel manipulator applications—a survey. Modern Mech Eng 2:57–64 2. Bai S, Hansen MR, Andersen TO (2009) Modelling of a special class of spherical parallel manipulators with Euler parameters. Robotica 27:161–170 3. Wu G (2012) Multiobjective optimum design of a 3-RRR spherical parallel manipulator with kinematic and dynamic dexterities, modelling, identification and control. 33(3):111–122 4. Castri C, Messina A (2011) Vibration analysis of multilink manipulators based on timoshenko beam theory. J Robot 2011(890258) 5. Rong W, Luan Y, Qi L, Xie H, Sun L (2012) Stiffness analysis and modal analysis of precision parallel manipulator with flexure hinge. In: International conference on manipulation, manufacturing and measurement on the nanoscale, 29 August-1 September 6. Wu G, Dong H, Wang D, Bai S, (2013) A 3–RRR spherical parallel manipulator reconfigured with four-bar linkages, IEEE
Mode Based Crack Identification of Rotor Ridha Ali, T. Pooja Priya, V. Rashmi, C. V. Chandrashekara, and Suneel Motru
Abstract Rotor dynamics plays an important role in design and maintenance of all rotating machine. The presence of crack directly affects the dynamic characteristics hindering the productivity and performance of turbine. The precise projection of dynamic characteristics of turbine will facilitate the detection of faults in turbine during condition monitoring. The present paper reports an effective procedure, based on the mode shape characteristics to identify and locate a single and multiple transverse crack. The developed procedure is demonstrated with illustrative examples. Keywords Rotor · Mode shape · Crack
1 Introduction Rotating machines play an important role in almost every field of engineering, viz. power generation, transportation and machine tools etc. Rotor is an important part of the turbine that is subjected to bending and torsional vibrations during operating conditions. The presence of crack in rotor, directly affects the dynamic characteristics hindering the performance and productivity of turbine. The rotor failure accounts for large proportions of turbine failure downtime. Identifying the crack would potentially prevent serious damage and expensive repairs due to turbine failure. A review on dynamics of cracked rotor by Wauer [1] summarizes, finding the most appropriate procedure for detection of failure in the preliminary stages is crucial. Studying the behavior of non-rotating rotor also provides a relevant basis for crack modeling and detection of rotor to develop better diagnostic procedure. Darpe et al. [2] studied the response of Jeffcott rotor with a central transverse crack. The bending natural frequencies and side frequencies around impulse excitation frequency are used for crack detection. Agarwalla et al. [3] analyzed the effect of open crack on R. Ali (B) · T. Pooja Priya · V. Rashmi · C. V. Chandrashekara · S. Motru Department of Mechanical Engineering, PES University, Bengaluru, Karnataka 560085, India C. V. Chandrashekara e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_67
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modal parameters of free vibration of a cantilever beam. Change in natural frequencies and mode shape due to presence of crack can be studied to detect and determine the severity of the crack. Chandrashekara et al. [4] established a new formulation to determine the dynamic characteristics of cracked beam by considering the shift in the neutral axis of the cracked beam element. The approach effectively models discontinue structure providing more effective and accurate approach than previously adopted model. The study of dynamic characteristics of rotor with cracks will facilitate in understanding the effect of the crack on rotor to identify and locate the position of crack. In the present paper, an effective algorithm is developed to identify very precisely the location of the crack using the mode shape of the system.
2 Finite Element Formulation for Cracked Beam Element A rotor of length L, diameter d is considered and is discretized into n number of elements. A transverse crack of width x t and depth x d at a distance of x c from one end of the rotor is considered as shown in Fig. 1. An enlarged view of the crack location, representing an element of the rotor is shown in Fig. 2, with two degrees of freedom at each node, viz. vertical deflection, v and rotation, θ. Element stiffness matrix (k e ) is modified for a cracked rotor element as k ec , for which second moment of area of cross-section below the crack is derived to account for the change in the stiffness. The change in mass due to crack is assumed to be zero, as it is negligible when compared to the entire mass of the rotor. Figure 3 shows the portion of the area beneath the crack of the rotor. I c is the modified second moment of area of the portion below the crack to incorporate in the k ec . The modified second moment of area for rotor element with crack (I c ) is derived as, Crack
x
d
x L
Fig. 1 Rotor with a transverse crack
v1
y
v2
xt xd
x θ1
y
l
Fig. 2 Rotor element with crack and cross-section
θ2
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Fig. 3 Cross-section of rotor with crack
r4 × Ic = 8
θ ×π − sin θ 180
+
bh 3 36
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For an uncracked rotor element, I remain same and is given by, I =
π d4 64
(2)
The equation of motion for free vibration of the system is given by Thompson, [K ] − [M]ω2 {X i } = 0
(3)
For non-trivial solutions, the determinant of the coefficient of X i must be zero [K ] − [M]ω2 = 0
(4)
While assembling the element matrix, care is taken appropriately to place the crack element matrix in the global matrix.
3 Procedure for Identifying the Crack and Location Using Eq. (4), the eigenvector for both uncracked rotor, represented as [φ mn ] and for cracked rotor as [φ mn ]c is extracted. Procedure to identify the crack and location is demonstrated with the help of a flow chart shown in Fig. 8 (Annexure A).
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4 Illustrative Examples The developed procedure is demonstrated with three numerical examples. The rotor is discretized into 201 elements. First two examples demonstrate for single transverse crack at two different positions. Next one example demonstrates the multiple transverse cracks for different position. The material and geometrical properties of the rotor used are listed in Table 1.
4.1 Case 1: Single Crack at 42nd Element A transverse crack of depth (x d ) 2 mm and width (x t ) 1 mm is considered. The crack is introduced at 42nd element. Eigenvector for both cracked and uncracked rotor is obtained using Eq. (4). For the first mode, difference between the eigenvector of uncracked and cracked system is evaluated as deviation in the eigenvalues and is plotted as shown in Fig. 4. Table 1 Material mild steel 1018 property and dimension of the rotor
S. No.
Parameters
Notation
1
Length (m)
L
1
2
Diameter (m)
d
0.025
3
Young’s modulus (GPa)
E
198
4
Shear modulus (GPa)
G
80
5
Density (kg/m3)
ρ
7,870
Fig. 4 1st Mode deviation for crack at 42nd element
Value
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Fig. 5 Crack at 42nd element
Further, difference between the consecutive deviations of eigenvalues are evaluated and plotted as shown in Fig. 5. The peak in the plot indicates the location of the crack very precisely, i.e., at 42nd element.
4.2 Case 2: Single Crack at 111th Element In this case, crack is introduced in the 111th element. Similar procedure is adopted as described for case 1. The deviation in eigenvalues of uncracked and cracked rotor is not shown for further case (Fig. 6).
Fig. 6 Crack at 111th element
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Fig. 7 Crack at 45th and 161st element
4.3 Case 3: Multi-Crack at 45th and 161st Element A transverse crack of depth (x d1 ) 2 mm and (x d2 ) 3 mm at 45th and 161st element is considered, respectively. The width of crack is (x t ) 1 mm (Fig. 7). Using the procedure specified in Sect. 3 the crack is identified effectively for both single and multiple crack in rotor.
5 Conclusion In this paper, a new modal-based procedure to identify the location of the crack precisely is demonstrated. As the procedure is based on the finite element method, it is very simple and more suitable for computational-based dynamic analysis of rotor. Extensive illustrative numerical examples are considered to check the robustness of the procedure adopted. As the limitation of pages, only limited values are reported. The same procedure can be extended to simulation as well as experimental validation.
Annexure A See Fig. 8.
Mode Based Crack Identification of Rotor Fig. 8 Algorithm to locate the crack
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Define material, geometrical properties and crack parameter for the rotor
Define k and m element matrix for uncracked system and kec and mec matrix for cracked element
Assembly of Global matrix taking care of the crack element
Applying boundary condition
Extraction of eigen vector for both cracked and uncracked system
Deviation obtained by difference of eigen vector of uncracked and cracked system for a particular mode
Plot the difference between the consecutive deviations of eigen values. Highest peak in the plot indicates the location of the crack
References 1. Wauer J (1990) On the dynamics of cracked rotors: a literature survey. Appl Mech Rev 43(1):13– 17 2. Darpe AK, Chawla A, Gupta K (2002) Analysis of the response of cracked jeffcott rotor to axial excitation. J Sound Vibr 249(3):429–445 3. Agarwalla DK, Parhi DR (2013) Effect of crack on modal parameters of a cantilever beam subjected to vibration. Proc Eng 51:665–669 4. Chandrashekara CV, Pavan S, Dharani J, Himanshu A, Raj Arjun SI (2018) Formulation of effective stiffness for predicting natural frequency of cracked beams. Vibroeng Proc 19:135–140
A Note on Implementation of Raghavan–Roth Solution for Wrist-Partitioned Robots Rajesh Kumar, Alinjar Dan, and K. Rama Krishna
Abstract This paper focuses on providing a modification in the Raghavan and Roth algorithm for solving inverse kinematics. We show that this algorithm fails for wristpartitioned robots. The causes of failure are explored and the underlying architecture is modified to avoid the same. Other cases, where this algorithm does not provide the solution, are also discussed. The modification is illustrated through an example showing inverse kinematics of KUKA KR 5 robot. Keywords Inverse kinematics · Wrist-partitioned robots · Raghavan–Roth solution
1 Introduction Inverse kinematics solutions are an important part of robot control. The solutions serve as a map from the end-effector space to the joint space. Analytical expressions for a general six-degree-of-freedom system were a challenge until 1990s. During that time, an algorithm to compute the inverse kinematics solutions of a general six degrees of freedom serial manipulator was presented by Raghavan and Roth in [4]. Various algorithms based on the “Raghavan–Roth solution” were developed to efficiently compute the inverse kinematics solutions, like the one presented in [2]. The solution procedure is generalized and can be used in robotic simulation softwares. The solution procedure developed in [4] is based on utilizing the matrix equations as well as the power product equations and then, computing the determinant of a 12 × 12 matrix. R. Kumar (B) · A. Dan · K. Rama Krishna Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India e-mail: [email protected] A. Dan e-mail: [email protected] K. Rama Krishna e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_68
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In the present work, we show that the methodology fails for some special cases. This occurs when the 12 × 12 matrix is rank deficient. One of such cases is the occurence of wrist partitioning robots. We analyze the cause of failure of the “Raghavan and Roth” solution and present a modification in the same while computing solutions for the wrist-partitioned robots [1]. Section 2 presents a brief overview of the existing Raghavan–Roth method. The modification in the equations for wrist-partitioned robots are given in Sect. 3. Although this paper focuses on wrist-partitioned robots, other cases of failure are also possible. Further insights are presented in Sect. 5.
2 Raghavan and Roth Algorithm Preview Consider a general 6R serial manipulator. A set of 4 × 4 homogeneous transformation matrices establishes mathematical relationship between the configuration of any two links. End-effector transformation matrix is denoted by Aee , which denotes the position and orientation of a frame attached to the end effector. In order to compute inverse kinematic solutions, the Raghavan–Roth algorithm formulates a set of six equations by equating the first three rows of the last two columns of T and U (Eq. 1). A−1 A A−1 A A3 A4 A5 = A−1 2s 2v 1 ee 6 T
(1)
U
where Ai is the transformation matrix relating the coordinate frame moving with the i th joint to the coordinate frame fixed to the ground. In the solution algorithm, the matrix A2 is partitioned into A2s and A2v . A2s contains fixed DH parameters while A2v contains joint variables such that A2 = A2v × A2s , as done in [4]. Six equations are formed after equating last 2 columns of T and U are shown in Eqs. 2 and 3. ⎛
⎛ ⎞⎛ ⎞ ⎞ ⎛ ⎞ cos θ2 sin θ2 0 1 0 0 cos θ3 sin θ3 0 a2 ⎝ sin θ2 − cos θ2 0 ⎠ ˆl = ⎝ 0 −λ2 μ2 ⎠ ⎝ sin θ3 − cos θ3 0 ⎠ m ˆ + ⎝ 0 ⎠ (2) 0 μ2 λ2 0 0 1 0 0 1 b2 γ1
β1
⎛ ⎞⎛ ⎞ ⎞ 1 0 0 cos θ3 sin θ3 0 cos θ2 sin θ2 0 ⎝ sin θ2 − cos θ2 0 ⎠ cˆ = ⎝ 0 −λ2 μ2 ⎠ ⎝ sin θ3 − cos θ3 0 ⎠ dˆ 0 μ2 λ2 0 0 1 0 0 1 ⎛
γ2
β2
Expressions for different terms used in the above equations are given below.
(3)
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⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ l1 m1 c1 d1 ˆl = ⎝ l2 ⎠ m ˆ = ⎝ m 2 ⎠ cˆ = ⎝ c2 ⎠ dˆ = ⎝ d2 ⎠ l3 m3 c3 d3 m 1 = cos(θ4 ) j1 + sin(θ4 ) j2 + a2 m 2 = −λ3 (sin(θ4 ) j1 − cos(θ4 ) j2 + μ3 j3 m 3 = μ3 (sin(θ4 ) j1 − cos(θ4 ) j2 + λ3 j3 + b3 g1 = cos(θ4 )n 1 + sin(θ4 )n 2 g2 = −λ3 sin(θ4 )n 1 − cos(θ4 )n 2 + μ3 n 3 g3 = μ3 sin(θ4 )n 1 − cos(θ4 )n 2 + λ3 n 3 j1 = cos(θ5 )a5 + a4 j2 = − sin(θ5 )λ4 a5 + μ4 b5 j3 = sin(θ5 )μ4 a5 + λ4 b5 + b4 n 1 = sin(θ5 )μ5 n 2 = cos(θ5 )λ4 μ5 + μ4 λ5 n 3 = − cos(θ5 )μ4 μ5 + λ4 λ5 e = −l x a6 − (m x μ6 + n x λ6 )b6 + ρx f = −l y a6 − (m y μ6 + n y λ6 )b6 + ρ y g = −l z a6 − (m z μ6 + n z λ6 )b6 + ρz x = m x μ6 + n x λ6 y = m y μ6 + n y λ6 z = m z μ6 + n z λ6 i 1 = cos(θ1 )e + sin(θ1 ) f − a1 i 2 = −λ1 (sin(θ1 )e − cos(θ1 ) f ) + μ1 (g − b1 ) i 3 = μ1 (sin(θ1 )e − cos(θ1 ) f ) + λ1 (g − b1 ) c1 = cos(θ1 )x + sin(θ1 )v c2 = −λ1 (sin(θ1 )x − cos(θ1 )y) + μ1 z c3 = μ1 (sin(θ1 )x − cos(θ1 )y) + λ1 z where, λ = cos(α) and μ = sin(α). The six equations (Eqs. 2 and 3) are written as two vector equations as presented in Eq. 4. γ1 = β1 , γ2 = β2
(4)
The other eight equations arise from the equivalence of power product terms. Power products are the terms from a polynomial excluding constant coefficients. (e.g., the power products of the polynomial 2x y + x + x 2 y = 0 are x y and x, x 2 y). A detailed explanation for the power products can be found in [3]. So, for the Raghavan–Roth solution, eight additional equations obtained from the power product are given in Eqs. 5 and 6. γ1 T γ2 = β1 T β2 , γ1 T γ 1 = β1 T β1 , γ1 × γ2 = β1 × β2
(5)
(γ1 T γ1 )γ2 − 2(γ1 T γ2 )γ1 = (β1 T β1 )β2 − 2(β1 T β2 )β1
(6)
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So, the total no. of equations obtained is 14. The 14 equations can be written in the form of Eq. 7. Any 8 of the 14 equations are taken from Eqs. 2, 3, 5 and 6 and can be written in form of Eq. 8, where Aα is a 8 × 9 matrix and Bα is a 8 × 8 matrix. ⎛
⎞ ⎛ ⎞ sin(θ4 ) sin(θ5 ) sin(θ1 ) sin(θ2 ) ⎜ sin(θ4 ) cos(θ5 ) ⎟ ⎜ sin(θ1 ) cos(θ2 ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ cos(θ4 ) sin(θ5 ) ⎟ ⎜ cos(θ1 ) sin(θ2 ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ cos(θ4 ) cos(θ5 ) ⎟ ⎜ cos(θ1 ) cos(θ2 ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ sin(θ4 ) (A) ⎜ ⎟ ⎟ = (B) ⎜ sin(θ1 ) ⎜ ⎟ ⎜ ⎟ cos(θ4 ) ⎜ ⎟ ⎜ ⎟ ) cos(θ 1 ⎜ ⎟ ⎜ ⎟ sin(θ5 ) ⎜ ⎟ ⎝ ⎠ ) sin(θ 2 ⎝ ⎠ cos(θ5 ) cos(θ2 ) 1
(7)
where A is a 14 × 9 matrix, which contains terms as linear combination of sin(θ3 ), cos(θ3 ), 1, whereas B is a 14 × 8 containing all constant terms. Bα should be chosen in such a way that all rows and columns of it are linearly independent, as in future step, inverse of Bα is required. ⎛
⎞ ⎞ ⎛ sin(θ4 ) sin(θ5 ) sin(θ1 ) sin(θ2 ) ⎜ sin(θ4 ) cos(θ5 ) ⎟ ⎜ ⎜ sin(θ1 ) cos(θ2 ) ⎟ ⎟ ⎟ ⎜ cos(θ4 ) sin(θ5 ) ⎟ ⎜ ⎜ ⎜ cos(θ1 ) sin(θ2 ) ⎟ ⎟ ⎟ ⎜ cos(θ4 ) cos(θ5 ) ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ α ⎜ α ⎜ cos(θ1 ) cos(θ2 ) ⎟ ⎟ sin(θ4 ) (A ) ⎜ ⎟ ⎟ = (B ) ⎜ sin(θ1 ) ⎟ ⎜ ⎜ ⎟ cos(θ4 ) ⎟ ⎜ ⎜ ⎟ cos(θ1 ) ⎟ ⎜ ⎜ ⎟ sin(θ5 ) ⎜ ⎟ ⎠ ⎝ sin(θ2 ) ⎝ ⎠ cos(θ5 ) cos(θ2 ) 1
(8)
From Eq. 8, terms containing θ1 , θ2 can be written in terms of θ3 , θ4 , θ5 . Now it can be used to eliminate θ1 , θ2 from other six equations to form Eq. 9. ⎛
⎞ sin(θ4 ) sin(θ5 ) ⎜ sin(θ4 ) cos(θ5 ) ⎟ ⎜ ⎟ ⎜ cos(θ4 ) sin(θ5 ) ⎟ ⎜ ⎟ ⎜ cos(θ4 ) cos(θ5 ) ⎟ ⎜ ⎟ ⎟=0 sin(θ4 ) (R) ⎜ ⎜ ⎟ ⎜ ⎟ ) cos(θ 4 ⎜ ⎟ ⎜ ⎟ ) sin(θ 5 ⎜ ⎟ ⎝ ⎠ cos(θ5 ) 1
(9)
where R is 6 × 9 matrix having terms as a linear combination of sin(θ3 ), cos(θ3 ) and 1. Now, performing half tan angle substitution of θ4 , θ5 , θ3 as per Eq. 10 results in Eq. 11.
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sin(θi ) =
2xi 1 + xi2
1 − xi2 cos(θi ) = 1 + xi2
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⎛
⎞ x42 x52 ⎜ x 2 x5 ⎟ ⎜ 42 ⎟ ⎜ x ⎟ ⎜ 42 ⎟ ⎜ x4 x ⎟ 5 ⎟ ⎜ ⎟ x x (σ ) ⎜ 4 5⎟=0 ⎜ ⎜ x4 ⎟ ⎜ 2 ⎟ ⎜ x ⎟ ⎜ 5 ⎟ ⎝ x5 ⎠ 1
(11)
where σ has terms containing x3 only. Equation 11 can be rearranged [4] to form Eq. 12 with determinant equal to zero. σ 6×9 06×3 06×3 σ 6×9 = 0
(12)
Now, LHS of Eq. 12 is a 16 degree polynomial in x3 which can be solved to find x3 and subsequently θ3 .
3 Modified Architecture Wrist-partitioned robots have special characteristics such that the last three axes intersect at a particular point (Fig. 1). A set of general DH parameters for a wristpartitioned 6R robot is given in Table 1. The DH parameters are defined as in [5]. By using generalized DH parameters for wrist-partitioned robots, we can prove the failure of Raghavan–Roth solution.
3.1 Failure of Raghavan–Roth Solution Using the definitions of variables in Sect. 2 and [4], the subequations reduce to the following set of Eqs. 13–18 for DH parameters presented in table 1. j1 = 0
(13)
j2 = 0
(14)
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Fig. 1 Example of a wrist-partitioned robot Table 1 DH parameters for a general wrist-partitioned robot Joint no. Link length (a) Joint offset (b) Joint angle (θ) 1 2 3 4 5 6
a1 a2 a3 0 0 a6
θ1 θ2 θ3 θ4 θ5 θ6
b1 b2 b3 b4 0 b6
Twist angle α1 α2 α3 α4 α5 α6
j3 = b4
(15)
m 1 = a3
(16)
m 2 = μ3 b4
(17)
m 3 = λ3 b4 + b3
(18)
So, the initial set of six equations (Eqs. 2–3) (Sect. 2) reduces to the set of Eqs. 19–24 cos(θ3 )(λ3 b4 + b3 ) + sin(θ3 )(μ3 b4 ) = cos(θ2 )i 1 + sin(θ2 )i 2 − a2
(19)
sin(θ3 )a3 − cos(θ3 )μ3 b4 = −λ2 (sin(θ2 )i 1 − cos(θ2 )i 2 ) + μ2 (i 3 − b2 )
(20)
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λ3 b4 + b3 = μ2 (sin(θ2 )i 1 − cos(θ2 )i 2 ) + λ2 (i 3 − b2 )
(21)
cos(θ3 )d1 + sin(θ3 )d2 = cos(θ2 )c1 + sin(θ2 )c2
(22)
sin(θ3 )d1 − cos(θ3 )d2 = −λ2 (sin(θ2 )c1 − cos(θ2 )c2 ) + μ2 c3
(23)
d3 = μ3 (− cos(θ2 )c2 + sin(θ2 )c1 ) + λ2 c3
(24)
The other eight equations are to be formed from the power product equations. It should be observed that the left-hand side of Eqs. 19–21 is independent of any trigonometric function of θ4 and θ5 . However, Eqs. 22–24 have terms on the left-hand side containing coefficients (cos(θ4 ) sin(θ5 ), cos(θ5 ), sin(θ4 ) sin(θ5 )). So, the power product terms have terms with coefficients (cos(θ4 ) sin(θ5 ), cos(θ5 ), sin(θ4 ) sin(θ5 ), 1). While synthesizing the matrix A, the rows corresponding to the coefficients (sin(θ4 ) cos(θ5 ), cos(θ4 ) cos(θ5 ), sin(θ4 ), cos(θ4 ), sin(θ5 )) are null column vectors. So, A matrix is a rank 4 system. If we partition matrix A into Aα with 8 rows and Aβ with 6 rows, then the net matrix σ reduces to Eq. 25. It should be noted that wrist partitioning does not affect any element of matrix B, so under normal circumstances, the submatrix Bα is of full rank and remains invertible. σ = Aβ − Bβ (Bα )−1 Aα
(25)
So, the matrix σ is a matrix of rank 4 for the case of wrist-partitioned robots. So, we from σ6×9 to formulate a new submatrix P of the form in Eq. 26 to extract a σ6×4 ensure that the determinant is not trivially zero and we are able to gain a characteristic equation out of the same. However, in case, the submatrix σ consists of redundant rows, then a submatrix with independent set of rows need to be selected (σ ) needs to be selected appropriately. This might happen for some selected robots with some special set of DH parameters making other equations redundant, like for the case of KUKA KR 5 (presented in Sect. 4) which has equal b2 and b3 . σ 6×4 02×4 02×4 σ 6×4 = 0
(26)
P
The solution to Eq. 26 results in a polynomial which has inverse kinematics solution of wrist-partitioned robots. The determinant of 8 × 8 matrix gives a characteristic polynomial (degree 16) for wrist-partitioned robots. The methodology has been applied for a case of KUKA KR 5 robot, which is wrist partitioned and also has orthogonal axes. However, the proof presented does not ensure failure of Raghavan– Roth solution when three intersecting axes are not terminal.
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Table 2 DH parameters for a KUKA KR 5 Joint No. Link length (mm) Joint offset (mm) Joint angle (rad) 1 2 3 4 5 6
180 600 120 0 0 0
400 135 135 620 0 115
θ1 θ2 θ3 θ4 θ5 θ6
Twist angle (rad) π 2
π −π 2 π 2 −π 2
0
4 Numerical Example Case of KUKA KR 5 Robot KUKA KR 5 robot is an industrial robot with DH parameters presented in Table 2. The end-effector transformation matrix for the example is given in Eq. 27. The idea is to use modified Raghavan and Roth solution to find values of θ3 . ⎡
−0.8086 ⎢ 0 Aee = ⎢ ⎣ −0.5883 0
⎤ 0 0.5883 80 1 0 100 ⎥ ⎥ 0 −0.8086 1200 ⎦ 0 0 1
(27)
For the case of KUKA KR 5, the set of Eqs. 19–24 reduce to the set of Eqs. 28–33. 120 cos(θ3 ) − 620 sin(θ3 ) = cos(θ2 )(12.34 cos(θ1 ) + 100 sin(θ1 ) − 180
(28)
120 sin(θ3 ) + 620 cos(θ3 ) = sin(θ2 )(12.34 cos(θ1 ) + 100 sin(θ1 ) − 180) − 892.81 (29) (30) 0 = 100 cos(θ1 ) − 12.3455 sin(θ1 ) − cos(θ3 ) cos(θ4 ) sin(θ5 ) − cos(θ5 ) sin(θ3 ) = 0.5883 cos(θ2 ) − 0.8086 sin(θ2 ) (31) − sin(θ3 ) cos θ4 sin(θ5 ) + cos(θ3 ) cos(θ5 ) = 0.588 sin(θ2 ) cos(θ1 ) + 0.8086 cos(θ2 ) (32) (33) sin(θ4 ) sin(θ5 ) = −0.8086 cos(θ2 ) − 0.5833 sin(θ2 ) cos(θ1 ) Equations 28–30 do not contain θ5 and θ4 at all. The other eight equations need to be created from the power product. The left-hand side of Eq. 30 does not contain any term because KUKA KR 5 has equal b2 and b3 , which further reduces the rank of σ . This later results in a row of σ to be a zero row vector. Hence, the we use a 5 × 4 matrix of σ and a 3 × 4 zero matrix in Eq. 26. The determinant results in eight feasible solutions of θ3 given by {−165.66◦ , −165.66◦ , −160.14◦ , −160.14◦ , 2.03◦ , 2.03◦ , 7.55◦ , 7.55◦ }. The final solution is verified from the KUKA KR 5 inverse kinematics presented in [5].
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5 Discussion The Raghavan–Roth solution has a guaranteed failure for wrist-partitioned robots because the 14 underlying equations do not have all the coefficients as in Eq. 8. So, the matrix A is guaranteed to be rank deficient, which causes the determinant of σ to be trivially zero. Hence, we could not extract x3 values from the characteristic equation. However, if any other three axes intersect other than the last three, then the failure of the solution is not guaranteed. But, there can be examples where the solution fails because the matrix A or B turns out to be rank deficient. Such cases are under investigation. Equations 19–24 ensure failure of the Raghavan–Roth solution. It is well known that the forward kinematics of a closed-loop 7R spatial manipulator is equivalent to the inverse kinematics of a spatial 6R serial robot. Consider a spatial 7R closed-loop mechanism (1 DOF), with a known joint angle. Let us say the first joint angle (φ1 ) is known, then the equivalence is as shown in Fig. 2. So, we can use Raghavan–Roth algorithm for forward kinematics of closed-loop 7R spatial mechanism. However, this method fails for some cases of 7R spatial parallel mechanism. The Raghavan–Roth algorithm fails for the above-said mechanism when the second joint axis (τ2 ), the third joint axis (τ3 ), and the fourth joint axis (τ4 ) intersect at the same point (Fig. 3). The failure for the case of wrist-partitioned robots can be prevented if the algorithm is modified in such a way one of A4 or A5 is partitioned, instead of A2 . However, the solution methodology might not remain similar if partitioning matrix is changed, which is a subject of further study.
Fig. 2 Equivalence of forward kinematics of the parallel robot to the inverse kinematics of the serial robot
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Fig. 3 Condition of failure of kinematics of Raghavan–Roth solution for serial robots
6 Conclusion This paper is based on modifying the Raghavan and Roth solution to implement on wrist-partitioned robots of general geometry. A submatrix is detected and utilized to formulate a polynomial whose roots lead to inverse kinematic solutions. This method augmented with the original Raghavan and Roth solution can be used for a larger class of general robots, including wrist-partitioned robots, which are prevalent in industrial uses. There can be other cases of failure of the solution which can be investigated using a similar way and is a subject of further study.
References 1. Hollerbach JM, Sahar G (1984) Wrist-partitioned, inverse kinematic accelerations and manipulator dynamics 2:152–161. https://doi.org/10.1109/ROBOT.1984.1087172 2. Manocha D, Canny JF (1994) Efficient inverse kinematics for general 6R manipulators. IEEE Trans Robot Autom 10(5):648–657. https://doi.org/10.1109/70.326569 3. Raghavan M, Roth B (1990) Kinematic analysis of the 6r manipulator of general geometry, pp 314–320 4. Raghavan M, Roth B (1993) Inverse kinematics of the general 6R manipulator and related linkages. J Mech Des 115(3):502–508 5. Saha SK (2014) Introduction to robotics. Tata McGraw-Hill Education, New York
Experimental Identification of Residual Unbalances for Two-Plane Balancing in a Rigid Rotor System Integrated with AMB Gyan Ranjan, Rajiv Tiwari, and Harshal B. Nemade
Abstract An experimental dynamic balancing procedure is presented for a rigid rotor system supported on conventional bearings and integrated with active magnetic bearing (AMB). The residual unbalances present in the rotor system are the major causes of fatigue failure of the system at high speeds. So, it is necessary to balance the system to avoid any damage to the system. AMBs are utilized to suppress the vibration of the system through controlled magnetic forces. In this work, AMB is utilized to suppress the vibration of the system using active control through a PD controller. The balancing procedure can be carried out with less vibration in the presence of AMB. Also, after balancing, the AMB in the system can take care of abrupt rise in system response with less power consumption in vibration control. An eight-pole actuator is utilized for generating magnetic force in the rotor system. The PD controller is developed on the dSPACE and Simulink platform for the generation of control current. The measured displacement response through eddy current probes is utilized in the influence coefficient method (ICM) to obtain the balance masses at two balancing planes for dynamic balancing of the system. The obtained balance masses are placed at the respective balancing planes, and the reduction in vibrational response is observed. In comparison with the displacement responses before and after balancing, the displacement is found to be reduced after balancing. Keywords Residual unbalances · Dynamic balancing · Active magnetic bearing · Rigid rotors
G. Ranjan · R. Tiwari (B) Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, Assam, India e-mail: [email protected] G. Ranjan e-mail: [email protected] H. B. Nemade Department of Electronics and Electrical Engineering, IIT Guwahati, Guwahati 781039, Assam, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_69
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1 Introduction The rotating machinery, such as generators, gas turbines, machine tools, and industrial turbomachinery, is commonly used in industries. Vibration due to residual unbalance is a major concern as it limits the performance and fatigue life of the rotating components. The residual unbalances (inherent unbalances) in machineries occur due to manufacturing errors, improper commissioning, wear and tear, and so on. The amount of vibration induced by residual unbalance is proportional to square of rotating speed which can cause serious damage to the rotor system passing the critical speed. Therefore, the balancing of rotor system is mandatory to reduce the vibration occurrence in the system at high speed. To reduce excessive vibration, many balancing methods such as single-plane balancing, cradle balancing, and influence coefficient method have been developed by practicing engineers and researchers. The influence coefficient method is widely used approach for balancing of rotor systems in industrial applications. The balancing procedure involved is easy to carry out on a computer once measurement is available and requires very little knowledge of the rotating machineries mathematical modeling [1, 2]. Kang et al. [3] formulated influence coefficient matrices for unsymmetrical rotors from equations of motion using the finite element method. It was found that two trial operations for the measurement of unbalance responses are required for determination of influence coefficients and obtained the correction masses from influence coefficient, which reduced the residual vibration. Pennacchi et al. [4] investigated several robust regression methods for rotor balancing along with influence coefficients. They tried to develop an automatic balancing procedure for estimation of balancing masses based on the above method. Seve et al. [5] presented a balancing procedure of machines composed of a flexible rotating part (rotor) and a non-rotating part (stator) mounted on suspensions. The procedure was based on a numerical approach using rotor dynamics theory coupled with the finite element and influence coefficient method. Lin et al. [6] used finite element analysis to simulate the balancing of flexible rotor systems under various arrangements of sensors and planes. They showed that accuracy of balancing can be improved by selecting sensor locations and balancing planes which reduce the condition number. Lots of researches have been carried out to actively control the rotor system vibration, and it can be done effectively by the use of active magnetic bearing (AMB), which provides contactless actuation and helps in reduction of wear and tear. AMB reduces excessive vibration by applying magnetic force in the opposite direction to unbalance forces. However, instead of applying this corrective force all the time, if this corrective force from AMB can be used for finding the residual unbalance state of the rotor that will save unnecessary power consumption of the AMB. Chougale and Tiwari [7] proposed an algorithm for the estimation of residual unbalances and dynamic parameters of active magnetic bearing for flexible rotor system. The algorithm uses the AMB current and unbalances responses for estimating the dynamic parameters (for each AMB) and unbalances in flexible rotor. Kim and Lee [8] developed an active balancing device which is of an electromagnetic type and proposed an
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active balancing method using the influence coefficient method. They also verified the effectiveness of balancing method by experiments. Moon et al. applied an active balancing program using the influence coefficient method and an active balancing device of electromagnetic type with both simple and reliable structures to the developed high-speed spindle system. Gyan and Tiwari [9] described a generalized influence coefficient method (GICM) to estimate the unbalances present in the flexible rotor-AMB system supported on conventional bearings. However, it is beneficial to utilize AMB in the system while estimating the balancing mass in order to suppress the excessive vibration reducing fatigue stress. Also after balancing with reduction in unbalance, the requirement of controlling current to suppress vibration reduces resulting in less power consumption. AMB in the system after balancing can be utilized to avoid any unexpected increase in response of the rotor system. So, in the present work, an experimental simulation is described to perform the dynamic balancing of the rigid rotor system integrated with an AMB.
2 Influence Coefficient Method The influence coefficient method is used to predict residual unbalances in a rigid rotor system with the use of displacement values at two measuring planes. Displacements values are obtained for different trial unbalances at balancing planes for estimation of influence coefficients. Then, these are used in obtaining residual unbalances at balancing planes. A typical rotor system consisting of a shaft, rigid disks (balancing planes), and sensors (measuring plane) mounted on conventional bearings is shown in Fig. 1. In the ICM [2], two balancing planes m and n are selected for trial unbalances, and two measurement planes a and b are considered for the measurement of displacements of the rotor system. Apart from this, a reference signal is measured at a convenient shaft location to have measurement of all phases with respect to this signal. Displacements of the shaft are considered as r ia and r ib at the left and right measuring planes, respectively, and are shown below as ria = A L ejφL , rib = A R ejφ R Fig. 1 Schematic of a rotor-bearing system
(1)
a
b
CB-1
CB-2
m
n
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with r (t) = x(t) + jy(t) where AL and AR are the amplitudes of measurement, √ and φ L and φ R are the phase angles with respect to a reference signal, and j = −1. The subscript i (=1 to 3) represents different rotor runs, i.e., for i = 1, no trial unbalance is taken, and for i = 2 and 3, trial unbalance is taken at right and left balancing plane, respectively. In the second rotor run, the trial unbalance T R is taken at the right plane, and in the third run, the trial unbalance T L is taken at the left plane. The displacements for different trial runs are related as shown below αan αam Wn r1a = (2) r1b αbn αbm Wm r2a αan αam Wn + T R = (3) r2b αbn αbm Wm r3a αan αam Wn = (4) r3b αbn αbm W m + TL where W n and W m are the residual unbalances at the right and left planes, respectively; α an , α bn , α am, and α bm are the influence coefficients. The influence coefficients obtained are given as [2] αan =
r2a − r1a r2b − r1b , αbn = TR TR
(5)
αam =
r3a − r1a r3b − r1b , αbm = TL TL
(6)
The correction unbalances W R and W L are obtained at the balancing planes as given below α α WR = − an am αbn αbm WL
(7)
r1a αbm − r1b αam r1b αan − r1a αbn , WL = αbn αam − αan αbm αbn αam − αan αbm
(8)
WR =
r1a r1b
The correction unbalances are then added at the balancing planes to reduce the vibrational response of the system.
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3 Working of AMB An active magnetic bearing system supports a rotating shaft, without any physical contact by suspending the rotor in the air, with the help of magnetic force. AMB has unique characteristics of the absence of wear, lubrication, and seals due to free of contact between the rotor and the stator. The component of an AMB is actuator, amplifier, proximity sensor, and controller. Whenever air gap changes between the rotor and the AMB, the change is sensed by the sensor, and subsequently, the current is fed into the actuator through controller to apply magnetic force and maintain the equilibrium position of the rotor. A schematic diagram of AMB has been shown Fig. 2. The AMB uses displacement stiffness and current stiffness in two orthogonal directions. A proportional-derivative (PD) controller has been used in the system. The controller uses the transverse displacement at the node to generate the controlling current which is fed to the actuator to produce adequate magnetic force. The controlling force is expressed as [2] f mb = − K s ηm + K i i c
(9)
With Ks =
k sx x k sx y , ks yx ks yy
Ki =
Fig. 2 Schematic of an active magnetic bearings
ki x x ki x y ki yx ki yy
MAGNETIC ACTUATOR
ic
COILS
N
S
AMPLIFIER ROTOR
CONTROLLER
fm
PROXIMITY SENSOR
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⎧ ⎫ x⎪ ⎪ ⎪ ⎨ ⎪ ⎬ y x kP 0 kD 0 ηm = , ic = x˙ ⎪ 0 kP 0 kD ⎪ y ⎪ ⎩ ⎪ ⎭ y˙ where K s is the displacement stiffness matrix, K i is the current stiffness matrix, ηm is AMB displacement, and ic is the controlling current. Controller parameter k P represents the proportionality constant, and k D represents the derivative. The dot on the displacement represents differentiation, respectively.
4 Experimental Simulation The experimental setup for the balancing of flexible rotor using AMB is developed in the present work. The experimental setup available in the laboratory is a standard Bently Nevada Rotor Kit RK4 as shown in Fig. 3. The rotor kit consists of a 10 mm diameter and 560 mm long shaft supported on two journal bearings. The shaft weighs 350 gm and has two detachable disks of 800 gm each mounted on it. The disks have slots for the addition of unbalance in the system to predict the balance masses using influence coefficient method. The additional components used are magnetic actuator, PD controller (dSPACE), and amplifier.
Fig. 3 Rotor-bearing test rig (Bently Nevada Rotor Kit RK4)
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4.1 Physical and Electrical Connections The physical connection of the experimental setup for the vibration control problem is shown in Fig. 4. In the setup, change in position of the shaft is sensed by the eddy current sensors, and the analog signal is sent to input panel of the dSPACE I/O (input–output) panel as shown in Fig. 4. The analog signal is converted into a digital signal, and then, controlling current is generated though the PD controller (Simulink model based). The controlling current is transferred to the I/O panel to get an analog output for the amplifier. The current from the amplifier is fed to a magnetic actuator to reduce the vibration of the rotor system. In Fig. 5, the electrical circuit developed for working of differential-type actuator is shown. The circuit consists of a dSPACE platform, dc voltage source, a magnetic actuator, eddy current sensor, and amplifier. The bias voltage is supplied to two opposite poles in equal amount, i.e., eight-pole ib1 for upper and lower poles and ib2 for right and left pole as shown in Fig. 5. The displacement signal measured through the eddy current sensors from the rotor is utilized to generate the controlling current through a PD controller in dSPACE platform. The controlling current, i.e., ic1 flowing through the poles gets added in upper pole bias current, and at the same time, it gets subtracted from lower pole. As the shaft moves downward, the overall current in upper pole, i.e., ib1 + ic1 and that in lower pole, i.e., ib1 − ic1 allows the shaft to move back in its original position. In similar manner, control currents are generated for both the right and left poles.
Rotor system
Position sensor
Analog to Digital input dSPACE Controller
Magnetic actuator
Amplifier
Digital to Analog output
Fig. 4 Block diagram showing the physical connections for the vibration control problem
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Amplifier
N
ib 2 ic 2 ic 2
S
Sensor
Amplifier
Dspace
ic1
ib1 ic1
Sensor
ic1 y
Amplifier
704
ic 2
Amplifier
x
ib 2 ic 2
ib1 ic1
Fig. 5 Block diagram of the electrical circuit for working of differential-type actuator
4.2 Testing of the Physical System The following steps are involved in the installation and working of experiment setup for the reduction of vibration due to unbalance. 1.
2.
3.
4.
The active magnetic bearing was mounted on the shaft, and the shaft was rotated manually to check the proper alignment of bearing with the axis of shaft. The bearing was mounted in between two balancing disks, as shown in Fig. 3. The piezoelectric sensors were mounted and were maintained at a gap of approximately 30 mils (0.8 mm) or −5.75 ± 0.5 Vdc on the voltmeter. The gap was maintained to avoid nonlinearity arising in the displacement data received from the sensors. The controller was designed for the system, using Simulink and MATLAB and is executed with the help of the dSPACE interface. The connection was established between magnetic bearing, amplifier, and dSPACE, as shown in Fig. 5. In the dSPACE control desk, the controlling parameters (k P and k D ) are changed while the real-time simulation of the Simulink model. The real-time plot of displacement of rotor system along vertical axis vs. time was seen using data acquisition instruments available.
Experimental Identification of Residual Unbalances for Two-Plane …
5.
6.
7.
8.
705
The value of k P was varied from 0 to 14,000 A/m and that of k D from 0 to 15 A-s/m to see the gain limit of the controller. The fluctuation of the shaft was seen and stored with the use of data acquisition toolbox. The values of k P and k D are kept within the stable limit to see the reduction in vibration of the shaft with a variation of controlling parameters than without the use of magnetic bearing. The system is applied with an impulsive force at the standstill condition, and the changes in the damping and stiffness are calculated for the system with or without the effect of magnetic bearing. Also, the system was found to be stable for the value of k P between 5000 and 14,000 and k D between 3 and 15.
The system response is obtained after the application of the impulsive force on the system at standstill condition without or with application of AMB as shown in Fig. 6. The increase in proportional gain increases the stiffness and the derivative gain increases the damping property of the magnetic bearing. As seen in Fig. 6d, with the application of PD control through AMB, the vibration response gets eliminated in fewer cycles than in case of without AMB (d). The change in stiffness and damping property of the system is observed after the application of AMB in the system in a vertical direction. The controller parameter is considered to be k P = 13,000 and k D = 14. First, the FFT of the displacement response is carried out to obtain the natural frequency of the system. With the help of the overall mass of the system and natural frequency, the stiffness and damping of the system are calculated as shown in Table 1.
Fig. 6 In response to a impulsive force, b displacement of the rotor system without AMB, c impulsive force, and d displacement of the rotor system with AMB
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Table 1 Estimated properties of the rotor system
Mass of the system
1.93 kg
property
Without AMB
With AMB
First natural frequency
26.15 Hz
33.3 Hz
Stiffness (overall)
5.21 × 104 N/m
8.44 × 104 N/m
Damping (overall)
7.19 Ns/m
38.28 Ns/m
a
b
Coupling Motor
CB-1 m
AMB
CB-2 n
Fig. 7 Schematic of the experimental rotor-AMB setup
4.3 Dynamic Balancing The dynamic balancing of the rigid rotor system is carried out, including AMB in the system. The two disks (m and n) act as balancing planes, whereas eddy current sensors at two different locations (a and b) act as measurement planes as shown in Fig. 7. The AMB suppresses the vibration, and the vibration response is then utilized in the influence coefficient method to predict the balance masses.
The following steps are involved in the dynamic balancing of the rotor system. 1. 2.
3.
4. 5. 6.
7.
The controller parameter is considered to be k P = 13,000 and k D = 14. The reduction in displacement with the application of AMB at 900 rpm is checked as shown in Fig. 8. The half spectrum plots obtained after FFT for the displacement data with and without AMB are also shown in Fig. 9. The displacement amplitude without AMB at 1X harmonic is found out to be 9.5 × 10–5 m and with AMB is equivalent to 1.3 × 10–5 m, which gives a percentage reduction of 86.2%. The zero degree axis on the balance planes, i.e., disks is made coincident with the key phasor on the coupling of the system. The system is rotated to obtain the displacement along with both x and y directions at the measurement planes due to residual unbalance in the system. A reference signal is also obtained from the key phasor location to obtain the desired displacement signal with phase compensation from the reference signal. A trial mass is then added to the right plane, and displacement is obtained at the measurement planes.
Experimental Identification of Residual Unbalances for Two-Plane …
707
-4
1.5
x 10
With AMB Without AMB
Displacement (m)
1 0.5 0 -0.5 -1 -1.5 0
0.1
0.3
0.2
0.4
0.5
0.6
0.7
1
0.9
0.8
Time (s)
Fig. 8 Displacement of the rotor system with and without consideration of AMB at 900 rpm -4
1
x 10
Displacement amplitude (m)
With AMB Without AMB 0.8
0.6
0.4
0.2
0 0
500
1000
1500
2000
2500
3000
3500
4000
Spin speed(rpm)
Fig. 9 Displacement amplitude of the rotor system with and without consideration of AMB versus spin speed in rpm
8. 9. 10.
Similarly, the next set of displacement is obtained for trial unbalance at the left plane. The corrected unbalances are obtained at the two balancing plane using ICM. The obtained corrected unbalances are then added at the balancing planes, and displacement responses are obtained without AMB to see the reduction in vibration.
The identification of residual unbalances is done for different spin speed, i.e., 900, 1000, and 1200 rpm considering AMB and for 900 rpm without considering AMB. The balance masses obtained at left (m) and right (n) balancing planes are
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Table 2 Estimated balance masses at the left and right balancing planes for cases with or without considering AMB Balance mass at right plane
Balance mass at left plane
Cases
Speed (RPM)
Magnitude (kg-m)
Phase (degree)
Magnitude (kg-m)
Phase (degree)
Without AMB
900
2.4 × 10–4
– 11.6
3.5 × 10–4
129
900
2.6 ×
1000
2.2 × 10–4
1200
1.5 × 10–4
With AMB
10–4
– 3.8
3.1 ×
10–4
134.8
– 12.5
2.9 × 10–4
131.9
– 23.9
2.1 × 10–4
125.7
shown in Table 2 for both the cases. To balance the system, the balance masses of 2 gm at a radial distance of 3 cm from the center are added at 0° at the left plane and 135° at the right plane. After balancing, the displacement obtained is compared with the initial displacement response (without balancing) at 900 rpm as shown in Fig. 10. The FFT of both the displacement data is carried out to see the % reduction of vibrational displacement. The displacement amplitude before balancing at 1X harmonic is found out to be 9.5 × 10–5 m and after balancing equivalent to 8 × 10–5 m which gives a percentage reduction of 15.78%. As the displacement response consists of displacement due to shaft bow and residual unbalances, the slow run (250 rpm) response of the system is also obtained. The slow run amplitude at 1X harmonics is found out to be 6.8 × 10–5 m which is then subtracted from the amplitude of the response before and after balancing as mentioned above. After slow run removal, the percentage reduction in amplitude of vibration due to residual unbalance is found out to be 55%. -4
1.5
x 10
After balancing Before balancing
Displacement (m)
1 0.5 0 -0.5 -1 -1.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Fig. 10 Displacement of the rotor system before and after balancing at 900 rpm
0.9
1
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5 Conclusions In the present work, experimental identification of residual unbalance is carried out for the rigid rotor system integrated with AMB. The PD controller is applied to establish closed-loop control for the working of the system. The vibrational response of the system is effectively reduced with the use of AMB. The correction unbalances estimated with and without AMB are found to be approximately at the same orientation and magnitude. Also, the corrected unbalance in an addition to respective balancing planes shows a reduction in vibrational response, which in turn reduces the power consumption in control of the system. It shows AMB actively suppresses the vibration and also allows the estimation of unbalances. Therefore, the addition of AMB in the system allows to periodically monitor the unbalance in the system as well as maintain the abrupt increase in vibrational amplitude to a particular level for the safe operation of the rotor system. The work can be extended to balancing of the flexible rotor at higher speeds (i.e., above critical speeds).
References 1. Rao J (1996) In: Rotor dynamics. New Age International 2. Tiwari R (2017) In: Rotor systems: analysis and identification. CRC Press 3. Kang Y, Liu C-P, Sheen G-J (1996) A modified influence coefficient method for balancing unsymmetrical rotor-bearing systems. J Sound Vib 194:199–218 4. Pennacchi P, Chatterton S, Ricci R (2010) Rotor balancing using high breakdown-point and bounded-influence estimators. Mech Syst Signal Process 24:860–872 5. Sève F, Andrianoely M-A, Berlioz A, Dufour R, Charreyron M (2003) Balancing of machinery with a flexible variable-speed rotor. J Sound Vib 264:287–302 6. Kang Y, Lin T-W, Chang Y-J, Chang Y-P, Wang C-C (2008) Optimal balancing of flexible rotors by minimizing the condition number of influence coefficients. Mech Mach Theory 43:891–908 7. Tiwari R, Chougale A (2014) Identification of bearing dynamic parameters and unbalance states in a flexible rotor system fully levitated on active magnetic bearings. Mechatronics 24:274–286 8. Kim J-S, Lee S-H (2003) The stability of active balancing control using influence coefficients for a variable rotor system. Int J Adv Manuf Technol 22:562–567 9. Ranjan G, Tiwari R (2019) Application of active magnetic bearings for in situ flexible rotor residual balancing using a novel generalized influence coefficient method. Inverse Problems Sci Eng 27:943–968
Determination of Steering Actuator Mounting Points of a Load Haul Dump Machine for Optimum Performance SreeHarsha Rowduru , N. Kumar , and Vinay Partap Singh
Abstract Articulated steering system of a load haul dump (LHD) machine is made possible with the help of one/two hydraulic cylinders mounted between the two body sections. The present article investigates the variables affecting the steering performance like maximum possible steering angle (MPSA) and minimum steering force at articulated joint. In this respect, the geometrical approach is adopted to study the steering system of the LHD machine, and how the actuator mounting points influence the steering performance. Based on the study, MATLAB® code has been developed for finding the optimum mounting points of the steering actuators for MPSA. 3D model of the LHD machine is developed based on the optimum points obtained from the MATLAB® program. The force analysis of the 3D model is carried out in ADAMS environment. Based on the results, it is found that the steering cylinders with inward type offer MPSA and minimum steering force. Also, the optimization technique illustrates that the maximum steering angle may be achieved up to 58°, which leads to angle increase by 28% of the existing LHD machine. Keywords Steering actuators · Mounting points · Optimization · Geometrical approach
1 Introduction Articulated vehicles are having two or more body sections connected by a joint called articulated joint. Wheel loaders and LHD machine are the examples of articulated vehicles employed in mining industries as a trackless loading equipment to transport the mineral from ore site to the dumping site in narrow constrained pathway. Steering system of the LHD machine is provided by means of hydraulic steering cylinders. They are mounted symmetrically between the two body sections of the vehicle about the articulated joint of the LHD machine. Depending upon the size, S. Rowduru (B) · N. Kumar · V. Partap Singh Mining Machinery Department, IIT(ISM) Dhanbad, Dhanbad, Jharkhand 826004, India N. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_70
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capacity of the machine and its working conditions, manufacturers developed articulated steering system (ASS) using single or double hydraulic cylinders. In the current scenario, the MPSA achieved by the LHD machine is in the range of 40–45° . The mounting arrangements of the steering cylinders to achieve the maximum steering angle, minimum steering force, better response and steering performance are the wider field of research. The literary works on the steering performance and optimization of steering actuator mounting points of the articulated vehicle are discussed below: Zhu et al. [1] adopted a multidisciplinary design optimization (MDO) method for the steering system of the large-wheeled harvesting equipment. Almost, all the design parameters were considered to find optimal objective function and validated the results. Lei-yan et al. [2] applied genetic algorithm (GA) on steer-by-wire (SBW)type steering mechanism. Dynamic modeling of SBW steering mechanism was developed in MATLAB, and only vehicle parameters like tire relaxation length, front and rear tire cornering stiffness are optimized through GA. Lei et al. [3] established optimum design model of steering mechanism using mathematical equations. Using MATLAB optimization toolbox, optimized pivoting points of the steering actuator of a HT25J wheel loader is determined. Thulsiraman et al. [4] discussed about the mounting points of steering actuators using CAD package optimization tool and validated experimentally. The results show that with optimized parameters of the mounting points and steering actuator dimensions’, stresses at the mounting points are reduced, and steering force required at the bore end side is also reduced. Li et al. [5] in his article discussed about a new method to optimize the wheel loader digging force which combines sensitivity analysis with genetic algorithms to reduce design variables and to improve the optimization efficiency. The results show that approximately six percent increase in loader digging force is observed after the optimization. Jin et al. [6] optimized the steering properties of a hydrostatic steering system in simulation environment using combination of PID controller with GA. The required oil flowing through the cylinder is controlled by electro-hydraulic proportional valve using PID controller, and to reduce pressure, fluctuations constants of PID are optimized using GA. This resulted in rapid dynamic response and good steering performance. Yin et al. [7] report based on kinematic and dynamic analysis of the articulated vehicle suggest that length of the steering actuators and steering actuators geometrical arrangement influences the maneuverability and steering efficiency. Optimized results using analytic hierarchy process (AHP) model suggest that reduction in steering cylinder length provides more compact steering frame design and better steering performance of the vehicle. Zhou et al. [8] proposed GA-based optimization of steering mechanism, and its effectiveness is verified using automated dynamic analysis of mechanical systems (ADAMS) software. Cao et al. [9] worked on the depth optimization of hinged position of the articulated steering system using GA. None of the above articles discussed on the prediction of MPSA based on the steering cylinder mounting points. This article focuses the maximization of steering angle and minimization of steering force with respect to the geometric parameters of the steering arrangements to obtain the optimum steering performances of the LHD
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machine. This type of study has not been carried out to the best of authors’ knowledge, which is the novelty of the present article. This also facilitates the manufacturer to estimate the optimized mounting points of the steering cylinders used for any articulated vehicle.
2 Geometric Modeling of the Articulated Steering Arrangement Geometrical approach is a mathematical concepts-based approach developed to achieve a better and clear insight into the most salient features of multivariable dynamical systems. MPSA is dependent on various factors like steering cylinder arrangement/orientation, distance between steering cylinder from the articulated joint and stroke length. Various possibilities of the articulated steering cylinder arrangements are shown in Fig. 1. Inward case-type steering cylinder arrangement is currently employing in the articulated vehicles. Inward case steering cylinder arrangement type is considered for the geometric analysis. Referring to Fig. 2, points A and B are the hinged steering cylinder mounting positions on front part, and C and D are the fixed steering cylinder mounting positions on the rear part of the LHD vehicle. Point A and point B refer to the mounting position when the vehicle is in straight or neutral position, whereas points A’ and B’ refer to the mounting points when vehicle is performing steering action. Length of the steering actuator when vehicle is in neutral condition is ‘L.’ Point ‘O’ is the articulated joint. The angle of inclination of the bore-side mounting point with horizontal line passing through ‘O’ is the angle ‘θ.’ The angle of inclination of rod-side mounting points with the horizontal line passing through ‘O’ is the angle ‘β.’ When cylinder is fully extended, i.e., distance A’C in the above figure is ‘L + e,’ and when cylinder is in fully retracted, i.e., distance DB’ is ‘L − c.’
Fig. 1 Schematic diagram of various possibilities of the cylinder arrangement
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Fig. 2 Geometric arrangement of the steering cylinders
The angle subtended with circle of radius ‘r’ between the points AA’ or BB’ with articulated joint as center of the circle is the steering angle. As shown in the below figure, ‘α’ is the steering angle considered in this case. The angle ‘φ’ is the inclination of the steering actuator with a line parallel to reference line along in ‘x’ direction. The angles ‘γ ’ and ‘ψ’ are the subtending angles with respect to the initial inclination of the steering actuators when vehicle is taking turn. Overall, the steering cylinder mounting arrangement can be expressed in terms of four parameters, namely ‘L,’ ‘d,’ ‘r’ and ‘θ.’ Maximum extending length and minimum retracting length will be same for a particular type of actuator. Dimensions of the 912E LHD model is considered, and the possible range of values for the steering cylinder mounting points are determined as shown in Table 1: With the help of geometric relationships x1 = d cos θ, x2 = r cos(180 − β), x3 = r cos(180 − (β − α)), x4 = r cos(180 + β + α) Table 1 List of parameters and their possible range for mounting
Name of the varying parameter
Range (Min value–max value)
Neutral length of the steering actuator (‘L’ in mm)
400–700
Distance of fixed point from articulated joint (‘d’ in mm)
400–680
Distance of hinged point form articulated joint (‘r’ in mm)
160–300
Angle of inclination of the fixed point (‘θ’ in degree)
28–34
(1)
Determination of Steering Actuator Mounting Points …
y1 = d sin θ,
715
y2 = r sin(180 − β),
y3 = r sin(180 − (β − α)), y4 = r sin(180 + β + α)
(2)
AC = B D = L = r 2 + d2 − 2.r.d. cos(180 − (θ + β))
(3)
A C = L e = r 2 + d2 − 2.r.d. cos(180 − (θ + β + α))
(4)
r 2 + d2 − 2.r.d. cos(180 − (θ + β − α))
(5)
D B = Lc =
From the triangle A OC r d Le = = sin(180 − (θ + β + α)) sin(θ − ϕ − γ ) sin(β + ϕ + γ − α)
(6)
From the triangle AOC L r d = = sin(180 − (θ + β)) sin(θ − ϕ) sin(β + ϕ)
(7)
r d Lc = = sin(180 − (β + θ − α) sin(θ − ϕ + ψ) sin(β + ϕ − ψ − α)
(8)
sin(θ + β − α) sin(θ + β) − = sin(β − α) − sin(α) sin(θ − ϕ − γ ) sin(θ − ϕ)
(9)
sin(θ − ϕ + γ ) L +e = L −c sin(θ − ϕ + ψ)
(10)
Similarly,
tan(ϕ) =
d∗ sin θ − r ∗ sin β d∗ cos θ + r ∗ cos β
stroke length = (L e − L c )
(11) (12)
2.1 Flowchart The flowchart depicted in Fig. 3 explains the logic for the program code developed in MATLAB platform. This program determines the parametric values of cylinder arrangement for which the desired steering angle is obtained.
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Fig. 3 Logic employed for the MATLAB program developed
2.2 Results Observed from MATLAB Programming Based on the flowchart mentioned in Fig. 3, the MATLAB code has been developed, where the ranges for the steering arrangement parameters are varied as shown in Table 1. The set of the possible steering arrangement with parametric values for the desired range of MPSA (i.e., 58–60°) is shown in Table 2. MATLAB program provides the results for only MPSA for the steering cylinder mounting parameters. Only finding the MPSA is not sufficient for the optimum
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Table 2 Parametric values of steering arrangement (L, d, r and θ) for maximum range of MPSA L (mm)
d (mm)
r (mm)
θ (degrees)
Steering angle (α), (degrees)
650
585
165
28
58.8067
660
575
175
28
58.4201
660
580
170
28
59.1150
660
585
165
28
59.8538
650
585
165
30
58.8067
660
575
175
30
58.4201
660
580
170
30
59.1150
660
585
165
30
59.8538
650
585
165
32
58.8067
660
575
175
32
58.4201
660
580
170
32
59.1150
660
585
165
32
59.8538
650
585
165
34
58.8067
660
575
175
34
58.4201
660
580
170
34
59.1150
660
585
165
34
59.8538
performance, determining the minimum steering force is also required. Hence, 3D models are developed by varying each parameters of steering mounting points (L, d, r and θ ) based on the results obtained through MATLAB programming. The force analysis has been carried out using MSC ADAMS simulation software.
3 Force Analysis Using MSC ADAMS In this study, force experienced at the articulated joint is taken into account. This is because of two reasons. First one is the load acting on the machine is assumed constant, i.e., only machine self-weight is considered. So therefore, steering force required will be constant. Secondly, articulated joint is the weakest point on the machine, and therefore the dynamic force experienced at the articulated joint should be minimum. The force analysis has been carried out with the help of ADAMS environment, and all the real ground parameters acting on the vehicle when employed will be same. The following forces are considered for the simulation analysis as shown in Fig. 4: • Force due to machine weight, • Pressure acting across the cylinders according to the load (assumed constant load) and • Friction forces acting along with the tires (assumed constant friction force).
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Fig. 4 Schematic diagram showing forces acting on the LHD vehicle
Figure 5 explains the neutral and maximum steered position of the LHD vehicle in ADAMS simulation environment. The variation in the articulation angle with time and the force experienced at the articulated joint has been analyzed with each configuration.
4 Results and Discussion Combined results based on MATLAB program for MPSA and through ADAMS for force analysis have been discussed for better readability and understanding purpose. Mounting positions can be identified with the help of four parameters (L, d, r and θ ), and hence, four cases have been developed for the analysis part. In each case, one parameter is kept varying, and other three parameters are kept constant. The effect of variation of MPSA and force experienced at articulated joint with each parameter varying is discussed below in detail: Case (i) With varying theta (‘θ ’) (L = 650 mm, d = 585 mm and r = 165 mm are constant). Discussion Referring to Fig. 6, MPSA is independent of individual ‘θ ’ value, whereas the force at the articulated joint is having increasing nature with the increase in the value of ‘θ.’ The range of forces is also increasing with increasing value of ‘θ.’ Hence, based on the results, it can be concluded that the value of ‘θ ’ should be the possible minimum value for the better steering performance. The minimum possible value of theta is decided by the selection of ‘L,’ ‘d’ and ‘r’ values.
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Fig. 5 Simulation of LHD vehicle in ADAMS a neutral case and b steered to extreme right
Case (ii) With varying ‘d’ (L = 650 mm, r = 165 mm and θ = 28° are constant). Discussion By keeping all the other parameters constant, from Fig. 7, it can be observed that MPSA follows (approximately) second-order polynomial function by varying distance ‘d.’ But, the sum of angles ‘θ +β’ value is increasing linearly with increase
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Fig. 6 Variation of MPSA and force with varying ‘θ’
in distance ‘d.’ This is because the sum of ‘θ +β’ is dependent majorly on lengths ‘d’ and ‘r.’ This can be referred from Eq. 3. As the length of ‘d’ is increasing, sum of ‘θ +β’ value is also increasing. No significant impact on the force is experienced at the articulated joint with varying value of ‘d.’ Therefore, from the graph, it can be observed that with increase in the value of ‘d’ there is only a slight variation in the force and MPSA decreases with increasing value of ‘d.’ Case (iii) With varying ‘r’ (For L = 650 mm, d = 585 mm and θ = 28° are constant). Discussion By referring to Fig. 8, it can be observed that similar pattern is observed as that of the case with varying ‘d.’ Value of MPSA is decreasing with increasing value of ‘r’ but the sum of ‘θ +β’ value is increasing. But the slope of ‘sum (θ +β)’ is less compared to the slope of ‘sum (θ +β)’ with the earlier case. In case of force experienced at the articulated joint, it is decreasing with the increase in the value of ‘r.’ Higher value of
Fig. 7 Variation of MPSA and force with varying ‘d’
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Fig. 8 Variation of MPSA and force with varying ‘r’
‘r’ is useful for experiencing least force, and lower value of ‘r’ provides the higher MPSA value. Hence, there is a need to optimize the value of ‘r.’ Case (IV) With varying ‘L’ (For r = 165 mm, d = 585 mm and θ = 28° are constant). Discussion Referring to Fig. 9, sum of ‘(θ +β)’ value is decreasing with increase in value of ‘L.’ But the value of MPSA initially increases and achieves a maximum value, and thereafter, MPSA value decreases with increase in value of ‘L,’ whereas force experienced at the articulated joint behaves differently when compared to MPSA. With increasing value of ‘L,’ force range also increases which is not desirable. The value of length ‘L’ at which maximum value of MPSA is obtained experiences the least force. By observing the graph, the value of length L = 650 and at length L = 660, MPSA is attaining maximum value. But force experienced at L = 650 is less than L = 660. Further, by increasing the value of ‘L,’ force is increasing and MPSA is also decreasing which is not desirable. Hence, by considering both parameters, i.e., MPSA and force experienced at articulated joint, L = 650 favors the most desirable value of ‘L’.
5 Conclusion and Future Scope of Work 5.1 Conclusion Optimization of steering actuator mounting points of an LHD machine has been carried out using geometrical approach method. MPSA has been determined using MATLAB programming, and force experienced at the articulated joint has been analyzed using ADAMS simulation environment. The optimum steering performance of the LHD machine with respect to steering actuator mounting points is the points
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Fig. 9 Variation of MPSA and force with varying ‘L’
which can provide higher MPSA value and lower value of the force experienced at the articulated joint. Based on the results, it can be concluded that: • The MPSA obtaining is independent of the value ‘θ ’ but the force experienced at the articulated joint increases with increasing value of ‘θ ’. • The value of MPSA obtained decreases with the increase in value of ‘d,’ whereas there is only a slight change or no significant change on the force experienced at the articulated joint with increase in value of ‘d.’ • The value of MPSA with varying ‘r’ follows the same pattern as that of the ‘d,’ i.e., the value of MPSA obtained decreases with increasing value of ‘r.’ However, the force experienced at the articulated joint also decreases with the decreasing value of ‘r’ which is desirable. • With the varying value of ‘L,’ MPSA initially increases and reaches a maximum value for certain value of ‘L,’ and thereafter, MPSA decreases with increasing value of ‘L.’ However, force curve increases with increasing value of ‘L.’ • In the above-considered configuration, the MPSA value achieved is around 58°, and force experienced at the articulated joint is minimum for L = 650, d = 585, r = 165 and θ = 28°, respectively, the mentioned steering actuator parameter values. • The increase in MPSA is approximately 28% when compared with the existing steering arrangement of the LHD machine.
5.2 Future Scope of Work This current work can be extended in future in following areas: • Full dynamic analysis of the LHD model is to be carried out in the MSC ADAMS environment to determine the actual steering force instead of force experienced at the articulated joint.
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• The results obtained from the dynamic analysis of the LHD vehicle need to be validated analytically or experimentally.
References 1. Zhu Z, Du Y, Li H, Song Z (2015) Multidisciplinary design optimization method for steering system of large wheeled harvester. In: 2015 International conference on intelligent systems research and mechatronics engineering. Atlantis Press 2. Lei-yan Y, Ping-li Y, Feng L (2009) Research on dynamics modeling and genetic algorithm optimization of automobile steer-by-wire system. In: 2009 Second international conference on information and computing science, vol 3. IEEE, pp 78–81 3. Lei D, Kaiping D, You-ping G (2011) The optimum design for pivot points of steering mechanism on HT25J wheel loader. In: 2011 International conference on consumer electronics, communications and networks (CECNet), IEEE, pp 2760–2763 4. Thulasiraman BKS, Arumugam G, Sadali NG, Neelamegan I (2013) Steering linkage optimization of articulated construction equipment. In: Proceedings of the 1st international and 16th national conference on machines and mechanisms (iNaCoMM2013), vol 18, no. 20. pp 12 5. Xing L, Zhou Chen Y (2012) Wheel loader working device optimization design based on the sensitivity analysis. Adv Mater Res 479:1745–1749. Trans Tech Publications 6. Tian Xu J, Ling Lu Y (2015) Energy control and optimization of hybrid load-haul-dump. Appl Mech Mater. 727:729–732. Trans Tech Publications 7. Yin Y, Rakheja S, Yang J, Boileau P-E (2018) Design optimization of an articulated frame steering system. Proc Instit Mech Eng Part D: J Automobile Eng 232(10):1339–1352 8. Chen Z, Liu X, Xu F (2018) Design optimization of steering mechanisms for articulated off-road vehicles based on genetic algorithms. Algorithms 11(2): 22 9. Bing-wei C, Liu X, Chen W, Zhang Y, Li A-M (2019) Depth optimization analysis of articulated steering hinge position based on genetic algorithm. Algorithms 12(3):55
Dynamic Analysis of Helicopter Boom with Different Payload Configurations R. Gopikrishna, K. Kishore Kumar, and Y. R. Janarthanan
Abstract The armament layout of every military helicopter is decided by the aviation department based on a variety of factors. Amidst a variety of other constraints, the structural behavior plays a vital role in deciding the layout on the helicopter boom. It is necessary to avoid interaction of the rotor blade frequency with that of the natural frequency of the weapon system layout. Vibration analysis was carried out for various configurations, to determine the natural frequencies which are close to that of the rotor blade and therefore avoid them during flight. The finite element analysis followed by impact hammer tests was carried out to confirm the configurations. A passive damping technique is suggested to be adopted for attenuation of the vibrations due to resonance, if any. Keywords Dynamics · Helicopter boom · Vibration testing · Damping
1 Introduction The dynamic behavior of any vehicle is a major design criterion of interest for every structural designer. The vibrations of the structure have mostly led to failure of the structure or its functionality during operation. A military helicopter is no exception. Additionally, vibrations are even more critical owing to the frequency of the rotor blades of the helicopter. The dynamics of the structure of the helicopter is not sufficient enough to ensure the non-interaction of the structure with the rotor blade frequency. The arrangement of weapons on the boom of the helicopter is also a major concern, as it may exhibit a natural frequency close to that of the rotor blade. The helicopter boom is a slender tubular structure attached to the helicopter body, on to which the different weapon systems are mounted during flight. The helicopter’s arsenal can be chosen from a wide variety of weapons including air-to-air missiles, armed projectiles, anti-ground missiles, anti-tank missiles, anti-ship missiles, etc. The functioning of the weapon’s detecting unit is largely dependent on the structural R. Gopikrishna (B) · K. Kishore Kumar · Y. R. Janarthanan Defence Research and Development Laboratory, Kanchanbagh, Hyderabad, Telangana 500058, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_71
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dynamics of the helicopter. The visual output of an imaging device can be corrupted due to vibrations of the boom during flight. It is therefore necessary to verify the dynamic response of the helicopter for various weapon configurations on the boom. Similar exercise may be trivial for an aircraft as it does not involve the frequency of the rotor blade. The resonance behavior can be controlled using various damping techniques, provided we have the liberty to change the design of the helicopter boom or the helicopter itself.
2 Helicopter Boom and Its Arsenal The helicopter boom is a hollow tube with a length-to-outer diameter ratio (L/D) of 20 and outer diameter to thickness ratio (D/t) of 44. The boom is running through the body of the helicopter and supported to the cockpit of the helicopter, and the distance between the supports is 25% of the boom length. Additionally, two links connect the boom to the external helicopter fuselage body on either side. The schematic of the boom with the links and weapon stations is shown in Fig. 1. The helicopter boom has four stations along its length, where the armament is attached, termed as left-hand in-board (LH IB), left-hand out-board (LH OB), right-hand in-board (RH IB), and right-hand out-board (RH OB). The helicopter arsenal includes airto-air missile (ATAM), anti-tank missile (ATM), projectiles with ammunition and anti-radiation missiles (ARM). Each station can hold a launcher with two anti-tank missile or a launcher with two ATAMs or an ARM or a round of armed projectiles. The weapons can be arranged in the stations in any of the permissible configurations as recommended by the aviation department. It is therefore necessary to determine the natural frequency of all these permissible configurations, both analytically and experimentally.
Fig. 1 Schematic of boom with links and weapon stations
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3 Configurations of Weapons Systems and Their Dynamic Behavior A finite element analysis (FEA) was carried out to determine the natural frequency of various weapon configurations [1]. The weapon stations may hold empty ATM launcher, launcher with 1 ATM, launcher with 2 ATMs, launcher with projectile or empty projectile launcher in different combinations. A modal analysis was carried out using a commercial FEA software (ANSYS 17), considering the armament at each station as a lumped mass in a beam model of the helicopter boom. The attachment points of the boom with the helicopter and that of the links with the helicopter are idealized to be simply supported, as rotations of the boom and links are allowed at these locations. A beam model of the boom with the links and supports is shown in Fig. 2. The mass, center of gravity, and moment of inertias of different weapon systems are given as input to the FE software. The results of modal analysis of various configurations are given in Table 1. The first mode is of primary concern as the higher modes are far away from the rotor blade frequency. Interestingly, few of the configurations were found to have a natural frequency very close 20 Hz, the frequency of the rotor blades. Figure 3 shows the mode shape of configuration 7, which has empty in-boards, launcher with 2 ATMs at out-board LH and empty projectile launcher at out-board RH. A harmonic analysis was carried out for a frequency sweep of 0 25 Hz to verify resonance 20 Hz, on either side of the boom. The response of the boom is captured at the origin of the boom as well as between the armament points on either side of the origin. Figure 4 shows the response at the origin and on the boom, respectively. It is seen that effect of resonance is prominent on the stations between the armaments, while the origin does not experience high amplitude vibrations. Impact hammer tests were carried out on various configurations of weapons mounted on the helicopter boom, with the helicopter.
Fig. 2 Beam model with lumped mass and boundary conditions
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Fig. 3 Mode shape of first mode of configuration 7
Fig. 4 FRF at different nodes from harmonic analysis (0–25 Hz)
4 Impact Hammer Test Experimental determination of the natural frequencies of various weapon configurations on the helicopter boom was carried out using an impact hammer test. One accelerometer each was positioned on the port side and star board side of the helicopter boom. A modal testing hammer was used to give the short duration excitation input to the system, for each weapon configuration [2]. The natural frequency recorded was compared with the analytical frequency of the boom. The experimental frequencies (vertical direction) are given in Table 1. As observed from the analytical solution, the experimental data verifies the configurations which operate close to the rotor blade frequency of the helicopter 20 Hz. If flight trial is carried out in one of these configurations, resonance is expected to occur leading to high grms levels, greater than the acceptable limits of the onboard weapon system.
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Table 1 Natural frequencies of various configurations Configuration FE prediction Exp. result Configuration FE prediction (Hz) (Hz) (Hz) 1 2 3 4 5 6
17.29 17.37 8.85 8.18 32.6 31.1
18.9 19.2 9.3 8.6 34.1 33.0
7 8 9 10 11 12
18.87 24.53 12.94 11.67 20.17 32.77
Exp. result (Hz) 19.6 26.3 14.2 15.1 20.9 34.6
5 Post Flight Data In order to verify the phenomenon of resonance, a captive flight trial of the helicopter with the weapon systems in configuration 7 was carried out. On-board triaxial accelerometers were positioned on the port and star board sides to pick up the amplitude during flight. The PSD data of the port side accelerometer in the vertical direction captured a grams of 2.01 during the captive flight. Furthermore, the seekers of the on-board missiles were not able to detect the target due to high grms levels. Consecutively, captive trials were also carried out in configuration 5 and the measured maximum grms level was 0.65. Therefore, it is always advisable to fly with weapon configurations with frequencies away 20 Hz.
6 Sensitivity Analysis The easiest solution to avoid resonance during flight is to fly in configurations not close 20 Hz. However, the weapon system comprises of various missiles and armed projectiles and controlling the sequence of release at all times may not be feasible. It is hence advisable to consider damping options, even if the helicopter boom enters resonance with the main rotor blade. Therefore, the effect of structural damping was studied by carrying out a damping analysis using finite element software. The effect of amplitude of vibration on the LH side of the boom has been plotted in Fig. 5. It is observed that a 0.8% structural damping causes an 80% reduction in the amplitude of vibration. The damping can be achieved using passive or active techniques.
7 A Passive Damping Solution Stiffening the boom or the links to the helicopter shall help in altering the natural frequency of the structure. However, owing to the constraints on the modifications
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Fig. 5 Effect of structural damping on the amplitude of vibration of LH side
of the existing helicopter design, this solution was not feasible to be carried out. The alternate solution is to dampen the resonant vibrations. Viscoelastic materials are commonly used as a passive damping solution for vibration and noise control in structures. The structure is impregnated with layers of viscoelastic material to achieve damping. The layer treatment enhances damping resulting in control of excessive resonance vibrations. Layered damping may be constrained or free layer damping. Mathematical models are available for characteristics of free, constrained, and partial constrained layer damping [3, 4]. From experimental results, it was found that constrained layer damping provides higher attenuation [5]. This technique can be employed in the design of the helicopter boom and link rods, to dampen the high amplitude vibrations, if any, during flight.
8 Conclusion The configuration of armaments attached to the helicopter therefore plays a vital role in the performance of the weapon system. It is necessary to consider the rotor blade frequency in design of the helicopter structure as well as the attachment of weapon systems to it. Passive damping techniques may also be considered during the design of the helicopter structure.
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References 1. Heylen W, Lammens S, Saus P (1997) Model analysis theory and testing. Department of Mechanical engineering, Ku Leuven 2. Avitable P (2018) Modal testing a practitioner’s guide. University of Massachusetts, Lowell 3. Khalfi B, Ross A (2013) Transient response of a plate with partial constrained viscoelastic layer damping. Int J Mech Sci 68(1):304 4. Bangarubabu P, Kishore Kumar K, Krishna Y (2012) Damping effect of viscoelastic material on sandwich beams. In: International conference on trends in industrial and mechanical engineering, pp 171–173 5. Kishore Kumar K, Krishna Y, Bangarubabu P (2015) Damping in beam using viscoelastic layers. J Mater: Des Appl 229(2):117–125
Characterization of Composites Made with In-House Prepregs at Different Curing Cycle Piyush Sute, P. R. Krishna Mohan, M. Anil Kumar, P. M. Mohite, and Mahesh
Abstract The main objective of this work is to improve the properties of 90° carbon/epoxy laminates made with prepregs B-staged at different curing cycles. The effect of prepreg curing at 80 °C for 15 min and 100 °C for 5 min on the transverse modulus was analyzed for the epoxy system used. A series of tensile test were conducted to study the strength of material. Simultaneous Differential scanning calorimetry and Thermogravimetric analysis (SDT) is used to find the fiber volume fraction of the laminate. Void volume fraction was calculated by using ASTM D2734. Keywords 90° carbon/epoxy laminates · Transverse modulus · Fiber volume · Void content · Strength and stiffness
1 Introduction Composites are used in several industries because of its low weight and good mechanical properties like high strength, high stiffness, no corrosion, and good aging. Apart from that, optimizing weight and easy processing are another advantages of prepregs. The major reason behind the selection of prepreg for any applications is its cost and performance. Kulkarni [1] has worked in modifying the prepreg making machine to improve the properties of the unidirectional prepreg laminate and in the estimation of fracture toughness. The quality of the prepreg laminate has been checked by performing the chemical and mechanical test by Ghiorse et al. [2]. It was reported that the epoxy system used is highly sensitive to moisture, and the decrease in service temperatures. Krishna Mohan et al. [3] have conducted a series of modification and experimentation in the prepreg making machine to improve the quality of inhouse prepregs. Piyush et al. [4] have studied the effect of temperature on B-stage
P. Sute (B) · P. R. Krishna Mohan · M. Anil Kumar · P. M. Mohite Indian Institue of Technology, Kanpur, UP 208016, India e-mail: [email protected] Mahesh Punjab Technical University, Sector 14, Chandigarh 160014, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_72
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curing of prepregs and reported tensile characterization of 0° laminate made of inhouse unidirectional carbon/epoxy prepreg. The current study is the extension of the previous work. The primary objective of this study is to improve the properties of 90° carbon/epoxy laminates made with prepregs B-staged at two different curing cycles, i.e., (1) at temperature of 80 °C for 15 min and (2) at temperature of 100 °C for 5 min. Failure mechanism is analyzed by microscopy test after conducting the tensile test.
2 Material Making 2.1 Prepreg Making Process The prepreg making process involves matrix impregnation into the carbon fiber. Diglycidyl ether of bisphenol-A (DGEBA) as epoxy resin (LY556), methyl tetra hydrophthalic anhydride (MTPHA) as hardner, 2,4,5-tris [(dimethylamino)methyl]phenol (DMP-30) as accelerator and TC36s, 12 k grade carbon fiber are the constituents of the prepreg. Few modifications on the in-house prepreg making machine were done by Krishnamohan et al. [3], and prepreg sheets are made using the same. Prepreg sheet can be seen in Fig. 1. Fig. 1 Prepreg sheet
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Fig. 2 Heat is provided to prepreg sheet by silicon thermal pads
2.2 B-Stage Curing B-staging is introduced to improve the quality and property of the prepreg. B-stage curing cycle has been chosen by considering drap and tacky as the parameters. So, B-stage curing has been conducted at following curing cycles: (1) 80 °C for 15 min and (2) 100 °C for 5 min. Temperature for B-staging is provided uniformly to the prepreg sheets by the silicon thermal pads as shown in Fig. 2. Later, the prepreg is stored in freezer which is maintained at −18 °C to preserve the properties of the material.
2.3 Laminate and Test Specimen The prepreg sheet was cut in a required dimensions, and hydraulic press machine is used to make laminate. Laminate is made by adapting [3] the curing cycle of 85 °C for 3 hr and followed by 140 °C for 12 hr under 0.7 MPa pressure as shown in Fig. 3. Water jet machine is used, and tabs are applied to the specimen as shown in Fig. 4. Tensile test of 90° specimens is performed according to ASTM 3039 [5] as shown in Table 1. Fig. 3 90° laminate
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Table 1 Specimen dimension according to ASTM 3039 Layup
Length (mm)
Width (mm)
Thickness (mm)
Ga ge length (mm)
Tab material
Tab length (mm)
Tab thickness (mm)
0o
250
15
1
50
Al
56
1.5
90o
175
25
2
50
Al
25
1.5
3 Experimentation 3.1 Fiber and Void Volume Fraction Fiber and void volume fraction plays important role in mechanical behavior of laminates. To find the volume fraction of the composite, Eqs. (1) and (2) are used, and thermogravimetric analysis (TGA) ‘SDT Q600 apparatus is used by burning the matrix for this purpose. vf = wf × ρc =
wf ρf
ρc × 100 ρf
(1)
1 +
(2)
wm ρm
where vf is the total volume of fiber, ρ f and ρ m are the density of fiber and matrix, ρ c is experimental density of composite, and wf and wm are weight fraction of the fiber and matrix, respectively. The composite manufactured with void content below 1% is treated as the well fabricated composite [6]. To find void volume fraction, Eq. (3) is used, where ρ c is theoretical density of the material, and ρ e is the experimental density of the material. Weight versus temperature relation is obtained from TGA ‘SDT Q600 apparatus and the same can be seen in Fig. 5. Fiber content and void volume are reported in Table 2. vv =
ρc − ρe × 100 ρc
(3)
3.2 Tensile Testing A series of testing were performed on Instron-1195 universal testing machine, and extensometer is used to know the level of improvement in the properties of material. Crosshead rate was chosen 0.5 mm/min, and properties of material are calculated
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Fig. 4 Tensile specimen
Table 2 Fiber volume fraction
S. no.
Sample
Fiber content (%)
Void volume (%)
1
80 °C/15 min
35.72
0.86
2
100 °C/5 min
60.76
0.92
from data provided by the machine. Elastic modulus and strength are obtained from the stress–strain data as shown in Figs. 6 and 7.
4 Result and Discussion 4.1 Tensile results Figures 6 and 7 shows the strength and modulus of the laminates made from different B-staged prepreg material and the same are tabulated in Table 3. Three samples for both the cases are tested to check the repeatability of the result. Maximum and minimum value of the modulus of laminate with B-stage of 80 °C/15 min are found to be 24.75 GPa and 17.77 GPa, respectively, with very low value of variation 3.59 GPa. Similarly, strength values of B-staged 80 °C/15 min are observed in between 58.8 MPa and 39.4 MPa with variation of 10.97 MPa. For B-staged 100 °C/5 min case, maximum and minimum modulus values are observed to be 10.54 GPa and 9.39 GPa, respectively. Variation in modulus is 0.76
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a
80 oC/15min
100o C/5min
b
Fig. 5 Weight dropping versus temperature relation for a 80 °C/15 min and b100 °C/5 min
GPa, while minimum strength value of 16.95 MPa and maximum value of 36.56 MPa is observed, with variation of 10.27 MPa.
4.2 Comparison of Strength and Modulus with Available Literature From the work of Krishnamohan et al. [3] and Reddy et al. [7], the strength and modulus of 90° carbon/epoxy laminate can be seen. However, the fiber content of
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Fig. 6 Stress–strain data for 80 °C/15 min
Table 3 Strength and modulus values of 90° layup for different B-stages Sample
80 °C/15 min Modulus
100 °C/5 min Modulus
Strength (MPa)
Modulus (GPa)
Strength (MPa)
Modulus (GPa)
1
40.2
17.77
36.56
9.105
2
58.8
24.75
21.45
9.39 10.54
3
39.4
19.77
16.95
Average
46.133
20.76
24.98
9.68
Standard deviation
10.97
3.59
10.27
0.76
these laminates was different. A tabular and graphical comparison of these values with the current work has presented in Table 4 and shown in Fig. 8, respectively. From Fig. 8, it can be seen that among all the cases, the laminate made from 80 °C/15 min B-staged prepreg has significantly good transverse strength and modulus. Table 4 Comparison of strength and modulus of laminate made of in-house prepreg with present literature
Specimen
Strength (MPa)
Modulus (GPa)
80 °C/15 min
46.133 ± 10.97
20.76 ± 3.59
100 °C/5 min
24.98 ± 10.27
9.68 ± 0.76
Reddy et al. [7]
11.25
8.12
Krishnamohan et al. [3]
8.36 ± 1.64
–
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Fig. 7 Stress–strain data for 100 °C/5 min
4.3 Failure Analysis Using Microscope Using Zeiss optical microscope, the failure mechanism of the material is studied. Failure mechanism is primarily due to the delamination and matrix breaking. A typical image of failure is shown in Fig. 9.
5 Conclusions The main objective of this work is to improve the properties of 90° carbon/epoxy laminates made with in-house prepregs B-staged at two different curing cycles. The effect of prepreg curing cycle on the transverse modulus of the laminate is investigated. The main observation of this study is as follows: (1) (2) (3)
(4)
The fiber volume fraction of laminate was found to be 35.72% for 80 °C/15 min case and 60.72% for 100 °C/min. The void volume fraction is between 0.86 and 0.92% which is considerably good. B-stage curing of prepreg effects the mechanical properties of material. Among all the investigated and literature reviewed cases of 90° layup, the laminate made with 80 °C/15 min B-staged curing has better mechanical properties with strength and modulus of 46.133 MPa and 20.76 GPa, respectively. The failure mechanism is primarily due to delamination and matrix cracking.
The qualitative and quantitative analysis was performed to reveal the goodness of manufacturing process adopted.
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Fig. 8 Graphical comparison of laminate with present literature a Modulus and b Strength
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Fig. 9 90° failed specimen microscopy image a Front view and b Side view
References 1. Feraboli Kulkarni AP (1995) Modifications in unidirectional prepreg making machine and characterization of products. Master’s thesis, IIT Kanpur 2. Ghiorse SR, Foley GE, Zukas WX (1990) Pilot prepreg line: a comprehensive analysis of prepreg quality, 28 3. Krishnamohan PR, Kumar AM, Goutham, Mohite PM (2018) Development of In-house unidirectional carbon/epoxy prepregs and its characterization for aerospace applications. In: Proceeedings of SICE 2018. 4. Prakash SP, Krishnamohan, Kumar MA (2019) Effect of temperature on B-stage curing of prepregs and characterization of composites. In: Sixth international conference on recent advances in composite material, ICRACM-2019, February 25–28 5. ASTM D3039/D3039M-14 (2014) Standard test method for tensile properties of polymer matrix composite materials.Standard, ASTM International, West Conshohocken, PA 6. Mechanical testing of advanced fiber composites In: Hodgkinson JM (ed) Wodhead Publishing Limited 7 Reddy CV, Babu PR, Ramnarayanan R, Da D (2017) Mechanical characterization of unidirectional carbon and glass/epoxy reinforced composites for high strength applications. Mater Today: Proc 4(2):3166–3172
A Parametric Approach to Detect Isomorphism and Inversion in the Planar Kinematic Chains Kunal Dewangan
and Arvind Kumar Shukla
Abstract This work deals with the problem of detection of isomorphism and inversions among planar kinematic chains. To detect isomorphism and inversions a least distance matrix (LDM) is formed using the values on the basis of the parameter. Isomorphism is detected on the basis of chain string (CS) of least distance matrix (LDM). An inversion is generated by grounding a different links in the kinematic chain. Here, link string (LS) is generated for the different kinematic inversions of the chains so that we can detect kinematic inversions with the help of link string (LS). Keywords Isomorphism · Inversion · Kinematic chain · Least distance matrix
1 Introduction The main objective of this paper is to propose and develop a general method for detection of isomorphism and inversions among planar kinematic chains. Two planar kinematic chains are said to be isomorphic to each other if their type of links, link assortment and connectivity of links is similar. An inversion is created by grounding a different link in the kinematic chain. However some of the inversions may be structurally identical depending on the type of link fixed. The detection of distinct inversions is as important as detection of distinct kinematic chains, since various distinct inversions will produce so many structurally distinct mechanisms for use. Isomorphism in kinematic chains has been the subject of intensive research, but detection of distinct inversions has received somewhat lesser attention. Isomorphism or topological equivalence of planar kinematic chains has been studied, by Zeng et al. [1] uniquely represented kinematic chain by a graph, and a fast deterministic algorithm called the Dividing and Matching Algorithm (DMA) is proposed. Rai [2] proposed binary code method in which link labeling algorithm is used to detect isomorphism of kinematic chains. This method is test for simple and multiple joints also in epicyclic K. Dewangan (B) Kalinga University, Naya Raipur 492101, Chhattisgarh, India A. K. Shukla Lakshmi Narain College of Technology, Bhopal 462021, Madhya Pradesh, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_73
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gear trains for its efficiency. Kamesh et al. [3] presented graph theory to detect isomorphism in planar and geared kinematic chains. Mruthyunjaya [4] started with a multiple jointed binary chain and transformed them gradually in stages until all the joints became simple joints. Rai and Panjabi [5] proposed work with different structural invariants, i.e., primary structural invariants for isomorphism identification and secondary structural invariant provides power transmission and transmission efficiency. Sun et al. [6] joint-joint matrix proposed to describe the kinematic chain, which can uniquely represent the kinematic chain structure. The links and joints information were extracted from joint-joint matrix and the link code and joint code were use to represent the link and joint attributes isomorphism detection was done by comparing links, joints and matrices. Sun et al. [7] proposed improved hamming number method to detect isomorphism for multiple joints kinematic chains. Jointjoint matrix is used to represent kinematic chain uniquely then improved hamming number matrix is obtained by improved hamming number method. Joint and chain hamming number and joint hamming string are used to determine isomorphism. Shukla and sanyal [8] proposed a gradient concept used to identify isomorphism and inversion of kinematic chains. Gradient analogy is used to prepare gradient matrix in that first gradient chain string and second gradient chain string is formed is used for identification of isomorphism and inversion. The closest any method has come so far in terms of directness of approach is the contracted link adjacency matrix method proposed by Hwang and Hwang [9].
2 Methodology The method involves a parametric approach for detection of isomorphism and inversions. The method is explained using Watt and Stephenson’s chain. A least distance matrix (LDM) is formed using the values on the basis of the parameter. The following terminologies have been taken from “prajapati et al.” [10] (1)
Node Value—A node value is defined as the ratio of the number of parameters at particular node and the number nodes node at particular link. Generalized equation for node value will be NV =
(n − 1) n
(1)
n = number of node in each link (Fig. 1). (2)
Joint value (JV)—A joint value is defined as a sum of node connecting at a particular joint in the planar kinematic chain (Fig. 2).
(3)
Least distance matrix (LDM)—A least distance matrix is a square matrix and it is formed on the basis of a sum of least distance between joint values of the two links.
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Fig. 1 Node values of all possible links, a Binary link, b Ternary link, c Quaternary link, d Pentnary link, e Hex nary link
Fig. 2 Joint values of all possible combination of links
(4)
(5) (6)
(7)
Link value (LV)—A link value is the sum of the elements of each row of least distance matrix (LDM). It will give a particular value and this value is termed as link value (LV) for that particular link. Link string (LS)—A link string refers to elements of a particular row (or column) of least distance matrix (LDM) taken in ascending order. Chain string (CS)—A chain string is the sum of all the link value (LV) of a particular planar kinematic chain along with all the elements of the least distance matrix will give the chain string (CS) for that particular planar kinematic chain. Total chain value (TCV)—A total chain value is the sum of the all elements of least distance matrix.
Watt chain is shown in Fig. 3 Link A & D are ternary link and link B, C, D & E are binary link. Each node of binary link is assigned a node value 1/2 and for ternary
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Fig. 3 Watt chain
link joint value 2/3 is assigned. Also quaternary link and pentnary link had assigned the node values 3/4 and 4/5 respectively. For simplicity of calculations, a non-fractional value is assigned to each joint value on the basis of taking least common multiple of joint values by which can be taken same as the denominator for all joint value for various links connectivity. A least distance matrix (LDM) for Fig. 3 is given below:
As shown in the matrix, for Watt chain, ternary link A is connected to binary link B and the node value is assigned “7”. For connectivity between link A and link C a shortest distance is considered i.e., 7 + 6 = 13. Similarly for connectivity between link A and link D the joint value will be “8” and so on. A 6*6 matrix will be formed for a 6 link watt chain. Link string (LS) for link A, B, C, D, E & F are—[0, 2 (7), 8, 2 (13)], [0, 6, 7, 13,14,20], [0, 6, 7, 13, 14, 20], [0, 2 (7), 8, 2 (13)], [0, 6, 7, 13, 14, 20] and [0, 6, 7, 13, 14, 20] respectively. Watt chain as shown in Fig. 3 the link value (LV) for link A, B, C, D, E & F are—48, 60, 60, 48, 60, and 60 respectively and chain string (CS) for Watt chain are 336, 2 [2 (6), 4 (7), 8, 6 (13), 4 (14), 2 (20)].
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3 Isomorphism A method based on the “parameter” Concept is presented to test isomorphism among planar kinematic chains and kinematic inversions. The method is unique in the sense that the planar kinematic chains and numerical strings presented represent the planar kinematic chains uniquely. At this stage of research, though it is necessary to develop new and efficient methods to test isomorphism, it is desirable that such methods should also be able to reveal topological characteristics of the chains and inversions. This paper attempts to fulfill this need. Thus at a node joining a binary link and binary link, binary link and ternary link a node value will be assigned as 6/6, 7/6 (2/3, joint value of ternary link and 1/2, joint value of binary link) as shown in Fig. 4. A least distance matrix (LDM) for Fig. 4 is given below:
As shown in the matrix, for Stephenson chain, ternary link A is connected to binary link B and the node value is assigned “7”. For connectivity between link A and link C a shortest distance is considered i.e., 7 + 7 = 14 & connectivity between link A and link D a shortest distance is considered i.e., 7 + 6 = 13. Similarly connectivity between link A and link E the joint value will be “7” and so on. A 6*6 matrix will be formed for a 6 link Stephenson chain. Fig. 4 Stephenson chain
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Fig. 5 Eight link one degree of freedom kinematic chain
Link string (LS) for link A, B, C, D, E & F are—[0, 3(7), 13, 14], [0, 2(7), 3(14)], [0, 3(7), 13, 14], [0, 6, 7, 13,2(14)], [0, 6, 7, 13,2(14)], [0, 2(7), 3(14)] respectively. Stephenson chain as shown in Fig. 4 the link value (LV) for link A, B, C, D, E & F are—48, 56, 48, 54, 54 and 56 respectively. Also the sum of all the link values (LV’s) of a Stephenson chain will give the chain string (CS) is 316, 2[6, 6(7), 2(13), 6(14)]. Watt chain as shown in Fig. 3 the link value (LV) for link A, B, C, D, E & F are—48, 60, 60, 48, 60, and 60 respectively and chain string (CS) for Watt chain are 336, 2[2(6), 4(7), 8, 6(13), 4(14), 2(20)]. Watt chain Fig. 3 and Stephenson chain Fig. 4 have a different chain string (CS) hence they are non-isomorphic. A least distance matrix (LDM) for Fig. 5 is given below:
Link value (LV) for link A, B, C, D, E, F, G, H & I are—176, 224, 224, 176, 176, 176, 224 & 176 respectively and chain string (CS) for eight link one degree of freedom kinematic chain are 1600, 2[2(12), 4(14), 2(16), 4(26), 4(30), 2(32), 4(42), 2(44), 2(56)]. A least distance matrix (LDM) for Fig. 6 is given below:
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Fig. 6 Eight link one degree of freedom kinematic chain
Link value (LV) for link A, B, C, D, E, F, G, & H are: - 164, 164, 200, 200, 164, 164, 200 & 200 respectively and chain string (CS) for eight link one degree of freedom kinematic chain are 1456, 2[2(12), 4(14), 4(16), 4(26), 8(30), 2(32), 4(44)]. Eight link one degree of freedom chain as show in Fig. 5 the link value (LV) for link A, B, C, D, E, F, G, H & I are—176, 224, 224, 176, 176, 176, 224 & 176 respectively and chain string (CS) for eight link one degree of freedom kinematic chain are 1600, 2[2(12), 4(14), 2(16), 4(26), 4(30), 2(32), 4(42), 2(44), 2(56)]. Eight link one degree of freedom chain Figs. 5 and 6 chain have different chain string (CS) hence they are non isomorphic. Ten link one degree of freedom kinematic chain is shown in Fig. 7. Link A, D, E, H, I & J are ternary link and link B, C, F & G are binary link. Each node of binary link is assigned a node value “1/2” and for ternary link joint value “2/3” is assigned. Also quaternary link and pent nary link had assigned the node values 3/4 and 4/5 respectively. Thus at a node joining a binary link and binary link,binary link and ternary link and ternary link and ternary link a node value will be assigned as 60/60,70/60,80/60 (2/3, joint value of ternary link and 1/2, joint value of binary link) as shown in Fig. 7. A least distance matrix (LDM) for Fig. 7 is given below:
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Fig. 7 Ten link one degree of freedom kinematic chain
Link value (LV) for link A, B, C, D, E, F, G, H, I & J are—1280, 1500, 1500, 1280, 1280, 1500, 1500, 1280, 1320 & 1320 respectively & chain string (CS) for ten link one degree of freedom kinematic chain are 13,760, 2[2(60), 4(70), 7(80), 4(130), 8(150), 6(160), 3(210), 2(220), 2(230), 2(240), 2(280)]. A least distance matrix (LDM) for Fig. 8 is given below: Fig. 8 Ten link one degree of freedom kinematic chain
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Link value (LV) for link A, B, C, D, E, F, G, H, I & J are—1300, 1580, 1500, 1200, 1200, 1220, 1220, 1300, 1580 & 1500 respectively & chain string (CS) for Ten link one degree of freedom kinematic chain are 13,760, 2[2(60), 4(70), 7(80), 4(130), 8(150), 8(160), 4(210), (220), 4(230), (300)]. Ten link one degree of freedom chain as show in Fig. 7 the link value (LV) for link A, B, C, D, E, F, G, H, I & J are—1280, 1500, 1500, 1280, 1280, 1500, 1500, 1280, 1320 & 1320 respectively & chain string (CS) for ten link one degree of freedom kinematic chain are 13,760, 2[2(60), 4(70), 7(80), 4(130), 8(150), 6(160), 3(210), 2(220), 2(230), 2(240), 2(280)]. Ten link one degree of freedom chain Figs. 7 and 8 chain have different chain string (CS) hence they are non isomorphic. Consider Counter Example appeared in Refs. [4] and [9] planar kinematic chains shown in Figs. 9, 10, 11, 12, 13 and 14 have a different chain string (CS) hence they are non-isomorphic. A least distance matrix (LDM) for Fig. 9 is given below: Fig. 9 Ten link one degree of freedom planar kinematic chain
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Fig. 10 Ten link one degree of freedom planar kinematic chain
A least distance matrix (LDM) for Fig. 10 is given below:
Chain string (CS) for Fig. 9 = 2[60, 6(70), 2(75), 2(80), 2(85), 2(130), 3(140), 145, 5(150), 3(155), 5(160), 2(200), 210, 3(220), 225, 5(230), 290]. Chain string (CS) for Fig. 10 = 2[60, 6(70), 2(75), 2(80), 2(85), 130, 135, 4(140), 145, 7(150), 2(155), 3(160), 170, 200, 205, 2(210), 4(220), 2(230), 240, 280].
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Fig. 11 Twelve link one degree of freedom planar kinematic chain
Ten link one degree of freedom planar kinematic chain Figs. 9 and 10 have a different chain string (CS) hence they are non isomorphic. A least distance matrix (LDM) for Fig. 11 is given below:
A least distance matrix (LDM) for Fig. 12 is given below:
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Fig. 12 Twelve link one degree of freedom planar kinematic chain
A least distance matrix (LDM) for Fig. 13 is given below:
A least distance matrix (LDM) for Fig. 14 is given below:
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Fig. 13 Twelve link one degree of freedom planar kinematic chain
Chain string (CS) for Fig. 11 is 2[8(70), 8(80), 7(140), 10(150), 9(160), 7(210), 5(220), 4(230), 3(240), 3(290), 2(300)]. Chain string (CS) for Fig. 12 is 2[8(70), 8(80), 7(140), 10(150), 9(160), 3(210), 9(220), 8(230), 3(290), (300)]. Chain string (CS) for Fig. 13 is 2[8(70), 2(75), 4(80), 2(85), 6(140), 2(145), 8(150), 2(155), 5(160), 1(170), 2(210), 10(220), 2(225), 5(230), 4(290), (295)]. Chain string (CS) for Fig. 14 is 2[8(70), 2(75), 4(80), 2(85), 6(140), 2(145), 9(150), 2(155), 5(160), 1(170), 6(210), 8(220), 4(230), 2(225), 4(290), (295)]. Twelve link one degree of freedom chain 11, 12, 13 and 14 chain have different chain string (CS) hence they are non isomorphic. Twelve link one degree of freedom planar kinematic chain Figs. 11, 12, 13 and 14 have a different chain string (CS) hence they are non-isomorphic. Nine link two degree of freedom chain is shown in Fig. 15. Link C, D, E & I are ternary link and link A, B, F, G & H are binary link. Each node of binary link
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Fig. 14 Twelve link one degree of freedom planar kinematic chain
Fig. 15 Nine link two degree of freedom kinematic chain
is assigned a node value “1/2” and for ternary link joint value “2/3” is assigned. Also quaternary link and pent nary link had assigned the node values 3/4 and 4/5 respectively. Thus at a node joining a binary link and binary link,binary link and ternary link and ternary link and ternary link a node value will be assigned as 60/60,70/60,80/60 (2/3, joint value of ternary link and 1/2, joint value of binary link) as shown in Fig. 15. A least distance matrix (LDM) for Fig. 15 is given below:
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Fig. 16 Nine link two degree of freedom kinematic chain
Link Value (LV) for link A, B, C, D, E, F, G, H & I are—1360, 1180, 1010, 1030, 1010,1180,1280,1280 and 1030 respectively and chain string (CS) for nine link two degree of freedom kinematic chain are 10,360, 2[3(60), 4(70), 4(80), (120), 4(130), 8(150), 2(160), 2(190), 2(210), 4(220), 2(280)]. Nine link two degree of freedom chain is shown in Fig. 16. Link A, B, C & H are ternary link and link D, E, F, G & I are binary link. Each node of binary link is assigned a node value “1/2” and for ternary link joint value “2/3” is assigned. Also quaternary link and pent nary link had assigned the node values 3/4 and 4/5 respectively. For simplicity of calculations, a non fractional value is assigned to each joint value on the basis of taking L.C.M. of joint values by which we can take same denominator for all joint value for various links connectivity. Thus at a node joining a binary link and binary link,binary link and ternary link and ternary link and ternary link a node value will be assigned as 60/60,70/60,80/60 (2/3, joint value of ternary link and 1/2, joint value of binary link) as shown in Fig. 16. A least distance matrix (LDM) for Fig. 16 is given:
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Nine link two degree of freedom chain as show in Fig. 16 the link value (LV) for link A, B, C, D, E, F, G, H & I are—1030, 1020, 1070, 1360, 1180,1280, 1230, 990 and 1140 respectively and chain string (CS) for nine link two degree of freedom kinematic chain are 10,300, 2[2(60), 6(70), 3(80), (120), 2(130), 4(140), 7(150), (160), 2(190), (200), 3(210), 2(220), 2(280)]. Nine link two degree of freedom chain Figs. 15 and 16 chain have different chain string (CS) hence they are non isomorphic.
4 Inversion The process of fixing different links of a kinematic chain one at a time to produce distinct mechanisms is called inversion. Row and column strings obtained from the least distance matrix of a planar kinematic chain are useful to detect distinct inversions of a planar kinematic chain. The total number of inversion that can be obtained from a planar kinematic chain is equal to the number of the links in the planar kinematic chain. However, some of the inversions may be identical. The fact that the inversions are distinct or identical can be determined by comparing their link strings. Link value (LV) and link string (LS) for Table 1 (Watt chain) (Table 2). A comparative study of Watt chain & Stephenson chain shows that Watt chain Table 3 has two inversions and Stephenson chain Table 4 has three inversions. Table 1 Link value and link string of six link watt chain
Links
Links value
Link string (LS)
A
48/6
1/6[0, 2(7), 8, 2(13)]
B
60/6
1/6[0, 6, 7, 13, 14, 20]
C
60/6
1/6[0, 6, 7, 13, 14, 20]
D
48/6
1/6[0, 2(7), 8, 2(13)]
E
60/6
1/6[0, 6, 7, 13, 14, 20]
F
60/6
1/6[0, 6, 7, 13, 14, 20]
A Parametric Approach to Detect Isomorphism … Table 2 Link value and link string of six link stephenson chain
Table 3 Distinct kinematic inversions of six link watt chain
Table 4 Distinct kinematic inversions of six link stephenson chain
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Links
Links value
Link string (LS)
A
48/6
1/6[0, 3(7), 13, 14]
B
56/6
1/6[0, 2(7), 3(14)]
C
48/6
1/6[0, 3(7), 13, 14]
D
54/6
1/6[0, 6, 7, 13, 2(14)]
E
54/6
1/6[0, 6, 7, 13, 2(14)]
F
56/6
1/6 [0, 2(7), 3(14)]
Distinct links Link value Link string (LS)
Distinct Inversion
(B, C, E, F)
60/6
1/6 [0, 6, 7, 13, 14, 20] 2
(A, D)
48/6
1/6 [0, 2(7), 8, 2(13)]
Distinct links Link value Link string (LS)
Distinct Inversion
(A, C)
48/6
1/6 [0, 3(7), 13, 14]
3
(B, F)
56/6
1/6 [0, 2(7), 3(14)]
(D, E)
54/6
1/6 [0, 6, 7, 13, 2(14)]
5 Results The method is tested for six, eight, nine and ten links with simple joint. These results are verified by many researchers. Our method also verifies the result. The numbers of mechanisms are derived from the six link one degree of freedom kinematic chains with simple joints shown Tables 3 and 4 is not isomorphic. Eight link one degree of freedom kinematic chains with simple joints shown in Table 5 (appendix) are not isomorphic. Nine link two degree of freedom kinematic chains with simple joints shown in Table 6 (appendix) are not isomorphic. Ten link three degree of freedom kinematic chains with simple joints shown in Table 7 (appendix) are not isomorphic.
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6 Conclusion In this paper, a simple, efficient, and reliable method is proposed to identify isomorphism and inversions in planar kinematic chains. The method is simple and involves less mathematical calculations. Further, it is directly applicable to kinematic chains and does not require converting the kinematic chain to their graphs. This method can be easily implemented whenever the number of links and degree of freedom increases, as well as with some modification it can be suitable for n-number of link and f- number of degree of freedom. Such method can also computerize and coding can be developed to reduce time and acquire results faster.
Appendix See Tables 5, 6 and 7.
Distinct link
(A,D,E,H) (B,C,F,G)
(A,B,E,F) (C,D,G,H)
(A,C) (B,H) (D,G) (E,F)
Chain No
1
2
3
Kinematic chains
2(14),16,28,30,32,42 2(14),28,2(30),2(44) 14,2(16),26,2(30),32 12, 14, 26,30,42,2(44)
14,2(16),26,2(30),32 12,14,26,2(30),2(44)
14,2(16),26,30,32,42 12,14,26,30,42,44,56
Link string (LS)
2[12,6(14),3(16), 2(26),2(28),6(30),2(32),2(42), 4(44)]
2[2(12),4(14), 4(16),4(26),8(30),2(32), 4(44)]
2[2(12),4(14), 2(16),4(26), 4(30),2(32), 4(42), 2(44), 2(56)]
Chain string (CS)
Table 5 Eight link one degree of freedom kinematic chains distinct link, link string, chain String and distinct inversions
(continued)
4
2
2
Distinct inversion
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Distinct link
(A) (B) (C) (D) (E) (F) (G) (H)
(A,H) (B,E) (C,D) (F,G)
(A) (B,G) (C,F) (D,E) (H)
Chain No
4
5
6
Table 5 (continued)
Kinematic chains
2(14),16,2(28),2(42) 2(14),2(28),30,40,42 2(14),16,26,28,32,42 12,14,26,28,30,40,42 3(16),4(30)
14,2(16),28,2(30),42 2(14),16,26,28,30,40 12,14,26,28,30,40,42 14,2(16),28,2(30),42
2(14),16,26,28,30,32 2(14),2(28),30,40,42 2(14),16,2(28),30,42 14,2(16),26,2(30),32 12,14,26,2(30),40,44 12,14,26,28,30,40,44 2(14),28,2(30),2(44) 14,2(16),28,2(30),42
Link string (LS)
2[12,6(14),3(16),2(26),5(28),4(30),32,2(40),4(42)]
2[12,6(14),3(16),2(26),4(28),6(30), 3(40),2(42),44]
2[12,6(14),3(16),2(26),4(28),6(30),32,40,2(42), 2(44)]
Chain string (CS)
(continued)
5
4
8
Distinct inversion
762 K. Dewangan and A. K. Shukla
Distinct link
(A,B,D,E) (C,F,G,H)
(A,C,D,H) (B,E,F,G)
(A,E) (B,D) (C) (F) (G) (H)
Chain No
7
8
9
Table 5 (continued)
Kinematic chains
2(14),16,2(28),30,42 2(14),3(28),30,42 3(14),3(28),42 2(14),2(28),30,42,44 2(14),2(28),2(30),44 14,2(16),28,3(30)
2(14),16,2(28),2(30) 2(14),2(28),2(30),44
2(14),16,28,2(30),44 2(14),28,2(30),2(44)
Link string (LS)
2[8(14),2(16), 9(28),5(30), 3(42),44]
2[8(14),2(16), 8(28),8(30), 2(44)]
2[8(14),2(16), 4(28),8(30), 6(44)]
Chain string (CS)
(continued)
6
2
2
Distinct inversion
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Distinct link
(A) (B,H) (C,G) (D,F) (E)
(A) (B) (C) (D) (E) (F) (G) (H)
(A) (B,H) (C) (D) (E) (F) (G)
Chain No
10
11
12
Table 5 (continued)
Kinematic chains
3(15),17,27,29,31 14,15,28,2(30),32,42 2(14),15,28,29,42,44 2(14),3(28),31,40 2(14),17,26,28,2(32) 12,14,27,28,3(42) 12,15,26,2(30),40,44
3(15),17,2(27),29 12,15,26,2(30),32,42 12,14,27,28,30,42,44 2(14),16,26,29,30,42 14,16,17,26,2(30),32 12,14,27,30,2(42),44 12,15,26,2(30),2(42) 14,15,28,3(30),42
2(15),2(17),2(27),31 12,15,26,30,32,40,42 12,14,27,28,2(42),56 2(14),17,26,28,32,42 2(14),2(28),31,2(40)
Link string (LS)
2[12,5(15),17,26,27,4(28),29, 3(30),31,2(32), 40, 3(42), 44]
2[2(12),3(14), 3(15),16,17, 2(26),28,2(27), 29, 6(30), 32, 4(42),44]
2[2(12),4(14), 2(15),2(17), 2(26),2(27),3(28),30,31,2(32), 2(40),4(42),56]
Chain string (CS)
(continued)
7
8
5
Distinct inversion
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Distinct link
(A) (B) (C) (D) (E) (F,H) (G)
(A) (B,F,G,H) (C,E) (D)
(A,D) (B,C,E,F,G,H)
Chain No
13
14
15
Table 5 (continued)
Kinematic chains
3(15),18,3(27) 12,15,27,2(30),2(42)
4(15),2(29),43 14,15,2(28),2(30),42 3(14),28,29,2(42) 2(14),4(28),43
4(15),27,2(29) 12,15,26,3(30),42 12,14,27,28,30,2(42) 2(14),16,26,29,2(30) 2(14),16,29,2(30),42 14,15,28,3(30),42 14,15,28,4(30)
Link string (LS)
2[3(12),7(15), 6(27),6(30), 6(42)]
2[6(14),4(15), 7(28),2(29),4(30),4(42),43]
2[12,4(14),4(15),16,26,27, 2(28),2(29), 9(30),3(42)]
Chain string (CS)
(continued)
2
4
7
Distinct inversion
A Parametric Approach to Detect Isomorphism … 765
(A,D (B,C,E,F) (G,H)
16
Total Inversion
Distinct link
Chain No
Table 5 (continued)
Kinematic chains 4(15),2(27),30 12,15,27,3(30),42 2(15),5(30)
Link string (LS) 2[2(12),8(15), 4(27),12(30), 2(42)]
Chain string (CS)
71
3
Distinct inversion
766 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,F) (C,E) (D,I) (G,H)
(A) (B) (C) (D) (E) (F) (G) (H) (I)
Chain No.
1
2
Kinematic chains
70,2(80),140,3(150),210 70,2(80),130,2(150),160,200 2(70),80,2(140),150,2(210) 2(70),140,2(150),220,2(280) 60,70,120,2(150),190,2(220) 2(60),2(130),200,2(210),280 60,70,120,140,150,200,210,280 2(70),80,130,140,150,160,190 2(70),2(140),2(150),200,220
2(60),2(130),2(210),2(280) 60,70,120,2(150),190,2(220) 70,2(80),130,2(150),160,190 70,2(80),130,2(150),160,210 60,70,130,2(150),2(200),280
Link String (LS)
Distinct Inversion
2[2(60), 6(70), 3(80), (120), 2(130), 4(140), 7(150), (160), 2(190), (200), 2(210), 2(220), 2(280)]
(continued)
9
2[3(60), 4(70), 4(80), (120), (130), 5 8(150), 2(160), 2(190), 2(210), 4(220), 2(280)]
Chain String (CS)
Table 6 Nine link two degree of freedom kinematic chains distinct link, link string, chain string and distinct inversions
A Parametric Approach to Detect Isomorphism … 767
Distinct Link
(A,C) (B,D) (E,I) (F,H) (G)
(A,B) (C,D) (E,G) (F) (H,I)
Chain No.
3
4
Table 6 (continued)
Kinematic chains
70,2(80),130,2(150),160,210 70,2(80),130,150,160,210,230 60,70,120,2(150),220,230,280 2(60),2(130),2(210),2(280) 60,70,130,150,210,220,2(280)
2(70),80,140,150,160,210,230 2(70),140,2(150),2(220),280 70,2(80),130,3(150),160 60,70,120,2(150),2(220),230 2(60),2(130),2(210),2(280)
Link String (LS)
Distinct Inversion
(continued)
2[3(60), 4(70), 4(80), (120), 4(130), 5 6(150), 2(160), 4(210), 2(220), 2(230), 4(280)]
2[2(60), 6(70), 3(80), (120), 2(130), 5 2(140), 8(150), 2(160), 2(210), 4(220), 2(230), 2(280)]
Chain String (CS)
768 K. Dewangan and A. K. Shukla
Distinct Link
(A,D) (B,H) (C,I) (E,G) (F)
(A,E) (B,D) (C) (F,H) (G) (I)
Chain No.
5
6
Table 6 (continued)
Kinematic chains
2(70),80,130,140,160,190,210 2(70),2(140),150,200,210,260 2(70),80,2(140),2(210)270 60,70,120,140,150,190,210,260 2(60),2(130),2(200),210,270 3(80),4(150),210
2(70),80,130,140,150,190,220 2(70),140,2(150),200,2(220) 70,2(80),140,2(150),2(210) 60,70,120,140,150,190,210,220 2(60),2(130),2(200),(230)
Link String (LS)
Distinct Inversion
2[2(60), 6(70), 3(80), (120), 2(130), 5(140), 4(150), (160), 2(190), 2(200), 5(210), 2(260), (270)]
(continued)
6
2[2(60), 6(70), 3(80), (120), 2(130), 5 4(140), 6(150), 2(190), 2(200), 4(210), 4(220)]
Chain String (CS)
A Parametric Approach to Detect Isomorphism … 769
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I)
(A,B) (C,E) (D) (F,H) (G,I)
Chain No.
7
8
Table 6 (continued)
Kinematic chains
70,2(80),130,2(150),160,210 2(70),80,130,140,160,210,230 2(70),2(140),2(150),2(200) 60,70,130,140,2(210),2(280) 60,70,130,150,200,220,230,280
2(70),80,2(140),150,2(210) 2(70),2(140),200,210,220 2(70),80,130,140,150,200,210 60,70,130,140,150,2(200),210 60,70,130,2(140),200,2(210) 3(70),130,2(140),200,210 2(70),3(140),150,200,210 70,2(80),140,3(150),210 2(70),2(140),2(150),200,220
Link String (LS)
2[2(60), 6(70), 3(80), 4(130), 3(140),4(150), 2(160), 2(200), 4(210), (220), 2(230), 3(280)]
2[(60), 8(70), 2(80), 2(130), 8(140), 5(150), 4(200), (210), (220)]
Chain String (CS)
(continued)
5
9
Distinct Inversion
770 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C) (D) (E) (F), (I) (G) (H)
(A,G) (B,F) (C,E) (D) (H) (I)
Chain No.
9
10
Table 6 (continued)
Kinematic chains
2(70),80,130,140,150,210,220 60,70,130,140,150,200,210,260 60,70,130,2(140),2(200),220 3(70),2(130),2(150),220 2(70),2(140),150,2(200),220 70,2(80),2(140),3(150)
70,2(80),140,3(150),210 70,2(80),130,2(150),160,210 2(70),80,130,140,150,200,210 2(70),140,2(150),200,2(220) 60,70,130,2(150),200,2(220) 60,70,130,140,150,200,210,220 2(70),80,2(130),150,160,200 60,70,130,140,150,2(200),210
Link String (LS)
2[2(60), 7(70), 2(80), 4(130), 6(140), 5(150), 4(200), 2(210), 3(220), (260)]
2[2(60), 6(70), 3(80), 4(130), 3(140), 7(150), (160), 4(200), 3(210), 3(220)]
Chain String (CS)
(continued)
6
9
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 771
Distinct Link
(A,B) (C,F) (D,E) (G,H) (I)
(A,E) (B,D) (C) (F) (G) (H) (I)
Chain No.
11
12
Table 6 (continued)
Kinematic chains
2(70),80,2(140),150,2(210) 2(70),3(140),150,200,210 3(70),130,2(140),200,210 2(70),2(140),150,210,220,280 60,70,130,2(140),210,220,280 60,70,130,2(150),2(200),220 70,2(80),130,3(150),200
70,2(80),130,2(150),160,210 2(70),80,130,140,150,160,210 60,70,130,140,150,210,220,280 60,70,130,150,210,2(220),280 2(70),2(140),2(150),2(220)
Link String (LS)
Distinct Inversion
2[(60), 8(70), 2(80), 2(130), 8(140), 5(150), 3(200), 5(210), (220), (280)]
(continued)
7
2[2(60), 6(70), 3(80), 4(130), 3(140), 5 6(150), 2(160), 4(210), 4(220), 2(280)]
Chain String (CS)
772 K. Dewangan and A. K. Shukla
Distinct Link
(A,I) (B) (C) (D) (E) (F) (G) (H)
(A,C) (B,D) (E,H) (F,G) (I)
Chain No.
13
14
Table 6 (continued)
Kinematic chains
2(70),80,140,2(150),210,220 2(70),140,2(150),3(220) 2(70),80,130,140,2(150),220 60,70,130,140,150, 2(220),210 2(70),2(140),2(150)2(220)
2(70),2(140),150,210,220,280 3(70),2(140),210,220,270 2(70),3(140),150,200,210 2(70),80,130,140,160,2(210) 70,2(80),130,3(150),220 2(70),80,140,150,160,2(210) 60,70,130,140,2(210),2(280) 60,70,130,150,200,2(220),270
Link String (LS)
2[(60), 8(70), 2(80), 2(130), 5(140), 8(150), 2(210), 8(220)]
2[(60), 8(70), 2(130), 6(140), 4(150), (160), (200), 5(210), 3(220), (270), 2(280)]
Chain String (CS)
(continued)
5
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 773
Distinct Link
(A,D) (B,H) (C,I) (E,F) (G)
(A,E) (B,D) (C) (F,G) (H) (I)
Chain No.
15
16
Table 6 (continued)
Kinematic chains
2(70),80,130,140,150,160,210 2(70),3(140),150,200,210 3(70),3(140),2(220) 60,70,130,140,150,200,210,220 2(70),2(140),2(150),2(220) 70,2(80),140,4(150)
2(70),80,130,140,2(150),200 2(70),2(140),2(150),200,220 2(70),80,140,2(150),210 60,70,130,140,150,200,210,220 2(70),2(140),2(150),2(220)
Link String (LS)
2[(60), 8(70), 2(80), 2(130), 8(140), 6(150), (160), 2(200), 4(210), 2(220)]
2[(60), 8(70), 2(80), 2(130), 7(140), 8(150), 3(200), 2(210), 3(220)]
Chain String (CS)
(continued)
6
5
Distinct Inversion
774 K. Dewangan and A. K. Shukla
Distinct Link
(A,G) (B,H) (C,I) (D,F) (E)
(A,D,H,I) (B,C) (E,G) (F)
Chain No.
17
18
Table 6 (continued)
Kinematic chains
2(70),2(140),150,210,2(220) 2(70),80,140,2(150),210,220 3(70),2(140),2(210),220 2(70),2(140),150,210,2(220)
70,2(80),130,3(150),160 60,70,130,2(150),200,2(220) 60,70,130,140,150,200,210,280 2(70),80,130,140,150,160,210 2(70),2(140),2(150),2(200)
Link String (LS)
Distinct Inversion
2[10(70), (80), 9(140), 4(150), 6(210), 6(220)]
(continued)
4
2[2(60), 6(70), 3(80), 4(130), 3(140), 5 8(150), 2(160), 2(200), 2(210), 3(220), (280)]
Chain String (CS)
A Parametric Approach to Detect Isomorphism … 775
Distinct Link
(A,C) (B) (D,F,G,I) (E,H)
(A) (B) (C) (D) (E) (F) (G) (H) (I)
Chain No.
19
20
Table 6 (continued)
Kinematic chains
3(75),85,2(135),155,165 70,80,85,130,2(150),2(160) 60,70,120,150,155,220,2(230) 2(60),130,135,3(210),270 60,75,120,2(150),160,210,220 2(70),80,130,150,165,210,220 60,70,135,140,150,210,220,270 70,75,140,3(150),210,230, 60,75,130,2(150),160,210,230
3(70),3(140),2(210) 2(70),4(140),2(210) 2(70),3(140),150,210,220 2(70),80,2(140),2(150),210
Link String (LS)
2[3(60), 3(70), 3(75),(80),(85), (120), 2(130), 2(135), (140), 6(150), (155), 2(160), (165), 4(210),2(220), 2(230), (270)]
2[10(70), (80), 13(140), 4(150), 6(210), 2(220)]
Chain String (CS)
(continued)
9
4
Distinct Inversion
776 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,F) (C) (D) (E) (G) (H) (I)
(A,I) (B) (C) (D) (E) (F) (G) (H)
Chain No.
21
22
Table 6 (continued)
Kinematic chains
70,75,140,3(150),210,220 2(70),80,145,2(150),210,220 2(70),80,130,145,2(150),190 70,75,140,4(150),200 4(75),135,2(145),195 60,70,120,140,150,195,2(220) 2(60),130,135,200,3(210) 60,75,120,3(150),190,220
3(70),85,135,140,2(155) 70,75,2(140),150,160,210,230 3(70),2(140),210,220,290 2(70),3(140),155,200,230 2(70),85,130,140,150,2(160) 60,70,120,130,140,155,210,2(230) 2(60),130,135,200,2(210),270 60,75,120,3(150),220,230
Link String (LS)
Distinct Inversion
2[2(60), 4(70), 4(75), (80), (120), (130), (135), 2(140), 2(145), 9(150), (190), (195), (200), 3(210), 3(220)]
(continued)
8
2[2(60), 5(70), 3(75),(85),(120), (130), 8 (135), 5(140), (145), 2(150), 2(155), 3(160), (200), 3(210), (220), 3(230), (270)]
Chain String (CS)
A Parametric Approach to Detect Isomorphism … 777
Distinct Link
(A,D) (B,C) (E) (F) (G,I) (H)
(A,I) (B) (C) (D,F) (E) (G) (H)
Chain No.
23
24
Table 6 (continued)
Kinematic chains
2(70),2(140),155,2(230),290 2(70),85,140,3(160),220 3(75),85,135,145,2(155) 60,75,120,150,160,220,2(230) 2(60),135,210,220,280,2(290) 70,75,2(140),2(150),160,210 3(70),140,145,2(220),280
2(70),85,130,140,2(160),220 60,70,130,140,155,2(230),290 2(75),2(85),135,3(155) 2(70),2(140),155,2(230),290 60,75,120,2(160),3(230) 2(60),135,2(220),3(290)
Link String (LS) 6
Distinct Inversion
(continued)
2[2(60), 5(70), 3(75),(85),(120), (135), 7 4(140), (145), 2(150), 2(155), 3(160), (210), 3(220), 4(230), (280), 2(290)]
2[3(60), 4(70), 2(75), 2(85), (120), 2(130), (135), 3(140), 4(160), 2(220), 6(230), 3(290)]
Chain String (CS)
778 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,G) (C) (D) (E) (F) (H) (I)
(A) (B) (C) (D) (E) (F) (G) (H) (I)
Chain No.
25
26
Table 6 (continued)
Kinematic chains
2(70),85,130,140,2(160),210 2(70),2(140),155,2(200),230 2(70),85,140,2(160),210,220 60,70,135,140,2(210),270,280 60,75,150,2(160),200,210,230 2(85),2(75),2(135),2(155) 2(60),130,135,200,210,220,270 60,75,120,150,2(160),210,230 60,70,120,140,155,210,230,280
2(60),135„210,220,280,290,350 60,75,120,150,160,220,230,290 3(75),85,135,145,155,215 70,75,140,3(150),200,210 2(70),80,130,145,2(220),280 70,80,85,130,150,2(160),220 60,70,130,140,215,2(290),350 60,70,130,155,200,2(230),290
Link String (LS)
2[3(60), 4(70), 2(75), 2(85), (120), (130), 2(135), 3(140), (150), 2(155), 4(160), 2(200), 4(210), (220), 2(230),(270), (280)]
2[3(60), 3(70), 3(75),(80),(85), (120), 2(130), (135), (140), (145), 3(150), (155), 2(160), (200), (210), (215), 3(220), 2(230), (280), 3(290), (350)]
Chain String (CS)
(continued)
9
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 779
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I)
(A,F,H,I) (B,E) (C,D) (G)
Chain No.
27
28
Table 6 (continued)
Kinematic chains
70,75,2(140),2(150),200,220 3(70),130,145,200,2(220) 60,70,130,2(140),2(200),215 4(75),2(145),2(215)
2(70),80,130,145,160,192,210 70,75,140,3(150),200,210 3(75),85,2(135),145,195 60,75,130,2(150),2(210),270 60,70,135,160,2(210),220,270 70,80,85,130,2(150),160,210 60,75,120,2(150),160,190,210 2(60),130,135,200,2(210),270 60,70,120,140,150,195,220,270
Link String (LS) 9
Distinct Inversion
(continued)
2[(60), 6(70), 4(75), 2(130), 6(140), 4 2(145), 4(150), 5(200), 2(215), 4(220)]
2[3(60), 3(70), 3(75),(80),(85), (120), 2(130), 2(135), (140), (145), 6(150), (160), (190), (195), (200), 5(210), (220), 2(270)]
Chain String (CS)
780 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I)
(A,G) (B,F) (C,E) (D) (H,I)
Chain No.
29
30
Table 6 (continued)
Kinematic chains
2(70),85,2(130),160,170,220 60,70,135,140,200,210,220,270 60,75,130,150,160,200,210,230 2(75),2(85),2(135),2(155) 60,70,130,140,155,2(200),230
3(75),85,2(135),145,155 2(70),85,130,140,2(160),210 2(70),3(140),155,2(200) 3(70),130,140,145,210,220 60,70,135,2(140),2(210),280 60,75,130,2(150),160,200,210 60,70,135,140,3(210),280 60,75,130,2(150),200,210,220 70,75,2(140),2(150),160,210
Link String (LS)
2[3(60) 4(70), 2(75), 2(85), 4(130), 2(135), 2(140), (150), 2(155), 2(160), (170), 4(200), 2(210), 2(220), 2(230), (270)]
2[2(60), 5(70), 3(75),(85),2(130), 2(135), 5(140), (145), 3(150), (155), 2(160), 2(200), 5(210), (220), (280)]
Chain String (CS)
(continued)
5
9
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 781
Distinct Link
(A,F) (B) (C) (D) (E,I) (G) (H)
(A,I) (B) (C) (D) (E) (F) (G) (H)
Chain No.
31
32
Table 6 (continued)
Kinematic chains
70,75,2(140),2(150),2(210) 3(70),140,145,2(210),220 2(70),4(140),200,215 3(70),130,140,145,2(210) 60,70,135,2(140),3(210) 60,75,130,3(150),200,220 4(75),135,2(145),215 70,75,2(140),3(150),210
60,70,135,140,150,200,210,220 2(70),80,2(130),150,165,210 70,80,85,130,2(150),2(160) 3(75),85,3(135),165 60,75,130,2(150),160,200,210 60,70,135,150,2(210),2(220) 60,75,130,2(150),3(210)
Link String (LS)
Distinct Inversion
2[(60), 6(70), 4(75), (130), (135), 7(140), 2(145), 5(150), (200), 6(210), (215), (220)]
(continued)
8
2[3(60), 3(70), 3(75),(80),(85), 3(130), 7 3(135), (140), 6(150), 2(160), (165), 2(200), 5(210), 2(220)]
Chain String (CS)
782 K. Dewangan and A. K. Shukla
Distinct Link
(A,I) (B) (C) (D) (E) (F) (G) (H)
(A,I) (B) (C,G) (D,H) (E) (F)
Chain No.
33
34
Table 6 (continued)
Kinematic chains
70,75,140,3(150),2(210) 4(75),3(135),205 60,75,130,3(150),200,210 60,70,135,140,150,200,2(210) 2(70),80,2(130),2(150),205 2(70),80,135,2(150),2(210)
70,75,2(140),150,160,200,210 3(70),130,145,200,220,260 60,70,130,2(140),200,215,260 60,70,130,140,155,3(200) 2(70),85,2(130),2(160),200 60,70,135,140,2(210),200,260 60,75,130,2(150),200,225,260 3(75),85,135,145,155,215
Link String (LS)
2[2(60), 4(70), 4(75), (80), 2(130), 2(135), 2(140), (145), 9(150), 2(200), (205), 6(210)]
2[2(60), 5(70), 3(75),(85),3(130), (135), 4(140), (145), 2(150), (155), 2(160), 5(200), 2(210), (215), (220), (260), (270)]
Chain String (CS)
(continued)
6
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 783
Distinct Link
(A,B) (C,G) (D,F) (E) (H,I)
(A,C) (B,D) (E) (F,G) (H,I)
Chain No.
35
36
Table 6 (continued)
Kinematic chains
4(75),2(135),150,195 2(75),5(150),210 50,75,120,3(150),195,210 2(60),2(135),4(210) 60,75,135,3(150),2(210)
2(70),80,130,135,2(150),210 60,70,135,140,150,2(210),220 60,75,130,3(150),2(210) 4(75),4(135) 70,75,140,4(150),210
Link String (LS)
2[3(60), 8(75), (120), 4(135), 12(150), 2(195), 6(210)]
2[2(60), 4(70), 4(75),(80),2(130), 2(135), 2(140), 2(145), 10(150), 6(210), (220)]
Chain String (CS)
(continued)
5
5
Distinct Inversion
784 K. Dewangan and A. K. Shukla
Distinct Link
(A,E,F,G,H,I) (B,D) (C)
(A,G,H,I) (B,F) (C,E) (D)
Chain No.
37
38
Table 6 (continued)
Kinematic chains
60,75,120,2(150),165,2(210) 2(60),2(135),4(210) 3(75),90,3(135),165 60,75,135,2(150),3(210)
60,75,135,3(150),2(210) 4(75),3(135),150 2(75),6(150)
Link String (LS)
2[4(60), 6(75),(90),(120), 6(135), 6(150), 2(165), 10(210)]
2[3(60), 8(75), 6(135), 13(150), 6(210)]
Chain String (CS)
(continued)
4
3
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 785
(A) (B,F) (C) (D) (E) (G,I) (H)
(A) (B,I) (C,H) (D) (E,G) (F)
39
40
Total Inversion
Distinct Link
Chain No.
Table 6 (continued)
Kinematic chains
4(78),88,3(138) 60,78,130,3(156),200,216 60,70,138,140,200,2(216),276 2(70),88,2(130),2(166),266 60,78,120,2(156),166,2(216) 2(60),138,2(216),226,2(276)
5(78),2(138),148 70,78,2(140),3(156),216 3(70),130,148,2(226),286 60,70,138,2(140),2(216),276 60,78,130,4(156),216 60,78,120,3(156),216,226 2(60),138,3(216),276,286
Link String (LS)
2[4(60), 2(70), 4(78),(88),(120), 2(130), 3(138), (140), 5(156), 2(166), 2(200), 6(216), 2(276)]
2[3(60), 2(70), 5(78), (120), (130), 2(138), (140), (148), 8(156), 5(216), 2(226), (276), (286)]
Chain String (CS)
254
6
7
Distinct Inversion
786 K. Dewangan and A. K. Shukla
Distinct Link
(A,I) (B,H) (C,G) (D,F) (E) (J)
(A) (B,I) (C) (D) (E,H) (F,G) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A,C,DF,G,I) (B,E,H) (J)
Chain No
1
2
3
4
Kinematic Chains
60,70,130,140,150,2(200),230,260 2(70),80,2(130),2(160),2(230) 3(80),6(150)
2(60),2(130),2(200),210,260,270 60,70,120,140,150,190,210,260,280 2(70),80,130,140,160,190,210,230 2(70),2(140),150,2(200),230,260 2(70),80,130,140,160,210,230,270 60,70,130,140,150,200,210,260,280 60,70,130,140,150,2(200),230,260 2(70),80,2(130),2(160),190,230 60,70,120,140,150,190,200,230,260 3(80),5(150),210
2(60),130,210,2(290),3(360) 60,70,120,150,2(230),3(300) 2(70),80,130,2(160),3(230) 3(80),5(150),210 2(70),80,130,140,160,2(230),290 60,70,130,140,150,230,2(300),360 2(70),2(140),150,230,2(300),360
2(60),120,130,190,200,210,260,270 60,70,120,140,150,180,210,230,280 2(70),80,130,140,160,190,210,230 2(70),2(140),150,200,210,260,280 2(70),80,2(140),2(210),2(270) 3(80),4(150),2(210)
Link String (LS)
2[3(60), 6(70),3(80),6(130), 3(140), 6(150), 3(160), 6(200), 6(230), 3(260)]
2[3(60), 6(70),3(80), (120),4(130),4(140), 5(150), 2(160), 2(190), 4(200), 3(210), 3(230), 3(260),(270), (280)]
2[3(60), 6(70), 3(80),(120), 3(130), 3(140), 2(160),(210),7(230), 2(290), 6(300), 3(360)]
2[3(60), 6(70), 3(80),2(120),2(130), 5(140), 4(150),(160),(180), 2(190), 2(200), 6(210), 2(230), 2(260), 2(270), 2(280)]
Chain String (CS)
Table 7 Ten link three degree of freedom kinematic chains distinct link, link string, chain string and distinct inversions
(continued)
3
10
7
6
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 787
Distinct Link
(A,J) (B,I) (C,H) (D,G) (E,F)
(A,G) (B,F) (C,E) (D,J) (H,I)
(A,F) (B,E,G,) (C,D,H,I)
Chain No
5
6
7
Table 7 (continued)
Kinematic Chains
2(60),2(130),2(210),2(280),340 60,70,120,2(150),220,230,280,300 70,2(80),130,2(150),160,210,230
2(60),120,130,190,2(210),2(280) 60,70,120,2(150),180,2(220),230 70,2(80),130,2(150),160,190,230 70,2(80),130,2(150),160,2(210) 60,70,130,2(150),2(220),2(280)
2(60),120,130,190,210,270,280,340 60,70,120,2(150),180,220,230,280 70,2(80),130,2(150),160,190,210 70,2(80),130,150,160,210,230,270 60,70,130,150,210,220,2(280),340
Link String (LS)
2[4(60), 4(70), 4(80),2(120),4(130), 8(150), 2(160), 4(210), 2(220), 4(230), 4(280), 2(300), (340)]
2[4(60), 4(70),4(80),2(120), 4(130), 8(150), 2(160),(180),2(190), 4(210), 4(220), 2(230), 4(280)]
2[4(60), 4(70),4(80),2(120), 4(130), 6(150), 2(160),(180),2(190), 4(210), 2(220), 2(230), 2(270), 4(280), 2(340)]
Chain String (CS)
(continued)
3
5
5
Distinct Inversion
788 K. Dewangan and A. K. Shukla
Distinct Link
(A,H) (B,F,G,I) (C,D,E,J)
(A,D) (B,C) (E,H) (F,J) (G,I)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,J) (C) (D) (E) (F) (G) (H) (I)
Chain No
8
9
10
11
Table 7 (continued)
Kinematic Chains
2(60),130,210,230,290,350,360,420 60,70,120,150,220,230,290,300,360 2(70),80,130,150,160,220,230,290 70,2(80),140,3(150),2(210) 2(70),140,2(150),200,2(220),230 2(70),80,130,140,220,2(290),350 60,70,130,140,210,290,2(290),420 60,70,130,150,200,230,2(300),360 70,2(80),130,150,160,2(230),290
70,2(80),130,2(150),160,190,230 70,2(80),140,3(150),2(210) 2(70),80,130,140,150,160,190,230 60,70,120,140,150,180,210,220,230 2(60),120,130,190,200,210,270,280 2(60),120,130,190,2(210),260,280 60,70,120,2(150),180,2(220),230 2(70),140,2(150),2(220),2(280) 2(70),2(140),2(150),200,220,260 2(70),80,2(140),150,2(210),270
60,70,120,140,150,180,210,220,250 2(60),120,130,190,200,210,260,270 2(70),80,130,140,150,190,220,250 2(70),140,2(150),200,2(220),260 70,2(80),140,2(150),2(210),270
2(60),2(130),2(210),2(280),340 60,70,120,2(150),190,2(220),280 70,2(80),130,2(150),160,190,210
Link String (LS)
2[3(60), 6(70),3(80), (120), 3(130), 2(140), 5(150),(160),(200), 2(210), 3(220), 3(230), (280),4(290),2(300), (350), 3(360), (420)]
2[3(60), 6(70),3(80),2(120), 2(130), 4(140), 7(150),(160),(180), 2(190), (200),4(210),3(220), 2(230), (260),(270), 2(280)]
2[3(60), 6(70),3(80),2(120), 2(130), 4(140), 6(150),(180),2(190), 2(200), 4(210), 4(220), 2(250), 2(260), 2(270)]
2[4(60), 4(70), 4(80),2(120),4(130), 4(140), 8(150), 2(160), 4(190), 4(210), 4(220), 4(280), (340)]
Chain String (CS)
(continued)
9
10
5
3
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 789
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A,J) (B,I) (C,H) (D,G) (E,F)
Chain No
12
13
14
Table 7 (continued)
Kinematic Chains
2(70),140,2(150),2(220),2(280) 2(70),80,140,150,160,210,230,270 70,2(80),130,3(150),160,190 60,70,120,2(150),180,2(220),230 2(60),120,130,190,210,270,2(280)
2(70),140,2(150),3(220),280 70,2(80),140,3(150),2(210) 2(70),80,2(130),150,160,190,200 60,70,130,140,150,200,210,220,260 60,70,130,140,150,190,210,220,270 2(70),80,130,140,150,200,210,270 60,70,120,140,150,190,200,220,270 2(60),2(130),200,2(210),260,280 60,70,120,2(150),190,2(220),260 70,2(80),130,2(150),160,190,210
2(70),140,2(150),200,3(220) 70,2(80),140,2(150),3(210) 2(70),80,2(130),150,160,190,220 60,70,120,140,150,190,200,220,230 2(60),2(130),2(200),2(210),260 60,70,120,140,150,190,210,220,260 2(70),80,130,140,150,190,210,220 60,70,130,140,150,200,210,220,260 60,70,130,150,200,210,2(220),260 70,2(80),130,2(150),160,210,230
Link String (LS)
2[3(60), 6(70), 3(80),2(120),2(130), 2(140), 8(150), 2(160), (180),2(190),2(210), 4(220),2(230), 2(270), 4(280)]
2[3(60), 6(70), 3(80),(120), 4(130), 3(140), 7(150),(160),2(190), 3(200), 4(210), 4(220), 2(260),(270), (280)]
2[3(60), 6(70),3(80), (120),4(130),3(140), 6(150),(160),2(190), 3(200), 5(210), 5(220), (230), 2(260)]
Chain String (CS)
(continued)
5
10
10
Distinct Inversion
790 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,I) (C,H) (D,G) (E,F) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
15
16
17
Table 7 (continued)
Kinematic Chains
70,2(80),130,3(150),160,190 2(70),80,130,140,150,160,2(210) 2(70),2(140),2(150),2(200),220 2(70),80,130,140,150,160,190,210 60,70,120,140,150,190,210,220,280 2(60),2(130),200,2(210),2(280) 60,70,120,2(150),190,3(220) 60,70,130,140,150,210,220,2(280) 60,70,130,2(150),3(220),280 70,2(80),130,3(150),160,210
2(60),130,140,200,210,270,280,340 60,70,120,2(150),2(22),230,280 70,2(80),140,3(150),160,210 70,2(80),130,2(150),160,210,230 60,70,130,150,200,220,230,280,300 60,70,130,140,2(210),2(280),340 2(70),80,130,140,160,210,230,270 2(70),2(140),2(150),2(200),220 2(70),80,130,140,150,160,210,230 60,70,120,140,150,210,230,280,300
2(60),2(130),200,2(210),2(280) 60,70,120,140,150,190,220,230,280 2(70),80,130,140,150,160,190,210 70,2(80),130,2(150),160,210,230 60,70,130,150,210,2(220),2(280) 2(70),2(140),2(150),200,2(220)
Link String (LS)
2[3(60), 6(70),3(80), (120),4(130),3(140), 8(150), 2(160), 2(190), 2(200), 4(210), 4(220), 3(280)]
2[3(60), 6(70),3(80), (120),3(130),4(140), 6(150), 2(160), 2(200), 4(210), 2(220), 3(230), (270),3(280),(300),(340)]
2[3(60), 6(70), 3(80),(120), 4(130), 3(140), 6(150), 2(160), 2(190),(200),4(210), 4(220), 2(230), 4(280)]
Chain String (CS)
(continued)
10
10
6
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 791
Distinct Link
(A,I) (B,H) (C,G) (D,F) (E) (J)
(A,H, I,J) (B,G) (C,F) (D,E)
(A,H) (B,G) (C,F) (D,E) (I,J)
Chain No
18
19
20
Table 7 (continued)
Kinematic Chains
2(70),80,2(130),160,200,230,270 60,70,130,140,200,210,270,280,340 60,70,130,150,200,220,230,260,280 70,2(80),130,2(150),160,2(210) 60,70,130,140,150,2(200),210,260
60,70,130,140,150,200,210,220,280 2(70),80,2(130),150,160,200,210 70,2(80),130,2(150),160,2(210) 60,70,130,150,210,2(220),2(280)
60,70,130,140,150,210,220,2(280) 2(70),80,130,140,150,160,210,230 70,2(80),130,3(150),160,210 60,70,120,2(150),2(220),230,280 2(60),2(130),2(210),3(280) 2(70),2(140),2(150),2(220),280
Link String (LS)
2[3(60), 6(70),3(80),6(130), 2(140), 4(150), 2(160), 5(200), 4(210),(220),2(230), 2(260), 2(270), 2(280), (340)]
2[3(60), 6(70),3(80),6(130), 2(140), 6(150), 2(160), 3(200), 6(210), 4(220), 4(280)]
2[3(60), 6(70),3(80), (120),4(130),3(140), 8(150), 2(160), 4(210), 4(220), 2(230), 5(280)]
Chain String (CS)
(continued)
5
4
6
Distinct Inversion
792 K. Dewangan and A. K. Shukla
Distinct Link
(A,H) (B,I) (C,J) (D,G) (E,F)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,H) (C,G) (D,F) (E) (I) (J)
(A) (B,F) (C,E) (D) (G) (H) (I) (J)
Chain No
21
22
23
24
Table 7 (continued)
Kinematic Chains
3(70),130,2(140),190,210,220 2(70),3(140),150,200,210,220 2(70),80,2(140),150,3(210) 2(70),2(140),150,210,220,2(280) 70,2(80),130,3(150),190,220 60,70,120,2(150),190,3(220) 2(60),2(130),2(200),2(210),280 60,70,120,2(140),190,2(210),280
3(70),2(130),140,2(200),270 60,70,130,2(140),2(200),2(210) 60,70,130,140,150,2(200),210,260 2(70),80,130,140,150,200,210,270 2(70),2(140),150,2(200),220,270 70,2(80),140,3(150),2(210) 2(70),2(140),2(150),2(200),220
2(60),2(130),3(200),210,270 60,70,120,140,150,210,220,280 2(70),80,130,140,150,190,210,220 2(70),2(140),150,200,210,220,260 2(70),80,2(140),150,2(210),270 2(70),3(140),150,200,210,280 3(70),130,2(140),190,210,220 60,70,120,2(140),190,2(210),260 70,2(80),140,3(150),2(210) 2(70),2(140),2(150),200,2(220)
70,2(80),130,3(150),160,210 60,70.130,2(150),200,3(220) 60,70,130,140,150,200,210,220,280 2(70),80,2(130),150,160,200,210 60,70,130,140,150,2(200),210,220
Link String (LS)
2[2(60), 8(70),2(80), (120),2(130),8(140), 5(150), 2(190), 2(200), 7(210), 4(220), 2(280)]
2[2(60), 8(70),2(80),4(130), 7(140), 5(150), 8(200), 4(210), (220),(260), 3(270)]
2[2(60), 8(70),2(80), (120),2(130),8(140), 5(150), 2(190), 3(200), 6(210), 3(220),(260),(270),(280)]
2[3(60), 6(70),3(80),6(130), 2(140), 8(150), 2(160), 5(200), 4(210), 5(220), (280)]
Chain String (CS)
(continued)
8
7
10
5
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 793
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,I) (C) (D) (E) (F,J) (G) (H)
(A) (B) (C) (D) (E) (F) (G) (H) (J) (I)
Chain No
25
26
27
Table 7 (continued)
Kinematic Chains
3(70),130,2(140),200,210,270, 2(270),3(140),150,150,2(200),230 2(70),80,130,140,150,160,210,230 60,70,130,140,150,200,210,220,260 60,70,130,140,150,200,220,270 2(70),80,2(130),150,160,200,230 60,70,130,140,150,3(200),230 60,70,130,2(140),200,2(210),260 2(70),2(140),2(150),200,2(220) 70,2(80),140,4(150),210
2(60),130,210,280,290,350,2(360) 60,70,120,150,220,230,290,2(300) 2(70),80,130,150,160,220,2(230) 70,2(80),140,4(150),210 2(70),80,140,150,160,2(230),290 2(70),2(140),150,230,2(300),360 3(70),2(140),220,2(290),350 2(70),2(140),2(150),2(220),280
3(70),130,140,150,2(200),210 2(70),3(140),150,2(200),210 2(70),80,2(140),150,3(210) 2(70),2(140),150,200,210,220,230 2(70),80,130,140,150,200,2(210) 60,2(70),140,2(150),2(200),210 60,70,130,2(140),2(200),2(210) 70,2(80),130,3(150),200,210 60,2(70),130,2(150),2(200),220 60,70,130,2(140),200,2(210),230
Link String (LS)
2[2(60), 8(70),2(80),4(130), 7(140), 6(150), (160),6(200),3(210), 2(220), 2(230),(260),(270)]
2[2(60), 8(70),2(80), (120),(130),5(140), 6(150), (160),(210), 3(220), 4(230),(280),3(290), 4(300),(350), (360)]
2[2(60), 8(70),2(80),3(130), 7(140), 7(150), 7(200), 7(210), 2(220), (280)]
Chain String (CS)
(continued)
10
8
10
Distinct Inversion
794 K. Dewangan and A. K. Shukla
Distinct Link
(A,J) (B) (C) (D) (E) (F) (G) (H) (I)
(A,J) (B) (C) (D) (E) (F) (G) (H) (I)
(A) (B,H) (C,G) (D,F) (E) (I) (J)
Chain No
28
29
30
Table 7 (continued)
Kinematic Chains
3(70),3(140),2(210),270 2(70),3(140),150,200,210,260 2(70),80,130,140,150,160,190,210 60,70,120,140,150,190,210,220,260 2(60),2(130),2(200),210,270,280 70,2(80),140,4(150),210 2(70),2(140),2(150),2(220),280
2(70),80,140,150,2(210),230 2(70),80,140,150,160,2(210),230 70,2(80),130,3(150),210,220 60,70,130,150,200,2(220),260,290 60,70,130,140,200,210,270,2(280) 2(70),80,2(130),160,200,2(230) 60,70,130,140,150,3(200),230 60,70,130,2(140),200,2(210),260 3(70),130,140,200,220,270,290
2(70),2(140),150,210,220,2(280) 2(70),80,140,150,160,2(210),230 70,2(80),130,4(150),220 60,70,120,2(150),3(220),290 2(60),130,135,200,210,270,2(280) 60,70,120,140,150,210,230,2(280) 2(70),80,135,140,150,160,2(210) 2(70),3(140),150,200,210,220 3(70),2(140),210,220,270,290
Link String (LS)
2[2(60), 8(70),2(80), (120),2(130),8(140), 6(150),(160),2(190), 2(200), 5(210), 2(220), 2(260),(270),(280)]
2[2(60), 8(70),2(80),4(130), 5(140), 4(150), (160),5(200),3(210), 3(220), 3(230),(260),(270), 2(280), (290)]
2[2(60), 8(70),2(80), (120),2(130),6(140), 6(150),(160),(200), 5(210), 4(220),(230),(270), 4(280), (290)]
Chain String (CS)
(continued)
7
9
9
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 795
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,G) (C,F) (D,E) (H) (I) (J)
(A) (B,H) (C,G) (D,F) (E,I) (J)
Chain No
31
32
33
Table 7 (continued)
Kinematic Chains
2(60),2(130),200,2(210),2(280) 60,70,120,140,150,190,2(220),270 2(70),130,140,2(150),190,210,220 2(70),80,140,2(150),210,220,270 2(70),140,2(150),3(220),280 2(70),2(140),2(150),200,2(220)
3(70),130,2(140),200,2(210) 2(70),3(140),150,2(200),210 2(70),80,130,140,150,160,2(210) 60,70,130,140,150,200,210,220,280 70,2(80),130,3(150),200 60,70,130,2(150),2(200),2(220) 60,70,130,2(140),2(210),2(280)
60,70,130,2(140),2(210),2(280) 60,70,130,140,150,200,220,230,280 2(70),80,130,140,150,160,2(210) 70,2(80),130,3(150),210,220 60,70,130,150,200,2(220)270,280 60,70,130,140,2(210),3(280) 2(70),80,130,140,160,2(210),230 2(70),3(140),150,2(200),210 3(70),130,2(140),210,220,270 2(70),3(140),150,210,220,280
Link String (LS)
2[2(60), 8(70),2(80), (120),2(130),5(140), 8(150), 2(190), (200),2(210),8(220), 2(270), 2(280)]
2[2(60), 8(70),2(80),3(130), 7(140), 5(150), (160),5(200),6(210), 2(220), 2(280)]
2[2(60), 8(70), 2(80),4(130),7(140), 4(150), (160),2(200),6(210), 3(220), (230),(270), 4(280)]
Chain String (CS)
(continued)
6
7
10
Distinct Inversion
796 K. Dewangan and A. K. Shukla
Distinct Link
(A,G,I,J) (B,F) (C,E) (D,H)
(A) (B,G) (C,F) (D,E) (H,I) (J)
(A,E,F,J) (B,D,G,I) (C,H)
Chain No
34
35
36
Table 7 (continued)
Kinematic Chains
60,70,130,140,200,210,270,280,340 2(70),80,130,140,210,220,270 2(70),2(140),2(150),2(200),220
2(70),80,2(130),150,2(200) 60,70,130,140,150,2(200),210,220 60,70,130,140,150,200,210,2(220) 2(70),80,130,140,2(150),200,210 2(70),2(140),2(150),200,2(220) 2(70),80,2(140),2(150),2(210)
60,70,130,140,150,200,210,2(220) 2(70),80,140,2(150),2(210),220 2(70),80,140,2(150),2(210),220 2(70),140,2(150),4(220)
Link String (LS)
2[2(60), 8(70),2(80),4(130), 6(140), 4(150), 4(200), 4(210), 3(220), 4(270), 2(280), 2(340)]
2[2(60), 8(70), 2(80),4(130),6(140), 8(150), 6(200), 4(210), 5(220)]
2[2(60), 8(70),2(80),4(130), 4(140), 8(150), 3(200), 4(210), 10(220)]
Chain String (CS)
(continued)
3
6
4
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 797
Distinct Link
(A,D,E,H) (B,C,F,G) (I,J)
(A,F,G,H) (B,C,E,J) (D,I)
(A,H) (B,G) (C,E,I,J) (D,E)
Chain No
37
38
39
Table 7 (continued)
Kinematic Chains
60,70,130,2(140),2(200),210,270 3(70),140,200,220,270 2(70),2(140),150,200,2(220),270 2(70),80,140,2(150),210,220,270
60,70,130,140,150,200,210,220,280 2(70),80,130,140,2(150),200,210 2(70),2(140),2(150),2(200),220
60,70,130,140,150,210,2(220),280 2(70),80,130,140,2(150),210,220 2(70),2(140),2(150),3(220)
Link String (LS)
2[(60), 10(70),(80),2(130), 8(140), 4(150), 5(200), 2(210), 6(220), 6(270)]
2[2(60), 8(70),2(80),4(130), 6(140), 8(150), 6(200), 4(210), 3(220), 2(280)]
2[2(60), 8(70),2(80),4(130), 6(140), 8(150), 4(210), 9(220), 2(280)]
Chain String (CS)
(continued)
4
3
3
Distinct Inversion
798 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C,I) (D) (E) (F) (G) (H) (J)
(A,J) (B,I) (C,H) (D,G) (E,F)
(A,D) (B) (C,J) (E,G) (F) (H) (I)
(A,G) (B,F) (C,E,H,J) (D,I)
Chain No
40
41
42
43
Table 7 (continued)
Kinematic Chains
70,80,2(140),150,2(210),2(220) 3(70),2(140),150,2(210),220 2(70),3(140),150,210,2(220) 2(70),80,2(140),2(150),2(210)
2(70),80,130,140,2(150),190,220 2(70),2(140),2(150),200,2(220) 2(70),80,2(140),2(150),2(210) 60,70,120,140,150,190,210,2(220) 2(60),2(130),200,2(210),270,280 2(70),2(140),2(150),2(220),280 2(70),2(140),2(150),2(220),270
2(70),3(140),2(210),2(280) 3(70),2(140),2(210),220,270 2(70),3(140),150,200,210,220 3(70),130,140,150,2(210),220 60,70,130,140,200,210,270,2(280)
2(70),4(140),200,2(210) 3(70),2(140),2(210),220,270 2(70),2(140),150,210,2(220),280 2(70),80,140,2(150),2(210),220 2(70),80,130,140,2(150),210,220 60,70,130,140,150,200,2(220),270 60,70,130,2(140),2(210),2(280) 3(70),130,2(140),2(210),220 2(70),3(140),210,220,280
Link String (LS)
2[(60), 10(70),(80),2(130), 12(140), 4(150), 4(200), 8(210), 3(220)]
2[2(60), 8(70),2(80), (120),2(130),7(140), 8(150),(190),(200), 4(210), 6(220),(270),(280)]
2[(60), 10(70),(80),2(130), 10(140), 2(150), 2(200), 8(210), 3(220), 2(270), 4(280)]
2[(60), 10(70),(80),2(130), 10(140), 4(150), (200),7(210),6(220), (270), 2(280)]
Chain String (CS)
(continued)
4
7
5
9
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 799
Distinct Link
(A) (B) (C) (D,J) (E) (F) (G) (H) (I)
(A,E) (B,D,F,H) (C,G,I,J)
(A,C,E,H) (B,D,F,G,I,J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
44
45
46
47
Table 7 (continued)
Kinematic Chains
2(60),120,130,195,200,2(270),330 60,70,120,140,155,180,210,230,280 2(70),85,130,140,2(160),190,210 2(70),2(140),155,2(200),230,260 3(70),130,140,160,210,220,270 60,70,135,140,2(210),270,280,330 60,75,130,150,160,200,210,230,270 70,2(75),85,2(135),2(155),195 60,75,120,150,2(160),180,210,230 2(60),120,135,190,210,220,260,270
2(70),4(140),3(210) 2(70),4(140),2(210),280
2(70),4(140),2(210),280 3(70),2(140),3(210),280 2(70),3(140),210,280
3(70),130,2(140)200,2(210), 2(70),4(140),200,2(210) 3(70),3(140),3(210) 2(70),80,130,140,2(150),200,210 60,70,130,140,150,2(200)210,220 60,70,130,2(140),200,2(210),280 2(70),3(140),150,210,220,280 2(70),80,2(140),2(150),2(210) 2(70),80,2(140),2(150),2(210)
Link String (LS)
2[4(60), 4(70), 2(75),2(85), 2(120), 2(130), (135),3(140),(150), 2(155), 3(160),(180),(190),(195), 2(200), 4(210), (220),2(230),(260), 3(270), (280),(330)]
2[12(70), 18(140), 12(210), 3(280)]
2[12(70), 14(140), 12(210), 7(280)]
2[(60), 10(70),(80),2(130), 12(140), 4(150), 4(200), 8(210), 2(220), (280)]
Chain String (CS)
(continued)
10
2
3
9
Distinct Inversion
800 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,I) (C) (D,H) (E,G) (F) (J)
(A,I) (B,H) (C) (D,G) (E,F) (J)
(A,E) (B,D) (C) (F,J) (G,I) (H)
Chain No
48
49
50
Table 7 (continued)
Kinematic Chains
60,70,120,140,155,210,230,280,290 2(70),85,130,140,2(160),210,220 2(70),2(140),155,2(200),2(230) 2(60),130,135,200,210,220,270,290 60,75,120,150,2(160),210,2(230) 2(75),2(85),2(135),3(155)
2(60),120,135,2(220),3(290) 60,75,120,150,2(160),3(230) 2(75),2(85),2(135),3(155) 2(70),85,130,140,2(160),2(220) 60,70,130,140,155,2(230),2(290) 2(70),2(140),155,2(230)2(290)
3(60),135,2(220),2(290),350 60,75,120,2(160),3(230),290 2(75),2(85),135,3(155),215 2(70),85,130,140,2(160),190,220 60,70,120,140,155,190,2(230),290 2(60),2(130),200,215,2(290),350 60,2(70),2(140),155,200,2(230)
Link String (LS)
2[4(60), 4(70),2(75),2(85),2(120),2(130), 2(135), 3(140),(150),3(155), 4(160), 2(200), 4(210), 2(220), 4(230),(270),(280), 2(290)]
2[4(60), 4(70),2(75),2(85), 2(120), 2(130), 2(135), 3(140),(150),3(155), 4(160), 4(220) 6(230), 6(290)]
2[4(60), 4(70),2(75),2(85),2(120),2(130), (135),3(140),3(155), 4(160), 2(190),(200),(215), 2(220), 6(230), 5(290), (350)]
Chain String (CS)
(continued)
6
6
7
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 801
Distinct Link
(A) (B,J) (C) (D) (E,I) (F,H) (G)
(A,G) (B,F) (C,E) (D) (H,J) (I)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
51
52
53
Table 7 (continued)
Kinematic Chains
60,70,120,140,155,200,230,240,310, 2(70),85,2(130),2(160),170,240 60,70,130,140,155,2(200),2(230) 60,70,130,140,155,200,230,290 2(70),85,2(130),160,170,220,240 60,70,135200,210240,270,310 75,150,160,200,210,2(230) 2(75),2(85),135,3(155) 60,75,120,150,2(160),2(230) 2(60),130,135,200,210,220,270,290
60,70,135,140,200,210,220,260,270 2(70),85,2(130),160,170,190,220 60,70,120,140,155,190,200,230,260 2(60),2(130),2(200),215,2(260) 60,75,130,150,160,200,210,230260 2(75),2(85),2(135),2(155),215
2(60),130,215,2(290),300,2(350) 60,70,120,155,2(230),240,2(290) 2(70),85,130,2(160),170,2(220) 2(75),2(85),2(135),2(155),215 60,75,130,150,160,200,2(230),290 60,70,135,140,200,220,2(290),350 2(70),85,2(130),170,2(240),300
Link String (LS)
2[4(60), 4(70), 2(75),2(85), (120),4(130),2(135), 2(140),(150),3(155), 3(160), (170),4(200),2(210), (220), 4(230), 2(240),(270), 2(290)]
2[4(60), 4(70),2(75),2(85),(120),4(130), 2(135), 2(140),(150),2(155), 2(160),(170),2(190), 4(200), 2(210), (215),2(220),2(230), 4(260), (270)]
2[4(60), 4(70),2(75),2(85),(120),3(130), 2(135), (140),(150), 2(155), 2(160),(170),2(200), (215),2(220) 4(230), 2(240), 6(290),(300),2(350)]
Chain String (CS)
(continued)
10
6
7
Distinct Inversion
802 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,H) (C) (D,G) (E,F,I,J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
54
55
56
Table 7 (continued)
Kinematic Chains
2(60),120,130,195,210,270,2(280) 60,70,120,150,155,180,2(220),230 70,80,85,130,2(150),2(160),190 2(70),80,130,145,150,210,220,280 60,70,135,140,150,210,220,270,280 60,75,130,2(150),160,210,230,270 3(75),85,2(135),145,155,195 60,75,120,2(150),160,180,220 2(60),120,135,190,2(210),270,280 70,75,140,3(150),220,280
3(75),85,2(135),145,195,215 60,75,130,2(150),210,220,270,280 60,70,135,150,2(210),220,270,280 70,80,85,130,150,2(160),210,220 2(70),80,130,145,150,190,2(220) 60,70,120,140,150,180,215,220,280 2(60),120,130,195,200,210,270,280 2(60),120,135,190,2(210),220,270 60,75,120,2(150),160,180,210,220 70,75,140,2(150),160,200,2(210)
2(60),135,2(220),4(290) 60,75,150,2(160),4(230) 2(75),2(85),135,4(155) 2(70),85,2(130),2(160),220 60,70,130,140,155,200,230,290
Link String (LS)
2[4(60), 3(70),3(75), (80),(85), 2(120), 2(130),(135),(140),(145),6(150),(155),2(160), (180),(190), (195),4(210),3(220), (230),(270), 3(280)]
2[4(60), 3(70),3(75), (80),(85), 2(120), 2(130), 2(135),(140),(145), 5(150), 2(160),(180),(190),(195),(200),5(210),(215), 4(220), 2(270), 2(280)]
2[4(60), 4(70), 2(75),2(85), (120),4(130),(135), 2(140),(150),4(155), 4(160), (170),2(200),2(220), 8(230), 4(290)]
Chain String (CS)
(continued)
10
10
5
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 803
Distinct Link
(A) (B,J) (C) (D) (E) (F) (G) (H) (I)
(A,J) (B,I) (C) (D) (E) (F) (G) (H)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
57
58
59
Table 7 (continued)
Kinematic Chains
2(60),130,135,3(210),270,280 60,70,120,150,155,2(220),280 70,80,85,130,150,3(160),220 2(70),80,130,145,150,190,210,220 60,70,120,140,150,195,220,270,280 2(60),130,135,200,210,220,270,280 60,75,120,2(150),160,190,210,230 3(75),85,135,145,155,195 60,75,120,150,160,210,220,270 70,75,140,150,160,200,210,2(220)
2(60),120,135,210,220,280,350 60,75,120,2(150),220,230,290 3(75),85,2(135),145,155,215 70,75,140,150,160,200,2(210) 2(70),80,130,145,2(220),2(280) 60,70,130,140,215,2(290),2(350) 60,70,130,155,200,2(230),2(290) 70,80,85,130,3(160),2(220)
2(60),135,210,220,280,2(350) 60,75,120,150,160,220,230,2(290) 3(75),85,135,145,2(215) 70,75,140,2(150),160,200,210,230 2(70),80,130,145,150,2(220),280 60,70,120,140,150,2(290),350 2(60),2(130),200,215,2(290),350 60,70,120,150,155,3(230),290 70,80,85,150,3(160),220
Link String (LS)
2[4(60), 3(70),3(75), (80),(85), 2(120), 2(130), 2(135),(140),5(150), (155),3(160),(190),(195),(200),4(210),4(220), (230),2(270), 2(280)]
2[4(60), 3(70),3(75), (80),(85), 2(120), 2(130), 2(135),(140),(145), 3(150),(155),3(160), (190),(195), (200),2(210),(215), 4(220), 2(230), 2(280), 2(290), 2(350)]
2[4(60), 3(70),3(75), (80),(85), 2(120), 2(130),(135),(140),(145),4(150),(155),3(160), (200),(210),2(215), 3(220), 3(230), (280),5(290),2(350)]
Chain String (CS)
(continued)
10
8
9
Distinct Inversion
804 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B) (C) (D) (E,I) (F,J) (G) (H)
(A) (B,J) (C) (D) (E) (F) (G) (H) (I)
Chain No
60
61
62
Table 7 (continued)
Kinematic Chains
2(60),135,210,220,270,280,290,350 60,75,120,150,160,210,230,240,290 3(75),85,2(135),155,165,215 60,75,130,2(150),160,200,210,230 60,70,135,140,150,200,2(210),270 2(70),80,2(130),165,2(240),280 60,70,130,140,200,215,2(290),350 60,70,130,155,200,3(230),290 70,80,85,130,150,3(160),220
2(60),130,135,3(210),2(270) 60,70,120,150,155,2(220),2(230) 70,80,85,130,2(150),3(160) 2(70),80,2(130),150,165,210,240 60,70,135,140,150,2(210),220,270 60,75,130,2(150),160,2(210),230 3(75),85,3(135),155,165 60,75,120,2(150),160,2(210),240
60,70,135,150,2(210),2(220),270 70,80,85,130,150,2(160),220 2(70),80,2(130),150,165,195,210 60,70,120,140,150,195,200,220,270 2(60),130,135,200,2(210),220,270 60,75,120,2(150),160,190,2(210) 3(75),85,3(135),165,195 60,75,130,135,150,3(210),270 60,70,135,140,150,200,2(210),220 60,75,130,135,150,160,200,2(210)
Link String (LS)
2[4(60), 3(70),3(75), (80),(85), (120),3(130),2(135), (140),3(150),(155), 3(160), (165), 2(200), 3(210),(215),(220), 3(230), 2(240),(270),(280), 3(290), (350)]
2[4(60), 3(70),3(75), (80),(85), (120),3(130),3(135), (140),6(150),(155), 3(160), (165),7(210),2(220), 2(230),(240), 2(270)]
2[4(60), 3(70),3(75), (80),(85), (120),3(130),4(135), (140), 5(150), 2(160),(165),(190),(195),2(200), 7(210), 3(220), 2(270)]
Chain String (CS)
(continued)
9
8
10
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 805
Distinct Link
(A) (B) (C) (D) (E) (F,J) (G) (H) (I)
(A) (B) (C) (D,J) (E) (F) (G) (H) (I)
(A) (B,I) (C) (D) (E,J) (F) (G) (H)
Chain No
63
64
65
Table 7 (continued)
Kinematic Chains
2(60),130,215,3(290),350,360 60,70,120,155,3(230),290,300 2(70),85,130,3(160),220,230 3(75),85,135,145,2(155),215 70,75,140,2(150),160,2(230),290 3(70),130,145,230,2(300),360 60,70,135,2(140),200,2(290),350 60,75,130,2(150),160,2(230),290
60,70,135,140,200,2(210),260,280 60,75,130,2(150),200,220,260,290 3(75),85,135,145,155,2(215) 70,75,140,2(150),160,200,210,230 3(70),130,145,190,220,230,280 60,70,120,2(140),190,215,260,290 2(60),2(130),3(200),215,260 60,70,120,140,155,190,200,2(230) 2(70),85,2(130),2(160),190,230
2(60),120,130,195,200,3(270) 60,70,120,140,155,180,210,2(230) 2(70),85,130,140,3(160),190 2(70),3(140),155,200,230,260 3(70),140,145,210,220,270,280 70,75,140,2(150),160,210,230,270 3(75),85,135,145,2(155),195 60,75,120,2(150),160,180,220,230 2(60),120,135,190,2(210),260,280
Link String (LS)
2[3(60), 5(70),3(75), (85),(120), 2(130),(135),2(140), (145),3(150),2(155), 3(160),(215),(220), 7(230), 5(290), 2(300), (350), (360)]
2[3(60), 5(70),3(75), (85),(120), 3(130),(135),3(140), (145),3(150),(155), 2(160), 2(190), 4(200), 2(210), 2(215),(220),3(230), 2(260),(280),(290)]
2[3(60), 5(70),3(75), (85),2(120), (130),(135), 4(140),(145),3(150), 2(155), 3(160),(190),(195),(200), 3(210), 2(220), 3(230),(260),3(270), (280)]
Chain String (CS)
(continued)
8
9
9
Distinct Inversion
806 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C) (D,J) (E) (F) (G) (H) (I)
(A,I) (B,H) (C) (D) (E,J) (F) (G)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
66
67
68
Table 7 (continued)
Kinematic Chains
60,70,135,140,3(210),270,280 2(70),85,130,140,2(160),210,220 2(70),3(140),155,2(200),230 3(70),130,140,150,190,210,220 60,70,120,2(140),195,210,270,280 2(60),130,135,200,2(210),220,270 60,75,120,2(150),160,190,210,230 3(75),85,2(135),150,155,195 60,75,130,2(150),200,210,220,270 70,75,2(140),2(150),160,2(210)
2(60),120,135,210,220,280,2(290) 60,75,120,2(150),160,220,2(230) 4(75),135,2(145),2(155) 2(70),75,140,3(160),2(220) 2(70),2(140),155,2(230),2(290) 3(70),140,145,2(220),2(280) 70,75,2(140),2(150),160,2(210)
2(60),130,135,200,2(210),260,280 60,75,120,2(150),160,220,230,290 3(75),85,135,145,2(155),215 70,75,140,2(150),200,210,230 3(70),130,145,200,220,280,290 60,70,130,2(140),200,215,260,290 60,70,130,140,155,3(200),230 2(70),85,2(130),3(160),200 60,70,120,140,155,200,2(230),290
Link String (LS)
2[3(60), 5(70),3(75), (85),(120), 2(130), 2(135), 5(140), 4(150),(155),2(160), (190),(195), 2(200), 6(210), 2(220),(230),2(270), (280)]
2[3(60), 5(70),3(75), (85),2(120),2(135), 4(140), (145),3(150),2(155), 3(160), 2(210), 4(220), 4(230), 2(280), 4(290)]
2[3(60), 5(70),3(75), (85),(120), 3(130),(135),4(140), (145),2(150),2(155), 3(160), 5(200), 2(210),(215),(220), 3(230),(260),(280), 2(290)]
Chain String (CS)
(continued)
10
7
9
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 807
Distinct Link
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,I) (C) (D) (E) (F) (G) (H) (J)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
Chain No
69
70
71
Table 7 (continued)
Kinematic Chains
60,70,130,140,200,210,230,2(270) 2(70),85,2(130),2(160),200,220 60,70,130,140,155,3(200),230 60,70,130,2(140),2(200),205,260 3(70),2(130),145,200,220,270 60,70,130,2(140),200,210,220,270 60,75,130,2(150),160,200,210,230 (75),85,2(130),145,155,205 60,75,130,1(150),200,210,220,260 70,75,2(140),2(150),160,200,230
2(60),130,210,220,280,2(290),350 60,75,120,150,160,220,2(230),290 3(75),85,130,145,2(155),215 70,75,2(140),2(150),160,200,210 3(70),130,140,145,2(220),280 60,70,130,2(140),215,2(290),350 60,70,130,140,155,200,2(230),290 2(70),85,130,140,3(160),220 2(70),3(140),155,2(230),290
60,70,120,140,155,210,2(230),280 2(70),85,130,140,3(160),220 2(70), 3(140), 155, 200, 2(230) 3(70),130,140,150,210,220,270 60,70,130,2(140),210,220,270,280 60,75,130,2(150),160,210,2(230) 3(75),85,130,135,150,2(155) 60,75,120,2(150),160,210,220,230 2(60),130,135,200,2(210),2(270) 70,75,2(140),2(150),160,210,230
Link String (LS)
2[3(60), 5(70),3(75), (85),5(130), (135),4(140),(145), 3(150),(155),2(160), 6(200), 2(210), 2(220), 2(230),(260), 2(270)]
2[3(60), 5(70),3(75), (85),(120), 3(130), 5(140), (145),2(150),2(155), 3(160),(200),(210),(215),3(220), 4(230), (280),4(290),(350)]
2[3(60), 5(70),3(75), (85),(120), 3(130),(135),5(140), (145),3(150),2(155), 3(160),(200),4(210), 2(220), 4(230), 2(270), (280)]
Chain String (CS)
(continued)
10
9
10
Distinct Inversion
808 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B,I) (C) (D) (E,J) (F) (G) (H)
(A) (B) (C) (D) (E,I) (F,J) (G) (H)
(A,J) (B) (C) (D) (E) (F) (G) (H) (I)
Chain No
72
73
74
Table 7 (continued)
Kinematic Chains
70,75,140,3(150),210,220,270 2(70),80,145,2(150),210,220,270 2(70),80,130,145,2(150),190,220 60,70,120,140,150,180,215,2(220) 2(60),120,130,195,200,210,2(270) 2(60),120,135,190,3(210),270 60,75,120,3(150),180,2(220) 4(75),135,2(145),195,215 70,75,140,4(150),200,210
60,70,130,140,3(210),2(270) 2(70),85,140,3(160),2(220) 2(70),3(140),155,3(200) 3(70),2(130),140,205,210,270 60,70,130,2(140),200,210,220,270 60,75,130,2(150),160,2(200),210 3(75),85,3(130),155,205 60,75,2(150),200,2(210),270
2(60),130,210,220,270,2(290),340 60,75,120,150,160,210,2(230),300 3(75),85,2(130),2(155),205 2(70),85,140,3(160),2(220) 2(70),2(140),155,200,2(230),290 3(70),130,140,205,2(300),340 60,70,130,2(140),2(210),220,270 60,75,130,2(150),160,2(200),210
Link String (LS)
2[3(60), 4(70),4(75), (80),2(120), (130),(135), 2(140), 2(145), 9(150),(180),(190),(195),(200),4(210),(215), 4(220), 3(270)]
2[3(60), 5(70),3(75), (85),5(130),6(140), 3(150),(155),2(160), 5(200),(205),5(210), 2(220), 3(270)]
2[3(60), 5(70),3(75), (85),(120), 3(130), 4(140), 2(150), 2(155), 3(160), 2(200), (205),3(210),2(220)]
Chain String (CS)
(continued)
9
8
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 809
Distinct Link
(A,J) (B) (C) (D) (E) (F) (G) (H) (I)
(A) (B) (C) (D) (E) (F) (G) (H) (I) (J)
(A) (B,J) (C) (D,I) (E,H) (F,G)
Chain No
75
76
77
Table 7 (continued)
Kinematic Chains
2(60),135,2(210),2(280),2(350) 60,75,120,2(150),2(220),2(290) 4(75),130,2(145),2(215) 70,75,140,4(150),200,210 2(70),80,130,145,150,2(220),280 60,70,130,140,200,215,2(290),350
60,70,135,140,150,210,2(220),270 2(70),80,130,145,2(150),210,220 2(70),80,130,145,2(150),190,220 60,70,120,140,150,195,2(220),270 2(60),130,135,200,3(210),270 60,75,120,3(150),190,210,220 4(75),2(135),2(145),195 60,75,130,3(150),210,220,270 70,75,140,4(150),200,220 70,75,140,4(150),210,220
70,75,140,3(150),2(210),220 2(70),80,145,2(150),2(210),220 70(2),80,2(130),2(150),190,205 60,70,120,140,150,195,200,2(220) 2(60),130,135,200,4(210) 60,75,120,3(150),190,210,220 4(75),2(135),145,195,205 60,70,135,140,150,200,3(210) 60,75,130,3(150),200,2(210)
Link String (LS)
2[3(60), 4(70),4(75), (80),(120), 2(130),(135),2(140), 2(145), 7(150), 2(200), 2(210), 2(215), 4(220), 2(280), 4(290), 2(350)]
2[3(60), 4(70),4(75), (80),(120), 2(130), 2(135), 2(140), 2(145), 9(150),(190),(195),(200), 4(210), 5(220), 2(270)]
2[3(60), 4(70),4(75), (80),(120), 2(130), 2(135), 2(140),(145),9(150), (190),(195),2(200), 2(205), 8(210), 3(220)]
Chain String (CS)
(continued)
6
10
9
Distinct Inversion
810 K. Dewangan and A. K. Shukla
Distinct Link
(A) (B) (C) (D,H) (E,I) (F) (G) (J)
(A,E) (B,D) (C) (F,H,I,J) (G)
(A,J) (B) (C) (D) (E) (F) (G) (H) (I)
Chain No
78
79
80
Table 7 (continued)
Kinematic Chains
70,75,2(140),2(150),2(210),270 4(75),135,2(145),195,215 60,75,120,3(150),190,220,260 2(60),130,135,200,3(210),270 60,70,120,130,2(140),195,2(270) 3(70),130,140,145,190,2(210) 2(70),4(140),200,215,260 3(70),140,145,2(210),220,270 70,75,2(140),2(150),2(210)
3(70),130,145,190,2(220),260 60,70,120,2(140),190,215,2(260) 2(60),2(130),4(200),275 70,75,2(140),2(150),200,220,260 4(75),2(145),2(215),275
60,70,135,140,150,2(210),2(220) 2(70),80,130,145,2(150),2(210) 2(70),80,2(130),2(150),205,210 60,70,135,140,150,200,2(210),220 60,75,130,3(150),200,2(210) 4(75),3(135),145,205 60,75,130,3(150),3(210) 70,75,140,4(150),2(210)
Link String (LS)
2[2(60), 6(70),4(75), (120),(130), (135),7(140),2(145), 5(150),(190),(195),(200),7(210),(215),(220),(260), 3(270)]
2[2(60), 6(70),4(75), (120),(130),6(140), 2(145), 4(150), 2(190), 4(200), 2(215), 4(220), 5(260), (275)]
2[3(60), 4(70),4(75), (80),3(130),3(135), 2(140),(145),10(150), 2(200),(205),9(210), 2(220)]
Chain String (CS)
(continued)
9
5
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 811
Distinct Link
(A) (B,J) (C) (D) (E) (F) (G) (H) (I)
(A) (B,J) (C) (D,H) (E,I) (F) (G)
(A) (B,I) (C) (D,H) (E,G) (F,J)
Chain No
81
82
83
Table 7 (continued)
Kinematic Chains
2(60),135,2(210),2(280),2(350) 60,75,120,2(150),2(220),2(290) 4(75),135,2(145),2(215) 70,75,2(140),3(150),2(210) 3(70),140,145,210,2(220),280 2(70),3(140),215,2(290),350
3(70),140,145,2(210),2(220) 70,75,140,3(150),3(210) 4(75),2(135),145,205,215 60,75,130,3(150),200,210,220 60,70,135,2(140),4(210) 3(70),2(130),140,205,2(210) 2(70),4(140),2(200),205
3(70),130,145,200,2(220),270 70,75,2(140),1(150),200,210,220 4(75),135,2(145),2(215) 60,75,130,3(150),200,220,260 60,70,135,2(140),200,2(210),270 3(70),2(130),145,200,2(220) 60,70,130,2(140),3(200),210 60,70,130,2(140),2(200),210 70,75,2(140),3(150),200,220
Link String (LS)
2[2(60), 6(70),4(75), (120),(135),6(140), 2(145), 5(150), 4(210), 2(215), 4(220), 2(280), 4(290), 2(350)]
2[2(60), 6(70),4(75),2(130), 2(135), 6(140),(145),6(150), 2(200),(205),10(210), (215), 2(220)]
2[2(60), 6(70),4(75),2(130),(135),6(140), 2(145), 5(150), 6(200), 2(210), 2(215), 4(220), (260),(270)]
Chain String (CS)
(continued)
6
7
9
Distinct Inversion
812 K. Dewangan and A. K. Shukla
Distinct Link
(A,E) (B,D) (C) (F,H) (G) (I,J)
(A,H,I,J) (B,G) (C,F) (D,E)
(A,F,H,I) (B,E) (C,D) (G,J)
Chain No
84
85
86
Table 7 (continued)
Kinematic Chains
60,75,120,150,165,210,2(240),300 3(75),90,2(135),2(165),225 60,75,135,2(150),3(210),270 2(60),135,210,225,270,2(300),360
60,75,135,2(150),3(210),270 3(75),90,3(135),165,195 60,75,120,2(150),165,180,2(210) 2(60),120,135,195,2(210),2(270)
60,70,135,2(140),3(210),270 60,75,130,3(150),200,210,220 4(75),2(135),2(145),210 3(70),130,140,145,210,2(220) 2(70),4(140),2(200),210 70,75,2(140),3(150),210,220
Link String (LS)
2[5(60), 6(75),(90), 2(120), 4(135), 5(150), 4(165), 6(210), 2(225), 4(240), 2(270), 4(300), (360)]
2[5(60), 6(75),(90), 2(120), 6(135), 6(150), 2(165),(180),2(195), 10(210), 4(270)]
2[2(60), 6(70),4(75),2(130), 2(135), 7(140), 2(145), 6(150), 2(200), 6(210),(215),4(220), (270)]
Chain String (CS)
(continued)
4
4
6
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 813
Distinct Link
(A,E) (B,D,F,H) (C,G) (I,J)
(A,H) (B,G) (C,F) (D,E) (I,J)
(A,G) (B,H) (C,D,E,I) (F,J)
Chain No
87
88
89
Table 7 (continued)
Kinematic Chains
2(60),135,2(210),285,2(360),420 60,75,120,2(150),225,2(300),360 4(75),135,150,2(225),285 60,75,120,2(150),2(225),300,360
2(60),120,135,195,3(210),270 60,75,120,3(150),180,210,225 4(75),2(135),150,195,225 60,75,135,3(150),2(210),270 2(75),5(150),2(210)
2(60),2(135),4(210),270 60,75,120,2(150),165,2(210) 3(75),90,3(135),2(165) 60,75,135,2(150),4(210)
Link String (LS)
Distinct Inversion
2[4(60), 8(75), 2(120), 2(135), 10(150), 4(210), 5(225), 2(285), 3(300), 4(360), (420)]
2[4(60), 8(75),2(120),4(135), 12(150), (180),2(195),8(210), 2(225), 2(270)]
(continued)
4
5
2[5(60), 6(75),(90), 2(120), 6(135), 6(150), 4(165), 12(210), 4 2(240), (270)]
Chain String (CS)
814 K. Dewangan and A. K. Shukla
Distinct Link
(A,E) (B,D,F,H) (C,G) (I,J)
(A,G,H,I) (B,F) (C,E) (D) (J)
(A,D) (B,C,E,F,G,H,I,J)
Chain No
90
91
92
Table 7 (continued)
Kinematic Chains
4(75),4(135),210 60,75,135,3(150),3(210)
60,75,135,3(150),3(210) 4(75),3(135),150,195 60,75,120,3(150),195,2(210) 2(60),2(135),5(210) 2(75),6(150),210
2(60),2(135),4(210),270 60,75,120,3(150),195,210,270 4(75),2(135),150,2(195) 2(75),5(150),2(210)
Link String (LS)
2[4(60), 8(75),5(135),12(150), 13(210)]
2[4(60), 8(75),(120),6(135), 13(150), 2(195), 11(210)]
2[4(60), 8(75),2(120),4(135), 12(150), 4(195), 8(210), 3(270)]
Chain String (CS)
(continued)
2
5
4
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 815
Distinct Link
(A,J) (B,I) (C) (D,H) (E,G) (F)
(A) (B,J) (C) (D) (E) (F) (G) (H) (I)
(A,B) (C) (D) (E,J) (F) (G) (H,I)
Chain No
93
94
95
Table 7 (continued)
Kinematic Chains
2(60),120,138,3(216),1176,286 60,78,120,4(156),216,226 5(78),3(138),148 70,78,140,4(156),2(216) 3(70),130,148,2(226),2(286) 60,70,138,2(140),2(216),2(276) 60,2(78),130,3(156),2(216)
2(60),138,2(216),226,2(276),296 60,78,120,2(156),166,2(216),236 4(78),88,3(138),158 60,78,130,3(156),200,2(216) 60,70,138,140,200,3(216),276 2(70),88,2(130),3(166),226 60,70,120,140,158,200,2(236),296 2(60),130,138,200,6(216),276 60,78,120,2(156),166,2(216),236
2(60),120,138,2(216),226,2(276) 60,78,120,3(156),166,2(216) 4(78),88,4(138) 60,78,130,3(156),200,2(216) 60,70,138,140,200,2(216),2(276) 2(70),88,2(130),2(166),226
Link String (LS)
Distinct Inversion
2[4(60), 3(70),5(78),2(120),(130),3(138), 2(140),(148),10(156), 8(216), 2(226), 2(276), 2(286)]
2[5(60), 2(70),4(78), (88),2(120),2(130), 3(138), (140),5(156),(158), 3(166), 2(200), 8(216), (226),2(236),2(276), (296)]
(continued)
7
9
2[5(60), 2(70),4(78), (88),2(120),2(130), 6 4(138),(140),6(156), 2(166), 2(200), 8(216), 2(226), 4(276)]
Chain String (CS)
816 K. Dewangan and A. K. Shukla
(A) (B,I) (C) (D) (E) (F) (G) (H,J)
(A) (B,I) (C) (D,H) (E,G) (F) (J)
(A,C,E,G,H,J) (B,F,I) (D)
96
97
98
Total Inversion
Distinct Link
Chain No
Table 7 (continued)
Kinematic Chains
60,80,120,4(160),2(220) 2(60),140,4(220),2(280) 6(80),3(140)
2(60),138,3(216),2(276),286 60,78,120,3(156),2(216),226 5(78),3(138),148 60,78,130,4(156),200,216 60,70,138,2(140),200,2(216),276 3(70),2(130),148,2(226),286 70,78,2(140),4(156),216
2(60),138,3(216),276,286,336 60,78,120,3(156),216,226,276 5(78),2(138),148,198 60,78,120,4(156),190,216 2(60),130,138,2(200),2(216),276 60,70,120,2(140),198,2(276),336 3(70),130,148,190,2(226),286 70,78,140,4(156),200,216
Link String (LS)
2[6(60), 6(80),3(120),3(140), 12(160), 12(220), 3(280)]
2[4(60), 3(70),5(78), (120),2(130),3(138), 3(140),(148),8(156), (158),2(200),7(216), 2(226), 2(276), (286)]
2[4(60), 3(70),5(78),2(120), (130),2(138),2(140), (148),9(156),(190),(198),2(200), 5(216), 2(226), 3(276),(286), (336)]
Chain String (CS)
684
3
7
8
Distinct Inversion
A Parametric Approach to Detect Isomorphism … 817
818
K. Dewangan and A. K. Shukla
References 1. Zeng K, Fan X, Dong M, Yang P (2014) A fast algorithm for kinematic chain isomorphism identification based on dividing and matching vertices. Mech Mach Theory 72:25–38 2. Rai RK, Punjabi S (2019) A new algorithm of links labelling for the isomorphism detection of various kinematic chains using binary code. Mech Mach Theory 131:1–32 3. Venkata Kamesh V, Mallikarjuna Rao K, Balaji Srinivasa Rao A (2017) An innovative approach to detect isomorphism in planar and geared kinematic chains using graph theory. J Mech Des 139(12) 4. Mruthyunjaya TS, Balasubramanian HR (1987) In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains. Mech Mach Theory 22(2):131–139 5. Rai RK, Punjabi S (2019) Isomorphism detection of planar kinematic chains with multiple joints using information theory. J Mech Des 1–42 6. Sun W, Kong J, Sun L (2018) A joint–joint matrix representation of planar kinematic chains with multiple joints and isomorphism identification. Adv Mech Eng 10(6):1687814018778404 7. Sun W, Kong J, Sun L (2017) The improved hamming number method to detect isomorphism for kinematic chain with multiple joints. J Adv Mech Des Syst Manuf 11(5):JAMDSM0061JAMDSM0061 8. Shukla A, Sanyal S (2017) Gradient method for identification of isomorphism of planar kinematic chains. Australian J Mech Eng 1–18 9. Hwang WM, Hwang YW (1992) Computer-aided structural synthesis of planar kinematic chains with simple joints. Mech Mach Theory 27(2):189–199 10. Purushottamprajapati, Deshmukh PB, Shukla AK (2013) A novel approach for detection of isomorphism of kinematic chains. Inacomm, December 2013, pp 525–532
Feasibility of Tensegrity-Based Walking Robot P. K. Malik, Keshab Patra, and Anirban Guha
Abstract Constructing robots out of tensegrity mechanisms allows one to significantly reduce weight. This paper explores the feasibility of walking gait generation in such mechanisms. Four, five, six and eight-bar symmetric tensegrity mechanisms have been investigated, and cable actuations have been used for moving ground touching nodes. The node movements were assumed to be slow enough to ignore vibration, and Monte Carlo form-finding method was used to get the stable positions at intermediate positions of a gait trajectory. It was found that only the four-bar tensegrity mechanism allowed walking gait. All the other mechanisms had scraping at the ground nodes which made gait generation practically infeasible. It was surmised that increased level of connectivity between nodes was responsible for this, and an investigation of simultaneous cable and strut actuation was suggested for further investigation. Keywords Tensegrity · Gait generation · Discrete path · Walking robot
1 Introduction Robots have been traditionally constructed with rigid links. Electrical and hydraulic actuators are the traditional methods of actuation [1]. Designers of such robots have traditionally struggled with the problem of reduction of mass of links while maintaining structural integrity. In other hand, design of soft robots has been attempted to address this issue. Tensegrity mechanisms have the potential to address these drawbacks. Tensegrity mechanisms consist of tensile and compressive members as discussed by Zhang [2], the tensile members are called strings or cables, and the compressive members are called struts or bars. Change of shape of these mechanisms can be achieved by changing the length of struts or changing the length of cables. Change of length of a strut would require the strut to be replaced by a linear actuator while change of length of a cable can be achieved by reeling in or out a cable P. K. Malik (B) · K. Patra · A. Guha Mechanical Engineering Department, Indian Institute of Technology Bombay, Bombay 40076, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_74
819
820
P. K. Malik et al.
through a motor mounted at the end of a strut. Change of length in cables results change in positions of nodes which can be used to generate a desired motion. It is important to identify a sequence of cable actuations which create nodal displacement such that the tensegrity mechanism undergoes walking locomotion. Tensegrity-based locomotive robot was first introduced by Kanchanasaratool and Williamson [3, 4]. They provided a solution to the path tracking problems through a passive nonlinear particle dynamics model. Jager and Skelton [5] have developed a nonlinear model and linearized the nonlinear dynamics for a 2D tensegrity mechanism. But the method fails for 3D tensegrity mechanisms because the nonlinearity cannot be handled by their solution method. Aldrich et al. [6] have developed a dynamic model by considering the struts as fully rigid. But still, difficulty level is not reduced much due to nonlinearity in dynamics of the mechanism. To remove the above-mentioned difficulties, the proposed work considers the walking process as quasi-static. A user-defined path has been created and divided into small segments. Thereafter, a sequence of inputs (in terms of cable actuations) has been found out so that the tensegrity mechanism travels through the end points of each small segment step by step through quasi-static processes. This ensures that the tensegrity mechanism will be in equilibrium at the end of each small segment in following the path.
2 Guiding Principle of Gait Generation in Tensegrity In order to make a tensegrity mechanism walk, it is necessary to keep at least three nodes immobile and in contact with ground while allowing other to move. The nodal displacements are generated by actuating the cables. This paper searches for a sequence of cable actuations which generate motion along a predefined path. The actuation process has been considered quasi-static to avoid the problems occurring due to nonlinear dynamics. The path has been discretized into some finite segments, and the strategy for gait generation is very simple. Let the chosen path be denoted by ‘g’, and it is divided into ‘R’ small segments with end points (g0 , g1 , g2 , · · · , g R ). At each point ‘gi ’, the tensegrity mechanism is supposed to be in equilibrium and stable. Let the stable configuration at each state be denoted by ‘Si ’. We are interested in finding the sets of constant inputs (e1 , e2 , · · · , e R ) in such a way that when the mechanism moves from ‘gi−1 ’ to ‘gi ’, for each ‘ei ’ at ‘gi ’, the tensegrity will be still in equilibrium and stable. We have generated tensegrities using Monte Carlo-based method described Li et al. [7] for this study. A four-bar mechanism, shown in Fig. 1, has been taken as example to show the sequence of cable actuations. The lengths of the bars are equal to 15 units, and unstretched length of vertical and horizontal cables is 10 and 5 units, respectively, for the formfinding process. To avoid unnecessary cable actuations, a sensitivity analysis has been performed to identify the cables which are more sensitive towards a particular ground node displacement.
Feasibility of Tensegrity-Based Walking Robot
821
Fig. 1 Four-bar tensegrity mechanism obtained by Monte Carlo method
2.1 Sensitive Analysis for Cable Actuation The sensitivity analysis is being conducted to have an estimation of how the actuation(s) of a particular cable(s) affect the displacement of a particular ground node keeping all other ground node’s positions unchanged. During actuation, the cables are actuated up to 1.5% of unstretched length for both increase and decrease in step sizes of 0.1% of the unstretched length. Walking in the forward direction without turning requires the trajectory of the ground nodes to be only in the XZ plane. It is seen that cable numbers 5, 6, 7, 8 and 10 are the most sensitive ones for movement of node 6 in the XZ plane. Figure 2 shows the net magnitude of displacement of node 6 for actuation of the different cables. A similar analysis was done for other nodes.
2.2 Concept of Locomotion The concept of locomotion of four-bar tensegrity mechanism has been borrowed from the movement of a four-legged land animal. Four-legged animals have four discrete steps in one complete gait cycle. Four lower nodes (5, 6, 7 and 8) of the tensegrity mechanism are like the four legs of animals. The stepwise movement of the base of the tensegrity mechanism in X–Y plane has been shown in Fig. 3 and discussed as follows. In step 1, node 6 is moved forward keeping other nodes in ground; in step 2, node 8 is moved forward keeping other node in ground; in step 3, node 5 is moved forward keeping other node in ground and same has been followed for node 7.
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Fig. 2 Magnitude of displacement of node 6 in XZ direction vs cable actuations
Fig. 3 Stepwise translation of base of four-bar tensegrity in X–Y plane
2.3 Simulation of Gait of a Four-Bar Tensegrity Mechanism Each step of the gait cycle is divided into two sub-steps. In the first sub-step, a particular node is moved forward with a small upward movement (along Z-direction), and in the second sub-step, it is moved forward with a small downward movement
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so that the node will come to rest on the ground. A positive actuation value indicates increase in length and negative means decrease in length. In the first sub-step, node 6 will be moved forward (along X-direction) with some upward movement (along Z-direction) after giving the specified cable actuations as mentioned in Table 1. In the second sub-step, the node 6 will be moving further in forward direction and will rest on the ground after giving the cable actuations as mentioned in Table 2. The net translating motion after this step is 0.0406 unit in X-direction. Similarly, the actuations of step-up and step-down are performed for nodes 8, 5 and 7. Tables 3, 4, 5, 6, 7 and 8 describe the actuation of nodes 8, 5 and 7, respectively. From the above result, it is clear that the movement of all the nodes in X-direction is same (within 2.5% range). At the end fourth step, the mechanism regains its initial configuration but in a different position which is ahead of the initial position. Thus, we can conclude that a gait cycle has been completed successfully, and the mechanism has translated to a new position. These steps can be repeated to make the mechanism follow a predefined path. Figures 4 and 5 show the plots of X, Y and Z movements of each ground node vs normalized gait cycle time and gait cycle, respectively. The figures clearly show that the movement of each node is very close Table 1 Actuation of node 6 step-up Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0
−0.05
0.05
0.01
0
−0.04
0
0
Table 2 Actuation of node 6 step-down Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
−0.03
0.02
0
0.01
0
0
0
0
Table 3 Actuation of node 8 step-up Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0.05
0
0
0
0
0.005
0
0.02
Table 4 Actuation of node 8 step-down Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0.05
0
0
0
0
0.005
0
0.02
The net translating motion after this step is 0.0406 units in X-direction
Table 5 Actuation of node 5 step-up Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0.03
0.01
0
0
0
0
0.01
0.02
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Table 6 Actuation of node 5 step-down Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0.06
0.03
0
0.01
0
0.001
0.01
0
The net translating motion after this step is 0.0407 units in X-direction
Table 7 Actuation of node 7 step-up Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0.01
−0.01
−0.01
0.01
0
0.04
0.005
0
Table 8 Actuation of node 7 step-down Cable No.
1
2
3
4
5
6
7
8
9
10
11
12
Actuation
0
0
0
0
0
0
0.04
−0.01
0
0
0.015
0
The net translating motion after this step is 0.0417 units in X-direction
Fig. 4 Figure 12 X–Y –Z movement in of individual ground node
to the ideal movement. The small deviations seen in Y motion of node 5 and X motion of node 7 do not create a major change in trajectory of the gait.
2.4 Gate Generation of Other Tensegrity Mechanism It was observed that none of gait cycles in tensegrities having bar 5, 6 and 8 are not feasible because all of them led to significant unwanted movement of the stationary ground nodes. We consider it to be a matter of great significance that a walking gait
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Fig. 5 X–Y –Z directional movement in gait cycle
appears to be feasible only in a four-bar tensegrity mechanism. A tensegrity mechanism of any higher complexity seems to be incapable of allowing walking gait by only cable actuations. This is probably due to the increased level of interconnectivity in complex tensegrity mechanisms. However, it is possible that some combination of cable and strut actuation would allow walking gait even in these more complex tensegrity mechanisms. A more detailed search of the entire set of cable and strut actuations would be undertaken in future.
3 Conclusion This paper explores the feasibility of walking gait generation in simple tensegrity mechanisms by cable actuation where the movement of node has been discretized into small steps. Movement at every such step was assumed to be slow enough to avoid
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problems associated with vibration. It was found that a walking gait is possible only in four-bar tensegrity. For all the other mechanisms—5, 6 and 8 bar—an attempt to move one or more ground nodes led to significant unwanted movement. This would lead to ground scraping in real life. Thus, one is forced to conclude that high connectivity between nodes in complex tensegrity mechanisms makes walking gait infeasible. However, a study involving simultaneous actuation of struts and cables is necessary to prove or disprove this hypothesis, and such a study will be taken up in future.
References 1. Kasac J, Novakovic B, Majetic D, Brezak D (2006) Global positioning of robot manipulators with mixed revolute and prismatic joints. IEEE Trans Autom Control 51(6):1035–1040 2. Zhang J, Ohsaki M (2015) In: Tensegrity mechanisms, Springer 3. Kanchanasaratool N, Williamson D (2002) Modelling and control of class tensegrity mechanisms. Int J Control 75(2):123–139 4. Kanchanasaratool N, Williamson D (2002) Motion control of a tensegrity platform. Commun Inf Syst 2(3):299–324 5. De Jager B, Skelton RE (2005) Input-output selection for planar tensegrity models. IEEE Trans Control Syst Technol 13(5):778–785 6. Aldrich JB, Skelton RE, Kreutz-Delgado K (2003) Control synthesis for a class of light and agile robotic tensegrity mechanisms. In: American control conference, 2003. Proceedings of the 2003, June, vol 6. IEEE, pp 5245–5251 7. Li Y, Feng XQ, Cao YP (2010) Monte Carlo form–finding method for tensegrity mechanisms. In: AIP conference proceedings, vol 1233. No. 1, AIP, pp 1112–1116
Design and Development of a Climb-Free Telescopic Mechanism for Harvesting from Tall Trees Bhupinder Singh, Sekar Anup Chander, and Vhatkar Dattatraya Shivling
Abstract Harvesting fruits or nuts from tall trees have been a great challenge in recent times. There is a shortage of skilled manpower required to harvest, and also, the risk attached to this task is high (Punchihewa PG, Arancon RN (2006) Coconut: postharvest operations. Asian and Pacific Coconut Community). There is the absence of standard devices in the market for climb-free harvesting from tall trees likes coconut or areca nut. In this paper, a deployable mechanism concept capable of reaching tall tree heights is discussed. This design was achieved after comparing and evaluating continuous cable-type and discrete cable-type telescopic mechanisms. In addition to the telescopic mechanism design, a novel support system which can follow the morphology of the tree was also designed to avoid buckling of the telescopic structures. The overall mechanism was designed and prototyped as an efficient platform to enable accessibility to tall tree heights to which an end effector may be attached. Keywords Climb-free · Harvesting · Coconut · Buckling · Telescopic structures · Continuous cable-type telescopic mechanism · Discrete cable-type telescopic mechanism
1 Introduction India is one of the leading producers of coconut [1] in the world producing around 13 billion nuts per annum [2]. The leading coconut producer countries are Philippines, Indonesia, India, Sri Lanka, Thailand and Malaysia. Coconut trade is a major source of export revenues for these countries [3]. The livelihood of about one crore people in India is dependent on coconut [4]. The major problem with harvesting from tall trees (e.g. coconut trees) is that one cannot carry safety equipment while climbing the tree as there is no such provision B. Singh (B) · S. A. Chander (B) · V. D. Shivling CSIR-CSIO, Sector-30C, Chandigarh, India e-mail: [email protected] V. D. Shivling e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_75
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in the current practices. Moreover, there is a continuous decline in the availability of skilled workforce who can climb up those trees for harvesting. People are reluctant to enter into this field because of high risk factors involved [5]. Also, it is not ethically acceptable to climb up the tall trees without any safety provision. This situation provides the scope to mechanize the harvesting from such tall trees. The concept of the proposed device is based on ‘collapsible or foldable mechanisms’. Though there has been a lot of development in the field of collapsible or foldable structures, very few have been reported as research studies. Tilbert described the importance of deployable tensegrity structures and carried out the analysis of tensegrity structures [6]. The collapsible mechanism used in the present research work is known as ‘telescopic mechanism’ which is a series of telescoping tubes nested one within the other concentrically and operated with the help of cable and pulley systems. The telescopic mechanisms have been in use for various purposes. They have been in use for shipbuilding tasks for bidirectional workspace expansion [7]. The telescopic mechanisms have been in use for spacecrafts as well for antenna deployment [8, 9]. The research work proposes the application of this mechanism for harvesting purpose. As the mechanism is supposed to expand vertically, the buckling of the mechanism when deployed under its own weight becomes a factor to be carefully considered [10]. This buckling can be prevented with the help of lateral confinement using stiffeners [11]. In the presented research work, that lateral confinement has been introduced with the help of circular support.
2 Design Concept The design consists of a climbing mechanism and an end effector. The end effector depends upon the type and location of the fruit to be harvested. The paper describes climbing mechanism only which consists of two parts—the telescopic mechanism and a support system to prevent self-buckling [12] (Fig. 1).
3 Selection of Appropriate Telescopic Mechanism 3.1 Types of Telescopic Mechanism Continuous Rigging-Type Telescopic Mechanism. In this type of telescopic mast, only one cable covers all the pulleys. When cable is given a winch, the cable length passing over the pulleys shortens in length and thus lifting up of the concentric tubes becomes a lateral action [13]. The schematic diagram of continuous rigging-type telescopic mechanism is as shown in Fig. 2.
Design and Development of a Climb-Free Telescopic Mechanism … Fig. 1 Design concept of tree climbing mechanism Node 3
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Circular support system
Node 1
To Winch
Node 2
Fig. 2 Continuous rigging-type telescopic mechanism (a schematic of its functionality, not the actual construction of the mechanism)
The major problem with this mechanism is that the wire or cable may lose contact with the pulley when it retracts back at very high speed because of inertia effects. To overcome such situation, separate cables for lifting and retracting can be used, but this makes the system even more complex. One major drawback of this system is that slots are needed to be cut out on the concentric tubes which will in turn affect the stiffness or strength of the concentric tubes. Static Analysis of Continuous Rigging-type Telescopic Mechanism. The different concentric tubes can be termed as stages, viz. first stage, second stage and third stage, respectively. Considering S 1 , S 2 and S 3 as the weights of the different stages. The first stage with weight S 1 will be fixed to the ground. The cable is passed over to
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the pulleys present at various nodes, namely Node 1, Node 2 and Node 3, respectively. As the cable is going to support the weight to be lifted W, Stage S 2 and Stage S 3 , the tension in the whole cable would be the summation of W, S 2 and S 3 . T = W + S2 + S3
(1)
Dynamic Analysis of Continuous Rigging-type Telescopic Mechanism. The analysis is similar to that in the case of static analysis. The only difference is that the apparent weight of each concentric tube becomes the actual weight of the tube plus the pseudoforce generated in it due to acceleration effects. Let us say x¨ is the instantaneous acceleration provided at the winch end, then the acceleration of second and third stage ¨ respectively. These instantaneous accelerations at the different would be x2¨ and x, stages are responsible for pseudo-weights or pseudo forces. Therefore, the total tension in the cable will be given as follows. T = W + S2 + S3 +
(W + S3 ) S2 x¨ ( )+ x¨ g 2 g
(2)
Cascade Rigging-Type Telescopic Mechanism. In this type of telescopic mast, discrete cable elements are used for secondary actuation [14]. The primary actuation (actuation of the 2nd Stage) can be done by means of lead screw or winch mechanism. The advantage of this system over its continuous counterpart is that there is no need to cut out slots in the concentric shells. So, strength or stiffness of the members is retained in this case [15]. Likewise, this system can also have separate cables for lifting and retracting operations (Fig. 3). Fig. 3 Cascade rigging-type telescopic mechanism (a schematic of its functionality, not the actual construction of the mechanism)
2 ̈ W ̈
T2
Node 3
S3 T2
S2
Node 1 T1 T1 ̈
S1
Node 2
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Static Analysis of Cascade Rigging-type Telescopic Mechanism. Just like in the case of continuous rigging-type telescopic mechanism, it has three stages, namely Stage 1, Stage 2 and Stage 3, respectively. Likewise, it may have more number of stages depending upon the degree to which system must be made compact. Since individual cables are used, every cable has its own tension, namely T 1 and T 2 depending upon its static configuration. T1 = 2(W + S3 ) + S2
(3)
T2 = W + S3
(4)
The maximum tension T 1 is more in magnitude in comparison with continuous counterpart. So, at a cost of ‘lesser time to get deployed’ trait, the cable is going to bear more tension. Dynamic Analysis of Cascade Rigging-Type Telescopic Mechanism. The apparent weight of the mast will be the actual weight of the mast plus the pseudo-forces generated due to acceleration. When winch end is provided with an acceleration of x, ¨ the stage 2 will have the acceleration of x¨ and stage 3 will be having acceleration of 2 x¨ (motion of Stage 3 is because of two actuations—Primary actuation and the actuation due to the differentiation of length of cable passing over to the pulley). 4(W + S3 ) S2 (x) ¨ + S2 + x¨ g g
(5)
(W + S3 ) 2(W + S3 ) (2 x) ¨ = (W + S3 ) + (x) ¨ g g
(6)
T1 = 2(W + S3 ) + T2 = (W + S3 ) +
These tensions are more than those in case of static tensions and are dependent on the value of acceleration. So, the acceleration either has to be avoided or limited against the strength of the cable.
3.2 Comparison of Telescopic Mechanisms The comparison of Continuous Rigging and Cascade Rigging-type telescoping mechanism is based on the following parameters. End Effector-to-Winch Displacement Ratio. This term may be defined as the ratio of displacement at the end effectors or at the final stage to the displacement given at the winch end. For continuous rigging-type telescopic mechanism, the end effectorto-winch displacement ratio is always equal to one because the displacement given at the winch end gets distributed among the number of working stages.
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End Effector-to-winch Displacement Ratio = 1 Relative Displacement in between Stages =
x n−1
(7) (8)
The end effector-to-winch displacement ratio in case of the cascade rigging-type telescopic mechanism will be equal to (n−1), and total displacement will be (n−1)x, where x denotes the displacement given at the winch end and n−1 denotes number of working stages in a n-member telescopic mechanism. End Effector-to-winch displacement Ratio = n−1
(9)
Relative Displacement in between Stages = x
(10)
Time for Deployment. Due to greater amplification factor, the cascade rigging-type telescopic mechanism will take lesser time to deploy for a particular speed of winch. If τ is the time taken by the continuous rigging-type telescopic mechanism to deploy, then cascade rigging-type telescopic mechanism would take the time which is the time in case of continuous rigging-type telescopic mechanism divided by number of x . working stages, i.e. n−1 Tensions. In continuous rigging-type telescopic mechanism, there is only one tension which is uniform throughout the span of the cable. On the other hand, every discrete cable has different tensions in case of cascade rigging-type telescopic mechanism. The thing which is noticeable in the equations of tensions is that at an expense of more displacement amplification, the tension T 1 seems much more than the tension T in case of continuous rigging-type telescopic mechanism and tension T 2 is smaller in magnitude than T. It can also be noticed that different cables have different tensions, and different cables of different strengths can be used so as to optimize the material requirement of the mechanism. Tensions (Considering Pseudo Forces). Even though the fact that the system is safe against static forces, still system can fail because pseudo-forces come into play when mechanism is deployed with acceleration. So, acceleration either has to be avoided or limited against the strength of the cable. The tensions considering pseudo-forces are as under. (W + S3 ) S2 x¨ + x¨ (11) T = W + S2 + S3 + g 2 g 4(W + S3 ) S2 (x) ¨ + S2 + x¨ g g
(12)
(W + S3 ) 2(W + S3 ) (2 x) ¨ = (W + S3 ) + (x) ¨ g g
(13)
T1 = 2(W + S3 ) + T2 = (W + S3 ) +
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Strength of the Mast. The strength of the concentric masts is lesser in case of continuous rigging-type telescopic mechanism because slots have to be cut out of the masts so as to accommodate the continuous cable. Slots have adverse effect on the stiffness of the masts. Moreover, slots reduce fatigue strength as well as they introduce discontinuity in material. The strength of the concentric masts in case of cascade rigging-type telescopic mechanism is more because no slots need to be cut for cables.
3.3 Selection of Mechanism From the equations, it is evident that there is a tradeoff between time of deployment and tensions in cable. If the mechanism has to deploy faster for a particular winch speed, then the suitable mechanism is cascade rigging-type telescopic mechanism. But increased time for deployment will be on the expense of generation of higher tension in the cable. Likewise, opting for continuous rigging-type telescopic mechanism increases the time for deployment for a given winch speed but the tension in the cable will be lesser in magnitude. Considering that there will be a tradeoff between time for deployment and tensions in cable, cascade rigging-type telescopic mechanism was selected as the mechanism to go up the tall tree, as the mechanism is supposed to be speedy while going up for a particular winch speed. However, the mechanism should be designed carefully because cables have to bear more tension and the slot cutouts to accommodate the cables also have to be considered.
4 Concept of Support System A very long and tall structure tends to buckle under its own weight whenever its height is more than a certain threshold. The mechanism, when erected, may buckle under its own weight plus the weight that has been supported over it. Therefore, to avoid buckling, the telescopic mechanism can be provided a support of the tree. The mechanism consists of linkage systems (L and L’) and hinges (H and H’) to adjust according to the morphology of the tree. The support part is closed with the help of chain C which enables the support to adjust according to the diameter of the tree without opening up completely and leaving the contact with the tree. The torsion springs S and S’ make sure that there is positive contact with the tree trunk. The conceptual design and the top view of the mechanism are as shown in Fig. 4. The mechanism can compensate for varying radial distance from tree, tree shape and trunks going sideways as shown in Fig. 5.
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Fig. 4 Design of support system
a. Radial Adjustment
b. Tree shape compensation Fig. 5 Set of compensations by support system
c. Sideways adjustment
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5 Testing of Prototype A model coconut tree was built using PVC pipes. The telescopic mechanism was prototyped using pipes and pulleys. A weigh balance was used to measure the tension in the strings. The pulleys were configured as per the type of mechanism to be tested. The tests were conducted on the prototype with the following parameters (Table 1; Fig. 6). During the tests conducted, the theoretical and practical results were compared (Table 2). It was observed that cascade rigging-type telescopic mechanism is faster to deploy to its complete length when compared to continuous rigging-type telescopic mechanism. Table 1 Parameters for the prototype testing
Fig. 6 Prototype testing setup
Parameter
Value
S 1 (Weight of first stage)
1.073 kg
S 2 (Weight of second stage)
0.86 kg
S3
0.735 kg
W
0 kg
x˙ (winch speed)
0.05 m/s
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Table 2 Test results Continuous rigging-type telescopic mechanism
Cascade rigging-type telescopic mechanism
T = 1.595 Kgf (Theoretical)
T = 2.33 kgf (Theoretical)
T = 1.793 Kgf (Observed)
T = 2.49 kgf (Observed)
Time for deployment
15 s
7s
End effector-to-winch displacement ratio
1 (Theoretical)
2 (Theoretical)
0.96 (Observed)
1.95 (Observed)
Maximum tension
6 Conclusion In this paper, we have discussed the types of telescopic mechanism which are continuous rigging-type and cascade rigging-type mechanism. From the tests, we came to know that there is an inverse relation between speed of deployment and tension in the cable. If we have to go for the speed, then the cable will have to bear more tension. Likewise, with less speed of deployment, the tension in the cable would be lesser. The criterion for selection of mechanism was speed. Therefore, we opted for cascade rigging-type telescopic mechanism. Additionally, the concept of the support system has been described which can adjust according to the shape and trajectory of the tree. This simple, cost-effective and easily transportable system can form the platform for harvesting system from tall trees. A suitable end-effector can be attached depending on the type of fruit to be harvested. Acknowledgements The research work was supported by SERB (a statutory body of the Department of Science and Technology, Government of India) through grant EEQ/2017/000598, for which the authors are immensely grateful. The support of CSIR-CSIO for providing laboratory and prototyping facilities is also deeply acknowledged.
References 1. Punchihewa PG, Arancon RN (2006) Coconut: post-harvest operations. Asian and Pacific Coconut Community 2. Coconut Development Board. https://coconutboard.nic.in. Last accessed 19 Jul 2019 3. Burton J The world leaders in coconut production. https://www.worldatlas.com/articles/theworld-leaders-in-coconut-production.html 4. Singh RM India leading in coconut production and productivity in the world. https://pib.nic. in/newsite/PrintRelease.aspx?relid=149492 5. Megalingam RK et al 2017 IOP Conf Ser: Mater Sci Eng 225012201 6. Tilbert G (2002) Deployable tensegrity structures for space applications 7. Lee D, Chang D, Shin Y, Son D, Kim T, Lee K, Kim J (2011) Design and application of a wire-driven bidirectional telescopic mechanism for workspace expansion with a focus on shipbuilding tasks. Adv Robot 25:699–715 8. Leavy W, Griffin C (1979) Antenna deployment mechanism for use with a spacecraft
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9. Allen B, Butler D Hoop/column antenna deployment mechanism 10. Duan W, Wang C (2008) Exact solution for buckling of columns including self-weight. J Eng Mech 134:116–119 11. Hancock G, Kwon Y, Stefan Bernard E (1994) Strength design curves for thin-walled sections undergoing distortional buckling. J Constr Steel Res 31:169–186 12. Wei D, Yan S, Zhang Z, Li X (2010) Critical load for buckling of non-prismatic columns under self-weight and tip force. Mech Res Commun 37:554–558 13. Becchi P, Dellamico S (1989) Design and testing of a deployable, retrievable boom for space applications 14. Introduction to Manipulators. https://www.instructables.com/id/Introduction-to-Manipulat ors/. Last accessed 20 Jul 2019 15. Whitefield B (2012) Manipulators for FIRST FRC Robotics
Simulation Modeling of 37 Degrees-of-Freedom ICF Coach Bharath B. Mahadikar, Charanpreet Singh, Akarsh K. S., and C. V. Chandrashekara
Abstract The safety and comfort of the locomotives depend on the basic suspension design against the dynamic load, to meet the desired dynamic characteristics. The passenger’s comfort and goods safety in a locomotive are correlated with the performance of suspension system subjected to external excitation. In the present paper, a complex 37 degrees-of-freedom Integral Coach Factory (ICF) locomotive model is developed using discretized lumped mass system approach. The equations of motion are derived using Newton’s second law of motion. The scaled down simulation model is developed using multi-body dynamics software ADAMS. The first nine natural frequencies observed in simulation model are in good correlation with the analytical results. Keywords Discretized · Lumped mass · Multi-body dynamics · Simulation · ADAMS
1 Introduction The safety and comfort of the locomotives depend on the basic suspension design against the dynamic load, to meet the desired dynamic characteristics. The passenger’s comfort and goods safety in a locomotive are correlated with the performance of suspension system subjected to external excitation. Wickens [1] discussed the lateral stability of railway vehicles and compared with the experimental work. Chang et al. [2] studied the behavior of the vertical irregularities for a six-axle locomotive. For the first time, a system with higher degrees-of-freedom is considered and is used for determining different design parameters and dynamic response. Nishimura et al. [3] discussed the suspension design for high-speed railway vehicles for a 10 degrees-of-freedom of system. The design strategy includes the development of a B. B. Mahadikar (B) · C. Singh · Akarsh K. S. · C. V. Chandrashekara Department of Mechanical Engineering, PES University, Bengaluru, India e-mail: [email protected] C. V. Chandrashekara e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_76
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lumped parameter vehicle model, formulating the design objective and to achieve optimized key suspension parameters. Dikmen et al. [4] examined the vibrational characteristics of a rail vehicle considering a 19 degrees-of-freedom system. Critical speeds of the system are determined for yawing and pitching motions. Sharma [5] formulated a 37 degree-of-freedom system using Lagrangian dynamics to evaluate the influence of rail parameters on vertical and lateral ride behavior. Graa et al. [6] investigated the effects of vehicle speed and rail irregularity on ride comfort through numerical simulation. A seventeen degrees-of-freedom of system is developed using Lagrangian dynamics. A parametric study is carried out in order to improve the ride characteristics such as sterling index. In the present paper, a complex coupled 37 degrees-of-freedom Integral Coach Factory (ICF) locomotive model is developed using discretized lumped mass system approach. The equations of motion are derived using Newton’s second law of motion. The geometrical parameters for solid modeling of the components are obtained from field visit to Carriage Repair Workshop Hubballi, Karnataka. The overall dimensions of the coach are obtained from the open source. The scaled down solid model is developed using Solid Edge software. The simulation of the model is carried out in the ADAMS environment. The responses and natural frequencies are extracted for the coupled system and being reported for the first time. The rail car parameters taken into account for analysis are car body, bolster, bogie frame, primary and secondary suspension, and wheel–axle sets. The effects of disturbances/excitation and the relative motion of various parts on the model are studied. The first nine natural frequencies observed in simulation model are in good correlation with the analytical results.
2 Simulation Modeling Thirty-seven degrees-of-freedom rail car body is considered for study. Scaled down ICF coach system is modeled using Adams software. All masses are rigid in the model. Realistic values are incorporated for masses of each body and the values of stiffnesses and damping coefficients along different directions as listed in the analytical model. The model is constructed with following assumptions: • • • •
Friction between moving parts is neglected All masses are rigid and the system is symmetric about the longitudinal plane The wheel does not lose contact with the track Base excitations are considered as sinusoidal displacement–time functions that reflect the irregularities of the track profile.
It includes various types of translational and rotational motions like pitch, roll, and yaw for individual masses in the model. The track is made as ground and the profile of the track is modeled in terms of individual motions about three axes (x, y and z) given to the four wheel–axle assemblies as listed in Table 1. The motions are
Simulation Modeling of 37 Degrees-of-Freedom ICF Coach Table 1 Induced motion at wheel–axles
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Wheel–axle number
Direction
Excitation function (m)
1
Vertical
0.003 sin(2t)
Lateral
0.003 sin(2t)
Vertical
0.001 sin(2t)
2
Lateral
0.001 sin(2t)
3
Vertical
0.002 sin(2t)
Lateral
0.002 sin(2t)
4
Vertical
0.005 sin(2t)
Lateral
0.005 sin(2t)
defined as sinusoidal input. It is assumed that the wheel–axles are undamped with respect to the track at all 8 points of contacts. All masses have restricted translational motion along x-axis and allowed to freely vibrate along y-axis and z-axis. There are 8 spring-dampers along vertical direction (y-axis) and 8 spring-dampers along lateral direction (z-axis) connecting the 4 wheel– axles to the 2 bogie frames. Each bogie frame is connected to a bolster via 2 springdampers along vertical direction. There are 4 spring-dampers linking the 2 bolsters to the car body along vertical direction and 4 spring-dampers along the lateral direction linking the 2 bolsters to the car body. The four wheel–axle assemblies constitute 16 degrees-of-freedom (4 vertical, 4 lateral, 4 yaw, and 4 roll). The 2 bogie frames account for 10 degrees-of-freedom (2 vertical, 2 lateral, 2 pitch, 2 roll, and 2 yaw). The 2 bolsters constitute 6 degrees-offreedom (2 vertical, 2 lateral, and 2 roll). The car body is having 5 degrees-of-freedom (1 vertical, 1 lateral, 1 pitch, 1 roll, and 1 yaw). The total degrees-of-freedom for the lumped mass system adds up to 37. The parameters of the simulation model constructed are in reference to Sharma [5]. The model is shown in Fig. 1. y
y x
Fig. 1 37 degrees-of-freedom rail car body side view and front view
z
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3 Equations of Motion The equations of motion for all 37 degree-of-freedom are derived using Newton’s second law of motion considering the coupled motion for the first time. The equations of motion are reduced to matrix form as shown in Eq. (1). [M] X¨ + [C] X˙ + [K ]X = 0
(1)
where [M], [K ], [C] are the matrices extracted from the 37 equations. Using Eq. (1), natural frequencies and response of the system are evaluated.
4 Natural Frequencies A system with n degrees-of-freedom will have n vibration modes, having a natural frequency corresponding to it. If an undamped or mildly damped system is excited externally at or near to its natural frequencies, then the amplitude of the vibration increases exponentially. This phenomenon is known as resonance. It is therefore important to determine the natural frequencies of a system and its excitation frequencies in order to avoid resonance. In the ADAMS interface, the ‘Run an Interactive Simulation’ option is selected under the ‘Simulation’ tab. The ‘Simulation Control’ pop-up dialog box appears. The free vibration frequencies are extracted using the ‘Compute linear modes’ option. This option opens another dialog box showing eigen values of free vibration. The undamped natural frequencies of the system for the 9 vertical modes, from the dialog box, are listed in Table 2. The natural frequencies obtained by using analytical and simulation model are matching with a maximum error of 3.862%. Table 2 Natural frequencies of the system S. No.
Natural frequencies in ADAMS (Hz)
Natural frequencies in MATLAB (Hz)
Percentage error (%)
1
0.9190
0.8835
3.862
2
5.6827
5.6827
0
3
5.7131
5.7107
0.042
4
66.9819
66.9819
0
5
67.7252
67.6675
6
184.6276
184.6276
0
0.0864
7
184.6276
184.6276
0
8
184.6276
184.6276
0
9
184.6276
184.6276
0
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Fig. 2 Response curves plotted for vertical and lateral vibration of bogie frames
5 Response of the System The geometrical and physical parameters considered for the system are the realistic values for an ICF coach. To understand the behavior of car body as influenced by the excitation at wheel–axles, the simulation is conducted. In the ADAMS interface, the ‘Run an Interactive Simulation’ option is selected under the ‘Simulation’ tab. The ‘End Time’ and the ‘Step Size’ are now set in the ‘Simulation Control’ pop-up dialog box. These parameters are set for 20 s with a step size of 0.01. The responses are extracted and are plotted in time domain using the ‘Plot output’ option. Vertical and lateral responses are obtained for bogie frames as shown in Fig. 2. The response curves for car body in lateral and rolling modes are shown in Fig. 3. The response curves for car body in vertical and pitching modes are shown in Fig. 4. It is observed from the graphs that the responses are uniform and periodic for the given set of kinematic parameters of the system. It is also evident from the plots of the car body that the amplitude of vibration is reduced from that of the wheels. Along the vertical direction, the amplitude of car body is around 3 mm and along lateral direction it lies within 1 mm. The complex models considered in Sect. 4.8 considering higher degrees-of-freedom help in understanding of the behavior of each part subjected to vibration along different directions. The dynamic analysis of simulation reports shows that the vibration of the car body housing the passengers lies within the comfort range.
6 Conclusion The dynamic characteristics of the ICF coach rail vehicle, considered as a 37 degrees-of-freedom system, are obtained. The dynamic characteristics include
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Fig. 3 Lateral and rolling response curves of car body
Fig. 4 Vertical and pitching response curves of car body
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natural frequency and the responses of the system. The responses obtained are in good correlation with the mathematical modeling equations. The natural frequencies and responses of the system are also in good correlation with both the simulation and mathematical model for the basic model.
References 1. Wickens AH (1965) Dynamics of railway vehicles on straight track fundamental considerations of lateral stability. In: Proceedings of institution of mechanical engineers 1965–66, vol 180 2. Chang EH, Garg VK, Goodspeed CH, Singh SP (1979) Comparative study of the linear and non-linear locomotive response. J Dyn Syst Meas Control 101 3. Nishimura K, Perkins NC, Zhang W (2004) Suspension dynamics and design optimization of a high-speed railway vehicle. In: Proceedings of the 2004 ASME/IEEE joint rail conference 4. Dikmen F, Bayraktar M, Guclu R (2011) Vibration analysis of 19 degrees-of-freedom rail vehicle. Sci Res Essays 6(26):5600–5608 5. Sharma RC (2011) Parametric analysis of rail vehicle parameters influencing ride behavior. Int J Eng Sci Technol 3(8):54–65 6. Graa M, Nejlaoui M, Houidi A, Affi Z, Romdhane L (2015) Modeling and simulation for vertical rail vehicle dynamic vibration with comfort evaluation. In: Multiphysics and simulation for systems design and monitoring, applied condition monitoring. Springer International Publishing Switzerland, pp 47–57
Transmission Efficiency and Surface Damage of Polymer–Polymer Gear Pair Under Wet Lubrication Sarita Bharti
and Selvaraj Senthilvelan
Abstract Polyamide, with its low friction coefficient, high heat resistance and good moldability, is a promising polymer to meet the rising demand for lightweight, durable gears. This paper considered injection-molded polyamide 66-polyamide 66 gear pair wear performance under unlubricated and wet lubricated condition. The experiment was conducted using in-house developed power absorption gear test rig. The torque of 1.8 Nm, rotational speed 800 rev/s and lubricant SAE 75W85 was considered during testing. During the test, gear tooth temperature under unlubricated condition and lubricant temperature under wet lubricant condition was measured and monitored. The net surface temperature on the gear surface and lubricant was observed as 370 K and 303.4 K, respectively. The transmission efficiency was increased by 5% compared to the unlubricated condition. Testing confirmed that the gear exhibited surface wear at both the face and flank region in unlubricated condition. While testing under wet lubricant condition, the wear was observed on the face of the gear. Simulation result confirmed that the deflection of test gear is significantly higher in double tooth contact region shared by the face side as compared to flank side of the driven (test) gear. That could be the possible reason for test tooth wear in the face region. Keywords Polymer · Spur gear · Lubrication · Transmission efficiency
1 Introduction Polyamide gears are being extensively used for light-duty engineering applications such as robotics, automated teller machine, food processing industry and office equipments. Polymer material exhibits inferior wear and poor heat resistance, which affects the transmission performance of gears. Usage of reinforcement would reduce the S. Bharti (B) · S. Senthilvelan Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India e-mail: [email protected]
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amount of frictional and hysteresis heating [1, 8, 14]. Multiple works have been carried out to understand the behavior of polymer–polymer and metal–polymer gear pair under unlubricated condition. In polymer–polymer gear pair contact, gears fail due to local thermal softening. Extensive investigation has been carried out on metal–metal gear pair under wet lubricated condition. Hence, in the present research, unreinforced polyamide 66 spur gear was run under the wet lubricated condition. Breeds et al. [2] evaluated the wear characteristics of polyamide–polyamide and acetal–acetal gear pair. Result revealed that gear exhibited wear failure at low loading condition. In contrast, at higher loading, gear exhibited failure due to thermal failure. Pogaˇcnik and Tavˇcar [15] adopted an accelerated testing method on polyoxymethylene (POM) and polyamide 6 gear pair. Gear failed mainly due to fatigue and sudden melting. Hu and Mao [6] examined the misalignment effect on the acetal–acetal gear pair under unlubricated condition. Mertnes and Senthilvelan [13] have attempted to improve polymer gear performance by supplying compressed air on the contact surface. Results confirmed that air cooling reduced the temperature and improved the wear resistance. Bushimata et al. [3] investigated the wear characteristics of unreinforced PA 66, polyacetal, PEEK and glass fiber-reinforced polyamide, glassreinforced polyacetal and glass-reinforced PPS gears. Results confirmed that when similar material gear pair meshed, unreinforced PA 66-PA 66 gear pair exhibited highest wear. In contrast, acetal and PA/PEEK gear pair exhibited the lowest level of wear in comparison with other dissimilar material gear pair, due to less adhesion. Results suggested that wear failure may be avoided by using an appropriate combination of gear pair. Senthilvelan and Gnanamoorthy [18] investigated the performance of the polyamide 66 carbon fiber reinforced and polyamide 66 gear. Increase in gear tooth temperature worsens the gear stiffness during testing which causes lower transmission efficiency. Dearn et al. [4] evaluated the efficiency of polyether ether ketone (PEEK)–polyether ether ketone gear pair, and polyether ether ketone–steal gear pair lubricated with automotive engine oil. Results revealed that PEEK–PEEK gear pair exhibited higher efficiency compared PEEK–steel gear pair may be due to large tooth deformation occurred on the PEEK gear. Walton et al. [20] examined the behavior of the gear pair’s efficiency on speed and load. Polyamide composite- steel gear pair efficiency at low speed found to be less compared to polyamide composite– polyamide composite gear pair. On the other hand, at higher speed, both gear pairs showed similar efficiency. Mao et al. [12] investigated the performance of wire cut acetal gear pairs and compared it with injection-molded acetal gear pair. Results confirmed that both gears failed similarly. Kodeeswaran et al. [9] attempted to evaluate the performance of unidirectional and bi-directional load polyamide gear. Results concluded that thermomechanical and root crack were detected in the case of bi-directional loading. On the other hand, the only root crack was observed in the case of unidirectional loading. Kodeeswaran et al. [10] predicted the transmission error using finite element technique and compared with experimental data. The static transmission error with the linear material models was lower than that of the nonlinear material models, and it is further decreased with
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increased strain rate, for both single tooth contact and double tooth contact. Senthilvelan and Gnanamoorthy [16] investigated the effects of different filet radius on the performance of polyamide 66 gear. Higher fillet radius 0.75 mm gear exhibited a better performance compared to lower fillet radius 0.25 mm. Dearn et al. [5] studied the different solid lubricant coated on polymer gear. Results found that the frictional heating reduced significantly in case of PTFE coated gear. The failure of gear is due to mainly delamination of coated and abrasive wear. Kirupasankar et al. [7] investigated the polyamide 6 and clay-reinforced polyamide gear. Results concluded that reinforcement is improved gear efficiency. Senthilvelan and Gnanamoorthy [18] attempted to correlate the reinforcement on the profile of the polymer gear. Results confirmed that reinforcement increases lead deviation. Sarita and Senthilvelan [16] investigated the performance of polyamide 66–steel gear pair under lubricated condition. Results confirmed that the scuffing occurred on the flank region of test gear. Pitting, tooth breakage and tooth root crack are the failure modes of the PEEK gear run against steel under lubricated condition examined by Lu et al. [11]. Although from the above-mentioned literature, the gear performance of polyamide against steel gear and polyamide against polymer gear has been studied extensively, the amount of literature on polyamide–steel gear pair and polyamide–polyamide gear pair behavior under lubricated condition is still limited. In this paper, the effect of wet lubrication on the injection-molded polyamide 66– polyamide 66 spur gear pair transmission efficiency and temperature is discussed and compared with the unlubricated condition. Damage modes of gear observed with different test conditions are reported.
2 Methodology 2.1 Experimental Methodology In-house developed power absorption gear test rig was used to evaluate the contact load-bearing performance of injection-molded polyamide 66 spur gear. Gear tests were carried out at a torque of 1.8 Nm and 800 revs/min rotational speed under unlubricated and wet lubricated condition. Wet lubricated condition test was carried in a splash lubricated system using SAE 75W85 grade gear oil (kinematic viscosity at 373 K: (11–15) cSt). 8 and 4 mm face width polyamide gears were used as driving and driven gear, respectively. A personal computer-based data acquisition method was used to continuously monitor gear’s surface temperature (only unlubricated condition) for driven gear and torque for both driving and driven gears. Lubricant temperature was measured using J-type thermocouple. The detailed explanation of the test rig is described elsewhere, Sarita and Senthilvelan [16]. Test gear the post-damage morphology investigation was carried out by the digital microscope (celestron, digital microscope pro). In-house developed power absorption gear experimental setup and molded polyamide 66 gears are shown in Fig. 1a and b, respectively. Transmission
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Fig. 1 a In-house developed power absorption gear experimental setup and b Molded polyamide 66 gears
efficiency between the driver and the driven gear shaft was calculated using Eq. (1). Gear transmission efficiency(%) =
Torque measured at driven × 100 Torque measured at driver gear
(1)
Here, for simplification in a calculation, the bearing loss and gear inaccuracies were neglected (Table 1). Table 1 Test gear parameter details
Parameters
Driver gear
Material
Polyamide 66 Polyamide 66
Pressure angle
20°
Module
3
3
Number of teeth
18
18
Outer circle diameter (mm)
60
60
Face width (mm)
8
Hub inner circle diameter (mm) 15
Driven gear 20°
4 15
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Fig. 2 a Multipair contact model of the spur gear and b meshed model
2.2 Numerical Analysis To investigate the effect of mating gear material on the damage characteristics, it is imperative to characterize the tooth deflection behavior along the line of action. Since transmission error is a function of tooth deflection, it was studied as well. Gear tooth deflection causes rotational lag, which results in multiple issues such as increase vibration and premature failure. Tooth deflection was determined using finite element analysis (FEA) for polymer–polymer pair. The FEA ABAQUS model is shown in Fig. 2. The Young’s modulus of polyamide 66 is 1.071 GPa. The coefficient of friction between contacts assumed as 0.1. Torque was applied at the center of the driving gear. A kinematic coupling was established between the hub and center of driving gear to transfer the torque. All degrees of freedom were constrained at the hub of the driven gear to prevent the rotation of gear. The mesh of the driving and driven gear was identical. The contact region was discretized with four-noded quadratic elements of CPE4R 0.05 mm edge length. The deflection was determined for a torque of 1.8 Nm.
3 Results and Discussion 3.1 Temperature Measurement The net surface temperature of test gear tooth was measured under the unlubricated condition measured using the non-contact temperature sensor (Raytek, MID10LT). Under the wet lubricated condition, lubricant temperature was measured using a thermocouple. Figure 3 shows that the gradual increase in temperature was observed in the gear tooth surface during the initial period. The thermal balance was found to be stable between heat generated and heat dissipated after approximately 1.2 × 105 number of cycles. During the test, frictional heat was generated because the pair of gears experienced both rolling and sliding contact due to surface interaction. Since the thermoplastic material is viscoelastic in nature, material hysteresis due to
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Fig. 3 Shows the variation of gear tooth surface temperature and lubricant temperature with respect to no. of cycles run
repeated gear tooth deflection results in heat generation. A similar trend in temperature has been reported by several researchers. Sarita and Senthilvelan [16] observed around 360 K as stable temperature while testing PA 66 against steel gear at 1.8 Nm. Senthilvelan and Gnanamoorthy [17] also noticed stable temperature (340) K when PA 66 runs at 2 Nm. The increase in the lubricant temperature is very low, and this lubricant temperature plays a vital role in heat dissipation from the gear tooth surface.
3.2 Transmission Efficiency Measurement To quantify the effect of gear tooth wear performance, evaluation of gear transmission efficiency was carried out. The sliding and rolling motion of gear tooth which leads to frictional heating affect the power loss and transmission efficiency. In this work, transmission efficiency was evaluated only considering the loss due to friction and wear. Transmission efficiency under the unlubricated condition and wet lubricated condition is shown in Fig. 4. Transmission efficiency under lubricated condition was found to be higher as compared to the unlubricated condition. Under the unlubricated condition, test gear exhibited more frictional heating on the contact surface which led to less transmission efficiency. In the presence of lubricant, the generated frictional heat between the contact surfaces is less due to the lubricant present between the contact surfaces. This dissipates the heat generated and increases efficiency. The percentage (%) variation in transmission efficiency for unlubricated condition and wet lubricated condition is less than ± 5% for and ± 6%, respectively.
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Fig. 4 Shows the gear efficiency under the unlubricated and wet lubricated conditions
3.3 Test Gear Surface Wear Morphology Figure 5 shows the damage surface morphology of gear tooth surface subjected at a torque of 1.8 Nm under unlubricated and lubricated condition. Under the unlubricated condition, high surface wear was observed compared to the wet lubricated condition. Since the thermal conductivity is low in the polymer material, the damage in polymer gears is higher under unlubricated condition. In the presence of lubricant, the direct contact of tooth surfaces is prevented by the lubricant. The adsorption of lubricant between the contact surfaces smoothens the rotation between the surfaces and reduces the friction between them. The microscopic examination of gear tooth damage in unlubricated condition revealed the occurrence of excessive wear in both face and flank regions. In contrast, under lubricated condition, the wear was higher near the face region compared to the flank region. Similar wear pattern
Fig. 5 Damage surface of gear subjected at 1.8 Nm, a under unlubricated condition and b under lubricated condition
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Fig. 6 Predicted gear tooth deflection along the line of action
was observed by Pogaˇcnik and Tavˇcar [15] while testing the POM–POM polymer gear under unlubricated condition. Figure 6 depicts the variation of tooth deflection of driven gear simulated in ABAQUS. The maximum deflection is at the highest point of the single tooth contact region (HPSTC). In the double tooth contact region, tooth deflection progressively increases in the addendum (face) region of the test gear tooth, whereas it decreased in the dedendum (flank) region. Under the lubricated condition, the wear was observed in the face region only, as the lubricant minimized the gear tooth wear.
4 Conclusion In this work, polyamide 66–polyamide 66 spur gears were tested. Test gear exhibited higher transmission efficiency under the wet lubricated condition at 1.8 Nm loading condition. Under the unlubricated condition, the net gear tooth surface becomes stable at 370 K. Lubricant temperature was limited at 303.4 K under lubricated condition, and however, actual gear tooth surface temperature could be higher. Gear surface wear was more under unlubricated condition compared to the wet lubricated condition. Under wet lubricated condition, wear was observed at the gear tooth face region due to higher gear deflection in this region.
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References 1. Basavaraj E, Ramaraj B, Lee JH (2013) Polyamide 6/carbon black/molybdenum disulphide composites: friction, wear and morphological characteristics. Mater Chem Phys 138(2–3):658– 665 2. Breeds AR, Kukureka SN, Mao K, Walton D, Hooke CJ (1993) Wear behaviour of acetal gear pairs. Wear 166(1):85–91 3. Bushimata T, Sakuta H, Asai T (2001) Wear of plastic spur gears made by injection molding. Jap J Tribol 46(6):347–358 4. Dearn KD, Hoskins TJ, Andrei L, Walton D (2013) Lubrication regimes in high-performance polymer spur gears. Adv Tribol 2013 5. Dearn KD, Hoskins TJ, Petrov DG, Reynolds SC, Banks R (2013) Applications of dry film lubricants for polymer gears. Wear 298:99–108 6. Hu Z, Mao K (2017) An investigation of misalignment effects on the performance of acetal gears. Tribol Int 116:394–402 7. Kirupasankar S, Gurunathan C, Gnanamoorthy R (2012) Transmission efficiency of polyamide nanocomposite spur gears. Mater Des 39:338–343 8. Kodeeswaran M, Suresh R, Senthilvelan S (2019) Effect of strain rate on bending and transmission characteristics of injection molded polyamide 66 spur gears. Proc Inst Mech Eng Part L: J Mater: Des Appl 233(6):1145–1155 9. Kodeeswaran M, Verma A, Suresh R, Senthilvelan S (2019) Effects of frequency on hysteretic heating and fatigue life of unreinforced injection molded polyamide 66 spur gears. Proc Inst Mech Eng Part L: J Mater: Des Appl 233(5):781–789 10. Kodeeswaran M, Suresh R, Senthilvelan S (2016) Transmission characteristics of injection moulded polymer spur gears: experimental and numerical evaluation. Int J Powertrains 5(3):246–263 11. Lu Z, Liu H, Zhu C, Song H, Yu G (2019) Identification of failure modes of a PEEK-steel gear pair under lubrication. Int J Fatigue 125:342–348 12. Mao K, Langlois P, Hu Z, Alharbi K, Xu X, Milson M, Li W, Hooke CJ, Chetwynd D (2015) The wear and thermal mechanical contact behaviour of machine cut polymer gears. Wear 332:822–826 13. Mertens AJ, Senthilvelan S (2016) Durability enhancement of polymer gear using compressed air cooling. Proc Inst Mech Eng Part L: J Mater: Des Appl 230(2):515–525 14. Mertens AJ, Senthilvelan S (2018) Surface durability of injection-moulded carbon nanotube– polypropylene spur gears. Proc Inst Mech Eng Part L: J Mater: Des Appl 232(11):909–921 15. Pogaˇcnik A, Tavˇcar J (2015) An accelerated multilevel test and design procedure for polymer gears. Mater Des 1980–2015(65):961–973 16. Sarita B, Senthilvelan S (2019) Effects of lubricant on the surface durability of an injection molded polyamide 66 spur gear paired with a steel gear. Tribol Int 137:193–211 17. Senthilvelan S, Gnanamoorthy R (2006) Effect of gear tooth fillet radius on the performance of injection molded Nylon 6/6 gears. Mater Des 27(8):632–639 18. Senthilvelan S, Gnanamoorthy R (2008) Influence of reinforcement on composite gear metrology. Mech Mach Theory 43(9):1198–1209 19. Senthilvelan S, Gnanamoorthy R (2009) Efficiency of injection-moulded polymer composite spur gears. Proc Inst Mech Eng Part J: J Eng Tribol 223(6):925–928 20. Walton D, Cropper AB, Weale DJ, Meuleman PK (2002) The efficiency and friction of plastic cylindrical gears Part 1: influence of materials. Proc Inst Mech Eng Part J: J Eng Tribol 216(2):75–78
Effect of Acceleration of Moving Object During Collision with Stationary Object Pranav V. Deosant , H. T. Thorat, and Rupesh N. Tatte
Abstract In an application where a robotic arm hits an object, the effect of impact is required to be assessed. A classical approach determines the effect of impact based on the velocities of the objects. Magnitudes of acceleration are ignored in the analysis. In the robotic application, velocity shall always be associated with acceleration as it is a start and stop type of motion. This paper demonstrates the need of accounting for the acceleration magnitudes also to determine the effect of impact. Keywords Collision · Effect of acceleration · Impact · Coefficient of restitution
Nomenclature m1 m2 L θ1 θ θ θ α α α β
Mass of moving ball (kg) Mass of stationary ball (kg) Length of string (m) Initial angle of moving ball Actual angle of oscillation of moving ball after impact Maximum angle of oscillation of moving ball after impact assuming COR = 1 Maximum angle of oscillation of moving ball after impact using experimentally determined COR Actual maximum angle of oscillation of stationary ball after impact Maximum angle of oscillation of stationary ball after impact assuming COR =1 Maximum angle of oscillation of stationary ball after impact experimentally determined COR Angle of impact
P. V. Deosant (B) · H. T. Thorat · R. N. Tatte Visvesvaraya National Institute of Technology, Nagpur 440010, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_78
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1 Introduction A robot arm during motion is subjected to driving moments provided by motors, inertia forces acting on each link, and resistance to the motion because of friction. This results in, at any given time during the motion, some acceleration along with the velocity for any point on the arm. While assuming the effect of collision of the end effector of the robotic arm with any other stationary rigid body, the impact depends upon the velocity of the equivalent centroid and the mass. In the literature, well-established relations are available for finding out the effect of velocity on the moment transfer [1] and thus the effect of collision can be determined. However, in a robotic arm, the velocity of a point during collision also experiences some kind of acceleration or retardation. This acceleration and retardation are likely to affect the transfer of momentum. Effect of the collision between two spherical bodies in Centered Impact and Oblique Impact is deliberated in textbooks of Engineering Mechanics [1]. Colliding balls with an electronic counter are used to measure the collision time as a function of energy impact [2]. In the case of oblique collision, an instant of impact defines the normal direction, along the line that connects the two masses at the centers, and the tangential direction, along the line tangent to the surfaces at the point of contact. Momentum transfer is considered for both directions [3]. The momentum transfer in the practical condition is not fully achieved, but it is reduced by the Coefficient of Restitution (COR). This happens because of loss of energy due to vibration, plastic deformation, and viscosity of sphere [4]. When two balls collide, COR is always less than one [5]. In case of oblique impact, COR is sufficient to analyze using the principle of transfer of momentum, provided coefficient of friction is zero otherwise other parameters like frictional force, dynamic coefficient of friction, impulse ratio, etc. are required to be considered [6].
2 Coefficient of Restitution (COR) In the present study, COR is determined experimentally. In the experiment, lightweight plastic balls are hanged to untwisted string to form a pendulum using hooks. Figure 1 demonstrates a situation in which the balls make a contact with each other at a lowermost position where velocity is maximum and acceleration is zero. The arrangement in Fig. 1 is used to determine the coefficient of restitution (COR) for equal and unequal masses. In these experiments, mass of moving ball (m1 ) is 0.04635 kg, and that of stationary ball (m2 ) is 0.04635 kg. In case of unequal mass condition, m1 = 0.04635 kg and m2 = 0.01213 kg. String lengths are equal (L 1 = L 2 = 0.465 m) in both the cases. Table 1 shows the result of experiments, (θ 1 ) represents angle through which moving ball is raised, (α) represents maximum angle of oscillation for stationary ball after impact measured experimentally and (α ) represents theoretical angle of
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Fig. 1 Arrangement for central impact
Table 1 COR for equal and unequal mass condition Equal mass
Unequal mass
θ1
α average
α
30
28.5
30
0.8959
41
48.44
0.7700
40
39
40
0.9142
55.66
65.65
0.7906
50
48.5
50
0.9163
69.66
84.11
0.8058
60
58.25
60
0.9015
85.33
104.85
0.7596
e
α average
α
Average value of COR = 0.906
e
Average value of COR = 0.7815
oscillation of stationary ball when COR is 1. Theoretical angle of stationary ball is calculated using a standard theoretical approach of momentum transfer. For a given value of (θ 1 ), number of experiments have been performed and corresponding values of (α) have been obtained. The average value from the above readings has been calculated and this value is presented in the table and used to calculate the COR. Therefore, for given four values of (θ 1 ), four values of COR have been calculated. The final value of COR is obtained by taking the mean of the above four values of COR. The variation in COR is observed mainly because of the limitations of experiment such as resolution of angle measurement and non-conformity of the plane in which the swings take place, however, the variation is within 2%.
3 Effect of Acceleration on Momentum Transfer The effect of acceleration on the momentum transfer is investigated using the experimental method. Figure 2a demonstrates that in this situation as the vertical downward gravitational force is still acting on a mass its velocity is increasing. Hence it is having an acceleration in tangential direction and hence a component in horizontal direction.
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Fig. 2 Swinging ball contact positions
Thus, at the moment of contact the swinging ball is having velocity and acceleration both in the direction of impact. Velocity and acceleration in the direction normal to the impact are ignored as justified earlier. A conventional pendulum experiment is used for the experimentation since the approximation of point mass is possible in such experiments and the relation between the velocities, angle of oscillation, etc. are well established [1]. Figure 3 shows oblique impact between two smooth masses. The velocity vector components along x and y axes are resolved as u1x and u1y . The momentum of system is conserved along the line of impact (x axis) only [1]. Experiments are performed by lifting the moving ball through angle (θ 1 ) and releasing it. After collision, the stationary ball has maximum angle of oscillation of (α). The angle of 50°, 60°, 70° are used as (θ 1 ) and for each value of (θ 1 ) multiple experiments are performed. Average value of (α) for each (θ 1 ) is presented in Table 2.
Fig. 3 Oblique impact (acceleration)
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Table 2 Observations for effect of acceleration Equal mass
Unequal mass
θ1
u1x
θ
α
α
θ
α
50
1.9787
19.42
33.84
38
28.29
51.22
61.6
60
1.5472
25.22
44.13
47.4
36.78
67.83
75.4
70
1.8576
30.39
53.62
56.2
44.51
84.12
88.8
α
Theoretical value of maximum angle of oscillation of the stationary ball (α ) is calculated using oblique collision. Corresponding values of (α) are presented by the side of (α ) in the Table 2. Experiments are performed for equal mass as well as unequal mass conditions and COR are considered as 0.906 and 0.7815 for the two conditions respectively. For accelerated oblique impact, m1 = m2 = 0.04635 kg, coefficient of restitution (e) is consider as 0.906 as discussed earlier. In case of unequal masses, m1 = 0.04635 kg, m2 = 0.01213 kg, coefficient of restitution (e) is consider as 0.7815. The length of strings L 1 = 0.465 m, L 2 = 0.39 m and the angle of impact (β) = 31° has been considered. From Table 2, it has been observed that the effect of acceleration on stationary ball during oblique impact increases the angle of oscillation of stationary ball. (α is always greater than α ) indicating that the acceleration is increasing the effect of the velocity. The increase is to the tune of 6%.
4 Effect of Retardation on Momentum Transfer The effect of retardation on the momentum transfer is investigated using the experimental arrangement as shown in Fig. 4. It demonstrates a situation of contact where the swinging ball hits the stationary ball after crossing the bottom position. As tangential velocity starts reducing after this point the velocity is associated with retardation in this situation. For equal masses (m1 = m2 = 0.04635 kg), coefficient of restitution (e) has been considered as 0.906 as discuss earlier based on previous experiments. In case of unequal masses (m1 = 0.04635 kg, m2 = 0.01213 kg), coefficient of restitution (e) is consider as 0.7815. The length of strings L 1 = 0.465 m, L 2 = 0.4 m and the angle of impact (β) = 31°. Table 3 shows, different angles of swinging ball (θ 1 ), maximum oscillation angle of moving ball (θ ) and maximum oscillation angle of stationary ball (α ) after impact has been calculated for both equal and unequal masses. For each value of (θ 1 ) multiple experiments are performed to obtain maximum oscillation angle of stationary ball (α). Average value of (α) for each (θ 1 ) is presented in Table 3. From Table 3, it has been observed that the effect of retardation on stationary ball during oblique impact reduces the angle of oscillation of stationary ball. (α is always
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Fig. 4 Oblique impact (retardation)
Table 3 Observations for effect of retardation Equal mass
Unequal mass
θ1
u1x
θ
α
α
θ
α
α
50
1.1987
19.42
33.40
60
1.5472
25.22
43.55
29
28.29
50.538
45.88
37.2
36.78
66.869
70
1.8576
30.39
52.90
55.2
48
44.51
82.83
69
less than α ) indicating that the retardation is decreasing the effect of the velocity. The decrease is to the tune of 8%.
5 Conclusion When there is an impact between two free bodies transfer of momentum is dependent on both the velocity and acceleration of moving bodies. Acceleration increases the momentum transferred and retardation reduces the momentum.
References 1. Hibbeler RC (2010) Dynamics engineering mechanics, 12th edn. Pearson Prentice Hall, New Jersey 2. Hessel R, Perinotto AC, Alfaro RAM (2006) Force-versus-time curves during collisions between two identical steel balls. Am J Phys 74(3):176–179
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3. Peraire J, Windnall S (2009) Linear impulse and momentum collision. Dynamics MIT Open Source, pp 1–12 4. Kuwabara G (1987) Restitution coefficient in a collision between two spheres. Jpn J Appl Phys 26(8):1230–1233 5. Cross R (2013) The coefficient of restitution for collisions of happy balls, unhappy balls, and tennis balls. Am J Phys 68(11):1025–1031 6. Dong H (2003) Measurement of impact behavior between balls and walls in grinding mills. Miner Eng 16:543–550
Finite Element Modelling of the Human Lumbar Vertebrae for Dynamic Analysis Raj Arjun S. I., Parth Goplani, Pavan Suswaram, and C. V. Chandrashekara
Abstract Analysing and interpreting the dynamic behavioural characteristics of the human lumbar vertebrae are important in assessing symptoms related to lower back pain (LBP). Finite element (FE) modelling and analysis of the vertebral column assist in static and dynamic simulations subjected to various load conditions. Most available finite element models are either expensive and inaccessible, or inaccurate and contain errors. The present paper demonstrates the development of a simplified, refined and error-free solid model of the human lumbar vertebrae for FE analysis. Modal analysis is carried out to evince the integrity of the developed model in dynamic simulation and analysis. Keywords Lumbar vertebrae · Solid modelling · Finite element analysis · Modal analysis · Biomechanics
1 Introduction Human biomechanics is an interdisciplinary study of the structure and motion of the human body system using the principles of mechanics. Experimentation is a direct means to study the mechanical behaviour of any human subsystem. But, ethical restrictions limit researchers in conducting experimental investigations on the human body. The use of finite element analysis is known to be a valuable computational tool in studying the mechanics of the human body. Finite element analysis enables effective and easier determination of the biomechanical behaviour of human subsystems, without being subjected to ethical limitations. Moreover, simulation techniques potentially reduce the cost and time in the development and analysis of prosthetics. The human vertebral column is one of the most widely studied subsystems of the human body. Several solid models and anthropometric models of the human vertebrae are reported and available for various static and dynamic analysis. Most models are either inaccessible or too expensive to gain access. Cheaper and open-source Raj Arjun S. I. · P. Goplani · P. Suswaram · C. V. Chandrashekara (B) Department of Mechanical Engineering, PES University, Bengaluru, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_79
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solid models are rather not suitable for all types of analysis. They usually consist of many errors related to meshing due to geometrical discontinuities. Even few models are dimensionally inaccurate when compared to the anatomical dimensions of the vertebrae. These factors cause the limited availability of solid models of the vertebrae that are suitable for finite element analysis. Lavaste et al. [1] presented the development of a three-dimensional finite element model of the lumbar vertebrae through dimensions obtained from two X-rays, i.e. anteroposterior and lateral. The model is primarily developed using six limited parameters digitized from the X-rays, and static analysis is carried out using finite element method. Kasra et al. [2] presented the development of an axisymmetric model of only a single L2–L3 lumbar disc-vertebra unit. The modelling is carried out based on in vitro measurements of lumbar vertebrae specimens, assuming symmetry about the sagittal plane. Lodygowski et al. [3] developed a simplified threedimensional model of a single vertebral segment (L4–L5) based on data produced from computed tomographic (CT) scans and magnetic nuclear resonance (MNR). The study concluded that it is often impossible to generate an accurate FE mesh for models developed from CT data. Xu et al. [4] developed five nonlinear FE models of the lumbar vertebrae from converting data obtained from CT scans of healthy subjects. An additional smoothing process is performed to eliminate irregular surface complexities such as spikes and holes, which could have been avoided by using an appropriate modelling procedure. In reported literatures, the most commonly used method of modelling bodies with a complex geometry such as a vertebra is CT scans and converting them to generate the three-dimensional model. During the process of conversion of CT data into surface grids, many errors may arise in the geometrical reconstruction. This often leads to errors in the process of finite element mesh generation as well. The complexity of the geometry makes it impossible to generate a model with smooth surfaces, which is critical in dynamic analysis. Models with uneven surfaces are suitable for limited static analysis and visualization purposes only. Dynamic analysis of the vertebrae requires a more refined geometrical model, allowing the error-free simulation of relative motion between contact surfaces. The present paper addresses the challenges in developing a solid model of the human lumbar vertebrae. It presents the development of a more refined, simplified and error-free model. The developed model has better scope for finite element analysis and dynamic simulations. Section 2 outlines the modelling procedure of the human lumbar vertebral column. Section 3 presents the modal analysis carried out to provide evidence for the structural integrity of the developed model of the lumbar vertebrae.
2 Solid Modelling The lumbar spine refers to the region of the vertebral column between the thorax and sacrum. They are the largest segments of the vertebral column, consisting of five vertebrae designated as L1–L5 from the top. A typical lumbar vertebra consists
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of the vertebral body, pedicle, transverse processes, spinous process, vertebral arch (lamina), superior and inferior articular processes and the vertebral foramen. The upper and lower surfaces of the adjacent vertebral bodies are connected by means of an intervertebral disc. The vertebral body is connected to the vertebral processes by means of the pedicle. The superior articular process of a vertebra is attached to the inferior articular process of an adjacent vertebra by means of a thin layer of articular cartilage. The dimensions of each of these vertebral elements are considered for the modelling of lumbar vertebrae. A comprehensive study of the geometrical characteristics of the lumbar vertebrae suggests that the dimensions of each vertebra are approximately equal and does not show a significant variation in geometry. However, the dimensions of each vertebra vary from person to person depending on various factors such as age, gender, height, weight and country of origin. Therefore, to simplify the modelling procedure, dimensions of the middle vertebra, L3 is considered to model all five vertebrae. The dimensions of the L3 vertebra are borrowed from Zhou et al. [5], where the geometrical dimensions are measured and averaged from 55 computed tomographic (CT) images of male patients of mean age, 50 ± 13.60, and are shown in Table 1. The reference notations for the L3 vertebra [6] are indicated in Fig. 1. These dimensions only aid in the preliminary modelling of the vertebra and are not sufficient to entirely model the vertebra to a higher level of accuracy. The required supplementary dimensions of surface contours and posterior elements are measured manually from the specimens of actual lumbar vertebrae, as they are not reported in literatures. The required dimensions are measured with utmost precision by using the appropriate scaling factor with respect to the dimensions available in [5]. Surface contours and complex shapes are traced and measured using the three-point method and approximated appropriately to obtain an accurate model. The manually measured dimensions of the vertebra are shown in Fig. 2 and shown in Table 2. The solid model of the L3 vertebra is developed in Solid Edge software. The modelling process is initiated by developing the vertebral body. Subsequently, the posterior elements are modelled using lofted protrusions between angles planes. Several attempts and multiple iterations are made in developing irregular parts and Table 1 Dimensions of L3 vertebra [5]
Dimension
Notation
Value (mm)
Upper vertebral width (UVW)
a
46.1
Upper vertebral depth (UVD)
b
34.1
Spinal canal width (SCW)
c
25.2
Spinal canal depth (SCD)
d
16.1
Pedicle width (PDW)
e
10.7
Transverse process length (TPL)
f
96.1
Vertebral body height (VBH)
g
30.6
Intervertebral disc height (DH)
h
12.4
Pedicle height (PDH)
i
14.9
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f
g
d
h e
c
i
a
b Fig. 1 Dimensions of a lumbar vertebra [5]
k o l
m
j
n Fig. 2 Dimensions of a lumbar vertebra
Table 2 Supplementary dimensions of L3 vertebra Dimension
j
k
l
m
n
o
Value
55°
35 mm
15.36 mm
18.36 mm
25°
55°
complex surface contours of the vertebra. The final model is well-refined and dimensionally intact. Different views of the developed model of the vertebra are shown in Fig. 3.
(a) Front view (anterior)
(b) Rear view (posterior)
Fig. 3 Developed solid model of L3 vertebra
(c) Top view
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Articular cartilage Intervertebral disc Vertebral body
(a) Front view
(b) Back view
(c) Right view
(d) Isometric view
Fig. 5 Fully developed model of the lumbar vertebrae
Similarly, the solid models of the intervertebral disc and the articular cartilage are also developed. The developed models of the vertebra, intervertebral disc and articular cartilage are assembled together to form a single lumbar functional spinal unit (FSU), as shown in Fig. 4. Subsequently, the full model of the lumbar vertebrae is developed by assembling five lumbar FSUs. The curvature of the lumbar vertebrae is taken into consideration while assembling the models. A lumbar lordotic angle of 20° for a seated human in upright position is considered [7]. Figure 5 shows the fully developed model of the lumbar vertebrae in different views. This model is ready to be imported into FE analysis for dynamic simulations.
3 Results and Discussion The developed solid model shows scope for both static and dynamic simulation, under any load conditions. For the present study, finite element modal analysis is carried out to demonstrate the structural integrity and robustness of the developed models. The models are imported into ANSYS finite element analysis package. The appropriate material properties of each vertebral element are taken into consideration.
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(b) 2nd Mode = 281.51 Hz Lateral flexion
(c) 3rd Mode = 361.37 Hz Rotation
Fig. 6 Three fundamental modes of the lumbar vertebrae
Program controlled meshing with proximity and curvature size function is defined to generate the finite element mesh. An additional tetrahedron meshing method is defined to avoid errors in meshing complex faces and edges. No pre-stress conditions are given. The top surface of the L1 vertebra and bottom surface of the L5 vertebra are considered to have a fixed support. The analysis is carried out; the total deformations and mode shapes are generated, and results are compared. The three fundamental modes generated for the lumbar vertebrae are shown in Fig. 6. The obtained natural frequencies are in good correlation with analytical values. The generated mode shapes also correspond to the actual modes of deformation, as indicated in previously reported studies. A similar approach can be used to model other biological systems with complex morphologies. This study shows scope for further in-depth analysis of the lumbar vertebrae when subjected to complex loading conditions.
References 1. Lavaste F, Skalli W, Robin S, Roy-Camille R, Mazel C (1992) Three-dimensional geometrical and mechanical modelling of the lumbar spine. J Biomech 25(10):1153–1164 2. Kasra M, Shirazi-Adl A, Drouin G (1992) Dynamics of human lumbar intervertebral joints: experimental and finite-element investigations. Spine 17(1):93–102 3. Lodygowski T, Kakol W, Wierszycki M (2005) Three-dimensional nonlinear finite element model of the human lumbar spine segment. Acta Bioeng Biomech 7(2):17–28 4. Xu M, Yang J, Lieberman IH, Haddas R (2016) Lumbar spine finite element model for healthy subjects: development and validation. Comput Methods Biomech Biomed Eng 20(1):1–15 5. Zhou SH, McCarthy ID, McGregor AH, Coombs RRH, Hughes SPF (2000) Geometrical dimensions of the lower lumbar vertebrae–analysis of data from digitized CT images. Eur Spine J 9(3):242–248 6. Frank HN (2011) Atlas of human anatomy, 5th edn. Saunders Elsevier, USA 7. Baumgartner D, Zemp R, List R, Stoop M, Naxera J, Elsig JP, Lorenzetti S (2012) The spinal curvature of three different sitting positions analysed in an open MRI scanner. Sci World J 2012:1–7
Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR) Ankit Nakoriya and Vijay Kumar Gupta
Abstract Mobile robotics has received a lot of interest among the researchers in recent years, particularly for applications in the remote areas. Six wheels multiterrain robot (SW-MTR) is a robot with semicircular rocker and bogie mechanism developed at PDPM IIITDM Jabalpur. The robot moves on different terrain including sandy area, uneven rocky area, steps of 6 cm height, boulders of 8 cm height, inclined plane of 25°, etc. For steering of the SW-MTR, the hybrid steering system has been proposed with a combination of differential steering mechanism with worm-sector steering mechanism. This paper presents the dynamic model of the hybrid steering system with specific consideration on the wheel skidding effects and load distribution on each of the six wheels. A comparison of the turning radius, contact forces between wheels and surface, torque and power consumption of the proposed hybrid steering system with other individual steering systems is also presented through simulation. Keywords Mobile robots · Hybrid steering · Multiterrain robots
1 Introduction Various robotic systems are highly in demand for numerous purposes such as robotic arms on assembly lines, aerial drones such as hexa-copter and quad-copter, and multiterrain robots. Out of these, the mobile robotics has emerged out as one of the most popular interdisciplinary fields for the purpose of the planet and asteroid exploration, rescue operations, and path finding or planning in complex areas where humans are unable to reach and security surveillance. For accomplishing these tasks, the robot A. Nakoriya (B) Mechatronics Laboratory, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India e-mail: [email protected] V. K. Gupta Professor, PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_80
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should have terrain adaptability and effective steering system. Terrain adaptability for SW-MTR has already been discussed by Gupta and Gupta [1]. The steering system is a very important part of any wheeled robotic system. It is observed that two basic steering systems, caterpillar differential steering or worm-sector steering, have been used independently in many multiterrain robots. The robots like ARES III [2] and shrimp [3] show the use of worm-sector steering while caterpillar differential steering system has been used by robots like Packbot [4], Talon [5], etc. In case of caterpillar differential steering mechanism due to skidding of the wheels, the turning radius is smaller while friction between wheels and the skidding surface is very high hence reduces wheel life. The worm-sector steering mechanism provides the low frictional forces between wheel and surface but the radius of steering is larger. In this paper, it is proposed to use a hybrid system for steering to combine the advantages of both types of steering mechanisms. A dynamic model of proposed steering for SW-MTR is necessary to understand the analogy with the physical system. It is very helpful in the trajectory controlling of the robot. A trajectory tracking control based on differential steering is discussed by Caracciolo et al. [6]. Similar kind of work is presented by Kozlowski et al. [7], Zhang et al. [8], and Wu et al. [9]. This paper presents an analytical formulation of longitudinal rolling resistance, lateral forces on wheels, and moment of resistance on each wheel to estimate skidding effect while taking a turn with the proposed steering system. Simulations are carried out in MSC ADAMS® and results are compared.
2 The Steering System of SW-MTR The steering system is one of the important parts of the wheeled robotic system. Good steering has the characteristics of a smaller turning radius, less power consumption for turning and able to turn robot in a complex environment in a short time. In this work, a hybrid steering system is proposed to combine the advantages of both differential steering and worm-sector steering system. An individual multibody dynamic study of differential steering, worm-sector steering, and proposed hybrid steering by using MSC ADAMS® is presented in this paper.
2.1 Differential Steering System Differential steering is a simple concept of steering in which robot steers by creating a difference in speed of wheels on both sides of the robot. For the differential steering dynamic analysis, the speed of wheels is taken as 85 rpm and 47 rpm on respective sides. These speeds are corresponding to the 3:5 rocker hinge ratio and friction conditions of µs = 0.6 and µk = 0.4. Differential speed also has limits over which the rocker mechanism of the robots has Willy problems. Hence, the speeds of the robot are selected in such a way that the rocker mechanism does not cause a Willy
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Table 1 Simulation parameters µs
µk
V1
V2
V1/V2
(R)simulation
0.8
0.6
85
50
1.7
129
0.6
0.4
85
47
1.8
110
problem. The coefficient of friction between rubber and flooring material is taken as 0.6–0.9 [10]. Willy problem. It is the upward movement of the rocker link (any one out of two) depending on the differential speed of wheel while the robot in motion. Due to 3:5 rocker hinge pivoting ratio, the weight gets distributed unequally among the first four wheels. Therefore, middle wheels are taking more weight as compared to front wheels causing an upward movement of front wheel. If the robot is taking a left turn then right-side rocker link gets subjected to Willy problem. To avoid this problem, the differential speeds of the robot must not have a large difference in speed. It is necessary to calculate the differential speed range for various environmental conditions for which there is no Willy problem for SW-MTR. Hence, by successive simulation in MSC ADAMS®, for different friction values, for high speed of 85 RPM on one side, corresponding lower speed is found out and presented in Table 1. Due to the lower radius of steering, second parameters are taken for further simulation purpose. Figure 2 shows the Willy problem in simulation. Steering radius, contact forces, motion torque, and power consumption are calculated and compared with other steering system in the results section.
2.2 Worm-Sector Steering The worm-sector steering consists of independent high torque motor to steer the SW-MTR which is shown in Fig. 1c. The limitation of steering is set to 15° because beyond this value the torque required on rear wheels motors increases as well as
(a) Inclined plane climbing by SW-MTR
Fig. 1 Prototype of SW-MTR
(b) Stair Climbing by SW-MTR
(c) Hybrid steering of SW-MTR
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Fig. 2 Willy problem
mechanism gets stuck. In the simulation, the motion given to the motor is very well described by the step function of STEP (time, 0, 0, 1, −0.75D). It returns an array of y values, corresponding to x values [11]. Format : STEP (A, X 0 , h 0 , X 1 , h l ) where A = an array of x values. X 0 = Value of x at which the step starts ramping from h0 to h1 . h0 = Value of h when x is less than or equal to X 0 . X 1 = Value of x at which the step function reaches h1 . h1 = Value of h when x is greater than or equal to h1 . Steering radius, contact forces, motion torque, and power consumption are calculated and compared with other steering system in the result section. Figures 3 and 4 show the maximum steering torque of 77 kgf-cm and power consumption of 1 kgfcm/s is required for worm-sector steering to steer SW-MTR. The simulation run time is 20 s. The variation in the values of torque and power is due to resistive forces offered by steering surface, which opposes motion of turning. Contact forces between the wheels and steering surface make the values of torque and power to deviate.
2.3 Hybrid Steering The main objective of hybrid steering is to combine the advantages of both differential and worm-sector steering systems. The differential steering provides the low steering radius. While in case of worm-sector steering, the frictional forces acting on wheels are less hence the life of wheels is more. The hybrid steering system
Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR)
Fig. 3 Torque required for steering by worm-sector steering
Fig. 4 Power consumption during steering by worm-sector steering
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gives the advantage of low reducing steering radius with less amount of contact forces acting on wheels. Hybrid steering also reduces steering torque requirement and power consumption of high torque motor used in worm-sector steering to 55 kgf-cm and 0.65 kgf-cm/s, respectively, and is shown in Figs. 5 and 6.
Fig. 5 Torque required for steering by hybrid steering
Fig. 6 Power consumption during steering by worm-sector steering
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3 Dynamic Modeling of Hybrid Steering A dynamic model of the hybrid steering system is developed by neglecting the effects produced by the robot suspension system and tire deformation [6]. The following are the assumptions made during the development of the model. (1) (2) (3) (4) (5)
A rigid robot turning left on inclined terrain present toward left (angle 3°). Center of gravity of the robot body shifts laterally, longitudinally, and vertically. Speed of the robot is below 50 cm/s. Wheel slip along the length is neglected. Lateral force on wheel contact is the function of vertical load on corresponding wheel contact.
3.1 Motion Analysis The robots fixed reference frame is indicated by F(X, Y) and the local reference frame is f(x, y). Since the robot turning on 3° plane toward left is considered, the center of gravity of robot is located at a distance ‘a’, ‘b’ and ‘c’ from the front, middle, and rear wheel axes, respectively. The height of the center of gravity is ‘h’ and it is laterally positioned at a distance ‘t 1 ’ and ‘t 2 ’ from the right and left sides, respectively. Let X˙ , Y˙ , and θ˙ are longitudinal, lateral, and angular velocities of the robot in local reference frame, i.e., f(x, y). Now we can write the velocities in the global reference frame as given below.
X˙ Y˙
x˙ cos θ − y˙ sin θ = x˙ sin θ + y˙ cos θ
x˙ = R(θ ) y˙
(1)
By differentiating the above equation, we get the acceleration matrix of the robot.
X¨ Y¨
cos θ − sin θ x¨ − θ˙ y˙ = sin θ cos θ y¨ + θ˙ x˙ X¨ ax = R(θ ) ay Y¨
(2)
where the terms ax and a y indicate the acceleration in local reference frame, i.e., f(x, y). For an infinitesimally small time, the robot moves in an exact circle with the center of rotation as ‘i’. ‘i’ is called as the instantaneous center of rotation (ICR) along which the robots longitudinal velocity in f(x, y) frame is almost zero. The coordinates of ICR are given as follows.
xi yi
=
− y˙ /θ˙ x/ ˙ θ˙
(3)
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When the robot moves in a straight-line path, the angular velocity θ˙ and lateral velocity y˙ become zero. The position of ICR in this case is along the y-axis at the infinity. When the robot moves along the curved path, the instantaneous center of rotation moves forward by an amount |x i |. The loss of motion stability is observed when the position of x i goes outside the wheel base of the robot. The longitudinal and lateral velocities in the frame f(x, y) are expressed below. x˙1 = x˙4 = x˙6 = x˙ − (t1 + t2 )θ˙ (left side) x˙2 = x˙3 = x˙5 = x˙ + (t1 + t2 )θ˙ (right side) y˙1 = y˙2 = y˙ + a θ˙ (front wheels)
(4)
y˙3 = y˙4 = y˙ + bθ˙ (middle wheels) and y˙5 = y˙6 = y˙ − cθ˙ (rear wheels)
3.2 Equation of Motion Figure 7a shows the free body diagram of the robot in which all components of forces as well as velocities are shown. The robot is having instantaneous positive velocity components x˙ and θ˙ and negative velocity y˙ . When the power is transmitted through the wheels, there is a generation of tractive forces F xi and rolling resistance force Rxi (i = 1 … 6). For minimization of longitudinal slip, it is necessary to assume that the wheel actuation of each wheel is the same. Because if each motor which is connected to wheels is actuated at the same time there will be minimum possibility of Willy problem. And hence, this will result in F x1 = F x4 = F x6 and F x2 = F x3 = F x5 . The lateral skidding causes the F yi. There is a resistive moment M r caused by resistive forces, i.e., Rxi and F yi . In the local reference frame f(x, y), the equations of motion
(a) Top view of free body diagram of SW-MTR
(b) Side view of free body diagram of SW-MTR
Fig. 7 Free body diagram of robot steering
(c) Front view of free body diagram of SW-MTR
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Fig. 8 Pivoting ratio of the rocker link
for a robot of mass m and inertia I about its center of mass are written as: max = 3Fx1 + 3Fx2 − Rx
(5)
ma y = −Fy
(6)
I θ¨ = 3t2 Fx1 − 3t1 Fx2 − Mr
(7)
The magnitude of the weight distribution of robot among the six wheels is necessary. Since the magnitude on each wheel is used to find longitudinal rolling resistive force Rx , lateral resistive force F y and resistive moment M r. Here we introduce a Coulomb friction model for ground and wheel contact. For calculation of reactions on wheels, the front four wheels are assumed to be as a two-wheel system (one on each side). This assumption is valid since the front four wheels are attached to the rocker system through a single axle. This converts the six-wheel system into the four-wheel system. By referring to Fig. 7b, it can be stated that the position of two hypothetical wheels is in line with front axle axis (Fig. 8). Hence, front axle load is given as: FZf =
h c mg − max c+d c+d
(8)
h d mg + max c+d c+d
(9)
The rear axle load is given as Fzr =
The robot is taking a left turn on the steep surface which is 3° inclined. Due to this, the CG shifts occur. Figure 7c illustrates a half-robot model for the front axle, where the virtual mass of the front axle M f is calculated as follows:
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m f =
Fzfl + Fzfr Fzf = g g
(10)
Front axle right-side load is given as
Fzfr
h.a y c h t2 = mg − max × + c+d c+d g.(t1 + t2 ) t1 + t2
(11)
And similarly, left-side load is Fzfl =
h.a y c t1 h − mg − max × c+d c+d t1 + t2 g.(t1 + t2 )
(12)
The front axle left and right forces calculated above are further divided into 3:5 ratio. As the pivoting ratio in the rocker mechanism is maintained at 3:5. Fzfl is divided into Fz1 and Fz4 as well as Fz f r is in Fz2 and Fz3 . They are written below. c 3 mg − 8 c+d c 3 Fz2 = mg − 8 c+d c 5 mg − Fz3 = 8 c+d c 5 Fz4 = mg − 8 c+d Fz1 =
h.a y t1 h − max × c+d t1 + t2 g.(t1 + t2 ) h.a y h t2 max × + c+d g.(t1 + t2 ) t1 + t2 h.a y h t2 max × + c+d g.(t1 + t2 ) t1 + t2 h.a y t1 h max × − c+d t1 + t2 g.(t1 + t2 )
(13) (14) (15) (16)
For rear axle loading on each side, follow the same method as followed for the front axle. But in this case, the load on each side will not get divided into 3:5 ratio. The forces are written below. h.a y h t2 d (17) mg + max × + Fz5 = c+d c+d g.(t1 + t2 ) t1 + t2 h.a y t1 h d Fz6 = mg + max × (18) − c+d c+d t1 + t2 g.(t1 + t2 ) If the robot is moving on the curved path with lower speeds, its lateral load due to centrifugal force can be neglected. The contact patch between ground and wheel is assumed as rectangular. Hence, the uniform pressure distribution is produced by tire vertical load. The rolling resistance equation is given by the following [6]. Rxi = fr Fzi sgn(x˙i )
(19)
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where fr is the coefficient of rolling resistance which is independent from the velocity of the robot. The total longitudinal resistive force is given as Rx =
6
Rxi = fr Fz1 sgn(x˙1 ) + fr Fz2 sgn(x˙2 )
i=1
+ fr Fz3 sgn(x˙3 ) + fr Fz4 sgn(x˙4 ) + fr Fz5 sgn(x˙5 ) + fr Fz6 sgn(x˙6 )
(20)
The lateral force acting on wheels of the robot with μ as the coefficient of lateral resistance is given below. Fy =
6
Fyi = μFz1 sgn( y˙1 ) + μFz1 sgn( y˙2 )
i=1
+ μFz1 sgn( y˙3 ) + μFz4 sgn( y˙4 ) + μFz5 sgn( y˙5 ) + μFz1 sgn( y˙6 )
(21)
Now the equation of resistive moment is given as follows. Mr = a Fy1 + Fy2 + b Fy3 + Fy4 − c Fy5 + Fy6 + t1 (Rx2 + Rx3 + Rx5 ) − t2 (Rx1 + Rx4 + Rx6 )
(22)
The dynamic model can be written in global reference frame F(x, y), introducing the generalized coordinates q = (X, Y, θ ) and matrix notation. M q¨ + c(q, q) ˙ = E(q)τ,
(23)
⎤ ⎤ ⎡ m 0 0 Rx cos θ − Fy sin θ where M = ⎣ 0 m 0 ⎦, c(q, q) ˙ = ⎣ Rx sinθ + Fy cos θ ⎦ and E(q) = Mr 0 0 I ⎤ ⎡ cos θ/r cosθ/r ⎣ sinθ/r sinθ/r ⎦, τi = 2r Fxi (i = 1, . . . , 4). (t1 + t2 )/r −(t1 + t2 )/r ⎡
where r is the wheel radius of the robot and τ1 , τ2 , τ3 & τ4 are the torques produced by left- and right-side motors at load side, respectively.
4 Comparison of Simulation Results The parameters selected for comparison purpose are the radius of steering, contact forces on wheels in all three directions, torque and power consumption of wheel motors.
882 Table 2 Comparison of simulation radius
A. Nakoriya and V. K. Gupta Simulation time (s)
Differential steering (cm)
Hybrid steering (cm)
Worm-sector steering (cm)
2.5
111
110
4608
5
110.3
105
1459
7.5
110
99
839
10
110
95
587
12.5
110
90
446
15
110
86
373
17.5
110
81
315
20
110
78
272
4.1 Comparison of Steering Radius From Table 2, it is observed that the differential steering has a constant steering radius throughout the simulation time. Worm-sector steering has reducing steering radius with a large value of initial steering radius, while hybrid steering combines both the steering has reducing steering radius with the lower value of initial steering radius.
4.2 Comparison of Contact Forces Contact forces between wheels and ground surface decide the life of the tire, power consumption, and torque requirement. Tables 3, 4, and 5 show the contact forces in Table 3 Comparison of lateral direction contact forces
Wheel
Differential steering F x (kg-f)
Hybrid steering F x (kg-f)
Worm-sector steering F x (kg-f)
Left front wheel
3.3
1.9
−1.4
Left middle wheel
3.3
−3.9
−1.5
3.35
−1.6
Right front wheel
−3.5
3.6
1.3
Right middle wheel
−1.9
−1.9
2
Right rear wheel
−1.9
−1.9
−1.25
Left rear wheel
4.7
Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR) Table 4 Comparison of longitudinal direction contact forces
Table 5 Comparison of reactive forces
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Wheel
Differential steering F y (kg-f)
Hybrid steering F y (kg-f)
Worm-sector steering F y (kg-f)
Left front wheel
−3.3
−3
2
Left middle wheel
−4.0
−3
−2
Left rear wheel −4.0
−5
1.5
Right front wheel
−3.5
2.9
1
Right middle wheel
−2.0
−2.0
−3
Right rear wheel
−2.5
−2.5
2.5
Wheel
Differential steering F z (kg-f)
Hybrid steering F z (kg-f)
Worm-sector steering F z (kg-f)
Left front wheel
−8.0
−7
−7
Left middle wheel
−9.0
−10
−7
Left rear wheel −13.0
−10
−6
Right front wheel
−8.0
−6
−7
Right middle wheel
−6.0
−4
−7
Right rear wheel
−5.0
−5
−6
all three directions. The x-axis is taken along the lateral side, the y-axis is taken along the longitudinal side, and the z-axis is taken along the height of the robot. Table 3 shows the values of contact forces along lateral direction in hybrid steering. Longitudinal direction contact forces are shown in Table 4. Most of the magnitude of contact forces in case of the hybrid steering system is between the values of differential and worm-sector steering system. By observing Table 5, in hybrid steering, while taking the left turn, wheels present on the right side experience less reactive force than other two steering systems. Hence, the life of the tire increases.
884 Table 6 Comparison of motion torque
A. Nakoriya and V. K. Gupta Wheel
Differential steering torque (kgf-cm)
Hybrid steering torque (kgf-cm)
Worm-sector steering torque (kgf-cm)
Left front wheel
19
13.5
12
Left middle wheel
24
22.5
7
Left rear wheel
27
24.375
3
Right front wheel
12.5
10.625
8
Right middle wheel
12
8
12.5
Right rear wheel
8
7
11
4.3 Comparison of Motion Torque Table 6 presents the motion torque requirement for differential, worm-sector, and hybrid steering system. The observation from the table is that though motion torque requirement of worm-sector steering for each wheel are least, hybrid steering is preferable because it gives reducing steering radius starting from the lower magnitude of radius as compared to worm-sector steering.
4.4 Comparison of Power Consumption Table 7 shows the power consumption for each of the six wheels in all three types of steering studied. The magnitude of power consumption for each wheel in hybrid steering falls between the magnitude of power for differential and worm-sector steering. All the comparison parameters tabled above are the basis of selection of hybrid steering as a steering system for SW-MTR. Hybrid steering gives lower reducing steering radius which is very useful for steering in very complex areas.
5 Conclusion Through the work presented in this paper, a hybrid steering system is proposed for SW-MTR. An individual multibody dynamic study using MSC ADAMS® of differential steering, worm-sector steering, and hybrid steering is carried out. Various parameters such as steering radius, wheel–surface contact forces, motion torque and
Hybrid Steering System of Six-Wheel Multiterrain Robot (SW-MTR) Table 7 Comparison of power consumption
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Wheel
Differential steering power (kgf-cm/s)
Combined steering power (kgf-cm/s)
Worm-sector steering power (kgf-cm/s)
Left front wheel
170
120
90
Left middle wheel
215
210
50
Left rear wheel 240
217.5
30
Right front wheel
65
50
75
Right middle wheel
45
35
110
Right rear wheel
38
40
100
power consumption on each wheel are compared for each of three steering systems. Based on this, a hybrid steering system is selected. Dynamic modeling of proposed hybrid steering has been presented.
References 1. Gupta AK, Gupta VK (2013) Design and development of six-wheeled multi-terrain robot. In: CARE 2013—2013 IEEE international conference on control, automation, robotics and embedded systems proceedings, pp 1–6. https://doi.org/10.1109/CARE.2013.6733751 2. Gliedes M, Santana P, Deusdado P, Mendonca R, Marques F, Henriques N, Lourenco A, Correia L, Barata J, Flores L (2012) ARES-III: a versatile multi-purpose all-terrain robot. In: IEEE international conference on emerging technology factory automation (ETFA 2012). https://doi. org/10.1109/ETFA.2012.6489633 3. Dian S, Liu T, Liang Y, Liang M (2011) A novel shrimp rover-based mobile robot for monitoring tunnel power cables. In: Proceedings of the 2011 IEEE international conference on mechatronics and automation. ISBN: 978-1-4244-8115 4. Yamauchi BM (2004) PackBot: a versatile platform for military robotics. Unmanned Gr Veh Technol VI 5422:228–237. https://doi.org/10.1117/12.538328 5. Talon: Talon tracked mitary robot [Online]. Available: https://www.army-technology.com/pro jects/talon-tracked-military-robot/ 6. Luca C, Alessandro De Luca SI Trajectory tracking control of four wheel differentially driven mobile robot. In: Proceedings of 1990 international conference on robotics and automation. Detroid, Michigan 7. Kozłowski K, Pazderski D (2004) Modeling and control of a 4-wheel skid-steering mobile robot. Int J Appl Math Comput Sci 14:477–496. https://doi.org/10.1016/j.compag.2011.12.009 8. Zhang Y, Hong D, Chung JH, Velinsky SA (1998) Dynamic model based robust tracking control of a differentially steered wheeled mobile robot. Proc Am Control Conf 2:850–855. https:// doi.org/10.1109/ACC.1998.703528 9. Wu X, Xu M, Wang L (2013) Differential speed steering control for four-wheel independent driving electric vehicle. In: IEEE international symposium on industrial electronics, p 1. https:// doi.org/10.1109/ISIE.2013.6563667
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10. El-Sherbiny YM, Hasouna AT, Ali WY (2012) Friction coefficient of rubber sliding against flooring materials. ARPN J Eng Appl Sci 7:121–126 11. Norton RL (2013) Adams tutorial kit for mechanical engineering courses, pp 1–352
Flexible Coupling—A Research Review Harshanand P. Ramteke
and Girish D. Mehta
Abstract In mechanical transmission of power from motor to process unit, occurrence of misalignment between the shafts to be coupled is a well-known phenomenon. To mitigate the parallel, angular or combination of these two misalignment, wide range and types of Flexible Couplings are being used. This Research Review is intended to study past and present types of Flexible Couplings and gain insight regarding their working, applications, and performance under different variables and loading conditions. On the basis of this Research Review future work can be performed to develop a New Type of Flexible Coupling which can help to overcome shortcomings of existing Flexible Couplings. Keywords Shaft misalignment (offset and angular) · Flexible couplings · Vibrations and stress induced due to vibrations
1 Background of Present Work and Scope of the Research In many cases of industrial domain, two shafts are joined together with a coupling. One of the situations is joining a motor shaft with centrifugal pump. This case is considered to be complex as misalignment in shafts may hamper the performance of the centrifugal pump. There are many cases which are confronting same situation. A solution to such situation is the usage of flexible couplings, which are present in the market. But it also allows the flexibility to some extent. There are some centrifugal chillers which are directly placed on the motor shaft and thus there is no need of such couplings [1]. Of course, it is a special case, but in most of the applications, one has to couple the motor shaft with process unit. Study reveals that the maintenance cost of any machine increases as misalignment in the shafts increases. Misalignment in the coupling produces moment at the bearings which in turn pulls the shaft away from its centerline. This action is periodic and cyclic; therefore it produces vibrations in entire machine. Under these vibrations, the coupling and its component are set into stress phenomenon which could otherwise be coined as H. P. Ramteke (B) · G. D. Mehta Priyadarshini College of Engineering, Hingna Road MIDC, Nagpur 440019, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_81
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stress under vibrations. If this stress pattern is cyclic then localized area could be formed which may culminate into its failure. Therefore it is inevitable to come out with a proper solution by evolving a new design of Coupling to sustain the cyclic stress and will give more flexibility in both the directions, i.e., angular and offset.
2 Reported Work Pertaining to Flexible Couplings A rigorous Literature Review is executed for the present research. Study reveals that misalignment is unavoidable condition. Due to misalignment, stresses induced in components of a coupling are ascertained to be cyclic; therefore it compels to localized stress area and some form of combined stresses. For the present research, various papers are reviewed, but due to space limitations, few prominent research papers and their reviews are discussed as below. The research reports toward evaluation of vibration signatures induced by angular misalignment. The experimental setup is made with two shafts four journal bearing and one flexible disk coupling. Analysis is executed to find out the effect of misalignment on displacement spectrum of the bearing. The effect of forces on the journal bearing is also ascertained by considering the system dynamics and by the forces and moment generated by the flexible coupling while accommodating misalignment. In this paper, a structural analysis of disk type coupling is also reported by Finite Element Method [2]. Through this research, a study is executed in which the power transmission between the motor and load is performed by means of different types of coupling mainly those most frequently used in the industry. Results show that the conclusion drawn from a particular coupling is not necessarily applicable to others [3]. In this research, the author has tried to evaluate the accurate characterization of torsional stiffness of flexible disk coupling. Torsional stiffness is a key factor while studying torsional vibrations. Most of the literature reported the influence of coupling type and design selection parameters on the operating limits of various couplings. Flexible coupling while allowing the small amount of misalignment that may otherwise lead to equipment failure. This paper reports about the performance analysis of flexible disk coupling by comparing theoretical, experimental and computational methods [4]. A special paper is published by “World Pumps” is reviewed. This paper focused on the use of various types of flexible coupling for Industry. Their performance comparison including transmitting torque capacity, the amount of misalignment to be taken, bending moment, torsional stiffness, axial resisting force, lubrication required are nicely discussed [5]. Shaft Misalignment and rotor unbalance are major concerns in rotating machineries. To understand dynamic characteristics of these faults, a theoretical model of a complete Motor-Flexible Coupling–Rotor System capable of describing the mechanical vibration resulting from misalignment and unbalance is developed.
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Forcing frequencies due to shaft misalignment are even multiple frequencies of motor rotational speed, thereby creating resonance [6]. The rotor-bearing system is modeled using higher order finite elements by considering deflection/slope/shear force/bending moment with eight degrees of freedom per node. The reaction forces/moments developed due to flexible coupling misalignment are derived and introduced in the model. The imbalance response in two harmonics is evaluated. The location of the coupling with respect to the bending mode shape has a strong influence on the vibrations [7]. The effect that misalignment has on the machines is a function of the compliance of the flexible coupling. All torsionally loaded misaligned couplings have restoring moments which tend to bow the machine shafts. The amount of bowing, which can cause machine vibration, increases with higher speed shafts which are carrying higher torques for a given size machine. The reaction forces from the couplings depend upon how they accommodate misalignment. An analysis of various types of couplings was made. The gear coupling places the maximum bending moment on the machine shaft whereas the flexure coupling places lowest Bending Moment on the shaft. Diaphragm Coupling experiences least amount of axial forces as compared to other three types [8]. A classification of couplings as rigid, misalignment-compensating, torsionallyflexible, and combination purpose is proposed. Selection criteria for two basic subclasses of misalignment-compensating couplings are derived. A comparison of important characteristics of some commercially available types of combination purpose couplings is performed Coupling Design Index is devised [9]. Misalignments between hub and shaft in order to investigate the behavior of spline couplings in real working conditions, in particular, to investigate fretting wear phenomena. A new test rig for spline couplings has been designed. Its Peculiarity is the capability to apply an angular misalignment to the spline coupling in order to reproduce the real working conditions. Reactionary moments due to the spline coupling misalignment have been presented and an approximated global estimation of the fretting wear has been obtained by means of the Ruiz parameter [10]. The author had investigated the source of 2N (two times the running speed) vibration response in a misaligned system. For this purpose, he had modeled three disk type couplings (4 bolt, 6 bolt and 8 bolt coupling). Stiffness terms had 1, 2, 3N components. And again a tilt pad journal bearing was also tested under a load which did not show any 2N vibration component [11]. This paper analyzes the effects of the mathematical models of flexible couplings in rotating mechanical systems in terms of their vibrational behavior. The residual unbalance of the coupled shafts is considered to be the main source of vibration in the rotating system. The moments and the frequencies of the forces, which result from these effects, are close to the natural frequencies of the mechanical system.
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The models of Kramer and Nelson and Crandall presented here, take into account the inherent flexibility of the couplings. A parametric sensitivity analysis is of significant importance for the choice of the unbalance response and the d.o.f to be used in the process [12].
3 Summary and Conclusion From the above literature, it is ascertained that one may try a new type of coupling to ascertain its performance characteristics followed by optimized design parameters. Also one may look into the various patented flexible couplings which are available in the market (few are listed below in the table) to evolve with a new type of Flexible Coupling. To ascertain the response of the proposed new type of Flexible Coupling we have to test the coupling under various loading patterns with the help of an Experimental Setup. The experimentation would be planned considering the geometrical aspects of the coupling, the dynamic aspects of the experimental setup such as speed of the motor, rated power delivered, and load torque imposed. The main factor of dynamic condition is the load torque imposed. The load torque pattern is considered to be in various phases. The phases are: 1. 2. 3. 4.
Random Pattern Gradual Pattern Instantaneous Pattern Constant Pattern.
Under above loading pattern, the proposed coupling would be tested to ascertain various responses like: 1. 2. 3. 4.
The stress induced in different components of the coupling Vibration induced due to misalignment at the side of the bearing Temperature induced at the outer body of the Coupling Power consumption of the Motor.
The main intention of designing such coupling is to mitigate the stress under vibration, vibration induced at the bearing, temperature induced at the outer body of the Coupling, Power consumption of the motor. This Aim could be accomplished by gathering the Experimental Data. A Mathematical Model between the Response Variables and Design Variables would be formed. If one optimizes these models for the minimization of responses then the best set of designed variables could be accomplished. The data generated from the experimentation will now be useful to establish the empirical relationship or models between the design variables of the Coupling. The Quantitative and Qualitative Analysis of the established empirical models will be
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executed. The Quantitative Analysis will include sensitivity, reliability, and optimization of the established models while Qualitative Analysis will ascertain the influence of indices on response variables. The probable and tentative variables for the proposed coupling are detailed out as below: Output Design Variables: 1. 2. 3. 4.
Rated Power. Stress induced in the Swing Element of the Coupling. Temperature induced in the outer body of the Coupling. Vibration induced at the Bearings at the side of Coupling. Input Design Variables:
1. 2. 3. 4. 5.
Load Torque. Speed of Shaft etc. Atmospheric Variables. Geometrical Variables of Coupling. Time Dimensions and Gravitational Constants.
The forthcoming paper would be based on the proposed new design of a Flexible Coupling considering the discussions made in Summary and Conclusion.
References 1. John P (1995) Shaft alignment handbook 2. da Silva Tuckmantel FW, Cavalca KL (2019) Vibration signatures of a rotor-coupling-bearing system under angular misalignment. Mech Mach Theory 133:559–583. https://doi.org/10.1016/ j.mechmachtheory.2018.12.014 3. Verucchi C, Bossio J, Bossio G, Acosta G (2016) Misalignment detection in induction motors with flexible coupling by means of estimated torque analysis and MCSA. Mech Syst Signal Process 80:570–581. https://doi.org/10.1016/j.ymssp.2016.04.035 4. Francis A, Avdeev I, Hamann J, Ananthasivan S (2015) Accurate characterization of torsional stiffness of flexible disk couplings. J Eng Gas Turbines Power 137:082504. https://doi.org/10. 1115/1.4029392 5. Johnson CM (1996) An introduction to flexible couplings. World Pumps 6. Xu M, Marangoni RD (1994) Vibration analysis of a motor-flexible coupling-rotor system subject to misalignment and unbalance, Part I: theoretical model and analysis 7. Sekhar AS, Prabhu BS (1995) Effects of coupling misalignment on vibrations of rotating machinery. J Sound Vib 185:655–671. https://doi.org/10.1006/jsvi.1995.0407 8. Gibbons CB, TBC: Coupling misalignment forces by C. In: Proceedings of fifth turbomachinery symposium, pp 111–116 9. Rivin EI (2010) Design and application criteria for connecting couplings. J Mech Transm Autom Des 108:96. https://doi.org/10.1115/1.3260794 10. Cuffaro V, Curà F, Mura A (2013) Test rig for spline couplings working in misaligned conditions. J Tribol 136:011104. https://doi.org/10.1115/1.4025656 11. Avendano RD, Childs DW (2013) One explanation for two-times running speed response due to misalignment in rotors connected by flexible couplings. J Eng Gas Turbines Power 135:062501. https://doi.org/10.1115/1.4023232
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12. Tadeo AT, Cavalca KL (2003) A comparison of flexible coupling models for updating in rotating machinery response. J Braz Soc Mech Sci Eng 25:235–246. https://doi.org/10.1590/s1678-587 82003000300004
Design of Post-Curing Inflator Using Bistable Locking Mechanism Md Zafar Anwar , Jitendra Prasad Khatait, and Sudipto Mukherjee
Abstract Bistable systems have two distinct stable states, where the system stays without any application of external power. In this paper, stable states of the buckled beam are used along with a four-bar mechanism to design a locking for the postcuring inflator. Post-curing of tires is necessary to avoid deformation in nylon fabric during cooling. Current setups, available in the market, use external force loop for the application of pressure during the process causing significant bending stresses in the frame, hence making the design bulky. Further, such conventional setup limits the number of simultaneous operations on a shop floor. In this paper, we propose a bistable structure-based locking mechanism, where, loading elements are in tension instead of bending, which makes the whole setup compact. We also discussed the scope of pseudo-bistability for design consideration. Keywords Post-curing inflator · Bistable system · Pseudo-bistable system · Four-bar mechanism · Locking mechanism
1 Introduction In tire manufacturing industries, curing is the chemical cross-linking of rubber and vulcanizing agents, resulting in an elastomer. Temperatures of the tire after the curing process can go up to 150 °C. The tire contains nylon fibers, which have the property to deform upon cooling. If the hot cured tire is cooled in an open environment, it distorts and loses its characteristic shape [1]. Hence, the tire is pressurized to assume its forms and allowed to cool under inflated condition. For these purposes, post-curing inflators (PCI) are used to provide constraint cooling and avoid deformation of tire shape. The PCI setups used in the industries are quite bulky. The conventional design Md Z. Anwar (B) · J. P. Khatait · S. Mukherjee Indian Institute of Technology Delhi, New Delhi 110016, India J. P. Khatait e-mail: [email protected] S. Mukherjee e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_82
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available in the market uses an external force loop, which loads the components in bending. To design for this bending stress, lots of material is used, and the whole setup requires lots of space to operate. In addition to this conventional setup operates on hydraulics to move the sealing lids, which adds to the operational cost. To avoid these issues and reduce the cost, external force loop can be shorted using an internal force loop such that Instead of bending, the elements will be loaded in tension. This would reduce the space, materials as well as the manufacturing cost. In this paper, we propose a novel bistable mechanism-based locking design for creating an internal force loop in the post-curing inflator (PCI) setup. Section 2 of this paper discussed the design requirements for post-curing inflator. Section 3 deals with the bistable system using a simple buckled beam example. It further discussed pseudo-bistability, retractable mechanism, compliant mechanism and four-bar linkage. In Sect. 4, the design of a new locking mechanism using a bistable structure and four-bar linkage is proposed. At last, it discusses the prototyped setup and its functionality.
2 Design Requirements The primary design statement is to seal the axial ends of the tire without distorting the rubber during the process. The assembly of setup with tire cannot be that of car axle, and tire as the freshly cured hot tire will deform if it is mounted using tire changing machine. For this, there are two sealing lids mounted over the axial opening of the tire. One lid seals the bottom of the freshly cured tire, and another lid presses the tire from the top. The chamber formed between the two covers and the tire is pressurized using an external source. The cover at the top holds the pressed position up to 10 psi pressure and moves up with 10 mm (to allow the tire to assume its final shape) as the pressure builds up inside the tire. The top lid holds this position for the rest of the process. The locking should be self-stabilizing, the more pressure builds up the more lock should tighten up. The chamber should be Leak-proof for pressurization.
3 Bistable System 3.1 Bistable Beam Bistable mechanisms have two stable equilibrium position with at least one unstable equilibrium state in between [2, 3]. A simple buckled beam snaps from one stable state to another by applying external forces. It is one of the most straightforward bistable systems with limited motion range [4]. Similarly, a spherical shell can act as a bistable system [5].
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Fig. 1 a Buckled beam and its 4-link approximation b Potential energy diagram of bistable
Four-link approximation. A straight beam, with an application of axial force, buckles and assume one of the states. For energy analysis, a single buckled beam is modeled as a 4-link system with torsional springs at joints. The energy diagram of a buckled beam (four-link model) in Fig. 1b shows the two stable points with an unstable region in between them. For a location of mid-point, two energy states are there depending on the configuration of the links (elbow up and down). For the transition between stable states, external energy is required to overcome the unstable points.
3.2 Pseudo-Bistable System A perfect bistable system upon actuation never returns to its original state unless external actuation is provided. While a mono-stable structure returns instantly to its original state once the force is removed. Pseudo-bistability lies in between the two regions. Pseudo-bistable structures return to its original state without further actuation within a finite time [6]. This finite time depends on various parameters like elastic modulus, Poisson relations, boundary conditions, system dimensions etc. By varying these properties one can go from mono-stability to perfect bistability [6].
3.3 Retractable Mechanism Another form of bistability can be achieved using a complex mechanism such as retractable mechanism used in pen or in push-push button. These mechanisms do not use the minimum potential energy point instead employ slot to distinguish between the states. These mechanism changes there slot alternatively so that on one actuation it follows the first path and in the second actuation follows another.
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3.4 Compliant Mechanism A compliant mechanism is a device which relies on the deflection of flexible segments instead of using conventional joints [7]. With the advancement in additive manufacturing, synthesis of a mechanism as a single component has become a common trend. Replacement of joints with a compliant structure eliminate mechanical issues like friction, noise, or backlash, and they significantly reduce the total part count of the mechanism hence reducing the assembly cost [8].
3.5 Four-Bar Linkage The four-bar linkage is synthesized using four rigid members in a plane: the frame, the input link, the output link, and the coupler link. These members are joined together in a closed-loop kinematic chain with single-degree-of-freedom [9]. These members can be joined via revolute pair or can be synthesized as a single component with compliant flexible segments acting as joints. Compliant four-bar linkage will increase reliability and reduce manufacturing and maintenance cost.
4 Design Setup A locking mechanism is designed for post-curing inflator using a bistable system (a combination of 3 buckled beam in shape of an asterisk). The bistable system forms the driving links in a four-bar mechanism with lock hinge as the output link. Two stable states of the bistable system give two distinct positions of locking hinges as shown in Fig. 2. The transition between these positions is achieved by applying external force at the center of the beam. Another possibility for achieving bistability is the use of spherical shell instead of beams. Spherical shells can also snap to another state with the application of external force. Both the stable states of the buckled beam have the same potential energy while in the case of spherical shell the energy level is different. In a spherical shell, one state is the manufactured shape, and the other state is the collapsed shape. Hence in collapsed shape, there is always a restoring force in action making it a pseudo-bistable structure. Hence, the collapsed state of the spherical shell tends to snap back even without any application of external force [6]. Since the motion is limited, four-bar linkage can be made as a single component with compliant joints instead of revolute pairs. This will further reduce the assembly cost as well as the maintenance cost during operations. As per the safety requirement, the pseudo-bistable structure is better as it eliminates the danger that a worker pressurized the chamber without locking the setup.
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Fig. 2 Designed PCI in two states a Unlocked b Locked
As once the external actuation is removed by the worker, this spherical shell goes back to the lock state. Hence ensuring the safety of the shop floor.
4.1 Pressure Calculation To change the state of bistable beams, external actuation is required. This actuation pressure is calculated using Eq. (1) [10]. Et 3 2 k −1 3 12r
(1)
k tan α cot kα = 1
(2)
P=
where, P is the pressure (N/m), E is the elastic modulus, t is the thickness of the plate, r is the radius of curvature, 2α is the angle subtended by the arc. For 2α = 30◦ , t = 1mm, r = 0.7m, E = 200 ∗ 109 Pa. Pressure on each plate is calculated to be 170 kPa which gives a total load of 2.54 kN. With total 3 such plates in the structure required load will be 7.7 kN (Table 1).
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Table 1 Value of k for different angles [10] α (°)
15
30
45
60
75
90
120
k
17.2
8.62
5.8
4.37
3.50
3.00
2.36
5 Rapid Prototyped Setup Buckled beam and Spherical shell setups are prototyped using 3D printing facility at central Workshop, IIT Delhi to demonstrate the basic functionality of the designed bistable locking mechanism. Of the two designs, the buckled beam system is perfect bistable structure. Once actuated to a state, it remains there if no external load is applied. In case of the spherical shell design, which is a “pseudo-bistable” [4] structure, once actuated it tends to snap back to its original shape without any further actuation. The prototyped setup is shown in its configurations in Figs. 3 and 4.
Fig. 3 PCI with buckled beam locking (bistable structure)
Fig. 4 PCI with spherical shell locking (pseudo-bistable)
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6 Conclusion In this paper, we discussed a locking system designed for post-curing inflator using bistable structures coupled with four-bar mechanisms. Two stable states of the bistable structure correspond to the locked and unlocked case. Two bistable locking mechanism (Buckled beam and Spherical shell) based PCI had been prototyped using rapid prototyping to check the functionalities. Buckled beam system is found to be stable in both states while the spherical shell tends to snap back to its manufactured shape without any application of external force. Different Bistable structure such as retractable mechanism can also be tried to improve functionality. Further, a complete working setup demonstrating all functionality is to be developed.
References 1. Lim WW (2002) Thermal properties of tire cords and their effects on post-curing inflation of tires. Rubber Chem Technol 75(4):581 2. Zirbel SA, Tolman KA, Trease BP, Howell LL (2016) Bistable mechanisms for space applications. PLoS ONE 12(11) 3. Howell LL (2001) Bistable mechanisms. In: Compliant mechanisms. Wiley, New York, pp 355–384, Wiley pp 355–357 4. Camescasse B, Fernandes A, Pouget J (2014) Bistable buckled beam and force actuation: experimental validations. Int J Solids Struct 51:1750–1757 5. Taffetani M, Jiang X, Holmes DP, Vella D (2018) Static bistability of spherical caps. Proc Roy Soc A 474(2213) 6. Brinkmeyera A, Santerb M, Pirreraa A, Weavera PM (2012) Pseudo-bistable self-actuated domes for morphing applications. Int J Solids Struct 49(9):1077–1087 7. Howell LL, Magleby SP, Olsen BM (2013) Introduction to compliant mechanisms. In: Handbook of compliant mechanisms. Wiley, pp 6–9 8. Opdahl PG, Jensen BD, Howell LL (1998) An investigation into compliant bistable mechanisms. In: ASME design engineering technical conferences. Atlanta, GA 9. Pickarda JK, Carretero JA, Merlet J-P (2019) Appropriate analysis of the four-bar linkage. Mech Mach Theory 139:237–250 10. Young WC, Budynas RG (1989) Elastic stability-formulas for elastic stability of plates and shells. In: Roark’s formulas for stress and strain. McGraw-Hill, p 737
Stability Analysis of a Dual-Rate Haptics Controller Using Discrete-Time Root-Locus Method Suhail Ganiny , Majid H. Koul, and Babar Ahmad
Abstract In this paper, the classical discrete-time root-locus method is extended for analyzing the stability of a dual-rate haptics controller. The given controller involves two closed-loop feedback gains, damping and stiffness (of the virtual wall), implemented at different sampling rates. Owing to the multi-variable and multi-rate nature of such controllers, analyzing their stability by the direct application of the root-locus method is not feasible. At the outset, the characteristic equation of the controller is thereby set up in a suitable mathematical form amenable to the rootlocus analysis. Thereafter the analysis is carried out by sequentially considering the damping and stiffness as feedback gains. Besides helping in establishing the stability bounds readily, the classical discrete-time root-locus method provides qualitative information about the stability contours of the dual-rate haptics controller. Keywords Haptics controller · Discrete-time root-locus · Multi-rate controllers · Z-width
1 Introduction A haptic interface is a programmable mechatronic device that enables real-time physical interaction with virtual reality environments through touch. Devices that are configured to produce a force output for a given motion input are referred to as impedance-type interfaces. This class of interfaces is more prevalent than others in both the commercial as well as the research domain. Simplified virtual walls modeled as a spring-damper in mechanical parallel, form the building block of most virtual environments. A region of virtual wall stiffness S. Ganiny (B) · M. H. Koul · B. Ahmad Mechanical Engineering Department, NIT Srinagar, Hazratbal Srinagar, J&K, India e-mail: [email protected] M. H. Koul e-mail: [email protected] B. Ahmad e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_83
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and damping coefficients that a haptic interface can stably implement is referred to as Z-width [1]. Rendering stiff virtual walls while simultaneously maintaining the stability of the interface has been recognized as main limitation of impedance-type interfaces [1, 2]. High sampling rates are known to enhance the performance of haptic interfaces, but the fundamental trade-off between performance and stability places a constraint on the same [1, 2]. Nevertheless, several multi-rate control techniques have been presented that overcome this performance-stability trade-off to a certain extent [3– 6]. In particular, Koul et al. [5] have proposed a dual-rate controller strategy for improving the stability bounds of a one degree-of-freedom (1-DoF) haptic interface. The stability analysis reported here pertains to such a class of controllers. Haptic interfaces are desired to remain stable as they inevitably work in conjunction with a human operator. Stability of uniform-rate controllers has been studied extensively in the literature [2, 7–12]; as opposed to this, multi-rate controllers have received very limited attention [3–5]. Gil et al. [7], for instance, performed stability analysis of a 1-DoF haptic device by direct application of Routh’s criterion. This work was later extended by Hulin et al. in [8] to consider the effect of the human operator and by Gil et al. in [9] to study the effect of time delay and physical damping on stability, respectively. In [10] stability analysis of fractional-order haptic controllers was presented. Mashayekhi et al. [11] carried out stability analysis using frequency response function analysis; and more recently, the same group used Lyapunov-Krazuvskii functional approach in [12]. Lee and Lee [3] and Dai et al. [4] used Routh-Hurwitz method for obtaining stability bounds of their multi-rate controllers, respectively. Most of these works have however utilized analytical techniques that demand an extensive mathematical treatment and in addition yield non-intuitive results. Furthermore, based on the analytical conditions, evaluation of the stability limits is an arduous task as a large number of iterations have to be carried out to find them. Although, the iterative procedure could possibly be programmed, however, a knowledge of the stable range of virtual stiffness and damping is still needed, which is obviously not known beforehand. The classical discrete-time root-locus (DTRL) method promises to be a viable solution to this issue. This method not only provides a means for evaluating the stability bounds in a fast and reliable manner but also provides insight about the shapes of the stability curves obtained, owing to its graphical nature. In this work, we, therefore, carry out the stability analysis of a dual-rate haptics controller, as proposed in [5], using the classical DTRL method. The controller being a multi-rate and multi-variable sampled-data system, the direct application of the DTRL method for analyzing its stability is accordingly not feasible. A sequence of time-domain transformations and block-diagram manipulations are therefore applied to formulate its characteristic equation in forms amenable to the DTRL analysis. Stability ranges of the virtual damping and virtual stiffness are subsequently obtained and the Z-width contours are developed. DTRL method provides explanations concerning the shapes of the Z-width contours.
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2 Dual-Rate Haptics Controller Dual-rate haptics controllers are characterized by the fact that they render virtual wall damping and stiffness at two distinct (usually integer-related) sample rates, as opposed to uniform-rate controllers that utilize a single sample rate for the purpose. This decoupled sampling scheme enables rendering of stiff virtual walls and thereby leads to performance enhancement [5]. The schematic of a typical 1-DoF haptics controller subjected to dual-rate sampling scheme is depicted in Fig. 1. The rotational haptic device (plant) has been represented by a second-order model, with moment of inertia I , and viscous damping b, respectively. The virtual environment, on the other hand, is modeled as a spring-damper system in mechanical parallel with damping B (Nms/rad) and stiffness K (Nm/rad). The virtual stiffness K is rendered at a sample period of T /N via the stiffness loop, while the virtual damping B is rendered at T via the damping loop. Although, N can theoretically take any value in the range [0, ∞], here we assume that N ∈ Z+ . Furthermore, backward difference is used to obtain the velocity estimate. The resultant torque τVE from the virtual environment is comprised of two components; torque τK due to the virtual stiffness, and; the torque τB due to the virtual damping, respectively. τVE is fed to the operator every T /N instants, after passing through the zero-order-hold (ZOH) blocks. Although an actual haptics controller contains several non-linear phenomena like Coulomb friction, quantization, human operator dynamics, actuator saturation, and unilateral constraint, the same are avoided here. The reason being: quantization induced energy can be dissipated by Coulomb friction [13, 14]; and, in most cases human operator adds stability to a haptic device by reinforcement of his own damping [7, 8]. Haptic device
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3 Stability Analysis Using DTRL Method In this section, we employ the classical DTRL method for establishing the stability limits of the dual-rate haptics controller (Fig. 1). For a given haptic device and a basic sampling period of T , there are three variables; virtual damping B, virtual stiffness K and ratio of sample periods N, that can be used as a feedback gain for the DTRL method. However, due to the multi-rate nature of the controller it is not possible to formulate its characteristic equation with N as a feedback gain. For each value of N considered, two root-loci can thus be developed; one as a function of B and the other as a function of K. In both the cases, prior to the application of the root-locus method, the characteristic equation of the controller needs to be formulated in the following form: 1 + αψ(z) = 0 (1) where, α is the feedback gain, which in this case can be either B or K, and ψ(z) can be thought of as the open-loop transfer function of the controller.
3.1 Root-Locus as a Function of B The presence of two sampling rates in the dual-rate haptics controller makes it imperative to use two different discrete-time variables for the stiffness and damping loops. Referring to Fig. 1, it can be seen that z1 and z2 have been used in this case, respectively. To obtain the characteristic equation (with feedback gain B) that conforms to Eq. 1, several time-domain transformations and some block-diagram algebra are needed. A detailed derivation cannot be provided here owing to space constraints, the procedure is however outlined below, and afterwards the main result is given. Step Step Step Step Step
1: 2: 3: 4: 5:
Discretize plant using Zero-Order-Hold (ZOH) method. Close stiffness feedback loop. Resample transfer function obtained in Step 2. Close damping feedback loop. Obtain characteristic equation of the system.
The characteristic equation for this case is given as: 1+B
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positive value of K can be rendered, the value of K = 0.0001 Nm/rad was chosen. The rlocus command of MATLAB was used to develop the root-loci, and the same are depicted in Fig. 2. For all the cases, it can be seen that the system starts from being stable as no pole lies outside the unit circle. However, as the value of B increases, the poles start moving and at a certain critical value cross-over the unit-circle, as a result, the system becomes unstable. Stability is never regained afterwards. This suggests that for a particular value of N and T , virtual stiffness K can only be rendered for a limited range of B. The value of B at which the poles cross-over, which is equivalently the stable range of B, is higher for higher values of N , which is quite expected.
3.2 Root-Locus as a Function of K Following the similar steps as outlined in Sect. 3.1, the characteristic equation for this case is given by:
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Isolating K as a feedback gain is however not straightforward, and accordingly a little more mathematics is involved in this case. The polynomial coefficients are again provided in the Appendix. Once the stable range of B is known for a particular value of N and T , root-loci based on Eq. 3 can be used to obtain the stable range of K, for a specific value of B. Figure 3 shows several such root-loci for a particular value of N and T at four values of damping B. The value of B was restricted to B = 8, as for the particular value of N and T considered here, the stable range of B ∈ [0, 7.5] only (Fig. 2b). As expected for B = 1, 3 and 6, the system starts from being stable and as the value of K increases, the system becomes unstable. However, for B = 8, the system is unstable for the entire range of K ∈ [0, ∞]. Figure 3 also reveals that upto a certain value of B (3 ≤ B ≤ 6), the virtual stiffness K increases and afterwards starts decreasing and eventually drops to zero outside the stable range of B.
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4 Discussions and Conclusions In the previous section, the DTRL method was successfully applied to obtain the stability bounds for a dual-rate haptics controller. The obtained results revealed that for a particular sample period combinations B has a limited stable range, and for each value of B, the stable value of K is different. It was also observed that upto a particular value of B, the value of K has an increasing trend, but beyond that it starts decreasing and eventually reduces to zero. These results suggest that the Z-width boundary has the shape of a closed curve as there is a bound on the stable value of both B and K. Figure 4 shows several such Z-width plots for different values of T and N . A counterintuitive observation can be made based on these results; the range of B is different for different values of N . This range should have otherwise remained constant, as different values of N correspond to different sample rates of the stiffness loop, and have no bearing on the sample rates of the damping loop. Upon investigation, it was revealed that this behavior is not due to the root-locus method, but is attributed to the sequence of transformation steps that were carried to obtain the discrete-time characteristic equation in Sect. 3.1. In particular, we had employed Tustin method for the discrete-to-discrete transformation (Step 3). A different choice (say ZOH method) yields different results, but such a study is beyond the scope of the present work. The DTRL method is extremely useful in the identification of the most dominant poles of the controller and accordingly paves way for simplifying the stability analysis via techniques like the model order reduction. This is particularly helpful for haptics controllers that entail a temporal loop delay due to sampling effects and other computational or communicational delays. An analysis of such controllers will be the focus of our future work. In this paper, an attempt has been made to study the stability characteristics of a dual-rate haptics controller by extending the classical DTRL method. The method empowers one to obtain the stability bounds of the controller quantitatively as well
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as to get a qualitative idea about the trends in the stability curves. DTRL is fast and reliable and yields the stability boundaries with much ease as compared to the purely analytical techniques like the Routh-Hurwitz or Jury’s method.
5 Appendix The polynomial coefficients used in Eq. 2 (Section 3.1 ) are defined as: p˜3 = Ih2 h3 ; p˜2 = I (h1 h3 + h2 h4 − h2 h3 ); p˜1 = I (h1 h4 − h1 h3 − h2 h4 ) q˜2 = T [2b2 (h1 − en h2 ) + IKh2 h3 ]; q˜1 = T [2b2 (h2 − h1 − en h1 + en h2 ) + IK(h1 h3 + h2 h4 )]; q˜0 = T [−2b2 (h2 − en h1 ) + IKh1 h4 ] where, h1 = N + 1; h2 = N − 1; h3 = e˜ h1 + eˆ h2 ; h4 = e˜ h2 + eˆ h1 ; −bT ; eˆ = 1 − (1 + bT )en ; en = e IN e˜ = en − 1 + bT IN IN The polynomial coefficients used in Eq. 3 (Section 3.2 s) are defined as: pˆ2 = TIn2 ; pˆ1 = TIn1 ; pˆ0 = TIn0 qˆ3 = 2b2 Tm2 + BIn2 ; qˆ2 = 2b2 Tm1 + BI (n1 − n2 ); qˆ1 = 2b2 Tm0 + BI (n0 − n1 ); qˆ0 = −BIn0 where, m2 = N (1 − en ) + (1 + en ); m1 = −2(1 + en ); m0 = (1 + en ) − N (1 − en ) n2 = N 2 e˜ + N 2 eˆ − 2N eˆ − e˜ + eˆ ; n1 = 2N 2 e˜ + 2N 2 eˆ + 2˜e − 2ˆe; n0 = N 2 e˜ + N 2 eˆ + 2N eˆ − e˜ + eˆ .
References 1. Colgate JE, Brown JM (1994) Factors affecting the z-width of a haptic display. In: 1994 IEEE international conference on robotics and automation, 1994. Proceedings. IEEE, New York, pp 3205–3210 2. Abbott JJ, Okamura AM (2005) Effects of position quantization and sampling rate on virtualwall passivity. IEEE Trans Robot 21(5):952–964 3. Lee K, Lee DY (2004) Multirate control of haptic interface for stability and high fidelity. In: 2004 IEEE international conference on systems, man and cybernetics, vol 3. IEEE, New York, pp 2542–2547 4. Dai X, Zhang Y, Cao Y et al (2008) Stable multirate control algorithm for haptic dental training system. In: International conference on intelligent robotics and applications. Springer, Berlin, pp 27–35 5. Koul M, Manivannan M, Saha S (2017) Effect of dual-rate sampling on the stability of a haptic interface. J Intell Robot Syst pp 1–13 6. Kim M, Lee DY (2019) Multirate haptic rendering using local stiffness matrix for stable and transparent simulation involving interaction with deformable objects. IEEE Trans Ind Electron 7. Gil JJ, Avello A, Rubio A et al (2004) Stability analysis of a 1 DOF haptic interface using the Routh-Hurwitz criterion. IEEE Trans Control Syst Technol 12(4):583–588 8. Hulin T, Preusche C, Hirzinger G (2008) Stability boundary for haptic rendering: influence of human operator. In: IEEE, RSJ international conference on intelligent robots and systems (2008) IROS 2008. IEEE, New York, pp 3483–3488 9. Gil JJ, Sanchez E, Hulin T et al (2009) Stability boundary for haptic rendering: influence of damping and delay. J Comput Inform Sci Eng 9(1)
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10. Tokatli O, Patoglu V (2015) Stability of haptic systems with fractional order controllers. In: 2015 IEEE/RSJ international conference on intelligent robots and systems (IROS). IEEE, New York, pp 1172–1177 11. Mashayekhi A, Behbahani S, Ficuciello F et al (2018) Analytical stability criterion in haptic rendering: the role of damping. IEEE/ASME Trans Mechatron 23(2):596–603 12. Mashayekhi A, Behbahani S, Ficuciello F et al (2019) Delay-dependent stability analysis in haptic rendering. J Intell Robot Syst pp 1–13 13. Diolaiti N, Niemeyer G, Barbagli F et al (2006) Stability of haptic rendering: discretization, quantization, time delay, and coulomb effects. IEEE Trans Robot 22(2):256–268 14. Mashayekhi A, Boozarjomehry RB, Nahvi A et al (2014) Improved passivity criterion in haptic rendering: influence of coulomb and viscous friction. Adv Robot 28(10):695–706
Design of Compliant Iris Vinay Arora, Prakhar Kumar, Rajesh Kumar, and Jitendra Prasad Khatait
Abstract Stiffness-based approach has been used to design the compliant version of mechanical iris. The rigid links of the mechanism are replaced by sheet flexures. Iris has found use as grippers and also to carry out controlled opening of aperture. This paper is based on formulating an estimate of the stiffness based on geometrical parameters. Stiffness equation relates the input moment to the output deflection. The equations are utilized to design based on the required use case. Keywords Compliant iris · Stiffness equation · Design methodology
1 Introduction Mechanical iris has found use in many devices like iris diaphragm of a camera [2], iris valves [4], etc. It is also used in optical setups to control the amount of light passing through it [1]. Recently, iris structures have also found use in solar thermal receivers where the amount of light is controller mechanically. It is inherently a single input– output device, with many internal degrees of freedom. Many parts move relative to one another, but the net output of interest is the radius of the central aperture [5]. However, there are some limitations of the mechanical iris as it is composed of many plates moving relative to each other. So, a typical mechanical iris is composed of various moving parts resulting in issues like friction and backlash. However, the compliant iris is a single part mechanism which is based on large deflections of parts. The compliant counterpart of the iris is composed of a single part. This paper is based on prediction of bounds on stiffness of the iris system and a methodology to make design decisions. Compliant iris has found use in gripping and holding the objects. It can be used as an end effector gripper of a robotic arm. A closed-form expression of stiffness is required to relate design the compliant iris. The novelty of the paper lies in provid-
V. Arora · P. Kumar · R. Kumar (B) · J. P. Khatait Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_84
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Fig. 1 a Drawing, b 3D CAD model, and c 3D-printed prototype of the compliant iris
ing a methodology to design based on formulating stiffness equations. Although a closed-form expression is not delivered, stiffness bounds are conveyed for a particular set of design parameters. The structure of the compliant iris is shown in Fig. 1.
2 Iris Structure A mechanical iris basically is a rotational input–output mechanism, and the input rotation moves various plates in such a way that the free end of the plates rotates and moves in a radial direction, thereby creating an opening. Compliant iris is a compliant-based substitute of the same. The thin plates are replaced by flexible beams. The structure is in two halves. The curved beam is attached to one half and then to the other half via passing through the center. As each half is rotated in opposite directions, the flexible curved beams bend and create an opening in the center. The beams rotate and also create a central opening. As the compliant iris works on the basis of beam deflection, there is net input–output stiffness relationship that needs to be accounted for. The idea of distributed compliance is utilized where the beams deform largely to produce a relatively smaller output. The first known instance of compliant iris was developed by Flexsys Corporation, where they named it as ‘jointless iris.’ The motion of opening of a compliant iris is presented in Fig. 2. Although the flexible beams move relative to one another, the net output is the size of the aperture.
3 Stiffness Analysis In order to analyze the stiffness of the compliant iris arrangement in the compliant direction, stiffness expression of curved beam has been derived in terms of geometric dimensions and material properties and stiffness of multiple beams have been added to obtain overall stiffness. For a general beam, stiffness is derived using Castigliano theorem [3]. Castigliano’s theorem is one of the energy methods (based on strain
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energy) that can be used to solve a wide range of deflection problems. It relates the deflection in an elastic body in any direction (‘a’) to the partial derivative of the strain energy with respect to the input loads ‘Fi ’ in the direction ‘a’ (Eq. 1) δi =
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3.1 Derivation of Stiffness Expression With reference to the symbols presented in Fig. 3, consider, instantaneously, each of the curved beams has a radius of curvature R and makes an angle θ with the longitudinal axes. The whole of the iris is composed of 6 such curved beams. From Fig. 3, we get Fn = Fcos(θ ), Fθ = Fsin(θ ) and M = FRsin(θ ). Let ‘t’ be the thickness of the beam, and as Rt >> 1, so the strain energy terms take the form as presented in Eqs. 2–7. M 2R dθ 2EI F 2R dU2 = θ d θ 2AE MFθ dU3 = − dθ AE F 2R dU4 = 1.2 n d θ 2AG dU1 =
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Fig. 3 This is one of the curved beams used in compliant iris
where ‘A’ is the cross-sectional area of the beam, ‘G’ is the shear modulus of the material of the iris, ‘E’ is the elastic modulus of the material, and ‘I’ is the area moment of inertia for the curved beam. θ dU1 + dU2 + dU3 + dU4
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0
So, the net deflection δ is given by Eq. 8.
sin2θ δ = FR θ − 2
R2 1 1.2 − + 2EI 2AE 2AG
So, the stiffness of one beam of rectangular cross section A = bt and I = stiffness for each of the beams becomes Eq. 9. k=
bt 3 2 2 3R(2θ − sin2θ ) R −tE /12 +
t2 10G
(8) bt 3 ; 12
so, the
(9)
For a compliant iris with six beams, we have overall torsional stiffness as 11 utilizing the net deflection angle (α) and the net moment (Mt ). The net torsional stiffness is given by Eq. 10. 2 (10) Overall torsional stiffness (K) = 36 kriris
Design of Compliant Iris
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where riris is the net radius of the whole iris (the bigger structure). The net analytical expression of torsional stiffness of the iris ‘K’ is presented in Eq. 11. K=
2 bt 3 θ 3 12Eriris L3 (2θ − sin(2θ ))
(11)
However, the terms ‘R’ and θ are variables as the radius of curvature of the iris and the angle spanned by the flexure keeps on changing. So, we need to express both in terms of α which is the net rotation of the iris. The iris opens proportional to the rotation angle α. We assume that the net change in the length of each of the flexures is negligible. So, we have the geometric relation (Eq. 12). Rθ = L
(12)
Let αo be the initial rotation for the net iris output structure. Equation 12 is reduced to Eq. 13. θ α + αo = Rsin (13) riris sin 2 2 The equation reduces to Eq. 14.
α + αo sin 2
=
Graphically, nonzero solution exists when L > 2. riris
2riris θ sin L 2
2riris o sin α+α L 2
(14)
< 1, which results in
4 Discussion Although a closed-form expression for the net stiffness is not known, the bounds in the stiffness can be used for design purposes. It should be known that the net stiffness of the iris changes with change in the output angle. The upper bound and lower bound for the expression of stiffness are expressed in Eqs. 15–16. The expression of stiffness is increasing in θ for all values of L. So, the maximum stiffness occurs when each of the flexures is at extreme values of θ . Let the value of θ is limited by θo . The L and R = riris . So, in case we have a required minimum stiffness occurs when θ = riris stiffness during a complete operation of the opening of the aperture, then the various parameters in maximum stiffness and minimum stiffness expressions can be varied. Max stiffness =
2 bt 3 12Eriris − sin2θo )
L3 (2θo
(15)
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Fig. 4 Upper and lower bounds of stiffness variation with iris radius
Fig. 5 Upper and lower bounds of stiffness variation with flexure length
Min stiffness =
12Ebt 3 2L − riris sin r2L iris
(16)
The variation of the stiffness values with respect to the length of each of the beams as well as the variation of the analytical stiffness bounds with respect to the length of the flexural beam and radius of the iris is presented in Figs. 4 and 5. With the rise in the length of the flexural beam, both the upper bound and the lower bound of the stiffness expression reduce. The bounds also tend to converge to a particular value. The graphs presented in Figs. 4 and 5 correspond to the dimensions and material properties as riris = 0.1 m, b = 0.005 m, t = 0.001 m, E = 3 Gpa, G = 800 Mpa. Figure 4 does not have a constant iris radius, but the rest of the parameters are the same. With the change in iris radius, the bounds on the stiffness widen, so the discrepancy in design also raises. Hence, to ensure a proper design we might need to have large length of beam flexure and small radius of the iris. So, the set of equations provide a methodology to design compliant iris which is not yet available in the literature.
5 Conclusion We present a method for stiffness-based design of a compliant iris. Analytical stiffness equations are formalized, and bounds on stiffness are provided based on geometrical constraints of the iris. Though a closed-form expression for design is not feasible by the methodology, the bounds in stiffness give an idea of the required dimensions for the development of the iris structure.
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References 1. Fukuda T, Hattori S, Arai F, Matsuura H, Hiramatsu T, Ikeda Y, Maekawa A (1993) Characteristics of optical actuator-servomechanisms using bimorph optical piezo-electric actuator. In: Proceedings IEEE international conference on robotics and automation. IEEE, New York, pp 618–623 2. Hugo K (1925) Iris diaphragm. US Patent 1,547,658 3. Levinson M (1965) The complementary energy theorem in finite elasticity. J Appl Mech 32(4):826–828 4. Lubold RP (1968) Iris valve. US Patent 3,371,906 5. Ophoff C, Ozalp N (2017) A novel iris mechanism for solar thermal receivers. J Solar Energy Eng 139(6):
Investigation on the Effects of Nose Radius and Rake Surface of Cutting Tool for Machinability During Sustainable Turning of EN 31 Alloy Steel Sutanu Misra, Goutam Paul, and Asim Gopal Barman
Abstract Sustainable machining is an important issue in the modern manufacturing field. The use of minimum quantity lubrication during turning operation can lead to economical and hazard free process. In addition to these, the variation of tool nose radius can also lead to the reduction of temperature generation increasing the tool life. In this research study, the variation of tool nose radius was employed during turning operation on EN 31 (European Standard 31) alloy steel under dry and liquid CO2 -assisted minimum quantity lubrication (MQL) condition. Also, it has been investigated to obtain a better machinability circumstance. Keywords Tool nose radius · Orthogonal turning · MRR · Temperature · MQL
1 Introduction Metal cutting plays a substantial role in manufacturing industries. High-quality material with minimum cost is very much required. In the turning, extra material is removed from the cylindrical workpiece by a single-point cutting tool. Good machinability is an important issue in turning which depends on various factors like temperature generation, force generation, tool life, tool material, workpiece material, tool geometry and many. Cutting fluids are commonly used to decrease the friction-induced heat, which is generated during cutting. So, tool life is increased and surface finish is improved. Also, the formation of built-up edges is prevented. The use of minimum quantity lubrication (MQL) or machining in dry condition is getting priority in industries. Minimum quantity lubrication is an efficient method in S. Misra (B) · A. G. Barman Department of Mechanical Engineering, National Institute of Technology Patna, Patna 800005, India A. G. Barman e-mail: [email protected] S. Misra · G. Paul Department of Mechanical Engineering, University of Engineering and Management Kolkata, Kolkata 700160, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_85
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recent days to achieve better machining performance. In this process, a very small quantity of fine lubricant particles is sprayed between the tool–workpiece contact surfaces. Das et al. [1] investigated on hard turning operation on EN 31 alloy steel, and various output responses related to tool life, temperature generation and surface finish were optimized. Upadhyay et al. [2] made a review of the MQL machining to establish that the process is green and cost effective. A model on cutting force for machining of Inconel 718 in MQL condition is proposed by Behera et al. [3]. Surface roughness of MQL nozzle and air pressure affects the lubricant droplet size which is effectively reduced tool wear at the time of machining [4]. Effect of tool wear rate of tungsten carbide tools and surface roughness of EN 31 alloy steel can be predicted by developing models for getting best combination of design variables [5]. In this current research, coated inserts have been used to turn EN 31 alloy steel under two different cutting conditions, i.e. under dry and MQL conditions. Tool nose radius has been varied.
2 Experimentation The job selected for this research work is cylindrical EN 31 alloy steel of 16 × 300 mm. The material is appropriate for the use in various fields of automobile, e.g. axle, spindle, mandrels and also in moulding dies. These are being used by turning operation mainly [6]. Coated carbide insert with 0.03 and 0.06 mm nose radius has been used as a cutting tool. In this work, primary cutting parameters are spindle speed (N), feed (f ), depth of cut (d). The levels and the values of cutting parameters have been selected by prior experimentation within the range in the machine tool. This experiment has been carried out on the lathe machine (maximum spindle speed 1200 rpm) in dry condition and liquid CO2 -assisted minimum quantity lubrication. Figure 1 shows the photographic view of experimental setup. On the basis of preliminary experiments, the following process parameters with the level settings are shown in Table 1. Fig. 1 Experimental setup
Investigation on the Effects of Nose Radius … Table 1 Experimental process parameters and their level settings
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Input parameters
Level-1
Level-2
Level-3
Spindle speed (r.p.m.)
500
–
–
Feed rate (mm/min)
20
40
60
Depth of cut (mm)
0.1
0.3
0.5
The experiments have been conducted as per well-defined design of experiments (DOE). The values of material removal rate and temperature generation in principal as well as in auxiliary cutting face for various level settings of cutting speed (rpm), feed rate (mm/min) and depth of cut (mm) are shown in Table 1.
3 Results and Discussion Total 18 numbers of experiments with two replicates for each nose radius were carried out. The spindle speed was kept constant at 500 r.p.m., but feed rate and depth of cut were varied using the above level settings. The graphs are drawn (as depicted in Figs. 2, 3 and 4) based on the experimental results. The graphs not only illustrate the result but also provide the best regression equations as shown in the graphs.
3.1 Analysis on MRR It has been observed from Fig. 2 that MRR is maximum at 40 mm/min feed rate in MQL medium for both the tool nose radii, and it reduces with other feed rate settings. This is due to the fact the generated heat when the feed rate is 40 mm/min get much time to dissipate, and at the same time, the cool MQL medium helps to reduce the temperature of cutting. In case of 60 mm/min feed rate, the generated heat does not find much time to dissipate, and so, the MRR is also lower.
3.2 Analysis for Cutting Temperature The temperature at the tool was measured by placing the thermocouple at two different positions, i.e. at principal and auxiliary cutting edge, and it has been observed that the temperature at principal cutting edge (Tp) is quite higher than that of auxiliary cutting edge (Ta). This is due to the dissipation of heat through the tool material. The temperature generation in both the edges for dry medium is greater in comparison with MQL medium and well enough to decrease the tool life.
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S. Misra et al. 50 45
MRR_MQL AT 20 MM/MIN FEED RATE
MRR (gm/min)
40 y = -1.49x2 + 12.66x - 4.57 R² = .90
35
MRR_MQL AT 40 MM/MIN FEED RATE
30 25
MRR_MQL AT 60 MM/MIN FEED RATE
20
y = 2.61x2 - 6.93x + 10.54 R² = .91 Poly. (MRR_MQL AT 20 MM/MIN FEED RAT
15 10
y = 4.005x2 - 14.245x+14.6 R² = .92
5 0
Poly. (MRR_MQL AT 40 MM/MIN FEED RAT
0.1
0.3
0.5
0.7
0.9
Poly. (MRR_MQL AT 60 MM/MIN FEED RAT
depth of cut (mm)
b
y = 1.7892x3 - 18.526x2 + 55.445x - 32.948 R² = 0.7469
25
MRR _ MQL AT 20 MM/MIN FEED RATE
MRR (gm/min)
20 MRR _ MQL AT 40 MM/MIN FEED RATE
y=
15
1.4896x4
17.833x3
+ 63.91
+
72.365x2 R² = 0.97
10
- 113.71x MRR _ MQL AT 60 MM/MIN FEED RATE Poly. (MRR _ MQL AT 20 MM/MIN FEED RATE) Poly. (MRR _ MQL AT 40 MM/MIN FEED RATE)
5 0
Poly. (MRR _ MQL AT 60 MM/MIN FEED RATE)
0.1
y = 0.92x4 - 11.463x3 + 48.905x2 - 81.472x + 48.51 R² = 0.93
0.3
0.5
0.7
0.9
depth of cut (mm)
c
y = 8.16x2 - 34.44x + 42.97 R² =0.87
80
MRR_DRY AT 20 MM/MIN FEED RATE
70 MRR_DRY AT 40 MM/MIN FEED RATE
MRR(gm/min)
60 50
MRR_DRY AT 60 MM/MIN FEED RATE
40 30 y=
20
3.3133x3
-
33.087x2
Poly. (MRR_DRY AT 20 MM/MIN FEED RATE) + 95.29x - 58.294 R² = 0.85
10 0
0.1
0.3
0.5
0.7
depth of cut (mm)
0.9
Poly. (MRR_DRY AT 40 MM/MIN FEED RATE) Poly. (MRR_DRY AT 60 MM/MIN FEED RATE) y = -3.32x2 + 20.59x - 16 R² = 0.89
Fig. 2 Analysis for a MRR at 0.03 mm nose radius in MQL, b MRR at 0.06 mm nose radius in MQL, c MRR at 0.03 mm nose radius in dry medium, d MRR at 0.06 mm nose radius in dry medium, e comparison of MRR at dry and MQL conditions in 0.03 mm nose radius and f comparison of MRR at dry and MQL conditions in 0.06 mm nose radius
Investigation on the Effects of Nose Radius …
d
923 MRR _ DRY AT 20 MM/MIN FEED RATE
30
MRR ( gm/min)
25 20
y = 1.6367x3 - 18.536x2 + 60.007x - 36.452 R² = 0.9267
MRR _ DRY AT 60 MM/MIN FEED RATE
15 10
Poly. (MRR _ DRY AT 20 MM/MIN FEED RATE)
5 0
0.1
0.3
0.5
0.7
0.9
MRR (gm/min)
MRR_DRY AT 40 MM/MIN FEED RATE
60 MRR_MQL AT 40 MM/MIN FEED RATE
40
20
0.1
0.3
0.5
0.7
depth of cut (mm) 25
Poly. (MRR_DRY AT 40 MM/MIN FEED RATE) y = -1.49x2 + 12.66x - 4.57 R² = 0.98 0.9 Poly. (MRR_MQL AT 40 MM/MIN FEED RATE)
y = -1.2375x3 + 11.782x2 - 32.11x + 35.348 R² = 0.984
20
MRR(gm/min)
Poly. (MRR _ DRY AT 60 MM/MIN FEED RATE)
y = 8.16x2 - 34.44x + 42.97 R² =0.96
80
0
f
Poly. (MRR _ DRY AT 40 MM/MIN FEED RATE)
y = 1.5325x3 - 15.407x2 + 44.7x - 28.126 R² = 0.8265
depth of cut (mm)
e
MRR _ DRY AT 40 MM/MIN FEED RATE
y = -1.2375x3 + 11.782x2 - 32.11x + 35.348 R² = 0.984
MRR_DRY AT 40 MM/MIN FEED RATE
MRR_MQL AT 40 MM/MIN FEED RATE
15 10
Poly. (MRR_DRY AT 40 MM/MIN FEED RATE)
5 0 0.1
0.3
0.5
y = 2.450x4 - 27.61x3 + 102.9x2 - 144.0x + 72.91 R² = 0.98 Poly. (MRR_MQL AT 40 MM/MIN FEED RATE) 0.7 0.9
-5
depth of cut (mm) Fig. 2 (continued)
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300
Tp_MQL AT 20 MM/MIN FEED RATE
y = 16.65x2 - 50.45x + 87.3 R² = 0.92
250
Tp _ MQL AT 40 MM/MIN FEED RATE
Tp(0C)
200
Tp_ MQL AT 60 MM/MIN FEED RATE
150 y=
100
-2.45x2
+ 11.85x + 33.3 R² = 0.94
Poly. (Tp_MQL AT 20 MM/MIN FEED RATE)
50 y = -0.9x2 + 3.7x + 37.6 R² = 0.93
0 0.1
0.3
0.5
0.7
0.9
depth of cut (mm)
Tp (oC)
b
Poly. (Tp_ MQL AT 60 MM/MIN FEED RATE)
80
Tp _ MQL AT 40 MM/MIN FEED RATE
60
Tp _ MQL AT 60 MM/MIN FEED RATE
y = 4.9321x4 - 59.939x3 + 247.98x2 - 398.86x + 259.38 R² = .90 70
50 Tp _ MQL AT 20 MM/MIN FEED RATE
40 30
y = -0.3364x2 + 1.7976x + 44.982 R² = 0.89 Poly. (Tp _ MQL AT 40 MM/MIN FEED RATE) y = 1.2375x3 - 12.139x2 + 34.264x + 19.476 R² = 0.9038
20 10 0
0.1
0.3
0.5
0.7
depth of cut (mm)
c
Poly. (Tp _ MQL AT 40 MM/MIN FEED RATE)
140
Poly. (Tp _ MQL AT 60 MM/MIN FEED RATE) Poly. (Tp _ MQL AT 20 MM/MIN FEED RATE)
y = 12.125x4 - 143.83x3 + 576.66x2 892.55x + 522.59 R² = .9561
160
0.9
Tp_DRY AT 20 MM/MIN FEED RAT Tp _ DRY AT 40 MM/MIN FEED RAT
Tp (0C)
120 Tp _ DRY AT 60 MM/MIN FEED RAT y = 9.1625x3 - 91.313x2 + 262.39x - 127.86 R² = 0.8531 Poly. (Tp_DRY AT 20 MM/MIN FEE RATE)
100 80 60 40 20 0
y = 4.0767x3 - 38.762x2 + 105.45x - 23.546 R² = 0.9865
0.1
0.3
0.5
0.7
depth of cut (mm)
0.9
Poly. (Tp _ DRY AT 40 MM/MIN FE RATE) Poly. (Tp _ DRY AT 60 MM/MIN FE RATE)
Fig. 3 Analysis for a Tp at 0.03 mm nose radius in MQL, b Tp at 0.06 mm nose radius in MQL, c Tp at 0.03 mm nose radius in dry medium, d Tp at 0.06 mm nose radius in dry medium, e comparison of Tp at dry and MQL conditions in 0.03 mm nose radius and f comparison of Tp at dry and MQL conditions in 0.06 mm nose radius
Investigation on the Effects of Nose Radius …
d
925
120
Tp_DRY AT 20 MM/MIN FEED RATE
y = 8.9933x3 - 89.666x2 + 257.78x - 128.5 R² = 0.8519
100
Tp_DRY AT 40 MM/MIN FEED RATE
Tp(oC)
80 Tp_DRY AT 60 MM/MIN FEED RATE
y = 3.842x4 - 44.06x3 + 168.5x2 - 244.8x + 180.8 R² =0.97
60
Poly. (Tp_DRY AT 20 MM/MIN FEED RATE)
40
Poly. (Tp_DRY AT 40 MM/MIN FEED
20 0
- 15.952 y = 3.5667x3 - 32.804x2 + 86.749xRATE) R² = 0.9937
0.1
0.3
0.5
0.7
0.9
Poly. (Tp_DRY AT 60 MM/MIN FEED RATE)
depth of cut (mm)
e
y = 16.65x2 - 50.45x + 87.3 R² = 0.99
300 250
Tp(oC)
200
Tp_DRY AT 40 MM/MIN FEED RATE
Tp_MQL AT 40 MM/MIN FEE RTAE
150 100
Poly. (Tp_DRY AT 40 MM/MIN FEED RATE)
50 0 0.1
0.3
0.5
Poly. (Tp_MQL AT 40 MM/M y = 12.12x4 - 143.8x3 + 576.6x2 - 892.5x + 522.5 FEED RTAE) 0.7 0.9 R² =0.97
depth of cut (mm)
f
140
Tp_DRY AT 40 MM/MIN FEE RATE
120
y = 10.18x4 - 120.7x3 + 483.0x2 - 746.1x + 445.6 R² = 0.86
Tp(oC)
100
Tp_MQL AT 40 MM/MIN FEE RATE
80 60
Poly. (Tp_DRY AT 40 MM/M FEED RATE)
40 20 0
0.1
0.3
y = 4.932x4 - 59.93x3 + 247.9x2 - 398.8x + 259.3 R² = 0.91 Poly. (Tp_MQL AT 40 MM/M 0.5 0.7 0.9 FEED RATE)
depth of cut (mm) Fig. 3 (continued)
926 Fig. 4 Analysis for a Ta at 0.03 mm nose radius in MQL, b Ta at 0.06 mm nose radius in MQL, c Ta at 0.03 mm nose radius in dry medium, d Ta at 0.06 mm nose radius in dry medium, e comparison of Ta at dry and MQL conditions in 0.03 mm nose radius and f comparison of Ta at dry and MQL conditions in 0.06 mm nose radius
S. Misra et al.
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4 Conclusions In the research work, the turning at dry and MQL conditions with varying tool nose radius has been employed. The experiments have been set with full factorial design of experiments, and the main objective is to investigate on the improvement on machinability of EN 31 alloy steel at various machining conditions with variation of tool geometry. The following observations were made from the above research work: 1. 2. 3. 4.
MRR has dependency on the cooling medium, but in dry medium, the MRR increases with D.O.C. and feed rate as far as EN 31 alloy steel is concerned. MRR is higher in case of machining in MQL condition irrespective of nose radius at 500 rpm, 40 mm/min feed rate and 0.5 mm D.O.C. Temperature generation directly affects the tool life, and with increase of tool nose radius, it is increased. Generated heat is carried out by the cooling medium, so the temperature generation is quite lower in MQL medium compared to that in dry medium.
The research work has lots of industrial use as far as EN 31alloy steel material is concerned, and there is a scope to carry out analytical work on the material.
References 1. Anshuman D, Kumar PT, Hotta B, Bhushan B (2018) Statistical analysis of different machining characteristics of EN-24 alloy steel during dry hard turning with multilayer coated cermet inserts. Measurement 134:123–141 2. Vikas U, Jain PK, Mehta NK (2013) Machining with minimum quantity lubrication: a step towards green manufacturing. J Mach Mach Mater 13(4):349–371 3. BarczakL M, Batako ADL, Morgan MN (2010) A study of plane surface grinding under minimum quantity lubrication (MQL) conditions. Int J Mach Tools Manuf 50(11):977–985 4. Khan WA, Hoang NM, Bruce T, Hung Wayne NP (2018) Through-tool minimum quantity lubrication and effect on machinability. J Manuf Process. https://doi.org/10.1016/j.jmapro.2018. 03.047 5. Laxman A, Hameedullahb M (2015) Simultaneous optimization of multiple quality characteristics in turning EN-31 steel. Mater Today: Proc 2:2640–2647 6. Behera BC, Ghosh S, Rao PV (2017) Modelling of cutting force in MQL machining environment considering chip tool contact friction. Tribol Int. https://doi.org/10.1016/j.triboint.2017.09.015
Battery Performance Analysis of Static Temperature Variations for Medical Environment B. Banuselvasaraswathy
and R. Vimalathithan
Abstract Improving battery life is very essential need in all battery-based applications to improvise the component operation for longer period of time. But, the battery is easily influenced by surrounding temperature which leads to fast discharge and reduction of charge capacity. To analyze these problems, thermography instrument is used which provides better temperature measurement as well as heat flux variations in the battery. The temperature is varied at 10 °C interval for each 10 successful charging and discharging cycles. While increasing the temperature at 60 °C, the charging period is reduced to 2 h 10 min and discharging time to 2 h 18 min. Beyond this, continuous exposure of battery at different temperature makes the battery to get damaged and broken. Finally, the scanning electron microscope (SEM) images are analyzed to understand the temperature influence on anode and cathode plates. Keywords Lead–acid battery · Thermography · State of charge · Scanning electron microscope · Temperature
1 Introduction In medical field, the battery backup is very important to monitor the patient’s health status continuously. Nowadays, batteries are playing a major role in medical field to monitor the patient continuously from remote locations with the help of sensors through sensor communications where a regular power supply is unable to meet the demands and result in failure. Due to technology advancements, the doctors can able to easily access the patient’s health status directly from the hospital. All remotely operated components need power supply to monitor the health condition of patient. Hence, increasing the lifetime of battery is essential since it is the main source of B. Banuselvasaraswathy (B) Department of Electronics and Communications Engineering, Sri Krishna College of Technology, Coimbatore, India R. Vimalathithan Department of Electronics and Communications Engineering, Karpagam College of Engineering, Coimbatore, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_86
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power. So, maintaining the condition of battery is very essential to enhance the device life and performance. But, the battery lifetime and performance varies based on state of charge and environmental conditions. The major influence on the battery performance is environmental temperature. Therefore, it is necessary to understand the temperature influence on battery to predict the accurate state of charge and health of the battery. Batteries are highly liable to temperature variations, which change the effective charge capacity. In this circumstance, it is very difficult to estimate the behavior of batteries for longer periods. The overall performance of a battery depends on the individual cell performance. As cell temperature varies from one to another, the charging and discharging ratios also vary for each cell at every cycle. If the temperature difference prolongs, the cell module becomes imbalance, thereby reducing the cell performance [1]. Hu et al. [2] validated the performance of temperature on lead–acid battery and observed that the charge acceptance rate gets reduced while decreasing the temperature at sub-zero level. In embedded hardware, the operation of sensor node is significantly influenced by the thermal effect; specifically, the batteries get affected at different temperature variations [3, 4]. The electrochemical cells exhibit different effective charge capacities based on temperature [5, 6]. Hence, it is challenging to determine its behavior over time, especially the energy-aware algorithms such as voltage levels and state of charge (SoC) [7–9]. Many battery models are used in wireless sensor networks (WSNs), but most of the models do not consider the intrinsic behavior of battery such as thermal effects which can deteriorate the lifetime of battery [10–13]. Dougal et al. [14] introduced a dynamic model for Li-ion batteries which is widely utilized in portable systems. Streza et al. [15] analyzed the amount of heat being dissipated in electrode during discharge cycle by using IR camera with high resolution. In [16], a small variation in thermal homogeneities is detected easily. Additionally, variations in heat evolution are followed during discharging and charging processes. Thus, in this paper, the effect of temperature variation on battery has been analyzed by utilizing thermography imaging system. This method is very useful to identify the heat flux of cell as well as the surroundings of battery. Additionally, SEM images were taken to analyze the anode and cathode plate integrity.
2 Experimentation The experimental setup is utilized to carry out the observations which are discussed in the following section.
2.1 Implementation The battery used in this study is lead–acid battery and dimension of battery is 151 × 50 × 94 mm, battery capacity is 12 V, 7 Ah and cycle use of 14.5 V, 14.9 V at 25 °C.
Battery Performance Analysis of Static Temperature Variations …
931
Fig. 1 Experimental setup with thermography imager
Heating coils are used at both sides of the chamber to generate required amount of heat. At this temperature condition, batteries are frequently charged and discharged. K-type thermocouple sensors are used to measure the surrounding temperature of the battery with an accuracy of ±2.2 °C. The results obtained from thermocouple are processed with NI DAQ card. The final temperature is fed to LabView software to measure the temperature of the chamber continuously as shown in Fig. 1. Infrared thermography (IRT) is a non-contact-type surface utilizing gray scale (infrared color images) to measure temperature. The IRT image with brighter area typically represents a high-temperature zone, and darker area indicates a lower temperature distribution [17].
2.2 Thermography FLUKE Thermography TiX-580 is used to capture the temperature of the chamber and temperature of positive and negative terminal. It has infrared camera with 4 × pixel data at high resolution, which is capable of capturing multiple images. Later, the multiple images are combined into a single image (1280 × 960). The battery is kept at 45° with respect to the plane of thermal detector to eliminate the identification of radiation emitted or reflected by the camera and operator [17]. To eradicate reflection from other objects present in the environment as well as from the ceiling, a huge
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cardboard box was kept above the battery and stand. The camera was placed on tripod situated outside the enclosure and focused in such a way that it can view only the battery placed inside the box.
2.3 Scanning Electron Microscope Field emission scanning electron microscope (FE-SEM) images are used to analyze the change in alloy surface due to electrodeposition and corrosion during charging and discharging [9]. In this study, FE-SEM, Thermo-Fisher FEI Quanta 250 FEG is available in Sophisticated Instrumentation Center at VIT–Vellore. To prepare the FE-SEM sample, the anode and cathode plates were removed from the battery. The plates are washed with distilled water to naturalize the pH content. The samples are dried under vacuum at 50 °C for one full day. The dried samples were collected and made into triangular shape with a thickness of 2 mm.
3 Results and Discussion The results obtained using thermography image analysis and scanning electron microscope which were discussed in the following sections.
3.1 Thermography Image Analysis Battery failure occurs at different conditions like overcharging, undercharging, low electrolyte levels, high ambient temperature, vibrations and prolonged storage with insufficient recharging. As the temperature exceeds beyond the operating range, battery experiences unconditional accelerated chemical reactions resulting in high mobility of electrons between the anode and cathode plates. The change in electrodes has an influence on electrical resistance and impedance, which is caused due to the battery’s physical and chemical reactions. So, from the measurement of internal resistance, it is possible to determine the internal heat generated from the cells. The IR thermography camera is used to measure the IR radiation produced by the object due to heat and corresponding changes in anode and cathode parts. From the resultant images, the aging mechanism of electrodes can be easily analyzed. This in turn is helpful in identifying the nature of aging factors like uniform or non-uniform surfaces of the electrodes [18]. The analysis was considered for one full cycle (charging and discharging) of battery. While charging, the temperature of the battery is maintained at chamber temperature. Once the battery reaches 80% of the charge, the internal temperature variations are observed largely. The continuous charging of battery leads to overcharging state which may lead to battery failure. The same observation was
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measured using thermography images at initial charging and fully charged condition. Figure 2a shows the initiation of charging above room temperature (35.4 °C), and Fig. 2b shows the maximum charging capacity of battery. The temperature difference between the low charge capacity and high charge capacity is around 2 °C. Meanwhile, the time taken to reach the maximum capacity is 3 h and 43 min. Once the battery is completely charged, the battery is kept at resting state for about 30 min to reach the stable condition. After the resting period, the discharge is taken into consideration with constant load. Figure 3a shows the temperature of battery at resting position. The temperature gets reduced due to 30 min resting period. As the battery gets discharged, the temperature starts to increase due to ion exchange and internal resistance of the battery. If the battery’s internal resistance is high, the heat generation of the internal cell of the battery is also observed at high range. Figure 3b shows the condition of battery capacity at minimum level, and the temperature is measured at 39.6 °C. The heat dissipation rate varies from cell to cell, and it mainly depends on the cell capacity, cyclic period and temperature of the cell. This cycle of operation is carried out for 50 full charging and discharging cycles of battery.
(a)
(b)
Fig. 2 Charging of battery at room temperature a temperature of 35.4 °C at low capacity of the battery b temperature of 37.4 °C at full capacity of the battery
(a)
(b)
Fig. 3 Discharging of battery at room temperature a 36.8 °C initial discharging at room temperature, b 39.6 °C at high load discharging
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(a)
(b)
Fig. 4 Battery temperature at 50 °C a initial battery temperature is 49 °C, b at fully charged condition the battery temperature is around 57 °C
The battery’s surrounding temperature is raised to 50 °C with the help of heating unit. At this stage, the battery charging is initiated and temperature is measured with the help of thermal image depicted in Fig. 4. The recorded temperature is 45 °C and charging is continued until battery reaches its maximum capacity. After 2 h 48 min, the battery reaches its full range and temperature is found to be 57 °C. Compared with Fig. 2a and b and Fig. 4a and b, the charging time has been reduced considerably at high surrounding temperature. The heat generation rate increases from the start of discharge to end of discharge. The heat generation was observed to increase at steady state up to 80% of the discharge. The rate of heat generation varies rapidly and found to be high at the end of discharge. At the initial stage of discharging cycle, the spreading of temperature is less and remains contradiction at the end and intermediate of discharge cycle. Moreover, at the start of discharge cycle, different heating rates were observed across the cell. The cell heating is biased beneath the cathode, especially positive electrode. The positive electrode easily exhibits high temperature due to resistivity of electrode material of cathode. The battery chamber temperature is maintained at 50 °C. At this stage, the charging and discharging cycles were performed, and their corresponding IR thermal image was captured. During initial charging period, the temperature measured on battery is around 49 °C, and it gradually increases for further charging. This is due to the reduction of resistance and increase of electron mobility. Once the battery reaches its maximum capacity, the temperature of battery is raised up to 57 °C at 2 h 27 min. Likewise, the discharge capacity of the battery is performed and measured (Fig. 5) at 50 °C after 30 min of resting period. This resting period helps to maintain the battery chemical reactions to remain stable. From these results, it is inferred that the temperature variations influence the battery’s operating condition, battery capacity and lifetime. At high-temperature region, the battery gets charged and discharged rapidly when compared to normal operating temperature. If the battery is exposed to high temperature for long duration, it leads to explosion and failure. The result of this study indicates that an increase in the ambient temperature influences the rate
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(a)
(b)
Fig. 5 a Initial condition of discharge, the temperature range is 54 °C, b at fully discharged condition
of discharge and also reduces the battery capacity which is clearly depicted from the IR images.
3.2 Scanning Electron Microscope High-resolution SEM images were taken to analyze the integrity of anode and cathode plates in the scale of 30 µm range. The samples for imaging purpose are taken in two different areas of anode and cathode plates. The first sample is considered from the portion which is close to the electrical connector, and the second sample is obtained from the bottom side of each electrode. These kinds of samples help to identify the plate integrity at two different places. From Fig. 6, the orientations of anode plate at two different portions are clearly examined. It is evident that the temperature profile has an influence on the plate surface. Exposing the battery to higher temperature region will fasten the chemical
(a)
(b)
Fig. 6 a Anode plate analysis at near the terminal region b cathode plate analysis at near the terminal region
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Fig. 7 a Anode plate analysis—far from the terminal region b cathode plate analysis—far from the terminal region
reactions which produces more number of electron transportation within a short period of time. The number of voids observed in the image indicates the amount of materials dissolved during the chemical process. Due to fast charging and discharging of battery, the current density deposition does not take place; instead, the deposition starts to lift away from the substrate. So, the reduction of material in the plate will be one of the reasons for battery’s efficiency reduction. The reason behind is clear that the current collector substrate has an influence on the active material. The particle size of the anode plate (Fig. 6a) is varied from 100–200 nm at 37 °C [15]. Likewise, from Fig. 6b, the cathode material orientation also varies. The reason behind this is the temperature influence. At higher temperature, the charge carrier deposition rate is low because of stimulated chemical reactions. Figure 7a shows the anode plate orientation from bottom portion. Figure 6a result indicates that the voids are not visible in the region. Hence, it is inferred that the maximum influence of the temperature on the anode plate was observed at the nearest area of the terminals. This causes poor electrical conductivity due to reduction of plate material. Figure 7b shows the cathode plate orientation at bottom portion. The structural orientation is retained at certain level to produce ion exchange at proper values. But large grain size is observed which may lead to increased internal resistance of the plate.
4 Conclusion This paper discusses about the thermal influence on the battery’s anode and cathode materials at different temperature conditions. IR thermography analysis was performed to measure the cell and surrounding temperature of the battery. From the thermography analysis, it was observed that charging and discharging of battery at higher temperature cause a thermal effect which shortens the lifetime of the
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battery. Additionally, the sensor network lifetime and transmission efficiency also gets reduced. By maintaining at a room temperature, the battery’s thermal influence was observed only for a minimal range. At this condition, battery can able to withstand for long period of time ensuring better data transmission in WSN. Likewise, SEM analysis shows high-temperature influence on battery’s anode and cathode plates. The material removal rate from anode plate is observed high. This nature of material removal is varied at different areas of plates.
References 1. Pesaran AA, Vlahinos A, Burch SD (1997) Thermal performance of EV and HEV battery modules and packs. National Renewable Energy Laboratory 2. Hu HY, Xie N, Wang C, Wu F, Pan M, Li HF, Wu P, Wang XD, Zeng Z, Deng S, Wu M (2019) Enhancing the performance of motive power lead-acid batteries by high surface area carbon black additives. Appl Sci 9(1):186 3. Bannister K, Giorgetti G, Gupta SK (2018) Wireless sensor networking for hot applications: effects of temperature on signal strength, data collection and localization. In: Proceedings of the 5th workshop on embedded networked sensors, pp 1–5 4. Boano CA, Tsiftes N, Voigt T, Brown J, Roedig U (2009) The impact of temperature on outdoor industrial sensornet applications. IEEE Trans Ind Inf 6(3):451–459 5. Chen M, Rincon-Mora GA (2006) Accurate electrical battery model capable of predicting runtime and IV performance. IEEE Trans Energy Convers 21(2):504–511 6. Jaguemont J, Boulon L, Venet P, Dubé Y, Sari A (2015) Lithium-ion battery aging experiments at subzero temperatures and model development for capacity fade estimation. IEEE Trans Veh Technol 65(6):4328–4343 7. Hörmann LB, Glatz PM, Hein KB, Weiss R (2012) State-of-charge measurement error simulation for power-aware wireless sensor networks. In: 2012 IEEE wireless communications and networking conference (WCNC). IEEE, pp 2209–2214 8. Xie D, Wei W, Wang Y, Zhu H (2013) Tradeoff between throughput and energy consumption in multirate wireless sensor networks. IEEE Sens J 13(10):3667–3676 9. Wang CF, Shih JD, Pan BH, Wu TY (2014) A network lifetime enhancement method for sink relocation and its analysis in wireless sensor networks. IEEE Sens 14(6):1932–1943 10. Haase J, Molina J, Dietrich D (2011) Power-aware system design of wireless sensor networks: power estimation and power profiling strategies. Trans Ind Inf 7:601–613 11. Stehlík M (2011) Comparison of simulators for wireless sensor networks. M.S. thesis, Faculty Information, Masaryk University, Brno, Czech Republic 12. Musznicki B, Zwierzykowski P (2012) Survey of simulators for wireless sensor networks. Int J Grid Distrib Comput 5(3):23–50 13. Pereira RM, Ruiz LB, Ghizoni MLA (2015) MannaSim: A NS-2 extension to simulate wireless sensor network. Proc Int Conf Netw 2015:95–101 14. Gao L, Liu S, Dougal RA (2002) Dynamic lithium-ion battery model for system simulation. IEEE Trans Compon Packag Technol 25(3):495–505 15. Streza M, Nu¸t C, Tudoran C, Bunea V, Calborean A, Morari C (2016) Distribution of current in the electrodes of lead-acid batteries: a thermographic analysis approach. J Phys D Appl Phys 49(5):055503 16. Giess H (1997) Investigation of thermal phenomena in VRLA/AGM stationary lead/acid batteries with a thermal video imaging system. J Power Sources 67(1–2):49–59 17. Balaras CA, Argiriou AA (2002) Infrared thermography for building diagnostics. Energy Build 34(2):171–183 18. Lin C, Tang A, Mu H, Wang W, Wang C (2015) Aging mechanisms of electrode materials in lithium-ion batteries for electric vehicles. J Chem
Derivation of the Rotation Matrix for an Axis-Angle Rotation Based on an Intuitive Interpretation of the Rotation Matrix Roshan Kumar Hota and Cheruvu Siva Kumar
Abstract In this paper, we present the derivation of the rotation matrix for an axisangle representation of rotation. The problem is of finding out the rotation matrix corresponding to the rotation of a reference frame, by a certain angle, about an arbitrary axis passing through its origin. The axis-angle representation is particularly useful in computer graphics and rigid body motion. We have used an intuitive interpretation of the rotation matrix for this derivation. The intuitive interpretation is as follows: the columns of the rotation matrix are the coordinate axes unit vectors of the rotated frame as seen from the fixed frame. This interpretation of the rotation matrix helps in quick computation of the rotation matrices required to obtain the required rotation matrix. The required rotation matrix can be computed from just two rotation matrices which are easy to find and intuitive to understand. The derivation presented in simpler to understand than presented in most books and can be grasped with minimum geometric visualization. Keywords Rotation matrix · Angle-axis rotation · Kinematics · Rigid body motion
1 Introduction Axis-angle representation of rotation is useful in many scenarios in robotics, rigid body motion, and computer graphics. The problem addressed in this paper is finding the rotation matrix corresponding to the axis-angle representation. Leonhard Euler was the first to show that any set of rotations of a rigid body can also be achieved by a single rotation about an axis [1, 2]. The problem can be posed both ways, to find the rotation matrix corresponding to given axis-angle or to find the axis-angle corresponding to a given rotation. In this paper, we focus on the former problem. Efficient R. K. Hota (B) · C. S. Kumar Indian Institute of Technology, Kharagpur, West Bengal, India e-mail: [email protected] C. S. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_87
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solutions like Rodrigues’ formula [3] exist but its derivation requires the knowledge of exponential maps. The derivation is not easy to understand at an undergraduate level. In our work, we have proposed a method by which the rotation matrix can be derived in the matrix domain without the knowledge of exponential map or other concept. Very less or no graphical construction needed. The proposed method requires lesser number of computations, in terms of matrix multiplications, as compared to methods in standard textbooks [4, 5].
1.1 Problem Statement ˆ = [k x k y k z ]T about which the frame is Given a coordinate frame A and an axis K rotated by and angle θ to get the frame A , find the rotation matrix between the rotated frame A and A (refer Figs. 1, 2 and 3).
Fig. 1 Rotation matrix corresponding to the rotation of a frame about an arbitrary axis has to be found out
Fig. 2 Reference frame attached with z-axis attached along the axis of rotation
Fig. 3 Frames {A} and {K} rotate together
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2 Derivation of the Rotation Matrix We denote this rotation matrix between frame A and A as AA R. This rotation matrix transforms the expression of a vector in frame A to that in frame A. We go through the following steps to obtain the required rotation: ˆ 1. Consider a frame K whose z-axis is along the vector K. 2. Consider that the frame K and the frame A are rigidly connected. So, both the frames rotate together. 3. When the frame K is rotated about its z-axis, the frame A also rotates with it. By this rotation, K changes to the new frame K’ and frame A changes to a new frame which is our required frame A . Let us denote the rotation matrix from frame K and A as KA R. A rotation about the z-axis of frame K gives the new frame K . The net rotation matrix from frame K to frame A will be: A A (1) K R = K R R z K (θ ) Since we consider that the frame A and the frame K are rigidly connected, when K gets rotated to K , frame A gets rotated to A . One can also say that the rotation matrix from K to A remains the same for the rotation between K and A . Hence, we can write: A A (2) K R =K R Now using Eqs. 1 and 2, we can obtain our required rotation matrix AA R as:
= KA R KA RT
(3)
= KA R Rz K (θ ) KA RT
(4)
A A R
Or A A R
In Eq. 4:
⎡
⎤ cosθ −sinθ 0 Rz K (θ ) = ⎣ sin θ cosθ 0⎦ 0 0 1
(5)
For computing KA R, we use the interpretation of the rotation matrix that a column of the rotation matrix KA R represents a unit coordinate axes vector of frame K expressed ˆ j, ˆ kˆ are unit in terms of the unit coordinate axis vectors of frame A. If we say that i, coordinate axes vectors and we use a subscript to denote the frame for which they are the unit coordinate axes vectors, then iˆK , jˆK , kˆ K are unit vectors for frame K and iˆA , jˆA , kˆ A are unit vectors for frame A. Then the first column of the rotation matrix A ˆ ˆ ˆ ˆ K R represents i K expressed in i A , j A , k A . Similarly for second and third column.
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From the above discussion, we can see that for the rotation matrix KA R the last ˆ = [k x k y k z ]T which is kˆ K expressed in iˆA , jˆA , kˆ A . Now column will simply be K for the frame K we need to specify the x and y-axis. We know that the x-axis will be perpendicular to kˆ K . For getting the x-axis, we chose an arbitrary vector (say [0 0 1]T )and take its cross product with the z-axis: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ kx 0 0 −k z ky 0 ky ⎣k y ⎦ × ⎣0⎦ = ⎣ k z 0 −k x ⎦ ⎣0⎦ = ⎣−k x ⎦ 1 kz −k y k x 0 1 0
(6)
From Eq. 6, we obtain the unit vector along the x-axis of frame K expressed in the frame A as: ⎡ ⎤ k y / k x2 + k 2y ⎢ ⎥ ⎥ (7) iˆK = ⎢ ⎣−k x / k x2 + k 2y ⎦ 0 And we can also obtain jˆK . ⎤ k x k z / k x2 + k 2y ⎢ ⎥ ⎢ ⎥ jˆK = kˆ K × iˆK = ⎢ k y k z // k x2 + k 2y ⎥ ⎣ ⎦ −(k x2 + k 2y )/ k x2 + k 2y ⎡
(8)
Therefore, the rotation matrix becomes: ⎤ k y / k x2 + k 2y k x k z / k x2 + k 2y kx ⎥ ⎢ ⎥ ⎢ A 2 2 k y k z // k x2 + k 2y ky ⎥ K R = ⎢−k x / k x + k y ⎦ ⎣ 0 −(k x2 + k 2y )/ k x2 + k 2y k z ⎡
(9)
Now, if we put the values from Eqs. 9 and 5 into Eq. 4, we can obtain the terms of the required rotation matrix AA R: ⎤ ⎡ r11 r12 r13 A ⎦ ⎣ A R = r 21 r 22 r 23 r31 r32 r33
(10)
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We will show the detailed steps, for simplicity, for only the first term here. r11 = = =
(k 2y + k x2 k z2 )cosθ k x2 + k 2y
+ k x2
(k 2y + k x2 (1 − k x2 − k 2y ))cosθ k x2 + k 2y
+ k x2
((k 2y + k x2 ) − k x2 (k x2 + k 2y ))cosθ k x2 + k 2y
+ k x2
= cosθ − k x2 cosθ + k x2 = k x2 (1 − cosθ ) + cosθ r12 =
−k x k y cosθ −
k x2 k z sinθ k x2
(11) − k 2y k z sinθ + k 2y
+
k x k y k z2 cosθ
+ kx k y
= k x k y (1 − cosθ ) − k z sinθ
(12)
r13 = k x k z cosθ + k y sinθ (−k x cosθ + k y k z sinθ )k y + (k x sinθ + k y k z cosθ )k x k z r21 = + kx k y k x2 + k 2y
r22
(13)
= k x k y (1 − cosθ ) + k z sinθ (−k x cosθ + k y k z sinθ )(−k x ) + (k x sinθ + k y k z cosθ )k y k z = + k 2y k x2 + k 2y
(14)
= k 2y (1 − cosθ ) + cosθ
(15)
r23 =
(k x sinθ +
k y k z cosθ )(−k x2 k x2 + k 2y
−
k 2y )
+ k y kz
= k y kz(1 − cosθ ) + k x sinθ
(16)
r31 = k x k z (1 − cosθ ) − k y sinθ r32 = k x k y (1 − cosθ ) + k x sinθ
(17) (18)
r33 = k z2 (1 − cosθ ) + cosθ
(19)
The terms of the required rotation matrix are given in Eqs. (11) through (19). The complete rotation matrix is: ⎡
⎤ k x2 νθ + cθ k x k y νθ − k z sθ k x k z cθ + k y sθ A ⎣ k x k y νθ + k z sθ k 2y νθ + cθ k y kzνθ + k x sθ ⎦ A R = k x k z νθ − kysθ k x k y νθ + kxsθ k z2 νθ + cθ
(20)
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3 An Example Following is the example for computing the rotation matrix corresponding to axisangle representation of rotation. The expression for the rotation matrix in the simplest form is: A A A T (21) A R = K R R z K (θ ) K R We need to compute two rotation matrix. One of them is the standard rotation about Z-axis given by Eq. 5. The second is KA R which is given by Eq. (9). √ √ √ Example Compute the rotation of a matrix about axis [2/ 14, 3 14, 1 14]T by an angle π/6. From Eq. (9)
And from Eq. (5)
And,
⎡
⎤ 0.832 0.148 0.534 A ⎣ ⎦ K R = −0.554 0.222 0.801 0.000 −0.963 0.267 ⎤ 0.866 −0.500 0.000 Rz K (θ ) = ⎣0.500 0.866 0.000⎦ 0.000 0.000 1.000
(22)
⎡
⎤ 0.904 −0.076 0.420 A ⎣ 0.191 0.952 −0.238⎦ A R = −0.381 0.296 0.875
(23)
⎡
(24)
4 Conclusions In this work, we have derived the rotation matrix corresponding to axis-angle representation of rotation. The method used employs an intuitive interpretation of the rotation matrix which is that the columns of the rotation matrix represent the coordinate unit vectors of the rotated frame expressed in the coordinate unit vectors of the fixed frame. Our method requires knowledge of two rotation matrices. One of them is the standard rotation matrix about the z-axis, and the other is the rotation matrix between the initial frame and the frame attached to the axis of rotation. The second rotation matrix is obtained by performing two cross products. So, effectively the computations we are performing are two cross-products and two matrix multiplications. Further, the rotation matrix is intuitively constructed directly from the data given, that is, the unit vector of the axis of rotation.
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References 1. Euler L (1776) Formulae generales pro translatione quacunque corporum rigidorum. Novi Commentarii academiae scientiarum Petropolitanae 189–207 2. Bob P, Richard P, Stephen R (2009) A disorienting look at Euler’s theorem on the axis of a rotation. Am Math Monthly 116(10):892–909 3. Murray Richard M (2017) A mathematical introduction to robotic manipulation. CRC Press, Boca Raton 4. Craig JJ (2009) Introduction to robotics: mechanics and control, 3rd ed. Pearson Education, India (2009) 5. Fu KS, Gonzalez RC, George Lee CS (1987) Robotics: control sensing. Vis. Tata McGraw-Hill Education
Resolving Hyper-Redundant Planar Serial Robots to Ensure Grasp Rajesh Kumar and Sudipto Mukherjee
Abstract This paper provides a methodology to use hyper-redundant serial robots to ensure grasp in a planar case. Hyper-redundant serial robots are resolved to mimic behavior of a snake holding onto the objects. The algorithm focuses on caging a hard object and can be extended on squeezing a soft object. The methodology requires a bound on object boundary and location as input. The tractrix-based solution results in the tail of the manipulator to helically move toward the center of the circular bound within which the object is kept. Keywords Resolution of hyper-redundant robots · Caging · Planar grasps
1 Introduction Caging and grasping planar objects have been a widely addressed problem in the literature. Caging is referred as the process of restricting motions of an object using a manipulator. A cage set [8] is defined as a set of contacts such that an object cannot move out of the contact, though it still has some unrestricted motions. Process of caging has been extended to grasping and then to manipulation of the objects [9]. The methods to cage and then to manipulate the objects have been explored using various methods like using multiple mobile robots [12] and [7]. Various other methods have been studied in detail in the literature. Algorithms to search for the objects in order to cage and grasp them have been developed. Optimal contact points are important to ensure proper grasp. Two of such methodologies are presented in [1] and [2]. We intend first to cage the object and then grasp it using hyper-redundant serial robots. We use a hyper-redundant serial robot to search for a single object within a known boundary and grasp it. There should be only a single convex-shaped object in the boundary to grasp it. Hyper-redundant serial planar manipulators have been R. Kumar (B) · S. Mukherjee Department of Mechanical Engineering, Indian Institute of Technology Delhi, Delhi, India e-mail: [email protected] S. Mukherjee e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_88
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utilized to carry out operations imitating behavior of a snake. One such operation is grasping objects [3]. It is known that a hyper redunant serial robot is comprised of much more degrees of freedom in the joint space than in the end effector space. So, a methodology needs to be developed to resolve the redundancy such that resolution is used to grasp an object. One such methodology is utilization of “Tractrices.” Ghosal et al. [11] showed that resolving redundancy using tractrix curves can be used to avoid obstacles in a plane and also to carry out operations like tying a knot. Here, we show that the method of resolution can be used to cage a rigid object in plane, under knowledge of a bound containing the object boundary and the object location.
2 Tractrices and Tractors “Tractrices” are historical curves which are used to denote the motion of the nondriven end of a geodesic curve on a surface when the other end of the geodesic curve is driven along a known trajectory. Consider a particle moving on a frictional surface. The surface need not be planar and can be a generalized one. A curve is attached to the particle such that it represents a geodesic on the surface the particle is moving. Consider that the surface is modeled using parameters (u(s), v(s)) with the a function f(u,v) mapping the surface parameters to the Cartesian coordinate in R3 . A geodesic curve is made by a particle on a surface when it starts at a constant velocity and moves exactly under normal acceleration. Mathematically, the statement is represented by Eq. 1. Consider the geodesic curve to be γ (s), which follows a second-order differential equation. Detailed analysis of the rise of the geodesic curve on surfaces is found in [5]. It is also the shortest curve between two points on the surface. β μ d2 γ α α dγ dγ =0 (1) + βμ ds 2 ds ds α where βμ is written as per Einstein’s convention for brevity and γ ν represents the th ν parameter representing coordinates on the manifold. For a surface, the set γ contains (u(s), v(s)) and α1,2 , β1,2 , μ1,2 are dummy variables with γ 1 = u(s) and γ 2 = v(s). A detailed explanation can be found in [10]. The tractrix curve is the locus of the endpoint of this curve when the initial point is moved along a curve. By definition of geodesic, in case the surface happens to be a plane, then the geodesic between any two points is the line joining the two points. Evolution of tractrices on different submanifolds is explained in [6], but we will restrain only to planar tractrices. Numerous equivalent definitions are presented in [4]. In this paper, we use the fact that when one end of a straight line is moving on a frictional surface, the instantaneous velocity of the other end is along the link length. The statement is mathematically represented in Eq. 2.
X head − X tail Yhead − Ytail
=
X˙ tail Y˙tail
(2)
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Fig. 1 Tractrix curve when tractor is a straight line on planar surface
Fig. 2 a Tractrix curve to a quarter circle tractor b Trajectory of a redundant robot trying to squeeze an object with a known bound based on developed method
where (X head and Yhead ) are the start coordinates, and (X tail , Ytail ) represents the end coordinates of the line. An example tractor–tractrix combination is shown in Fig. 2a. An example of the tractrix curve on the plane surface (straight line geodesics) with the head following a straight line is shown in Fig. 1.
3 Methodology Incidentally, we can assume these straight lines to be links of a hyper-redundant robot. The end of each link behaves as a tractrix for the head of each link. Considering Eq. 2, the velocity for each of the end point on ith link is directed along the ith link itself. Ghosal et al. [11] showed how resolving redundancy of hyper-redundant robots using tractrices enhances the capabilities of tying a knot. Consider that the manipulator is composed on “n” links with n >> 3, and the head point of the last link is directed to follow bounded curve of the object (Fig. 3). If the end of the manipulator is not connected to the ground by a revolute pair but by a planar joint (two orthogonal prismatic joints) with variable coordinates as xend and yend , then the manipulator has “n + 1” degrees of freedom. The last link has two orthogonal prismatic pairs and has zero link length. Let the redundant planar manipulator has joint variables given by
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Fig. 3 Example of hyper-redundant serial robot used to grasp the object
θ = (θ1 , . . . , θn−1 ) and the end point’s position (xend , yend ). The end effector needs to move on a particular boundary curve resulting in two equations. So, the manipulator has “n − 1” extra degrees of freedom in the joint space that needs to be determined exactly. Consider the case when the head of the robot is following a closed convex curve y(s) = 0, where “s” is the path parameter. The initial orientation of the first link is assumed to be tangential to the curve. The initial velocity of the head is given . We need to resolve the redundancy in such a way by v 1 = σ y˙ (s), where σ = ds dt that the other part (tail) of the manipulator starts making a convex shape to hold any object within the bounded curve. We use inspiration from tractrix-based algorithm to tying a knot to hold onto an object. So, the velocity of the tail of the first link is given from Eq. 2. As the tail of the ith link and the head of the i + 1th link have the same motion, so this continues till the head of the last link. Hence, we get “n − 1” more equations for the complete trajectory. The velocity of the head of the second link is given by Eq. 3. i−1 ˙ X s − X ei−1 X si −1 = 1 (3) Ysi−1 − Yei−1 Y˙si The first-order differential equation in Eq. 3 needs to be solved along with the constraint that the length of the link is constant.
4 Simulation Showing Caging and Grasp The head tractor curve is a curve which bounds the object to be grasped. This is based on the assumption that the object has a convex boundary. Consider a known bound based on the prediction of the location and the boundary of the object, the
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Fig. 4 Various instances of the snake-like robot trying to cage and then grasp an object kept near the center of the bounded curve, using resolution methodology described
head of the manipulator follows a known curve, and the head moves on the bound curve, whereas the rest of the curve tries to tie a knot within itself. The methodology uses a circular-bounded curve as an input. The curve (circular here) should contain the object. The true object boundary (though unknown) should be convex in nature as the manipulator while interacting with the object should not get stuck within concave-shaped regions. The end effector follows the bounded curve, whereas the rest of the manipulator works under tractrix resolution. As shown in Fig. 4, the rest of the manipulator moves helically toward the center of the bounded curve. The “helical search” of the tail of the manipulator occurs until the links cannot move the object. The object is free to move when one or two links hit it. The links of the manipulator tend helically toward the center of the circular boundary, so if links indeed hit the object, such that the set of links hitting the object can give motion to it, then the manipulator tends to bring it toward the center and cage it. Then, all the links apply a constant torque on the object to ensure grasp of the object and its immobility. The methodology works because the tail of the manipulator moves helically toward the center of the bounded curve while maintaining a convex curve in its shape. The manipulator is stopped when at least three links hit a maximum torque limit, which suggests the fact that at least three links have hit the object and are rendered immovable, suggesting object grasp. In case, the object is soft then the torque limit can be extended to squeeze the object as much as possible. For simulation purposes, the motion of the object is assumed only along the instantaneous normal of the contact region, and the manipulator is assumed to not interact with the object tangentially. Only the component of the motion of the point in contact which is instantaneously normal to the object is transferred to the object.
4.1 Example Consider an object to be caged, and we know a bound to the object shape and position, as shown in Fig. 6. The trajectory generated from the methodology is shown in
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Fig. 5 Various instances of the snake-like robot trying to cage an object kept away from the center of the bounded curve, using resolution methodology described
Fig. 2b. The tractrix curve was initially tangential to the boundary curve. The head of the snake-like system follows the boundary, whereas the rest of the curve tries to hold the object. If the object is not in the center of the curve, then one or two links hit the object repeatedly until the object reaches to the center of the curve or a stage where at least three links can cage the object. Figure 4 shows stages of an ellipse, which was present at the center of the boundary curve to be caged by the manipulator. The series of figures show how the object is caged and then grasped. Each link is of length 0.2 units (except the last link) in the simulation example. In case the object to be grasped is present in a non-central location, then the helically moving manipulator tries to bring it to a point where at least three links can simultaneously come into contact with the object, thereby caging it. The example manipulation for an elliptical object is shown in Fig. 5. The links of the manipulator move the object to bring it near the center where it can be properly grasped. The motion of the center of the object is shown in Fig. 6. When the object is near the boundary, the manipulator hits the object successively in a near parabolic trajectory in the x − y plane. The near parabolic trajectory is representative of the component of the motion of the contacting link normal to the object to be held. The method works even for a larger object placed eccentrically within the boundary, away from the center (Fig. 7). The manipulator first brings the object near to the center and then tries to cage the object. In case the object is soft, the robot’s links will squeeze the object. However, in case of a rigid object, the robot’s links apply constant torque to the object.
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Fig. 6 a x − y motion of the center of the object as the manipulator tries to cage the object with time b x − y trajectory of the center of the object
Fig. 7 a x − y motion of the center of the object as the manipulator tries to cage the object with time b x − y trajectory of the center of the object
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5 Conclusion This paper presents a methodology to cage and then grasp, planar smooth convexshaped objects with unknown shape and location. However, there needs to be a knowledge of bound on the shape as well as the location of the object. A tractrixbased method is used to resolve redundancy where the manipulator’s head moves on a circular bound, and the rest of the manipulator performs a search for the object and brings it to the center and then cages it. The methodology is useful when there is a known loose bound on the position of the object, and we want to get hold of the object without vision feedback.
References 1. Allen TF, Burdick JW, Rimon E (2015) Two-finger caging of polygonal objects using contact space search. IEEE Trans Rob 31(5):1164–1179. https://doi.org/10.1109/TRO.2015.2463651 2. Allen TF, Rimon E, Burdick JW (2014) Two-finger caging of 3d polyhedra using contact space search. In: 2014 IEEE international conference on robotics and automation (ICRA), pp 2005–2012. https://doi.org/10.1109/ICRA.2014.6907125 3. Chirikjian GS, Burdick JW (1993) Design and experiments with a 30 dof robot. In: Proceedings IEEE international conference on robotics and automation, vol 3, pp 113–119. https://doi.org/ 10.1109/ROBOT.1993.291862 4. Gander W, Bartoˇn S, Hˇrebíˇcek J (2004) The tractrix and similar curves. In: Solving problems in scientific computing using maple and MATLAB®. Springer, pp 1–25 5. Kumar GR, Srinivasan P, Holla VD, Shastry K, Prakash B (2003) Geodesic curve computations on surfaces. Comput Aided Geomet Des 20(2):119–133 6. Madsen JJ, Markvorsen S (2017) Tractors and tractrices in riemannian manifolds. ArXiv preprint arXiv:1707.09532 7. Pereira GA, Kumar V, Spletzer JR, Taylor CJ, Campos MF (2003) Cooperative transport of planar objects by multiple mobile robots using object closure. In: Experimental robotics VIII. Springer, pp 287–296 8. Rimon E, Blake A (1999) Caging planar bodies by one-parameter two-fingered gripping systems. Int J Rob Res 18(3):299–318 9. Rodriguez A, Mason MT, Ferry S (2012) From caging to grasping. Int J Rob Res 31(7):886–900 10. Sen D (1960) On geodesics of a modified riemannian manifold. Can Math Bull 3(3):255–261 11. Sreenivasan S, Goel P, Ghosal A (2007) Redundancy resolution using a tractrix and its application to real-time simulations of hyper-redundant manipulators, snakes and tying of knots. In: 12th IFToMM world congress, Besancon, Paper. No. 524-1, Citeseer 12. Wang Z, Hirata Y, Kosuge K (2005) An algorithm for testing object caging condition by multiple mobile robots. In: 2005 IEEE/RSJ international conference on intelligent robots and systems, pp 3022–3027. https://doi.org/10.1109/IROS.2005.1545378
Boom Packaging with Yoshimura Pattern: Geometrical and Deformation Analysis Hemant Sharma , Omkar Raj, and S. H. Upadhyay
Abstract A number of folding mechanisms have been proposed for spaceborne deployable booms/masts based on origami patterns. The packaging and deployment of a cylindrical boom with Yoshimura pattern are analyzed in this paper. At first, the detailed design of the Yoshimura pattern is discussed, and geometrical expressions for various important parameters like radius of packaged boom, packaging efficiency and inside residual volume after complete folding are obtained to investigate the packaging behavior of the boom. Then, the deployment of a single story Yoshimura cylinder is simulated using FE software ABAQUS. The folded shape of the boom is modeled using pin-jointed wireframe technique, and a uniform nodal displacement is provided to the top polygon, and corresponding axial strains in the fold lines are analyzed numerically. The influence of the number of origami units on packaging and deformation behavior is also discussed. The results show that the fold line deformation decreases as the number of origami units increases, but this also decreases the packaging efficiency of the boom. Moreover, the expansion coefficient is defined to relate the circumferential strain with applied longitudinal strain. Keywords Deployable boom · Space structures · Inflatables · Folding patterns
1 Introduction Space structures equipped with large sized deployable reflectors and booms, often known as gossamer structures, have made groundbreaking space missions possible [1]. Deployable spaceborne structures offer the potential for exceptional packaging efficiency, lightweight low-cost hardware over conventional space structures.
H. Sharma (B) · O. Raj · S. H. Upadhyay Smart Materials and Structures Laboratory, MIED, IIT Roorkee, Roorkee, Uttarakhand 247667, India e-mail: [email protected] S. H. Upadhyay e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_89
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Foldable booms/masts are truly an essential part of any deployable space structure. A number of folding methodologies and mechanisms have been proposed for spaceborne deployable booms/masts, such as Z-folding, spiral folding, origami folding techniques, and telescopic folding [2]. The origami folding techniques include Yoshimura pattern, bellows pattern, Miura-ori pattern, helically triangulated folds, etc. Yoshimura pattern is an inextensional post-buckling solution for axially compressed thin-walled cylinders [3]. A folding geometry similar to Yoshimura pattern was obtained for the buckling pattern of a thin cylinder under torsional loading [4]. Senda [5] analyzed several models of inflatable tubes folded with Yoshimura and Miura-ori pattern. The type of folding pattern and the number of fold vertexes/origami units repeated around the circumference largely affects the packaging behavior and the deployment deformations of the boom. The air entrapped in the vacant space of the folded structure can cause an uncontrolled expansion of the structure, which led to numerous failures. The proper management of residual gas is still a significant issue in the deployment of inflatables. Till date, no study appears to have been reported which has analyzed the geometry of the bellow folding pattern to obtain the residual space inside the folded boom. The influence of different geometrical parameters of the bellows pattern on the packaging and deployment behavior of the boom has not been investigated in the literature. Moreover, the amount of deformation along the fold lines needed to be quantified clearly with respect to different geometrical parameters. In this study, the packaging and deformation behavior of a cylindrical boom folded with Yoshimura pattern is investigated. This paper is organized as follows. Section 2 explains the geometry of the Yoshimura folding pattern and mathematical expressions for various important parameters like radius of packaged cylinder, packaging efficiency, and residual space after complete folding. In Sect. 3, the influence of number of origami units on packaging behavior of the boom is investigated. Section 4 discusses the modeling and simulation of deployment of a single story Yoshimura cylinder. A pin-jointed wireframe technique [7] is used to model folded configuration of the boom. The expansion coefficient is defined and calculated for the boom deployment. At last, Sect. 5 concludes the findings of this study.
2 Geometry of Yoshimura Pattern A partially folded Yoshimura patterned cylinder is shown in Fig. 1. Figure 3 shows a folding pattern, which has three stories and four basic Yoshimura units in each story. Here, the solid lines denote mountain folds, and dashed lines denote valley folds. In a stress-free stowed configuration of the boom, the successive folds must form a closed cross-section. Figure 2 depicts a half story segment unit of Yoshimura pattern, which can be mirrored and patterned along the axial and circumferential directions to get the complete folding pattern (see Fig. 3). The resulting folded boom will have a m-fold rotational symmetry, and the cylindrical closure condition is given by:
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Fig. 1 Partially folded boom (from Tarnai 1994 [6])
Fig. 2 Dimensions of the Yoshimura unit
Fig. 3 Yoshimura pattern for foldable cylinders (m = 4)
mα = (m − 1)π
(1)
where m = number of origami units around the circumference and α = 2β. The length of the folding pattern (L) is equal to the perimeter of the boom, as shown in Fig. 3. If every story has m basic origami units, then all the dimensions of the Yoshimura unit can be described as a=
L m
(2)
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Fig. 4 Yoshimura polygon
b=
a sec β 2
(3)
h=
a tan β 2
(4)
2.1 Radius of Packaged Cylinder The fold polygon of Yoshimura pattern is shown in Fig. 4. The radius of the circumscribed circle of the fold polygon is termed as the radius of packaged cylinder (Rc ). This plays a vital role in the study of the packaging behavior of any cylindrical folding pattern. The radius of the packaged boom can be obtained as: RC =
4 sin
a α π sin 2m 2
2.2 Packaging Efficiency The packaging efficiency of the folded boom is given by:
(5)
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PE =
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Vinitial − Vfolded × 100 Vinitial
(6)
Volume of folded configuration Volfolded = Area of outer polygon(A) × Folded height H f
(7)
Area of outer polygon (A). In Yoshimura folding pattern, the outer polygon is a regular polygon, which consists of 2 m similar triangles ( AOD) as shown in Figs. 4 and 5 shows the details of AOD. The outer polygon area is obtained as A = m RC2 sin
π m
(8)
Folded height (H f ). In s single story of the Yoshimura pattern, four layer overlaps; hence, the total folded boom will be Hf = 4 × t × S
(9)
where t = thickness of the material S = stories required to fold the complete length = height of the boom/single story height (2 h). Volume of unfolded configuration Volunfolded = π ×
Fig. 5 Detailed view of different triangles
L 2π
2 (10)
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2.3 Residual Volume Calculations The total residual volume of the folded boom consists of inside residual volume Rvol(inside) residual volume between layers Rvol(b/w the layers ) . Inside residual volume Rvol(inside) . The inside residual volume in the folded boom is calculated by multiplying the inside polygon area (A ) with the folded height H f . Inside residual area (A ). It is the area of inner polygon, which consists of 2 m similar triangles, POQ (see Fig. 5). The inner polygon area is obtained as A =
1 m 2 P 2 tan(π/2m)
(11)
where b sin(α − 90) = PQ sin(α/2) = Residual area A × Folded height H f P=
Rvol(inside)
(12) (13)
Residual volume between the folded layers Residual area between layers. It can be seen that the folded configuration has triangular ( PAQ) free space between the layers with height 2t. 1 P × AR 2
(14)
Rvol(b/w the layers ) = 2mt A (2S − 1)
(15)
Rvol = Rvol(inside) + Rvol(b/w the layers )
(16)
A = Area of A P Q =
Total residual volume.
3 Packaging Behavior To study the packaging behavior of the boom, the influence of number of origami units (m) on the packaged boom radius (RC ), packaging efficiency (PE), and residual volume (Rvol ) is investigated using respective geometrical formulas. The perimeter and length of the boom are taken as 600 mm and 800 mm, respectively. The thickness of the material is taken as 0.2 mm in this section.
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Fig. 6 Variation of Rc with m
Fig. 7 Variation of PE with m
The perimeter (L) and length ratio (R) of the boom are kept constant for all values of m, and all other dimensions vary according to Eqs. 1 and 2 with respect to m. The variation in radius of packaged cylinder (RC ) and packaging efficiency (PE) with m is shown in Figs. 6 and 7, respectively. Figure 8 shows the variation of the residual volume with m; it can be seen that inside and total residual volume curves are nearly equal; the variation of residual volume between layers with m is shown in Fig. 9.
4 Deployment of Single Story Yoshimura Cylinder A single story Yoshimura folded cylinder (m = 4) is used to analyze the deformation behavior during deployment. The fold line deformations during the unfolding of the boom were obtained using nonlinear finite element analysis with ABAQUS. The folded geometry of the boom was modeled using wireframe technique, in which the fold lines are modeled as single truss elements. The Young’s modulus of all the elastic bars is taken as 1.05 × 106 MPa, and the initial cross-section area is 10 mm2 for all the simulations. The bottom polygon was made up of AB-AB, the middle
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Fig. 8 Variation of residual Vol with m
Fig. 9 Variation of R Vol b/w layers with m
polygon was made up of DC-DC, and the top polygon was made up of AB-AB fold lines (see Figs. 2 and 3), and AD, AC, and BC fold lines acts as connection between these polygons. The nodes of the bottom polygon were restricted in z and θ directions and allowed to move in radial direction only. The nodes of top polygon are given a displacement (d = 0.75 × 2 h) in axial direction. Then, the axial strains of different fold lines during the deployment were obtained numerically. The deployment of the folded boom is shown in Fig. 10. Since the story height of cylinders having different m will be different, so the applied displacement d (0.75 × 2 h) will also be different. Hence, to compare the deformation behavior, the variation of axial strains is shown in graphical form with respect to a normalized displacement, which is (Fig. 11) Normalized displacement =
Instantaneous displacement Total displacement (d)
(17)
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Fig. 10 Deployment of single story cylinder (m = 4)
a) bar AD (Connection)
b) bar DC (Middle Polygon)
Fig. 11 Deformation of a single story cylinder during the deployment
Expansion coefficient. It is defined as the ratio of the resulting circumferential strain to the applied longitudinal strain. The variation of expansion coefficient with longitudinal strain, for various m values, is shown in Fig. 12.
Fig. 12 Variation of expansion coefficient
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5 Summary The geometric design and packaging behavior of the Yoshimura pattern folded cylindrical boom were investigated. The fold line deformation analysis was carried out using numerical simulations. The influence of the number of origami units on the packaging behavior and fold line deformation during the deployment was studied. The results indicate that: 1.
2.
3.
As the number of origami unit increases, initially, the radius of packaged cylinder decreases rapidly; later on, the decrement becomes very less. The packaging efficiency decreases with the increment in the number of origami units around the circumference. The residual volume inside the boom and between the layers increases with the increases in number of origami units. Under the action of a specific nodal displacement, the axial strains in the folds increase with number of origami units (m). Also, for lesser value of m, as the nodal displacement increases, the axial strains of the fold lines increase rapidly. Expansion coefficient is negative for Yoshimura pattern folded boom, which indicates the lateral contraction of boom during deployment. As the boom expands, radius of packaged boom (RC ) decreases; for the same value of longitudinal strain, expansion coefficient will be higher for the boom with lesser m value.
In the case of inflatable booms, wireframe modeling approach can only predict the initial deployment strains. For a better estimation of the deformation, the folded boom should be modeled with a continuous surface using shell elements. This pattern can be used as a folding mechanism to pack the inflatable/deployable booms for space applications.
References 1. Jenkins CH (ed) (2001) Progress in astronautics and aeronautics: gossamer spacecraft: membrane and inflatable structures technology for space applications, vol 191. Aiaa 2. Schenk M, Viquerat AD, Seffen KA, Guest SD (2014) Review of inflatable booms for deployable space structures: packing and rigidization. J Spacecr Rockets 51(3):762–778 3. Yoshimura BY (1955) On the mechanism of buckling of a circular cylindrical shell under axial compression. Natl Advis Commun Aeronaut 4. Hunt GW, Ario I (2005) Twist buckling and the foldable cylinder: an exercise in origami. Int J Non Linear Mech 40(6):833–843 5. Senda K et al (2006) Deploy experiment of inflatable tube using work hardening, May 2006, pp 3–8 6. Tarnai T (1994) Folding of uniform plane tessellations. In: Origami science and art. Proceedings of the second international meeting of origami science and scientific origami, pp 83–91 7. Cai J, Deng X, Feng J, Zhou Y (2015) Geometric design and mechanical behavior of a deployable cylinder with Miura origami. Smart Mater Struct 24(12):
Multimodal Medical Image Fusion Based on Interval-Valued Intuitionistic Fuzzy Sets T. Tirupal, B. Chandra Mohan, and S. Srinivas Kumar
Abstract Multimodal medical image fusion is the process of combining two multimodal medical images to increase the quality and to extract maximum information from the output image for better treatment and precise diagnosis. The fused image obtained from non-fuzzy sets lags with complementary information. Compared with fuzzy set theory, intuitionistic fuzzy sets (IFS) are determined to be more suitable for civilian and medical image processing as more uncertainties are measured. In this paper, an algorithm based on an interval-valued intuitionistic fuzzy set (IVIFS) is presented for efficiently fusing multimodal medical images and the final fused image is passed through a median filter to remove noise. Simulations on few sets of multimodal medical images are performed and compared with the existing fusion methods, such as an intuitionistic fuzzy set and fuzzy transform. The superiority of the proposed method is presented and is justified. Fused image quality is additionally checked with different quality measurements, for example, entropy, spatial frequency (SF), average gradient (AG), etc. Keywords Image fusion · Fuzzy set · IFS
1 Introduction With the latest developments in the field of technology, digital image processing systems have turned into a reality in developing the number of fields, for example, machine vision, medical imaging, and military applications. The consequence of the utilization of these strategies is an awesome increase of the amount of data available. To extract all the valuable information from the source images and to reduce the increasing volume of data, a powerful method is used in image processing called T. Tirupal (B) Department of ECE, GPCET, Kurnool, Andhra Pradesh 518452, India B. Chandra Mohan Department of ECE, BEC, Bapatla, Andhra Pradesh 522101, India S. Srinivas Kumar Department of ECE, JNTUA, Ananthapuramu, Andhra Pradesh 515002, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_91
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image fusion. The main aim of the image fusion is to produce new images that are more appropriate for the purpose of human/machine perception and also to reduce the amount of data. Medical images provide different types of information: Computed tomography (CT) images embed with less distortion and provide details regarding dense structures, such as bones, magnetic resonance imaging (MRI) provides information about pathological soft tissues, magnetic resonance angiography (MRA) easily detects abnormalities in the brain, X-ray detects fractures and abnormalities in bone positions, vibro-acoustography (VA) provides the depth and thickness of the disease object, and positron emission tomography (PET), and single photon emission computed tomography (SPECT) provides functional and metabolic data about the human brain. By combining these in multimodal medical image pairs such as CT-MRI [1], MRI-MRA, X-ray-VA [2], MRI-PET [3], and MRI-SPECT, extra clinical data is extracted that is complimentary in nature. Accordingly, we can say that a single image will give all applicable data, and henceforth multimodal medical image fusion is fundamentally required to acquire all probable data in a single composite image called a fused image [4].
2 Literature Review Fuzzy set theory plays a vital role in the image processing proposed by Zadeh [5] to remove the ambiguity and vagueness present in images. Fuzzy sets in image processing increase contrast, flat the regions of interest, and refine the edges and fine erections of the image. Fuzzy sets use the membership function to remove the vagueness present in the image. While describing the membership function, an uncertainty due to lack of knowledge or personal error is found, which primes to another uncertainty called hesitation degree. The combination of membership, nonmembership, and hesitation degree defines the intuitionistic fuzzy set introduced by Atanassov [6] in 1986. Medical images are poorly illuminated, contain many uncertainties in the form of noise, have vague boundaries, have overlapping gray levels, have invisible blood vessels, and it is difficult to extract objects from the image. Many uncertainties exist in every phase of image processing, and in using IFS [6–9] and fuzzy transforms [10], these uncertainties can be removed. The most medicinal images are foggy for two reasons: one is that the noise signal obscures the high frequency signal of an image edge, the other one is the edge of the tumor with ordinary tissues cannot be exceptionally all around characterized by the images, in this manner, it is troublesome for radiology specialists to outline an image. There are a couple of utilizations of fuzzy logic-based image fusion, for instance, image segmentation and integration, deep brain stimulation, image retrieval, brain tumor segmentation, ovarian cancer diagnosis, and gene expression. Atanassov and Gargov [11] stretched out the intuitionistic fuzzy set to the intervalvalued intuitionistic fuzzy set (IVIFS), which is portrayed by a membership function
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and a non-membership function whose values are intervals rather than real numbers. This methodology not just extends the capacity of the intuitionistic fuzzy set to deal with uncertain data yet in addition improves its capacity to tackle practical decision-making problems. The notion of IVIFS is as follows: + Let μ Z (y) = [μ−Z (y), μ+Z (y)] and v Z (y) = [v − Z (y), v Z (y)] are intervals, − + − μ Z (y) = inf μ Z (y), μ Z (y) = sup μ Z (y), v Z (y) = inf v Z (y), v + Z (y) = sup v Z (y), then + Z I V I F S = {y, [μ−Z (y), μ+Z (y)], [v − Z (y), v Z (y)]|y ∈ Y }
(1)
with 0 ≤ μ+Z (y) + v + Z (y) ≤ 1. The hesitation degree also lies in an interval which is given as π Z (y) = [π Z− (y), π Z+ (y)] + + = [1 − μ−Z (y) − v − Z (y), 1 − μ Z (y) − v Z (y)]
(2)
+ If μ Z (y) = μ−Z (y) = μ+Z (y) and v Z (y) = v − Z (y) = v Z (y), then IVIFS becomes IFS. Finally, an interval-valued intuitionistic fuzzy set image (IVIFSI) is molded as: + − + F I V I F S I = {y, [μ−Z (y), μ+Z (y)], [v − Z (y), v Z (y)], [π Z (y), π Z (y)]|y ∈ Y }
(3)
Yue Entropy [12] The degree of fuzziness of a fuzzy set can be measured by defining different magnitudes of a fuzzy set. The same thing can be done for IFSs, that is, the degree of intuitionism of an intuitionistic fuzzy set is measured by defining different magnitudes of IFSs. From this idea, the intuitionistic entropy is defined as: 1 α p ln piαj , α ∈ T (1, 2, . . . , t) ln(mn) i=1 j=1 i j m
Eα = − where piαj =
πα m i jn i=1
j=1
πiαj
n
(4)
and 0 ≤ E α ≤ 1.
3 Proposed Method This section discourses a proposed system for multimodal medical image fusion using IVIFS and contrast visibility. 1. 2.
The two input source medical images are initially read as I1 and I2 . The first image I1 of size M × N is fuzzified by means of the formula
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μ Z 1 (Ii j1 ) =
3. 4. 5. 6.
Ii j1 − lmin lmax − lmin
(5)
where Ii j1 defines the gray level of the first image and ranges from 0 to L −1 (L is the extreme gray-level value). lmin and lmax represent the least and extreme gray-level values of the first image. The optimum value of α is computed for the first image using the entropy Eq. (4). Find the fuzzified IVIFS image with the optimum value of α for the first input medical image by using the Eq. (3) and denote as IY 1 . The above procedure is repeated for from step 2 to step 4 for the second input medical image to find fuzzified IVIFS, IY 2 . The two fuzzified images IY 1 and IY 2 are decomposed into m ×n blocks—each block size is considered as 5 × 5 in this paper—and then compute the contrast visibility (CV) of each block separately using the below equation. CV =
|I (i, j) − μk | 1 p × q (i, j)∈B μk
(6)
k
7.
where μk and p × q are the mean and magnitude of the block Bk , respectively. Then, a decision map (DM) is constructed that determines the combination of pixels of two images. ⎧ I I ⎨ 1 if C Vi Y 1 > C Vi Y 2 IY 1 D M = −1 if C Vi < C Vi IY 2 ⎩ 0 if C Vi IY 1 = C Vi IY 2
8. 9.
C Vi IY 1 and C Vi IY 2 are the contrast visibilities of the ith block of images IY 1 and IY 2 , respectively. DM is the decision map built by taking the decision for each coefficient using the contrast visibility of the respective block. A new decision map (NDM) is created by refining DM with consistency verification by using the majority filter introduced by Li et al. [13]. The fused image (FI) without uncertainty is attained based on the NDM as ⎧ if N D M(i, j) = 1 ⎨ IY 1 (i, j) F I (i, j) = IY 2 (i, j) if N D M(i, j) = −1 ⎩ [IY 1 (i, j) + IY 2 (i, j)]/2 if N D M(i, j) = 0
10.
(7)
(8)
A crisp image is constructed by defuzzification of the fused image by using the inverse of Eq. (5) I (i, j) = (lmax − lmin ) ∗ μY I FC S (F I (i, j)) + lmin
(9)
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where lmin and lmax denote the lowest and extreme gray-level values of the fused image, FI. Finally, the fused image is filtered for noise using median filter.
4 Experimental Results The first example shown in Fig. 1 of the first row discourses CT and MRI images (www.metapix.de/toolbox.htm) that are complementary in nature. The CT image gives data about bones and hard tissues while the MRI image gives soft tissue data. Fusing these two images gives plenteous information in a single image, which better analyzes a sickness. The fifth image of the first row of Fig. 1 is the fusion result of the proposed method, which outwardly demonstrates that the image is of higher contrast and luminance than the fused image of prevailing methods, intuitionistic fuzzy set and fuzzy transform. Further examinations of the outcomes are performed utilizing a couple of objective criteria utilized in [14], for example, average gradient (AG), entropy, spatial frequency (SF), and edge strength preservation (QAB/F ), and the results are listed in Table 1. It is seen from the table that the proposed technique gives better execution with respect to contrast, luminance, and visibility of the fused image. The second example addresses a T1-weighted MR image and the MRA image with some illness as white structures that are shown in the second row of Fig. 1. The
CT
MRI
MR image
MRA image
MRI
SPECT
Fig. 1 Fusion results for multimodal medical images. The first and second columns represent the input source images. The third column represents the fused images by an intuitionistic fuzzy set. The fourth column represents the fused images by fuzzy transform. The fifth column represents the fused image by an interval-valued intuitionistic fuzzy set
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Table 1 Objective assessment of various image fusion methods with the proposed interval-valued intuitionistic fuzzy set method for Fig. 1 Modality
Fusion method
AG (change in intensity/pixel)
ENTROPY (bits/pixel)
SF (cycles/millimeter)
QAB/F
CT-MRI
IFS [9]
14.4342
5.3570
30.7839
0.7516
Fuzzy transform [10]
14.4624
6.7823
31.5100
0.7324
Proposed IVIFS
18.5263
6.8026
35.1702
0.8451
IFS [9]
17.5735
5.1670
35.8483
0.6406
Fuzzy transform [10]
16.7764
6.0653
33.5737
0.7042
Proposed IVIFS
18.701
7.1052
36.2217
0.7165
MR-MRA
MRI-SPECT
IFS [9]
5.6202
0.9789
32.5970
0.4743
Fuzzy transform [10]
6.0501
1.4000
32.6430
0.4994
Proposed IVIFS
17.1271
5.9753
35.3234
0.7868
T1-weighted MR image provides soft tissue information in a clearer way, but it is unable to detect abnormalities present in the image. The MRA image can easily detect abnormalities but not soft tissue information because of low spatial resolution. Hence, the fusion of these two images gives complementary information in a single image, which can be helpful for better medical diagnostics. The fifth image of the second row of Fig. 1 gives the fused image for the proposed method and in comparison with the existing methods of fused images; the proposed method fused image has the high spatial resolution with better enhancement of illness and also the edge transformation (QAB/F ) of the source images are more transferred to the fused image. Table 1 gives a further comparison of the results. The third set of images is MRI and SPECT brain tumor images of magnitude 256× 256 taken from the Harvard University website (www.med.harvard.edu), and they are shown in the third row of Fig. 1 for the evaluation of different fusion algorithms. It can be observed that from the MRI image, we obtain anatomical information, while the SPECT image gives physiological/functional knowledge of the human brain. To have both types of knowledge simultaneously, the images are to be fused. The fifth image of the third row of Fig. 1 gives the fused image by the proposed method, and the tumor is clearly enhanced when compared with other methods.
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5 Conclusion This paper grants a new method for fusing multimodal medical images using intervalvalued intuitionistic fuzzy sets. The proposed algorithm is tested on several pairs of medical images, and it is found that the proposed method gives improved visual and quantitative results, with high contrast and luminance. Better results are obtained using IVIFS because it considers a greater number of uncertainties and incorporates a hesitation degree. As medical images are a low contrast with vague regions and boundaries, IFS aids in solving these problems. This algorithm is also included with intuitionistic fuzzy entropy to optimize the best parameter for membership, nonmembership, and hesitation degree functions. Further work includes the usage of neuro-fuzzy logic for improved image quality.
References 1. Shanker HO, Mishra, Bhatnagar S (2014) MRI and CT image fusion based on wavelet transform. Int J Inf Comput Technol 4(1):47–52 2. Hosseini HG, Alizad A, Fatemi M (2007) Integration of Vibro-Acoustography imaging modality with the traditional mammography. Int J Biomed Imaging 1–8 3. Tirupal T, Chandra Mohan B, Srinivas Kumar S (2018) Multimodal medical image fusion based on fuzzy sets with orthogonal teaching–learning-based optimization. In: Verma N, Ghosh A (eds) Computational intelligence: theories, applications and future directions, vol II. Advances in Intelligent Systems and Computing, p 799 4. Tirupal T, Chandra Mohan B, Srinivas Kumar S (2015) Image fusion of natural, satellite, and medical images using undecimated discrete wavelet transform and contrast visibility. In: IEEE national conference on recent advances in electronics & computer engineering (RAECE), pp 11–16 5. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353 6. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96 7. Tirupal T, Chandra Mohan B, Srinivas Kumar S (2017) Multimodal medical image fusion based on Sugeno’s intuitionistic fuzzy sets. ETRI J 39(2):173–180 8. Tirupal T, Chandra Mohan B, Srinivas Kumar S (2019) Multimodal medical image fusion based on Yager’s intuitionistic fuzzy sets. Iranian J Fuzzy Syst 16(1):33–48 9. Balasubramaniam P, Ananthi VP (2014) Image fusion using intuitionistic fuzzy sets. Inf Fusion 20:21–30 10. Meenu M, Rajiv S (2016) A novel method of multimodal medical image fusion using fuzzy transform. J Vis Commun Image Repr 40:197–217 11. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349 12. Yue C (2017) Entropy-based weights on decision makers in group decision-making setting with hybrid preference representations. Appl Soft Comput 60:737–749 13. Li H, Manjunath BS, Mitra SK (1995) Multisensor image fusion using the wavelet transform. Graph Models Image Process 57(3):235–245 14. Jagalingam P, Hegde AV (2015) A review of quality metrics for fused image. Aquatic Procedia. Int Conf on Water Resources, Coastal and Ocean Engineering (ICWRCOE), March 12–14, Mangalore, Karnataka, India, vol 4, pp 133–142
The Influence of Ultrasound for the Protection of Animals on Highways Through Electronic Circuits T. Tirupal and S. Fowzia Sultana
Abstract While driving in regions where creatures are regularly present, it is not unexpected to wind up in a mishap. Both wild and residential creatures might be outside and can keep running into the street. Normally, a driver’s first nature is to swerve to abstain from hitting the creature; however, that can have wrecking results, such as losing control of the vehicle and enduring genuine wounds. Swerving can deliver a domino impact, making the driver strike another vehicle or object, which can prompt far more atrocious outcomes, similar to the vehicle moving over or genuine damage to different drivers out on the road. It is essential to avoid such accidents and protect animals as well. To overcome this, new method is proposed in this paper which includes a circuit generating ultrasonic waves. It can be used as pest repellents. For generating ultrasonic waves of high frequency, a generator using 555 timer can be employed. These waves are designed to produce an extremely high-frequency sound that is beyond what humans can hear. Ultrasound is used to bring about enough irritation in animals and make them stay away from highways. Keywords Ultrasonic · Humans · Animals
1 Introduction There are different reasons why individuals must repulse creatures from territories where they can damage individuals or devastate important goods and furthermore stay away from mishaps. This objective can be accomplished in various ways utilizing various techniques. We can recognize electrical, chemical, mechanical, optical, reflective bags, acoustical strategies, and so on. The benefit of the acoustical technique contrasting with others is: economical to utilize, not unsafe to creatures and safe for individuals utilizing it. This is valid under the presumption that ultrasound is utilized, which is indiscernible for individuals and does not cause any consultation harm, notwithstanding when presented to sound weight levels up to 120 dB. At the point when creatures hear these sounds they will just sit and gaze at the region where T. Tirupal · S. Fowzia Sultana (B) Department of ECE, GPCET, Kurnool, Andhra Pradesh 518452, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_92
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it has all the earmarks of being radiating. Along these lines, a clever hardware framework is essential which can be included vehicles to maintain a strategic distance from the potential outcomes of mishaps. Generally, creature–vehicle crashes have been tended to through signals cautioning drivers of potential creature intersections. In different cases, natural life cautioning reflectors, mirrors, or wildlife fences have been introduced to ward off creatures from the street [1]. Nonetheless, regular cautioning signals seem to have just constrained impact since drivers are probably going to habituate to them. Natural life cautioning mirrors or reflectors may not be powerful. Wildlife fencing has been joined with natural life intersection structures to address these impediments; at the same time, fundamentally because of their moderately surprising expense, such intersection structures are constrained in number and width.
2 Literature Review Various methodologies are overviewed to counteract the creatures to come on the street and in influence to stay away from the demise because of mishaps. To limit the occurrences of wild vehicle–creature impact along the state course, Dodd et al. [2] introduced the framework dependent on wild creature vehicle crashes by and large happen more around the evening time by bigger vehicles. So the capacity to stay away from an impact is diminished in every one of these circumstances because of decreased perceivability and expanded ceasing separations. A framework is intended to create the ultrasonic waves what is more with some wise tasks to detect the traffic of a vehicle. Such creature detecting and ready framework can be mounted on the two sides of streets. It will remain unusable until any vehicle is recognized over the street [3]. Creatures are kept from being controlled via prepares and crack in the track have distinguished utilizing ultrasonic sensors and send a message to determined individual with no mediation of the human [4]. The Supreme Court has restricted vehicles in the woods for the creature security [5]. Table 1 gives the data about various creatures murdered in street mishaps in India from 2010 to 2018. To spare the wildlife from a particular mishap, a framework is introduced here to create the consultation frequency for the particular creature. The creature can caution with the signal of threat and can flee from that spot like the street, railroad track, and Table 1 Animals killed in road accidents in India from (2010 to April 15, 2018)
S. No.
Name of animal
Number of animals killed
1
Leopard
40
2
Rusty spotted cat
14
3
Sambar
17
4
Hyena
16
5
Tiger
27
The Influence of Ultrasound for the Protection of Animals … Table 2 Hearing frequency ranges of different animals
S. No.
Name of animal
1
Tiger
2
Chinchilla
3
Raccoon
4
975 Minimum frequency (HZ)
Maximum frequency (HZ)
20
65,000
90
22,000
100
40,000
Ferret
16
44,000
5
Sheep
100
30,000
6
Dog
67
45,000
7
Cat
45
64,000
8
Cow
23
35,000
9
Rabbit
10
360
42,000
Elephant
16
12,000
11
Hedgehog
250
54,000
12
Horse
55
33,500
13
Guinea pig
54
50,000
so on bringing about mishap evasion [6]. Every creature has his very own capable of being heard scope of frequency range and not all sort of creatures found all over the place. Notwithstanding, we watched this reality with unmistakable fascination and proposed the framework which can recognize the creature/vehicle and caution them at their discernable range [7–9]. Table 2 demonstrates the scope of maximum point of confinement and the lower breaking point hearing the frequency of the various species. Any endeavor to evaluate the impacts of sounds on creatures must consider species contrasts in hearing capacities. In spite of the fact that the consultation scopes of most species cover to an enormous degree, significant variety happens in high and low frequency hearing just as in outright affectability. Accordingly, a sound that is effectively discernable to one animal type might be less capable of being heard, or even unintelligible, to another.
3 Proposed Method Essential thought behind the framework is to create the sound signal which will be indiscernible to human and irritating for creatures. The irritating frequency can be evaluated by explicit rationale and can be chosen and balanced by an experimentation approach. The irritating frequency generator of a particular loudness is planned furthermore with some intelligent tasks to detect the traffic of creature/vehicle. Intelligent and energy productive framework is created with disturbing signal generation. Such creature detecting and the ready framework can be mounted on the two sides of streets. It will remain unusable until any vehicle is distinguished over the street.
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Ultrasonic gadgets work by transmitting high frequency sound waves more noteworthy than 20,000 Hz, while a few creatures, for example, dogs, bats, rodents, feathered creatures, and creepy crawlies can hear well into the ultrasonic range. The human ear comes up short on the ability to hear such sound. Ultrasonic gadgets are structured and developed to radiate sound of this frequency, when focused at pests; they make them awkward inside the zone of inclusion along these lines repulsing them away from the zone without influencing nature and non-target life forms, including man. Electromagnetic gadgets are fitted into home wirings and produce electromagnetic waves which are unfriendly to bothers. The advantages of this strategy over other bug control techniques incorporate the way that they are cheap, eco-framework neighborly, earth amicable, and have no known hazard to people.
3.1 Generation of Ultrasound Waves Using 555 Timer The 555 timer IC is an integrated circuit (chip) utilized in an assortment of timer, pulse generation, and oscillator applications [10]. The 555 can be utilized to give time delays, as an oscillator, and as a flip-flop component. Subsidiaries give two (556) or four (558) timing circuits in a single package. The 555 is the most wellknown coordinated circuit at any point made. 555 Timer IC can be utilized with a couple of straightforward segments to assemble an astable circuit which delivers a ‘square wave’ [11]. This is an advanced waveform with sharp advances between low (0 V) and high (+Vs), the spans of the low and high states might be unique. The circuit is called an astable on the grounds that it is not steady in any state: the yield is constantly evolving among ‘low’ and ‘high’ [12] (Fig. 1).
Fig. 1 Block diagram of the 555 timer [12]
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Time period and frequency The timespan (T) of the square wave is the time for one complete cycle, yet it is frequently better to think about the frequency (f) which is the quantity of cycles every second. T = 0.7(R1 + 2R2 )C1
(1)
F = 1.4/(R1 + 2R2 )C1
(2)
and
T = time period in seconds (s). f = frequency in hertz (Hz). R1 = resistance in Ohms. R2 = resistance in Ohms. C 1 = capacitance in Farads (F). Choosing C1 , R1 and R2 R1 and R2 ought to be in the range 1 K ohm to 1 M ohm. It is ideal to pick C1 first since capacitors are accessible in only a couple of qualities. Choosing C1 to suit the frequency range according to necessity. Choosing R2 to give the frequency (f ) which is required. Except that R1 is much smaller than R2 , at that point utilizing: R2 = 0.7/fC1 for R1 R2
(3)
Choose R1 to be about a tenth of R2 (the base is 1 K) except if the imprint time Tm to be essentially longer than the space time Ts. Table 3 gives the information of astable frequencies. Table 3 555 timer astable frequencies
C1 (µF)
R2 = 10 K R1 = 1 K
R2 = 100 K R1 = 10 K
R2 = 1 M R1 = 100 K
0.001
68 kHz
6.8 kHz
680 Hz
0.01
6.8 kHz
680 Hz
68 Hz
0.1
680 Hz
68 Hz
6.8 Hz
1
68 Hz
6.8 Hz
0.68 Hz
10
6.8 Hz
0.68 Hz
0.068 Hz
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Fig. 2 Circuit for a generation of ultrasound using 555 timer
4 Simulation Results Using the above circuit, ultrasonic waves of required frequency can be obtained by varying external capacitance and resistance values (Figs. 2 and 3).
5 Conclusion Because of the sound signal, the creature can flee from the street and thus there are least odds of mishaps. The fundamental thought behind the framework is to generate the sound signal utilizing 555 timer which will be quiet to human and irritating for creatures. Their irritating frequency can be assessed by explicit logic and can be chosen and balanced by experimentation procedure. Irritating frequency generator of a particular loudness is planned moreover with some shrewd tasks to detect the traffic of creature/vehicle. Intelligent and energy efficient framework is created with irritating signal production. Such creature detecting a ready framework can be mounted on the two sides of streets. It will remain unusable until any vehicle is distinguished over the street. It is another methodology in social perspectives for wild creature passing shirking and mishaps anticipation. Creature explicit frequency range signals are produced. The particular creatures are cautioned with these signal of threat and effectively fled. The framework can be included in vehicles or trains as opposed to mounting poles on the roadside.
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Fig. 3 Generation of ultrasonic waves
References 1. Gunson KE, Chruszcz B, Clevenger AP (2003) Large animal-vehicle collisions in the central Canadian rocky mountains: patterns and characteristics 2. Dodd N, Gagnon J, Schweinsburg R (2007) Evaluation of measures to minimize wildlifevehicle collisions and maintain wildlife permeability across highways in Arizona. USA 3. Sonone DJ, Patil DA, Rane KP (2014) Irritating and hearing frequency identification and generation to avoid animals accident, July 2014 4. Saritha B, Elakiya P, Mathavi S, Monika M, Nivetha V (2017) To prevent the animals accident and track crack detection system for railways, Mar 2017 Wildlife Protection Society of India 5. Valitzski SA (2004) Evaluation of sound as a deterrent for reducing deer-vehicle collisions. University of Georgia 6. Tiwari DK, Ansari MA (2016) Electronic pest repellent: a review. In: International conference on innovations in information embedded and communication systems (ICIIECS’16) 7. Ibrahim AG, Oyedum OD, Awojoyogbe OB, Okeke SSN (2013) Electronic pest control devices: a review of their necessity, controversies and a submission of designal considerations. Int J Eng Sci 2:2319–1805 8. Enayati A, Hemingway J, Garner P (2010) Electronic mosquito repellents for preventing mosquito bites and malaria infection (review). Wiley, New York 9. Jhaver S, Singh R, Hiremani T (2009) Electronic pest repellent. EE318 Electronic Designal Lab, Project Report, EE Dept, IIT Bombay 10. Yu DS, Zheng CY, Iu HHC, Fernando T (2015) A memristive astable multivibrator based on 555 timer. In: 2015 IEEE international symposium on circuits and systems (ISCAS) 11. Everest F (2000) The characteristics and use of the 555 timer. Electron Educ 2000(3):34–40 12. Camenzind HR (1997) Redesignaling the old 555 [timer circuit]. IEEE Spectr 34(9):80–85
Workspace Analysis of a 5-Axis Parallel Kinematic Machine Tool Anshul Jain
and H. P. Jawale
Abstract Parallel kinematics machines (PKM) are interesting alternative designs for high-speed machining applications since they offer several advantages over their serial counterparts like high stiffness, improved dynamic characteristics, high accuracy and high structural rigidity. Conventional parallel kinematic machines are designed to have 3-DOF; however, in this work, two additional degrees of freedom have been incorporated with the movement of the base in X- and Y-direction. The configuration of 3-DOF parallel manipulator used here is 3-RPS. This paper presents a type of 5-DOF parallel kinematic machine tool architecture on which the parallel manipulator with 3-DOF sliding on the frame structure of machine tool is responsible for the translation along X-direction and Y-direction. Workspace analysis of this mechanism is proposed so as to know the reach of the tool for complex machining operations. Quantitative improvement in reachable volume over conventional 3-DOF is presented. Keywords PKM · 5-DOF PKM tool · Workspace analysis
1 Introduction Parallel mechanisms offer a number of advantages over their serial counterparts like higher stiffness, better rigidity, higher payload/weight ratio and higher accuracy. After NC machines, parallel kinematic machines have opened a new domain in the machine tool industry. They are different from the conventional machines, since they are based on variable geometry truss structures [1]. 6-DOF parallel manipulator like Stewart platform requires six linear actuators for controlling; however, only three linear actuators are required to control 3-RPS mechanism. Also, 6-DOF parallel manipulator has complex kinematic analysis and mechanical design. So, the paper presents a 5-DOF parallel kinematics machine tool architecture based on a 3-DOF, 3RPS parallel manipulator and two additional degrees of freedom of the movement of the base in X- and Y-direction. The parallel manipulator slides on the frame structure A. Jain (B) · H. P. Jawale Visvesvaraya National Institute of Technology, Nagpur 440010, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_93
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of machine tool and translates along X- and Y-direction in order to increase the workspace in both the directions. Along with this, its rigidity also increases because of the adaptation of gantry structure [1].
2 Description of Parallel Kinematic Machine (PKM) The PKM, as shown in Fig. 2 basically consists of a workbench, a 3-DOF parallel manipulator (3-RPS), as shown in Fig. 1 and the frame structure of machine tool. Px , Py , Pz , ψ x , ψ y are the respective five axis motions of the machine tool, in which x, y and z are the positioning parameters, and ψ x and ψ y are the orientation parameters. Five feed motions are given in total out of which the parallel manipulator sliding on the frame structure is responsible for the translation along X-direction and Y-direction. The other feed motions (Pz , ψ x , ψ y ) are for mobile platform of the parallel manipulator that means one translational and two rotational movements. Workspace of the mechanism enhanced greatly because of sliding motion of the parallel manipulator along X- and Y-directions. Fig. 1 3-RPS parallel manipulator
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Fig. 2 5-axis parallel kinematic machine [1]
3 Position Analysis 3.1 Inverse Position Analysis (IK) and Forward Position Analysis (FK) Before doing workspace analysis of PKM, inverse and forward kinematics of the 3-RPS parallel manipulator have been carried out. In IK the position vector is given and the problem is to find the actuated variables [2]. In another sense, it is focused on deducing the joint motions when the position of End-Effector is known [3]. Here IKis carried out to calculate the limb lengths (d 1 , d 2 , d 3 ) from the equation di2 = [qi −ai ]T [qi −ai ]
(1)
and from the given location of the moving platform, while in FK limb lengths are known to us and we have to find the location of the moving platform by using the equation T qi −qi+1 qi −qi+1 − 3h 2 = 0
(2)
where, qi is the position vector of points, Bi with respect to the fixed coordinate system, ai is the position vector of points, Ai in the coordinate system A, and i = 1, 2, 3 is used for the corresponding limb parameters, respectively. See Fig. 1. As a sample at random, we have selected the values as Pz = 965 mm, ψ x = 10° and ψ y = 0°. Radius of base platform, g = 500 mm and radius of moving platform,
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h = 255 mm. The values of limb lengths obtained in IK are d 1 = 995.14 mm, d 2 = 1033.8 mm and d 3 = 959.49 mm, respectively. Now, in FK, considering limb lengths as d 1 = 995.14 mm, d 2 = 1033.8 mm, d 3 = 959.49 mm and g = 500 mm, h = 255 mm, we obtained total 16 values of limb angle ϕ 1 , in total 8 pairs, mirror image of each other. Correspondingly, other limb angles ϕ 2 and ϕ 3 can be obtained from ϕ 1 by back substitution. Out of those values of ϕ 1 , we take our initial guess value of ϕ 1 = 74.83°, ϕ 2 = 74.52° and ϕ 3 = 74.34° from the geometric constraints. It has been found that the results obtained in forward kinematics for the task space parameters are matched with those obtained in inverse kinematics.
4 Workspace Analysis The reachable workspace of a manipulator is defined as the space that can be reached by the reference point by at least one orientation [4]. Two cases have been considered for workspace analysis to show the reach of the moving platform as follows: Case (I) Workspace of point P (centroid of B1 B2 B3 ) of the moving platform is plotted when one limb is increasing in dimensions while keeping the other two constants. Let the increment in the respective limbs are of 1 mm, 3 mm, 5 mm, 7 mm and 10 mm. Limb lengths are d 1 = 995.14 mm, d 2 = 1033.8 mm and d 3 = 959.49 mm. Figure 3 shows the workspace of point P of moving platform by keeping d 2 and d 3 constant and increasing d 1 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively. Figure 4 shows the workspace of point P of moving platform by keeping d 1 and d 3 constant and increasing d 2 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively.
Fig. 3 Workspace of point P by keeping d 2 and d 3 constant
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Fig. 4 Workspace of point P by keeping d 1 and d 3 constant
Figure 5 shows the workspace of point P of moving platform by keeping d 1 and d 2 constant and increasing d 3 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively. Figure 6 shows the combined effect of all three of the above in one plot. Case (II) Workspace of point Bi of moving platform is plotted when one limb is increasing in dimensions while keeping the other two constants. Let the increment in the respective limbs are the same as in case (I). Figure 7 shows the workspace plot of coordinates of B1 of moving platform by keeping d 2 and d 3 constant and increasing d 1 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively.
Fig. 5 Workspace of point P by keeping d 1 and d 2 constant
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Fig. 6 Workspace of point P by keeping d 1 , d 2 and d 3 constant
Fig. 7 Workspace of point B1 by keeping d 2 and d 3 constant
Figure 8 shows the workspace plot of coordinates of B2 of moving platform by keeping d 1 and d 3 constant and increasing d 2 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively. Figure 9 shows the workspace plot of coordinates of B3 of moving platform by keeping d 1 and d 2 constant and increasing d 3 by 1 mm, 3 mm, 5 mm, 7 mm and 10 mm, respectively. Figure 10 shows the combined effect of all three of the above in one plot. Now, after showing the workspace of the centroid and the workspace of the coordinates of the moving platform, Fig. 11 shows the complete work volume of the
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Fig. 8 Workspace of point B2 by keeping d 1 and d 3 constant
Fig. 9 Workspace of point B3 by keeping d 1 and d 2 constant
manipulator. As shown in figure, the manipulator is moving 500 mm in X- and Ydirection, thereby presenting a more effectiveness and usefulness of the mechanism over conventional machines for the complex machining operations.
5 Results Complete work on the position analysis (inverse and forward kinematics) and the workspace analysis is carried out and quantitative improvement over reachable work
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Fig. 10 Workspace of points Bi by keeping d 1 , d 2 and d 3 constant
Fig. 11 Translation of the manipulator 500 mm along X- and Y-direction
volume is estimated. Results of forward kinematics matched with those obtained in inverse kinematics. From the plots of workspace analysis, it is clear that the special mechanism architecture proposed has a larger workspace and higher rigidity than many of the other counterpart platforms. Translation of the manipulator along X- and Y-direction over the structure considerably increases its reach and so the movement of the tool thereby increasing its utility. The results lead to the conclusion of reasonable increment of work volume and it can be carried out for any dimensional configuration. It is also observed that, its rigidity also increases because of the adaptation of gantry structure.
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6 Conclusion and Future Scope We have suggested a new configuration of 5-DOF parallel kinematic machine which consists of a 3-DOF spatial parallel manipulator (3-RPS) sliding on the frame structure of machine tool is responsible for the translation along X- and Y-direction providing extra 2-DOF. Inverse kinematics is carried out. Complicated forward kinematics is carried out, and the respective limb angles have been obtained after solving rigorous equations. We got 16 values for ϕ 1, and from the back substitution, we got 32 values for ϕ 2 and ϕ 3. From the geometrical constraints, we have selected one values of each, and finally, our results matched with the inverse kinematics results. Workspace analysis is also carried out to show the reachable work volume of the end-effector attached to the moving platform, and the corresponding plots have been shown in this paper. The same methodology can be applied for any dimensional configuration. After getting the work volume, immediately, we can go for the accuracy analysis of the same. Effects of tolerances can be considered and its results can be obtained in future.
References 1. Chen W, Zhao L, Zhang J (2011) A 3-DOF parallel manipulator based 5-axis parallel kinematics machine tool. Adv Mater Res 317–319:698–702 2. Zhang D, Gao Z, Su X, Li J (2011) A comparison study of three degree of freedom parallel robotic machine tools with/without actuation redundancy. Int J Comput Integr Manuf 230–247 3. Gao Z, Zhang D, Ge Y (2010) Design optimization of a spatial six degree of freedom parallel manipulator based on artificial intelligence approaches. Robot Comput Integr Manuf 26:180–189 4. Li Y, Xu Q (2007) Kinematic analysis of a 3-PRS parallel manipulator. Robot Comput Integr Manuf 23:395–408
Reaction Solvability Analysis Using Natural Coordinates Shivam Sharma and Ashitava Ghosal
Abstract In over-constrained mechanisms, all the joint reactions cannot be solved uniquely based solely on rigid body assumptions. However, a few joint reactions may be uniquely solvable, and an approach termed as reactions solvability analysis (RSA), in this paper, can be used to find such uniquely solvable joint reactions. Existing work has implemented RSA algorithms using absolute coordinates. In this work, the RSA algorithm is used with natural coordinates and this is found to be more efficient for finding uniquely solvable joint reactions. To use natural coordinates for RSA, they need to be modified and this is discussed in this work. Keywords Reaction solvability analysis · Over-constrained mechanisms · Natural coordinates
1 Introduction Over-constrained mechanisms have actual degree of freedom (DOF) more than the number computed using the well-known Grubler–Kutzbach criterion [1]. This happens because of the presence of multiple joints, called redundant joints, constraining the same degrees of freedom. These redundant constraints cause linear dependency in constraint equations, leading to singular Jacobian matrices. The redundant constraints can be removed arbitrarily without changing its kinematics [2], and the kinematics of the system can be solved. However, one cannot solve for joint reactions uniquely in such a system solely on the basis of rigid body constraint equations, and one needs to include flexibility/the material constitutive equations to obtain the joint reactions. This issue of “joint reaction indeterminacy” is the topic of this work. The usual approach for handling redundant constraints is to solve for the system by eliminating any arbitrary set of dependent constraints equations [3, 4], or by using algorithms capable of dealing with dependent equations (e.g., minimum-norm solution and augmented Lagrangian) [5–7], or by using penalty-based and weighing S. Sharma · A. Ghosal (B) Indian Institute of Science, Bangalore, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_94
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factor-based methods [8, 9]. However, these approaches are good for kinematic and certain dynamic analyses and do not yield correct results for joint reactions. For calculating joint reactions, one needs to discard the rigid body assumption, introduce material constitutive relations to have the complete set of equations, and solve the system using a finite element approach (FEA) or analytically [10–12]. A new strategy was suggested based on the observation that in an over-constrained mechanism, some joint reactions may be solvable without considering the material constitutive equations [10]. Hence, if our joint reactions of interest lie in this set of “solvable joint reactions,” then there is no need to use an FEA solver. This can save tremendously in computation for complex mechanisms. In the subsequent works [2, 5, 10], algorithms were developed to find such “solvable joint reactions” in over-constrained mechanisms. This analysis for solvable joint reactions is termed as “reaction solvability analysis (RSA)”. In existing works, the coordinate system of choice for RSA has been absolute coordinates. In this work, we show that using natural coordinates can lead to substantial benefits in terms of computational simplicity and efficiency. The main reason is that the constraint equations using absolute coordinates are transcendental, while the constraint equations in natural coordinates are maximally quadratic, leading to a Jacobian matrix with linear terms. However, the application of natural coordinates for RSA is not straightforward, and we had to modify the natural coordinates to make them usable for RSA. We demonstrate the RSA application using our modified natural coordinates with a planar over-constrained mechanism.
2 Equations of Motion and the RSA Algorithm In this section, we discuss the necessary equations and representation that we have used in this paper, and the existing RSA algorithms that have been developed in prior works.
2.1 Equations of Motion of Multi-body Systems We will be using the standard representation used in [13] and [14] to represent the dynamics and constraint equations of multi-body systems. The equations of motion of a multi-body system can be written in a compact form as Mq¨ + f = Q
(1)
where Mnxn is the mass/inertia matrix, qnx1 = (q1 , q2 , . . . , qn )T is the vector of generalized coordinates, and Qnx1 is the vector of external forces and other inertia terms
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such as the Coriolis and centripetal terms. The constraint equations of a multi-body system can be written as ⎡
⎤ 1 (q) ⎢ 2 (q) ⎥ ⎢ ⎥ (q) = ⎢ . ⎥ = 0 ⎣ .. ⎦
(2)
m (q)
where (q) : R n → R m is the vector containing all scalar constraint equations. If some of the equations in (2) are dependent, it gives rise to a redundantly constrained system. It is convenient to check for this redundancy by checking the rank of the Jacobian matrix of this system where the m × n Jacobian matrix of this system can be written as ⎛ ∂ ∂ ⎞ 1 1 ··· ··· ∂q1 ∂q2 ⎜ ∂2 ∂2 · · · · · · ⎟ ⎜ ∂q1 ∂q2 ⎟ (3) q (q) = ⎜ .. . . .. ⎟ ⎜ .. ⎟ . . ⎠ ⎝ . . ∂m ∂m m · · · ∂ ∂q1 ∂q2 ∂qn The generalized force vector fn×1 can be written as f = q T λ
(4)
where λm×1 is the vector of Lagrange multipliers. Equation (4) is used to obtain constraint reaction forces from the given constraint equations.
2.2 The RSA Algorithm From Eqs. (1) and (4), we can write q T λ = Q − Mq¨ = f
(5)
The above Eq. (5) is a convenient representation in the familiar linear-algebraic form Ax = b. Based on the analysis of this equation, a simple RSA algorithm was proposed [10]. In this algorithm, first, we split the Jacobian matrix q into two matrices—one, which contains constraints acting on a particular joint and other which contains constraints not acting on that particular joint. To analyze the joint i, we can get two matrices for that joint as
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iq : Matrix with all the constraints (rows) acting on joint i −i q : Matrix with all the constraints (rows) not acting on joint i We can now find the ranks of the corresponding matrices as ri : Rank of Matrix iq r−i : Rank of Matrix −i q The algorithm states that the constraints forces for joints can be solved for uniquely for which the following relation holds: r = ri + r−i
(6)
3 The Optimal Coordinate Formulation for RSA To implement the RSA algorithm discussed in the previous section, one needs a formulation which has constraint equations for each joint. This is the key idea differentiating which coordinates are “usable” for RSA and which are not. The two commonly used types of coordinates are the absolute coordinates (or reference point coordinates) and natural coordinates (or fully Cartesian coordinates). In absolute coordinates, a coordinate frame is attached to every link, and joints are defined by the constraint equations between these frames. Since the constraint equations are for joints, absolute coordinates are directly usable for RSA and have been used in prior works. In natural coordinates, one defines points and unit vectors (which usually denote the links) and then constrain these points and vectors. In this formulation, the joints are encapsulated in the formulation itself and the constraint equations refer to the rigidity of the links. Hence, there are usually no constraint equations for the joints, making them unusable for RSA. In the next section, we show that natural coordinates can be made usable for RSA by introducing extra points and unit vectors. Natural coordinates offer the benefit that all the constraints are maximally quadratic [15, 16], while in absolute coordinates, we have transcendental terms. We illustrate the absolute and natural coordinates with a simple planar four-bar mechanism, as shown in Fig. 1.
3.1 Joint Reactions from Natural Coordinates The constraint equations for four-bar modeled using natural coordinates, as shown in Fig. 2, are:
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Fig. 1 Planar four-bar modeled with different coordinates Fig. 2 Four-bar modeled with natural coordinates
(x1 − xa )2 + (y1 − ya )2 = l12 (x1 − x2 )2 + (y1 − y2 )2 = l22 (x2 − x3 )2 + (y2 − y3 )2 = l32 In these equations, the first corresponds to the rigid body constraint for the first link, the second one for the rigidity of second link, and so on. The revolute joints in the mechanism are captured by the sharing of the “basic” points; hence, no additional constraint equations are required for them. We propose a modified formulation of natural coordinates which help us write constraint equations for other kinds of joints. This formulation requires more “basic” points (and “unit vectors” for spatial mechanisms).
3.2 Joint-Augmented Natural Coordinate Formulation Our modified natural coordinates formulation, which we call joint-augmented natural coordinate formulation, is shown in Fig. 3. The key innovation here is that we have added extra geometric points to the links. While previously, for example, points 2 and 3 of the second and third links were the same, now we have to add extra constraint equations to make them coincident. Owing to these extra constraint equa-
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Fig. 3 Four-bar modeled with modified natural coordinates
tions, we now have equations constraining the joints. Note that we have dismantled link ends in the figure to show clearly how the points between different links are not shared as before, rather we share them using extra constraint equations. We now have extra constraints that correspond to the revolute joints. They are captured by four vector equations—hence, a total eight scalar equations—given as rA = r1 ,
r2 = r3
r4 = r5 ,
r6 = rB
(7)
This makes natural coordinates usable for RSA.
4 RSA of a Planar Mechanism Using Natural Coordinates In this section, we will implement our RSA methodology on a planar over-constrained mechanism using the joint-augmented natural coordinates. This mechanism was analyzed in [10] using absolute coordinates. The mechanism is shown in Fig. 4.
4.1 Assumptions We assume the following for this mechanism: • All bodies are considered rigid with no flexibility, • All joints are considered ideal, having only holonomic constraints, and • No friction is present at the joint.
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Fig. 4 Planar over-constrained mechanism
4.2 Modeling Using Natural Coordinates We follow the reference [4] to obtain the constraints with natural coordinates. The prismatic (P) joint constraints are of two types: one, collinear constraint (making cross product of the axes-defining vectors zero), and two, constraint for fixing the angle between the two connected bodies. Hence, for each prismatic joint, the constraint equations are P1:
r1 × r3 = 0 (r3 − r10 ).r1 = l1l6 cos(φ1 )
P2:
r7 × r8 = 0 (r9 − r7 ).r8 = l5l4 cos(φ2 )
P3:
(r10 − r3 ) × (r3 − r9 ) = 0 (r10 − r3 ).(r9 − r7 ) = l5l6
Prismatic Joint 1
(8)
Prismatic Joint 2
(9)
Prismatic Joint 3
(10)
For the revolute (R) joint, the points lying on the two bodies are coincident and we get the constraint equations as: R1:
r2 = r3
Revolute Joint 1
(11)
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R2:
r4 = r5
R3:
r7 = r6
Revolute Joint 2
(12)
Revolute Joint 3
(13)
The rigid body and fixed (grounding) constraints are r1 = (0, l1 )T |r4 − r2 | = l2 |r5 − r6 | = l3
r8 = (l4 , 0)T |r7 − r9 | = l5 |r3 − r10 | = l6
(14)
4.3 Applying RSA Algorithm The RSA algorithm as discussed in Sect. 2.2 was applied on this mechanism using mathematica® . When using natural coordinates, the rank of Jacobian matrix is r = rank(q ) = 19 and the ranks ri and r−i are shown in Table 1. For all the prismatic joints, ri + r−i > r = 19. Hence, for none of the prismatic joints, the reactions can be found uniquely. However, for all revolute joints, ri + r−i = r = 19 and all the revolute joint reactions can be found uniquely. When using absolute coordinates, as was done in [10], the rank r of the Jacobian matrix is 11. Here, for all prismatic joints, ri + r−i > r = 11 and for all revolute joints ri + r−i = r = 11. Hence, all revolute joint reactions can be found uniquely, but none of the prismatic joints can be. This is the same result as when using the natural coordinates. Results from both analyses are shown in Table 1 for comparison.
Table 1 Methodology-2 results for the planar mechanism Natural coordinates
Absolute coordinates
i
ri = rank(iq )
r−i = ri + r−i rank(q−i )
Y/N
ri = rank(iq )
r−i = ri + r−i rank(q−i )
Y/N
P1
2
18
20
N
2
10
12
N
P2
2
18
20
N
2
10
12
N
P3
2
18
20
N
2
10
12
N
R1
2
17
19
Y
2
9
11
Y
R2
2
17
19
Y
2
9
11
Y
R3
2
17
19
Y
2
9
11
Y
whether the joint reaction can be found uniquely (Y) or not (N)
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4.4 Discussion As can be seen from Table 1, both approaches of RSA analysis— using modified natural coordinates and using absolute coordinates (as done by [10]) yielded the same results. Further, it can be observed that natural coordinates had larger matrices (with greater rank). However, this did not lead to increased computational complexity, as Jacobian matrix obtained using natural coordinates had no transcendental terms as seen when absolute coordinates are used. This led to a great increase in speed of the RSA algorithm when using natural coordinates. Three more mechanisms were analyzed apart from this planar over-constrained mechanism, including one spatial mechanism (Bennett mechanism), in which the computational advantage was found to be even more profound.
5 Conclusions In this paper, we did a comparative study of various coordinate formulations which are usable for “reaction solvability analysis (RSA).” It was found that only absolute coordinates are directly usable. However, a small modification in the natural coordinates was found to make them usable for RSA. Being able to use natural coordinates for RSA is good news as they offer simpler equations, making the RSA algorithm operate much faster on a given mechanism. This was demonstrated by performing the RSA using natural coordinates on a planar over-constrained mechanism. RSA yielded the same result using both coordinates, while being faster when using natural coordinates.
References 1. Gogu G (2008) [Part-1] Structural synthesis of parallel robots Pt. 1: methodology. Springer, Dordrecht 2. Marek W (2009) Joint reactions in rigid body mechanisms with dependent constraints. Mechan Mach Theo 44(12):2265–2278 3. Bayo E, Ledesma R (1996) Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn 9(1):113–130 4. Garcıa J, de Jalón Bayo E (1994) Kinematic and dynamic simulation of multibody systems. Mechanical Engineering Series, Springer, New York 5. Marek W, Janusz F (2013) Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints. Multibody Syst Dyn 30(2):153–171 6. Xu Y, Wenlan L, Jiantao Y, Yongsheng Z (2015) A method for force analysis of the overconstrained lower mobility parallel mechanism. Mech Mach Theo 88:31–48 7. Wenlan L, Xu Y, Jiantao Y, Yongsheng Z (2017) The weighted Moore-Penrose generalized inverse and the force analysis of overconstrained parallel mechanisms. Multibody Syst Dyn 39(4):363–383 8. Francisco G, József K (2013) Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems. Multibody Syst Dyn 29(1):57–76
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9. Bilal R, József K (2011) A penalty formulation for dynamics analysis of redundant mechanical systems. J Comput Nonlinear Dyn 6(2) 10. Marek W (2005) Joint reaction forces in multibody systems with redundant constraints. Multibody Syst Dyn 14(1):23–46 11. Bi ZM, Kang B (2014) An inverse dynamic model of over-constrained parallel kinematic machine based on newton–euler formulation. J Dyn Syst Meas Cont 136(4):041001–041001-9 12. Zahariev E, Cuadrado J (2007) Dynamics of over-constrained rigid and flexible multibody systems. In: 12th IFToMM world congress, Besançon, France 13. Nikravesh PE (1988) Computer-aided analysis of mechanical systems. Prentice-Hall, Englewood Cliffs, N.J 14. Negrut D, Dyer A (2004) ADAMS/Solver Primer., Ann Arbor 15. Jalón JG (2007) Twenty-five years of natural coordinates. Multibody Syst Dyn 18(1):15–33 16. Uchida T, Callejo A, de Jalón JG, McPhee J (2014) On the Gröbner basis triangularization of constraint equations in natural coordinates. Multibody Syst Dyn 31(3):371–392
Design, Analysis and Development of Sweep Arm Scanner for Scanning Fast Breeder Reactor Core Ashish Kumar, Y. V. Nagaraja Bhat, B. K. Sreedhar, S. I. Sundar Raj, S. Murugan, and P. Selvaraj
Abstract Fast breeder reactors (FBRs) utilize liquid metals as coolant. However, inservice inspection of components using conventional optical methods is impossible in sodium due to its opacity. Ultrasonic technique is therefore employed for this purpose which requires the deployment of ultrasonic transducers (UTs) inside the reactor using a carrier mechanism. The carrier mechanism should have a proper kinematic design to access critical locations of reactor internals and ensure that transducers are positioned within close tolerances. Sweep arm scanner (SAS), as a carrier mechanism, is being developed to inspect and map the complete core of FBRs. Kinematic and dynamic analysis of motions is carried out to estimate capacity of drive motors and linear actuators. Keywords Fast reactors · In-service inspection · Mechanism · Kinematic and dynamic analysis
1 Introduction A ‘Three-Stage Nuclear Power Programme’ was conceptualized by Dr. Homi Jehangir Bhabha in 1950s to suffice the long term energy requirements of India. Progressing steadily under this programme, the country has entered into the second stage which is characterized by setting up of ‘fast breeder reactors’ (FBRs), which in general, are sodium cooled. One such reactor, known as prototype fast breeder reactor (PFBR) is being commissioned. Once operational, in-service inspection of reactor’s internals is would be mandatory for ensuring its structural integrity and for grant of clearance for activities such as fuel handling. Sodium being opaque renders all A. Kumar (B) · Y. V. Nagaraja Bhat · B. K. Sreedhar · S. I. Sundar Raj · S. Murugan · P. Selvaraj Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu 603102, India e-mail: [email protected] Y. V. Nagaraja Bhat e-mail: [email protected] B. K. Sreedhar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_95
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optical methods of inspection worthless, calling for development of alternate ways of inspection such as ultrasonic techniques [1]. Such technique requires a carrier mechanism for taking ultrasonic transducers (UTs) to critical locations of reactor internals. A mechanism called as sweep arm scanner (SAS) is being developed to carry these UTs inside the reactor and perform inspection.
2 Sweep Arm Scanner (SAS) There are two access openings for insertion of carrier mechanism into the reactor, one is in-vessel transfer port (IVTP) access opening in large rotatable plug (LRP), and other is observation port in the small rotatable plug (SRP), as shown in Fig. 1 along with dimensions. In order to scan the reactor core, SRP and LRP have to be rotated, but even after maximum possible rotations, a gap of around 840 mm exists between the centre of observation port and central sub-assembly (SA) as shown in Fig. 2. This means that there should be some provision in the carrier mechanism with the help of which UTs can reach the central subassembly. This is achieved by providing an extendable part (sweep arm), which can be extended once it is inserted inside the reactor. Hence, this mechanism has a closed configuration in which overall dimension of SAS is less than 450 mm, so that it can be inserted and retracted through observation port and an open configuration to reach till the central SA. Also, in order to get continuous signal, UTs are required to be translated in radial direction of the reactor core [2, 3], or else only some points can be scanned on the SA top. These points need interpolation to obtain the approximate orientation of the SA which may lead to errors. Fig. 1 Location of access ports in home position
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Fig. 2 Orientation after maximum possible rotation of SRP
2.1 General Aspects of SAS SAS consists of L, R, Z and θ-motions. It has a foldable arm called as sweep arm with an array of 6 equi-spaced down viewing transducers (DVTs) attached to it. For scanning the core, SAS has to be lowered in to the reactor through the access openings, with sweep arm in closed configuration. Inside the reactor, sweep arm can be opened and closed; this motion is called as L-motion. After opening, to and fro motion of the sweep arm in radial direction is called as R-motion. θ-motion imparts rotation of SAS which is limited to a sector of 270 deg to avoid any mechanical interaction with control plug components. The purpose of Z-motion is to adjust the height of the entire mechanism above the top of the general level of SA. SAS consists of two parts, viz. upper part and lower part. Total height of upper part is around 3 m and enveloping diameter is 600 mm. All components of upper part are located well above the roof slab and it mainly consists of support structures and Z, θ, L and R-motion drive systems. Entire load of SAS will be taken by support structures. Geared motors are provided for Z, R and θ-motion while a linear actuator is proved for L-motion. In addition to geared motor, ball screw nut mechanism and slewing bearing are also provided to accomplish the task of Z-motion and θmotion, respectively. Lower part of SAS consists of sweep arm, spinner tube and shield plug subassemblies. Sweep arm subassembly (Fig. 3) consists of sweep arm, holder, frame, ball screw and nut, guide rods and bush, gear shaft, bevel gears, universal joint, connecting link and anti-friction bearings. Spinner tube subassembly is provided with translational shaft, rotational shaft, mechanical stopper and spinner tube. Sweep arm subassembly is assembled below spinner tube subassembly in such a way that during L-motion whole sweep arm subassembly rotates about the hinged connection provided between gear shaft and the rotational shaft which is rotated by the up and down motion of translation shaft. While a mechanical stopper prevents
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Fig. 3 Lower part of SAS
further rotation of sweep arm subassembly beyond horizontal position, a collapsible joint is provided to ensure safe retraction of SAS in case of stuck condition. R-motion is executed by ball screw and nut mechanism under fully opened condition of sweep arm. Rotation of the ball screw causes translation of the nut; sweep arm with holder and anti-rotation arrangements is connected to the nut, and hence, it translates along with nut. Rotational shaft transmits rotational motion from drive system to ball screw through bevel gears. Ball bearings are provided for the geared shaft, ball screw and rotational shaft for radial and axial support and also for smooth rotation. During Z and θ-motions, sweep arm subassembly and spinner tube subassembly translates and rotates together by respective drive systems. A shield plug shields radiation streaming from the reactor through observation port. Sodium compatible materials are chosen for the construction of scanner, and at required locations, static and dynamic seals are provided to arrest argon cover gas leakage. As θ and Z-motions are simple, kinematic and dynamic analysis were carried out only for L and R-motions.
3 Analysis of L-Motion 3.1 Kinematic Analysis L-motion of SAS is executed with the help of frame and gear shaft (link 2), connecting link (link 3), translational shaft (link 4) and rotational shaft (link 1) as shown in Figs. 4 and 5. The rotational shaft remains stationary during L-motion. To carry out L-motion, translational shaft is actuated by a linear actuator, placed in the upper part
Design, Analysis and Development of Sweep Arm Scanner … Fig. 4 Kinematic model of four-bar linkage
Fig. 5 Details of links for L-motion
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Fig. 6 θ versus time during closing
of the SAS. The capacity of this actuator has to be estimated for which dynamic analysis has been carried out preceded by kinematic analysis. As forces required for closing the sweep arm is higher than opening, these analyses have been done for closing. The position, angular velocity and angular acceleration of the frame (link 2) and connecting link (link 3) during closing are shown in Figs. 6, 7, 8, 9, 10 and 11. Fig. 7 β versus time during closing
Fig. 8 Angular velocity of link 2 versus time
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Fig. 9 Angular velocity of link 3 versus time
Fig. 10 Angular acceleration of link 2 versus time
Fig. 11 Angular acceleration of link 3 versus time
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Fig. 12 Force versus time for carrying out L-motion
3.2 Dynamic Analysis To estimate the force required to execute L-motion, so as to arrive at the capacity of linear actuator, dynamic analysis has been carried out. This analysis has been carried out for closing the sweep arm under three friction conditions which are, • Without friction • In air: coefficient of friction is 0.2 [4] • In sodium: coefficient of friction is 0.7 [4]. Dynamic analysis was carried out under above friction conditions, and the graph is of force required vs. time is shown in Fig. 12. The maximum force (~1550 N) for closing was required when sweep arm was in extreme right position of ball screw. Hence, the linear actuator of capacity greater than 160 kg may be selected for carrying out L-motion.
4 Analysis of R-Motion 4.1 R-Motion Using Bevel Gears For R-motion, it is required to transmit rotary motion from vertical rotational shaft to horizontal ball screw. For this, bevel gears will be used as shown in Fig. 13. The rotational velocity of rotary shaft is kept at 4 RPM while the gear ratio of bevel gear is 1.5. The total traverse of sweep arm (nut) is 140 mm. Maximum torque required for R-motion using bevel gears was estimated to be around 105 N-mm.
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Fig. 13 Bevel gears for R-motion
4.2 R-Motion Using Alternate Options Use of bevel gear in sodium has not yet been fully unveiled, and hence, there is a need to have other options available for use. Literature survey [5] suggests the possibility of cylindrical joints being used inside sodium. A concept using only cylindrical joint to transmit rotary motion from the rotary shaft to the ball screw is by the use of three elbows arranged symmetrically within 2 housings, one of which is connected to rotary shaft while the other is attached to the ball screw as shown in Fig. 14. For simulating this concept in air, the coefficient of friction of 0.2 was taken, while for sodium operation, it was chosen as 0.7. The corresponding torque versus time plot for execution of R-motion using elbow joint is shown in Fig. 15. Cyclic fluctuation is observed in torque required to carry out R-motion using elbow transmission which may be due to additional torque requirement to overcome geometric constraints which occur in each cycle in a way similar to toggle positions in a four-bar linkage. The details of analysis of variation of torque with using elbow joint need to be studied and analysed further including the possibility of carrying out such transmission under practical conditions. Torque required for executing various motions were estimated using simple hand calculations and they were found to be comparable with the analytical results.
Fig. 14 Elbow joint for R-motion
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Fig. 15 Torque versus time using elbow joint
5 Summary Kinematic and dynamic analysis using commercially available multi-body dynamics software has been carried out for various motions of the sweep arm scanner to estimate the capacity of motor for R-motion and linear actuator for L-motion. Dynamic analysis for R-motion has been carried out for alternate option, i.e. elbows transmission for motion transmission from rotational shaft to ball screw instead of bevel gears. From analysis, it was concluded that the maximum magnitude of force required for L-motion is around 1550 N. Similarly, for R-motion, a drive motor having the capacity to deliver 5 N-m torque is sufficient.
References 1. Barrett LM, McKnight JA, Fothergill JR (1984) Ultrasonic viewing in fast reactors. Phys Techno1 15 2. Patankar VH, Lalwani SK, Agashe AA et al (2014) Under sodium ultrasonic imaging system for PFBR, Technology Development Article, issue number 336 3. Patankar VH et al (2011) Instrumentation & control system for under sodium ultrasonic scanner of PFBR, NDE 4. RCC-MRx 2012 Section III—Tome 1—subsection K: examination, handling or drive mechanisms 5. Nagaraja Bhat YV, Asokane C et al (2016) Literature survey on sweep arm scanner, an internal report of Indira Gandhi Centre for Atomic Research
Kinematics of Three Segment Continuum Robot for Surgical Application Shailesh Bamoriya and Cheruvu Siva Kumar
Abstract We propose a design of a multi-segment continuum robot with inherent compliance and flexibility for minimally invasive surgical applications. The multisegment continuum robot can be guided, with high dexterity, through complex curvilinear paths. Thus, they are ideal for endoscopic and surgical procedures. In this paper, we are discussing the kinematics of a three-segment tendon-driven continuum robot. We derived the forward and inverse kinematics for the proposed continuum robot, show the workspace analysis and quantify the robot reach and dexterity. Further, we discuss the kinematic advantages and disadvantages of tendon-driven three-segment continuum robot versus double-segment continuum robot. Keywords Continuum robot · Kinematics · Workspace · Dexterity
1 Introduction Continuum robots have proved to be very beneficial in the field of medical robotics. They have made Minimaly Invasive Surgery (MIS) in surgical space safer and more efficient. It is possible due to their inherent compliance, hyper-flexibility, and dexterity. The platforms used for robotic surgical operation can be broadly categorized into two groups, based on the applied surgical tools: laparoscopy and flexible endoscopy [1–3]. Instruments used in laparoscopic surgery are rigid, long, thin tube instruments, with inbuilt high-resolution light and cameras for visual feedback. However, the flexible endoscopy makes use of a steerable endoscope with a camera and narrow passage, which allows flexible gripper or forceps in order to access the human anatomy via natural orifices or a small incision. Forward kinematics of the continuum robot is based on the constant curvature model, which can be analyzed by the Euler Bernoulli beam model [4]. Many instances S. Bamoriya (B) · C. S. Kumar Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: [email protected] C. S. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_96
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of literature proposed different varieties of kinematic models, where Hannan and Walker [5, 6] presented the basic kinematic model of a planar curvature and further extended it for continuum robot segment. Jones and Walker [7] derived the robot kinematic mapping (robot specific and robot independent mapping) by assuming constant curvature bending. The kinematic modeling of various types of constant curvature continuum robots (like, tendon driven, wire driven, and concentric tube) and approaches are discussed [4]. Some of the continuum robots are also designed with bending as well as the extension abilities [8]. In addition, workspace and dexterity comparison and analysis for a different kind of continuum are disused [9–11]. This type of analysis is required in order to design the robot with specific requirements (stiffness, dexterity, and geometric parameters) for typical surgical operation. Continuum robots-based laparoscopic instruments and endoscopes are more flexible as compare to conventional laparoscopic tools. This paper presents the kinematics (forward and backward kinematics) and workspace analysis for three-segment continuum robot. Additionally, the workspace and dexterity-based analysis and comparison between three-segment and twosegment continuum robots are discussed. This shows the better usability of a threesegment continuum robot instead of a two-segment for some surgical tasks. The Monte Carlo method is used to determine the dexterity distribution based on relative kinematic flexibility by getting the estimate of reachable positions within the workspace, then determine data density across the workspace.
2 Preliminary Design Most surgical treatments require reaching deep surgical space through a natural orifice and uneven pathway without disturbing the peripheral tissues. Surgeons generally use straight or bendable endoscopes, surgical forceps, and additional instruments entering through the natural orifices like the anus, nostril, mouth, ear canal, or a small incision. However, those instruments only provide inadequate dexterity, due to which some regions stay inaccessible or hardly reachable. In order to perform the treatment of disease in those regions, conventional surgery (open surgery) needs to be performed. Continuum robots and manipulators have the capability to provide the missing dexterity and required stiffness. This paper presents a continuum robot design for surgical applications based on pre-assumed robot parameters, and kinematic assumptions are discussed below.
2.1 Model and Design Specification Current medical endoscopes are fairly large-sized with large diameters, and also they are of passive-type. So it is difficult to insert into a narrow space like a sinus cavity. It requires a flexible endoscope with a smaller diameter with some required dexterity
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Fig. 1 Solid model of the continuum robot segment
in order to perform the surgical procedure. Thus, this paper proposes a prototype of a continuum robot with a spring backbone. The proposed continuum robot has threesegments; each segment has 50 mm length and 8 mm diameter of spacer discs. The presented continuum segment is four tendon-driven and has two degrees of freedom. Where tendons 1, 3, and 2, 4 have relative actuation. The CAD model of the designed continuum robot is shown in Fig. 1. The special features of the design are flexibility with required stiffness and better bending capability of the whole body by using a compliant spring backbone.
2.2 Kinematic Assumptions The constant curvature bending assumption has been taken to derive the kinematics for the continuum robot segment. Spring compression has not been considered until now. By discretizing the curvature bending movement in 3D space with four independent motions in order to find the modified Denavit–Hartenberg (D-H) parameters [7]. A schematic diagram of the continuum segment with local coordinate frames is shown in Fig. 2. Based on the above-modified DH parameters, a conventional homogenous transformation matrix approach can be utilized to derive the forward kinematics.
3 Continuum Robot Kinematics Continuum robot kinematics has been sub-categorized into two stages kinematic mapping: robot-specific mapping (first stage) and robot-independent mapping (second stage).
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Fig. 2 Frame configuration of the continuum robot represents modified D-H parameters
3.1 Forward Kinematics Forward kinematics is the kinematic relationship to calculate the end tip position and orientation for a given set of the tendon (actuator) lengths as input. Forward kinematics mapping is subdivided into two-stage kinematic mapping [4]. First stage kinematic mapping is between joint space parameters to the configuration space parameters called as the robot-specific kinematic mapping. Second stage kinematic mapping is defined between configuration space parameters to task space (3D workspace) parameters called robot-independent kinematic mapping. Robot-Specific Forward Kinematics. Continuum robots are basically tendon or cable-driven and pneumatically actuated. The robot-specific forward kinematics leads to calculating the configuration parameters of a continuum robot segment when corresponds to a single segment robot when tendon lengths are given as the input parameters. Hence these are robot-specific. The continuum robot model is tendon-driven. Any variation in the tendon actuators length i.e. l1 , l2 , l3 , and, l4 will yield the change in endpoint position and orientation of the segment. For any joint actuation, it deforms in a circular curvature with ‘S’ curvature length, bending plane angle ∅(from the +x axis), bending angle of curvature θ and a radius of curvature r . q = [l1 , l2 , l3 , l4 ]T
(1)
ri = r − d cos ∅i , where i = 1, 2, 3 and 4
(2)
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where d is the distance from the center of a section of the robot to the center of the tendon actuator (which is the same for all tendon actuators) and ∅i specifies the angle between the robot’s bending direction and the location of the i th actuator as shown in Fig. 3. li = S − θ d cos ∅i , where i = 1, 2, 3 and 4
(3)
Assumption: The spacer discs are supporting the backbone of the continuum robot, with minimal disc thickness. Hence the length of each robot segment is represented as: S=
l1 + l2 + l3 + l4 ; 4
Lc =
l1 + l2 + l3 + l4 4n
(4)
where ‘n’ is the no of sub-segments, ‘L c ’ is the length of subsegment (distance between two consecutive spacer discs along the curvature). The kinematics for the four tendon driven continuum robot segment’s mechanism has been derived as: ∅i =
(i − 1) π − ∅, where i = 1, 2, 3 and 4 2
(5)
For continuum robot with four tendons, ∅1 = ∅, ∅2 = π2 − ∅, ∅3 = π − ∅, and ∅4 = 3π − ∅. These steps are following in the case of four-tendon-driven robot. 2 Fig. 3 Top view of the base of the four tendons drove continuum robot segment
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Table 1 Modified D-H parameters for single-segment continuum robot [4] Link
θ
d
a
α
1
∅
0
0
−π/2
2
θ/2
0
0
π/2
3
0
d =S×
0
−π/2
4
θ/2
0
0
π/2
sin(θ/2) θ/2
l4 − l2 ∅(q) = tan l3 − l1 (l1 − 3l2 + l3 + l4 ) (l4 − l2 )2 + (l3 − l1 )2 θ (q) = S d(l1 + l2 + l3 + l4 )(l4 − l2 ) −1
(6)
(7)
Robot-independent kinematic mapping. Robot-independent kinematic mapping does not depend on robot architecture. It describes the direct relationship between the continuum robot segments (3D curvature) configuration parameters and the task space parameters. Robot independent kinematic mapping is derived by utilizing the modified D-H parameters, which are the function of the configuration parameters. Modified D-H parameters are shown in Table 1. By using the homogeneous transformation matrix approach, the total transformation has been calculated, which gives the position and the orientational (task space parameters) information for each segment. The homogenous transformation matrix is represented as; k Tk−1
=
k k Pk−1 Rk−1 0 1
⎡
k Tk−1
cos ∅k cos θk − sin ∅k cos ∅k sin θk ⎢ sin ∅ cos θ cos ∅ sin ∅ sin θ ⎢ k k k k k =⎢ ⎣ − sin θk 0 cos θk 0 0 0
(8) Sk cos ∅k (1−cos θk ) θk Sk sin ∅k (1−cos θk ) θk Sk sin θk θk
⎤ ⎥ ⎥ ⎥ ⎦
(9)
1
where T kk−1 represents the transformation consists Rkk−1 and P kk−1 as the orientation matrix and position vector of end tip of the k th continuum segment, respectively (see Eq. 8). whereas ∅k , θk and Sk are the configuration parameters of the k th segment of the continuum robot. It has been further extended for three segment continuum robot by successive multiplication of the transformation matrices corresponding to each section for (k = 1, 2, 3). T03 = T01 .T12 .T23
(10)
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The total transformation matrix T03 contains the position vector and the orientation matrix of the end tip of the distal segment concerning the reference frame, of the three-segment continuum robot.
3.2 Inverse Kinematics Inverse kinematics (IK) of single-segment continuum robot involves two stages; the first stage derives the expression for configuration parameters S(q), ∅(q), and θ (q) from robot end position x, y, and z (task space variables). The relationship between configuration space parameters and task space parameters are generally robot independent as: ∅(q) = tan−1
y
θ (q) = 2 tan
−1
(11)
x x 2 + y2 z
(12)
Further, a second stage robot specific IK relationship has been derived from robot configuration space to the joint space variables q = [l1 , l2 , l3 , l4 ]T . As we have discussed above in Eqs. (1–5), for four tendons driven continuum robot segment. Inverse kinematics for a single segment can be further extended to find the solutions for each segment of the continuum robot if the distal segment’s end tip parameters (task space parameters) are provided. Inverse Jacobian based IK approach has higher computation cost and also have singularity problems. Therefore, Forward and Backward Recursive Inverse Kinematics (FABRIK) can be used to find the IK solution [12–14].
4 Workspace and Dexterity Analysis Most of the surgical procedures like single port surgeries, Natural Orifice Transluminal Endoscopic Surgery (NOTES), and surgeries in deep anatomical space require high dexterity across the surgical field. Continuum robot-based architecture is used in the surgical domain due to its flexibility and dexterity. The workspace and dexterity are the most important characteristics in order to design the robot for specific surgical procedures.
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Table 2 Simulation criteria followed for four tendons-driven continuum robot workspace plot No. of segments
Segment length S (mm)
θ range (for each segment)
∅ range (for each segment)
Steps
3
50, 50 and 50
[−π, π ]
[0, 0] (planar case)
40
Fig. 4 Planar workspace plot of a three-segment continuum robot in X-Z plane
4.1 Workspace Analysis Workspace analysis is the basic criteria to design the robot manipulators, which decides the robot’s maximum and minimum reach and joint actuation range to restrict the link collision. In the case of the surgical robot, surgical space is the workspace where the robot needs to be operated with required dexterity. Hence, it is the required workspace for the designed robot. “Monte Carlo method” is used to get the workspace with defined simulation criteria’s as shown in Table 2. Three-segment continuum robot can bend uniformly in all the direction. Hence, its workspace is symmetric about the z-axis. A planar workspace for three segment continuum robot has been plotted in MATLAB software with simulation parameters (see Table 2) shown in Fig. 4.
4.2 Dexterity Analysis Robot dexterity can be measured by approaches, like condition number of the Jacobian matrix, multiplication of the singular value of Jacobian matrix [15], etc. In contrast, a more effective way is to use kinematic flexibility to define dexterity [16]. Dexterity is basically the measure of the possible number of configurations
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with which the robot end effector (tip of the distal segment) can reach a specific position within the workspace or, in other words, the number of possible solutions of inverse kinematics [11]. Absolute and relative kinematic flexibility is measured in order to find the dexterity for a specific position and workspace, respectively [11]. Hence in this paper, we analyze the workspace dexterity distribution based on relative kinematic flexibility by utilizing the workspace positional data density approach. Workspace is a set of all unique robot configurations; however, many specific positions can also be achieved by many configurations that can be visualized in the presented work. Following criteria, as shown in Table 3, has been followed to compute the workspace for dexterity analysis. Figures 5 and 6 are representing the relative dexterity distribution across the workspace for three segments and two-segment continuum robots, respectively. Dexterity across the workspace is color mapped (see Figs. 5 and 6) on the basis of the relative data density (normalized), which signifies the kinematic flexibility across the workspace in color bounded region. Table 3 Simulation criteria followed for four tendons driven continuum robot No. of segments
Segment length S (mm)
θ range (rad) (for each segment)
∅ range (for each segment) (planar case)
Steps
3
50, 50 and 50
[−π, π ]
[0, 0]
60
2
75 and 75
[−π, π ]
[0, 0]
465
Fig. 5 Dexterity distribution for three-segment continuum robot
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Fig. 6 Dexterity distribution for two-segment continuum robot
4.3 Result and Discussion Comparative analysis results in the three-segment continuum robot have higher workspace reachability around the inner region of the robot geometry as compared to the two-segment continuum robot. The uncolored middle region in Fig. 6 shows the unreachable space by two-segment continuum robot. The two-segment robot has higher dexterity in the outer region of workspace, but in case of three-segment has maximum dexterity nearer to the base region which may not be desirable during the surgical operation, but it can be enhanced by modifying the segment lengths. Long length segment causes lose in stiffness; hence, it has to be considered during robot design by optimal design.
5 Conclusion This paper has presented a comparative analysis of the workspace and its dexterity distribution for two-segment and three-segment continuum robots, respectively. Here we have found that the two-segment and three-segment continuum robot can be used in regions where dexterity is required. Still, shape optimization will be required to get higher dexterity in the minimum workspace during the robotic surgical operation in confined surgical spaces. A high dexterous region (high data density region) will be suitable for dual-arm motion coordination and operations, which requires high steerability like contact therapy (biopsy, suturing, etc.).
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References 1. Beasley RA (2012) Medical robots: current systems and research directions. J Robot 2. Hashizume M et al (2008) New real-time MR image-guided surgical robotic system for minimally invasive precision surgery. Int J Comput Assist Radiol Surg 2(6):317–325 3. Song HS, Chung JH, Kim KY, Lee JJ (2006) The development of human-arm like manipulator for laparoscopic surgery with force Sensing. In: Proceedings of the IEEE international conference on industrial technology 4. Webster RJ, Jones BA (2010) Design and kinematic modeling of constant curvature continuum robots: a review. Int J Rob Res 29(13):1661–1683 5. Hannan MW, Walker ID (2003) Kinematics and the implementation of an elephant’s trunk manipulator and other continuum style robots. J Robot Syst 20(2):45–63 6. Hannan MW, Walker ID (2011) Novel kinematics for continuum robots. Adv Robot Kinemat 227–238 7. Jones BA, Walker ID (2006) Kinematics for multisection continuum robots. IEEE Trans Robot 8. Garriga-Casanovas A, Rodriguez F, Baena (2018) Kinematics of continuum robots with constant curvature bending and extension capabilities. J Mech Robot 11(1):1–12 9. Wu L, Crawford R, Roberts J (2017) Dexterity analysis of three 6-DOF continuum robots combining concentric tube mechanisms and cable-driven mechanisms. IEEE Robot Autom Lett 2(2):514–521 10. Li Z, Wu L, Ren H, Yu H (2017) Kinematic comparison of surgical tendon-driven manipulators and concentric tube manipulators. Mech Mach Theory 107(September 2016):148–165 11. Li Z, Ren H, Chiu PWY, Du R, Yu H (2016) A novel constrained wire-driven flexible mechanism and its kinematic analysis. Mech Mach Theory 95:59–75 12. Aristidou A, Lasenby J (2011) FABRIK: a fast, iterative solver for the inverse kinematics problem. Graph Models 73(5):243–260 13. Zhang W, Yang Z, Dong T, Xu K (2018) FABRIKc: an efficient iterative inverse kinematics solver for continuum robots. In: IEEE/ASME international conference on advanced intelligent mechatronics, AIM, vol 2018-July, pp 346–352 14. Kolpashchikov D, Laptev N, Danilov V, Skirnevskiy I, Manakov R, Gerget O (2018) FABRIKbased inverse kinematics for multi-section continuum robots. In: Proceedings of the 2018 18th international conference on mechatronics 15. Yoshikawa T (1985) Manipulability of robotic mechanisms. Int J Rob Res 4(2):3–9 16. Lenarcic J, Bajd T (2013) Robot mechanisms. Springer, New York
Automatic Seed Cum Fertilizer Sowing Machine with Water Dripping on Seeds T. Tirupal and D. Rajasekhar
Abstract Agriculture has reliably been the establishment of India’s bolstered improvement. As the quantity of occupants in India continues building up, the interest for things grows too. Subsequently, there is a more noteworthy requirement for multiple cropping in the farms and this requires adequate and efficient machines. The wheel period of the rainstorm is antagonistically influencing the nation’s precipitation quantity between long stretches of June to July. The late arrival of the monsoon affects the yielding capability of the crop. In this regard, farmers should be trained to cultivate in the right season using modern technologies with the minimum usage of water. This paper discusses an automatic seed cum fertilizer sowing machine with water dripping on the seeds at the time of sowing. This proposed mechanism will help the farmers to cultivate the land even if the arrival of the monsoon is late. Keywords Agriculture · Monsoon · Fertilizer
1 Introduction The wheel period of the storm is antagonistically influencing the nation’s precipitation portion between June to July. Poor or no precipitation has furthermore influenced the nation’s precipitation share during the long stretch of June. In India, late monsoon is observed in certain parts of the country like; the west parts of Punjab, Haryana, Delhi, Rajasthan, Gujarat, Uttar Pradesh, south parts of Karnataka, Tamil Nadu, Andhra Pradesh, parts of Telangana, Madhya Pradesh, Chhattisgarh and the leeward side of Maharashtra. The late arrival of monsoon affects the yielding capability of the crop. In this regard, farmers should be trained to cultivate in the right season time using modern technologies and with the minimum usage of water. Groundnut, cotton, maize, jowar, bajra, sesamum, and dal are the major rainfed crops in the Rayalaseema region including Anantapur, Chittoor and parts of Kadapa and Kurnool districts, while the dry land legume is cultivated in large areas and in other areas also
T. Tirupal (B) · D. Rajasekhar Department of ECE, GPCET, Kurnool, Andhra Pradesh 518452, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_97
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in the country. Farmers often suffer a heavy loss due to crop failure because of inadequate or no rains while lack of the best practices in cultivation is also considered to be one of the reasons that are depriving farmers getting good revenue from the commercial crop. In the developing technique, much of the time used customary seeding movement takes extra time and more work. The seed feed rate is even more anyway the time required for the complete expense is more and the total cost is extended in view of the hiring of equipment, labour. The traditional seed planting machine is less proficient and tedious. The present period is walking towards the quick development of all areas including the agriculture sector. To fulfil the future nourishment needs, farmers need to actualize the new procedures which would not influence the soil surface yet will increase the total crop production. Agriculture in India has a noteworthy history. Today, India positions second worldwide in farm yield. Agriculture is demographically the broadest budgetary portion and plays a basic occupation in the general economy of India. For the development of Indian economy, automation is important. The primary reason for automation in agribusiness is to improve the general efficiency and production. Planting is ordinarily done physically, which includes both invigorate (people and draught creatures), and this results in a greater expense of development and deferral in planting. This paper talks about a programmed seed cum manure planting machine with water dribbling on seeds at the season of planting. This will assist the farmers with cultivating the land regardless of whether the entry of the storm is late. The essential goal of this proposed technique is to put the seed, required amount of water, compost in lines at an ideal depth and spread the seeds with soil and give suitable compaction over the seed. The endorsed amount of water, line to line dispersing, seed rate, seed to seed spacing and profundity of seed position change from harvest to harvest and for various agro-climatic conditions to accomplish optimum yields.
2 Literature Review Agriculture has consistently been the backbone of India’s sustained growth. As the quantity of occupants in India continues building up, the interest for produce grows as well. Accordingly, there is a progressively significant necessity for different crops in the farms and this hence requires profitable and productive machines. Roshan et al. [1] gave a technique managing different planting strategies utilized in India for seed planting and manure situation. The author likewise looked at the customary planting strategy and the new proposed machine as far as execution tasks and preferences. Amol et al. [2] exhibited a technique to contrast customary planting strategies and new proposed machines. The proposed machine gives detail depiction about line to line separation, the seed rate, seed to seed space and composts situation which shifts from harvest to harvest. This machine lessens the planting time, human endeavours
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and labour cost. Ramesh et al. [3] exhibited a concise survey of data about the different kinds of advancements done in seed planting equipment. As day by day, the labour accessibility turns into the incredible worry for the farmers and labour cost is expanding, this machine decreases the endeavours and total expense of planting the seeds and compost arrangement. Pranil et al. [4] proposed a technique which manages the different planting strategies utilized in India for seed planting and manure position. Robotized sunlight-based controlled seed planting machine has been proposed by Sateesh et al. [5]. Kunal et al. [6] proposed a technique where he interchanged the confounded gear framework by the Hall Effect sensor for simple seed planting and furthermore diminish the need of labour. The Hall Effect sensor changes over the pivot into separation for which seed planting at a specific distance. Additionally, there is customizable framework for planting at various distances. By utilizing this machine, the planting should be possible line by line and separation is kept up. So this machine lessens the endeavours and diminishes the expense of the seed planting process with incredible efficiency and precision with reduce in the labour prerequisite. Thorat et al. [7] proposed a machine that can plant different types and different sizes of seeds and it can also automatically vary the space between two seeds while planting. This increased planting efficiency and accuracy. Nageswara Rao et al. [8] made a design and fabrication of agriculture robot (AGROBOT). It consists of fire bird V robot and seed sowing mechanism. The seed sowing mechanism is operated by the rotor with four blades and rotor is driven with the help of motor. Krunal et al. [9] reviewed about seed sowing the robot and explained the advantages and applications of robots in agriculture. Thorat et al. [7] proposed a machine that can plant various sorts and various sizes of seeds and it can likewise consequently fluctuate the space between two seeds while planting. This expanded planting proficiency and precision. Nageswara Rao et al. [8] made a structure and fabrication of agriculture robot (AGROBOT). It comprises of fire bird V robot and seed planting component. The seed planting component is operated by rotor with four sharp edges, and the rotor is driven with the assistance of an engine. Krunal et al. [9] investigated about seed planting robot and clarified the points of interest and uses of robots in agribusiness. The existing methods say that there is an automatic seed sowing machine for all the variety of seeds, but there is no mechanism for water dripping on the seeds. Our research paper discusses an automatic seed cum fertilizer sowing machine with water dripping on seeds at the time of sowing.
3 Proposed Method The proposed strategy talks about a programmed seed cum manure planting machine with water dribbling on seeds at the season of planting. This will push the farmers to cultivate the land regardless of whether entry of the rainstorm is late. The essential goal of this proposed technique is to put the seed, the required amount of water,
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and manure in lines at wanted profundity and spread the seeds with soil and give appropriate compaction over the seed. The suggested amount of water, line to line space, seed rate, seed to seed dispersing and profundity of the seed position shift from harvest to harvest and for various agro-climatic conditions to accomplish ideal yields. This programmed seed cum compost planting machine with water trickling on seeds basically works on ‘vertical-discontinuous work principle’ which alludes to the vertical development which can be trailed by an individual body in an agricultural field and actualizes its broken activity in connection to the flat profession. According to name, this machine is utilized for planting seed, compost and dribbling water on the seed. The schematic diagram of the proposed technique is shown in Fig. 1 and the proposed structure with water tank is shown in Fig. 2. The proposed machine includes: • • • • •
Drill the ground Sow seed inside the drilled hole Put fertilizer in the hole Drip the water on the seed Cover the gap with the assistance of adjuster.
Firstly, a hole is drilled with the assistance of a 4 inch land boring apparatus having shaft distance across of 7 mm and diameter of edges—25 mm with a depth of cut of— 76.2 mm. This is kept running with the assistance of an engine of 300 rpm and 12 kgcm torque. This is associated with 12 V and 7 A DC battery. This is straightforwardly associated with the solar panel-based board through which it gets charged. This
Solar Panel
Microcontroller Unit
Digging Mechanism
Charging Controller
Seed Sowing Mechanism DC MOTOR Battery
Fertilizer Inserting Mechanism Water Dripping Mechanism
Fig. 1 Schematic diagram of the proposed method
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Fig. 2 Structure of the proposed machine
engine is constrained by an 8 bit microcontroller with the assistance of which it can begin and stop and we can likewise control the clockwise and anticlockwise movement of an engine. For dropping seed, manure and water, we are utilizing a container which is mounted behind engine and a switch plan is given on the handle when this switch is squeezed seed, compost and water will be dropped consequently from container travel into a pipe independently joined to it and drop in the opening. A movable iron plate is fitted in an uncommon side of the machine which will gather soil and spread the land which is bored. Along these lines, seed planting is done consequently by embedding manure and dribbling water with this machine. For example, groundnut requires the following parameters: Seed rate = 100 kg/h. Row to row spacing = 20 cm. Water used per seed = 1/4 L.
4 Conclusion The paper discusses an automatic seed cum fertilizer sowing machine with water dripping on seeds at the time of sowing. This will help the farmers to cultivate the land even if the arrival of the monsoon is late. The essential target of this proposed strategy is to put the seed, the required amount of water, compost in rows at wanted profundity and spread the seeds with soil and give legitimate compaction over the
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seed. The prescribed amount of water, line to line separation, seed rate, seed to seed spacing and profundity of the seed situation shift from harvest to harvest and for various agro-climatic conditions to accomplish ideal yields. The present proposed method project can be used for multiple applications in agriculture like seed sowing, inserting fertilizer and dripping water on to the seed at the time of sowing. This type of mechanism helps the farmers in some parts of our country to grow the crop with in time even if the late monsoon is observed. This type of system makes the crop to yield to a maximum extent. This type of mechanism is also an extension to the existing system where an existing system contains automatic seed and fertilizer sowing except the dripping water mechanism. If we include this water dripping mechanism, then we can try to help the cultivators to a maximum extent in and around our country.
References 1. Roshan VM, Gajanan PT, Swapnil KA (2013) Design and implementation of multi seed sowing machine. Int J Mech Eng Robot Res 2(4):422–429 2. Amol BR, Pavan DS, Sumit BP, Keshav KS (2014) A review on multi-seed sowing machine. Int J Mech Eng Technol 5(2):180–186 3. Ramesh D, Girishkumar HP (2014) Agriculture seed sowing equipments: a review. Int J Sci Eng Technol Res 3(7):1987–1992 4. Pranil VS, Amit W, Ashish S, Bhushan P, Rakesh B, Saurabh K (2015) Solar powered seed sowing machine. Glob J Adv Res 2(4):712–717 5. Sateesh M, Mahantesh T, Sujata H (2016) Automated solar powered seed sowing machine. J Adv Sci Technol 12(25):478–485 6. Kunal AD, Omkar RS, Megha SB, Achal AJ, Priyanka SC (2017) Design and development of automatic operated seeds sowing machine. Int J Recent Innov Trends Comput Commun 5(2):277–279 7. Thorat SV, Madhu LK, Patil GV, Patil R (2017) Design and fabrication of seed sowing machine. Int Res J Eng Technol 4(9):704–707 8. Nageswara Rao A, Pichi Reddy S, Raju N (2018) Design and development of seed sowing AGROBOT. J Emerg Technol Innov Res 5(5):783–787 9. Krunal G, Shubham P, Mahima C, Hetal P, Bhushan P (2019) Seed sowing robot—a review. Int J Adv Eng Res Dev 6(2):79–81
Automatic Drip Irrigation Control System for Paddy Fields in Depleting Water Resource Areas K. Saravanakumar , M. Karthigai Pandian , T. Chinnadurai , and J. Dhanaselvam
Abstract Agriculture needs major portion of freshwater available in the world to produce food for increasing population. In this proposed method, soil moisture sensor is placed in constant distances in the whole paddy field to measure the moisture content in the ground and measured values are sent to the controller. Humidity sensor, environmental light intensity sensor are the other sensors used in the field to measure all conditions in the paddy field to regulate the water flow rate in the irrigation system. The system also provides all data to the farmers using Internet of things to monitor the complete irrigation system in real time. 45–50% of water usage is saved because of the smart IOT-based irrigation system and the crop growth also increased because of controlled and regulated water usage. Keywords Moisture sensor · Humidity sensor · Light intensity sensor · Paddy field · Internet of things
1 Introduction Over decades, agriculture is one of the most primitive professions around the world. More than half of the Indian population resides on agriculture as their primary source of employment. Even in Indian economy, agriculture contributes a major part in calculating the GDP [1]. Water is the main substance in the world which is responsible for the health and wealth of all living things. Almost 80% of water present in earth need to be used for farming. The need for water increases with rapid growth of population. Overwatering and underwatering both the terms lead to ineffective farming [2]. Efficient usage of water for agriculture is very much important at present because of constantly decreasing rain fall. Drip irrigation is a very successful method of water usage for agricultural needs. Water and nutrient use efficiencies were found to be maximum in drip irrigated paddy with 52% water savings than traditional floodirrigated paddy [3]. Paddy fields are very common in south India which consumes K. Saravanakumar (B) · M. Karthigai Pandian · T. Chinnadurai · J. Dhanaselvam Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_98
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different amount of water at various stages of cultivation. Present world is in need of great changes and developments in agriculture in terms of using water resources. Southern India is mainly depending on monsoons than river water for agriculture. Agriculture is utilizing major portion of available freshwater resources [4]. Paddy fields in southern part of India need irrigation management system to control the water usage because paddy fields need variable amount of water in different stages of crop growth [4]. Continuous monitoring of water level and water availability in the field are difficult if we do it manually. Irrigation systems which are already available in the field are reducing the water wastage but still they are not effective water management systems for different crops. An automated irrigation management system is required at present to regulate water for crops like paddy and information about the irrigation also must be reported to farmers for monitoring using IOT technology. IOT can be expanded as Internet of things which may be defined as the connection between the networks of physical devices through the Internet [5]. It is found that the required water flow rate for traditional drip irrigation is 3321.75 l/h to cultivate an acre of paddy crop, whereas an automated drip irrigation system using neural networks utilizes only 2777.7 l/h [4]. It is evident that for the last 10 years, the groundwater level decreases tremendously and irregular climatic changes which make farmers to adopt automated drip irrigation system using neural networks for optimum utilization of available water resources. In this method, water and fertilizer in the form of water droplets are dripped directly to the root of the plants periodically. The design for water application varies according to the crop type. When compared to traditional method, it uses 30–50% less water [6]. It also gives the flexibility to farmers to interfere at any stage of this completely automated system and make manual control. Fertilizer feed mixing with water irrigation pipe is controlled by this system depending on the growth stages of the crops. The system is completely tested in 3 paddy fields to measure and control the water irrigation.
2 Proposed System In the proposed system, a completely automated irrigation management is tested for paddy field and results are analyzed using neural network backpropagation algorithm. Soil moisture sensor is placed in six different places, and environment humidity sensor and light intensity sensor are placed around the field in proper areas with equal distance. Water pump will be directly controlled depending on the moisture level in the soil. Humidity and light intensity also play a major role in maintaining water moisture level so that also taken into consideration to switch on or off the solenoid valve which are placed in the water flow path from tank to field. IOT-based system which is directly sending the data and current condition of the field to farmer’s mobile phone [7]. A manual control facility also provided to farmer to switch off the water pump from remote whenever it is necessary. The results are taken carefully into consideration. Moisture, humidity, light intensity levels, and water flow rate are fed
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Fig. 1 Proposed system experimental setup
into neural network backpropagation algorithm to analyze to find out the optimum irrigation conditions (Figs. 1 and 2). The system continuously monitors soil moisture using moisture sensor model FC28. Figure 3 shows the soil moisture sensor used in the automatic irrigation system. It takes input voltage of 3.3–5 V and generates output voltage of 0–4.2 V of both analog and digital signals. FC-28 gives the value from 0 to 1023 which can be measured in percentage from 0 to 100, respectively. Light intensity of the surrounding environment is measured using light-dependent resistor (LDR) module as shown in Fig. 4. The photo resistor of model 5528 is used for the qualitative detection of ambient light intensity. Maximum voltage (Vcc) of 5 V and maximum current of 6 mA. Humidity of the surrounding environment is measured using DHT11 humidity sensor. This sensor can measure relative humidity of 20–90% RH of accuracy ±5RH in the temperature range of 0–50 °C. Figure 5 shows the humidity sensor DHT11 used in this proposed irrigation system. Figure 6 shows the moisture sensors and solenoid valves placements in the paddy field. Humidity and light intensity sensors can be placed anywhere in the field,
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FARMER’S MOBILE PHONE
GSM MODULE
MANUAL CONTROL
MOTOR
MOISTURE SENSOR
LIGHT INTENSITY SENSOR
ARDUINO UNO ATmega328P MICROCONTROLLER
SOLENOID VALVE 1 SOLENOID VALVE 2
HUMIDITY SENSOR
SOLENOID VALVE 3 SOLENOID VALVE 4
Fig. 2 Block diagram of proposed system
Fig. 3 Soil moisture sensor
Fig. 4 Light intensity sensor
because the humidity and light intensity will not change much in a particular zone. But moisture content in a field which has to be analyzed throughout the field. Due to ups and downs in land area, soil of the paddy crop will not be equally moisturized. So moisture sensor should be placed in proper places after carefully surveying the field. Table 1 explains the scheduling of irrigation for different varieties of paddy. In our proposed automatic irrigation system, we are taking short and medium duration variety for water management analysis.
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Fig. 5 Humidity sensor
MOISTURE SENSOR 2
SOLENOID VALVES IN IRRIGATION LINE HUMIDITY AND LIGHT INTENSITY SENSOR
MOISTURE SENSOR 1
Fig. 6 Sensor and solenoid valve setup in paddy field
Table 2 indicates the moisture level variations to be maintained in the soil during the growth period of paddy.
3 Results and Discussion MATLAB neural networks toolbox (nntool) is used as the simulation tool in this paper. Sample data are collected from a normalized field without any pre-installation of higher end logics or neural networks. We are tabulating the various levels of
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Table 1 Scheduling of irrigation for paddy Short duration variety
Medium duration variety
Long duration variety
Days
No. of Water irrigation level (cm)
Days
No. of Water irrigation level (cm)
Days
No. of Water irrigation level (cm)
1–25
5–7
2–3
1–30
5–7
2–3
1–35
6–8
2–3
25
–
Thin film 30 of water
–
Thin film 35 of water
–
Thin film of water
28
–
Life 33 irrigation
–
Life 38 irrigation
–
Life irrigation
29–50
6
2–5
34–65
6–8
2–5
39–90 or 95
12–15
2–5
51–70
5–6
2–5
66–95
8–10
2–5
96–125
7–9
2–5
71–105
5–6
2–5
96–125
6–8
2–5
126–150 5–6
2–5
Table 2 Depth of submergence chart for low and medium duration paddy variety
Stages of crop growth
Depth of submergence (cm)
At transplanting
2
After transplanting for 3 days
5
Three days after transplanting upto maximum tillering
2
At maximum tillering (in fertile Drain water for three days fields only) Maximum tillering to panicle initiation
2
Panicle initiation to 21 days after flowering
5
Twenty one days after flowering Withhold irrigation
moisture, light intensity, and relative humidity. Corresponding flow rate changes also measured in terms of liters per hour (Figs. 7 and 8; Table 3). By this proposed method, total amount of water required for the paddy crop is limited to 600 mm.
Automatic Drip Irrigation Control System for Paddy Fields … Validation: R=0.55763
Output ~= 0.33*Target + 1.6e+03
Output ~= 0.36*Target + 1.3e+03
Training: R=0.68041 8000
Data Fit Y=T
7000 6000 5000 4000 3000 2000 1000
4000
2000
6000
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8000
Data Fit Y=T
7000 6000 5000 4000 3000 2000 1000
8000
4000
2000
Target
All: R=0.65543
Output ~= 0.38*Target + 1.3e+03
Output ~= 0.68*Target + 1e+03
Test: R=0.68681 8000
Data Fit Y=T
7000 6000 5000 4000 3000 2000 1000
4000
2000
8000
6000
Target
6000
8000
Data Fit Y=T
7000 6000 5000 4000 3000 2000 1000
8000
4000
2000
8000
6000
Target
Target
Fig. 7 Regression analysis using ANN 7
Best Validation Performance is 4620223.0259 at epoch 0 10 gradient
Train Validation Test Best
6
10
5
10
Mu = 10000, at epoch 6
5
mu
10 6
10
0
10
-5
10
Validation Checks = 6, at epoch 6 10 val fail
Mean Squared Error (mse)
Gradient = 397761.9789, at epoch 6
7
10
0
5
10
0
5
1
2
3
4
5
6 Epochs
Fig. 8 Validation performance and training state
6
0
1
2
3 6 Epochs
4
5
6
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Table 3 Stages of cultivation and various sensor data and flow rate variations Stages of cultivation
Moisture %
Light intensity (Lux)
Relative humidity (%)
Average flow rate (l/h)
Average irrigation depth (mm)
Field preparation
90.7
613
66
8258
198
Planting
87.85
652.75
64.95
6169
297
Flowering
88.75
653.175
64.925
1008
97
Maturity
89.1
639.6417
65.29167
190
10
References 1. Subathra MSP, Blessing CJ, Thomas George S, Thomas A, Dhibak Raj A, Ewards V (2016) Automated intelligent wireless drip irrigation using ANN techniques. IEEE, pp 1–14 2. Dela Cruz JR, Baldovino RG, Bandala AA, Dadios EP (2017) Water usage optimization of smart farm automated irrigation system using artificial neural networks. In: Fifth International conference on information and communication technology (ICoICT), pp 1–6 3. Bhardwaj AK, Pandiaraj T, Chaturvedi S, Singh TC, Soman P, Bhardwaj RK, Labh B (2018) Growth, production potential and inputs use efficiency of rice under different planting methods in drip irrigation. Curr J Appl Sci Technol 1–9 4. Barkunan SR, Bhanumathi V, Sethuram J (2019) Smart sensor for automatic drip irrigation system for paddy cultivation. Comput Electr Eng 73:180–193 5. Ananthi N, Divya J, Divya M, Janani V (2017) IoT based smart soil monitoring system for agricultural production. In: International conference on technological innovations in ICT for agricultural and rural development (TIAR2017). IEEE, pp 1–6 6. Rajalakshmi P, Mahalakshmi SD (2016) IOT based crop-field monitoring and irrigation automation. In: 2016 10th international conference on intelligent systems and control (ISCO). IEEE, pp 1–6 7. Kang MS, Kim SM, Park SW, Lee JJ, Yoo KH (2007) Assessment of reclaimed wastewater irrigation impacts on water quality, soil, and rice cultivation in paddy fields. J Environ Sci Health Part A 42(4):439–445 8. Gao Z, Ni J, Zhu Y, Jiang Q, Cao W (2018) Water-efficient sensing method for soil profiling in the paddy field. Int J Agric Biol Eng 11(4):207–216 9. Bhardwaj AK, Pandiaraj T, Soman P, Bhardwaj RK, Singh TC (2018) Drip irrigation scheduling for higher growth, productivity and input use efficiency of direct seeded basmati rice in IndoGangetic plains for climate resilient. Int J Environ Clim Change 332–340
Investigating the Ambient Thermal Loading Failure of Lead–Acid Battery Based on Thermal Analysis T. Chinnadurai, B. Banuselvasaraswathy , M. Karthigai Pandian , S. Saravanan , and K. Saravanakumar
Abstract The external (surrounding) temperature variation majorly influences the battery lifetime and performance. The temperature variations lead to failure of individual cells as well as performance of the battery. Lead–acid 12 V/ 7.2 Ah battery is used for the analysis. For heating purpose, two heating coils are fitted inside the wooden chamber. Three thermocouples are connected with DAQ card to measure the temperature. For every 10 full cycle periods, the temperature is increased by 10 °C up to 60 °C. At this stage, the battery starts to explode due to high chemical reactions, fast discharging, and charging cycles. The anode and cathode plates were removed from the battery and analyzed with the help of differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). From the DSC results, the change of the heat capacity of the anode and cathode plates is observed. Likewise, TGA result shows the degradation of both anode and cathode plates. Keywords Failure analysis · Lead–acid battery · DSC · TGA
1 Introduction In all aspect of electrical components, electrical energy is the main source of power for continuous operation. In case of power failure, the secondary power source is necessary to ensure that the component operation remains successful. Thus batteries are fulfilling the necessary demands. However, the performance of the battery is varied in different environmental and design factors. But the major influence on the battery depends on temperature. So, the analysis of temperature influence is very essential to improve the performance and lifetime of battery. The battery performance reduction depends on the plate performance as the number of cycling process increases. The T. Chinnadurai (B) · M. Karthigai Pandian · K. Saravanakumar Department of ICE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu 641042, India B. Banuselvasaraswathy Department of ECE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu 641042, India S. Saravanan Department of EEE, Sri Krishna College of Technology, Coimbatore, Tamil Nadu 641042, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_99
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main cause for the fast capacity decay is due to shedding and softening in positive active material, resultant of which the positive electrode’s cycling performance was significantly compromised. The author had performed the experiment to improve the performance of positive electrode with the help of additives (Sb2 O3 and SnSO4 ). As a result, the performance of plate was improved at 40 and 60% discharge [1]. Spanos et al. [2] analyzed the performance of plates at reverse charging. From the obtained result, it was inferred that reverse charging results in growth of discharge capacity on negative electrodes, whereas Pb–Ca–Sn positive capacity tends to decrease. Rand [3] mainly focused on anode and cathode plates of lead–acid battery and worked on different methods to lengthen the life time of battery. In addition, anode and cathode plate characteristics are also determined on rate of change of temperature. Catherino [4, 5] worked on thermal runaway effect on lead–acid battery. In brief, the observed effects are found to be similar to the electrolyte distribution in the separator. Thus, modifying the properties of AGM separator can yield a better method for controlling thermal runaway and decreases the failure mode. From all these analysis, it was inferred that battery efficiency and plate characteristics gets easily affected due to the thermal effect. This made the researchers to consider the effects of temperature influence. Thus, few researchers analyzed the thermal influence (DSC and TGA) on lead–acid battery materials. From this, it was noticed that the temperature rise will minimize the plate material, thereby deteriorates the total lifetime of the battery [6, 7]. In this study, the differential scanning calorimetry (DSC) method was used to measure the differences in energy inputs of a sample as a function of temperature. While the sample and reference materials were exposed to controlled temperature [7], the change of energy inputs was observed between the materials. Additionally, thermogravimetric analysis (TGA) was used to measure the mass of substances in a sample as a function of temperature variations [7].
2 Experimentation Lead–acid 12 V/7.2 Ah battery was utilized for this analysis. For heating purpose, two Ni–Cr heating coils were used inside the wooden chamber. The chamber was fully closed and equipped with fan to spread the generated heat uniformly in all four directions of the battery. To measure the temperature, three K-type thermocouples were used. It was connected with DAQ card and processed using LabView software as shown in Fig. 1. In this condition, the battery was charging and discharging for 10 full cycles and the temperature was increased by 10 °C till it reaches 60 °C. The battery gets heated due to internal resistance and surrounding environmental temperature destroy the battery completely and at the same time discharging the battery capacity with less time. Finally, the battery cover was removed, and both anode and cathode plates were separated and subjected to thermal analysis as shown in Fig. 2. Thermogravimetry analysis machine NETZCH (STA 449 F3 Jupiter) is available at PSG iTech Coimbatore. This equipment is capable of measuring the
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Fig. 1 Schematic of experimental setup
Fig. 2 Electrode plates after removing from the battery
mass changes and thermal effect between room temperature (27 °C) to 1550 °C. The equipment can able to measure the materials including inhomogeneous substances with good flexibility.
3 Result and Discussion For the analysis of anode and cathode plates with respect to temperature, it provides a deep knowledge about plate heat resistance and stability. Also, the battery behavior
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on high-temperature profile can be easily understood. This differential scanning calorimetry analysis is used to examine the reaction energy, latent heat, and glass transition temperature. Likewise, thermogravimetry analysis is used to measure the degradation behavior of anode and cathode materials with respect to thermal loading. From the obtained TGA results, it is observed that huge loss of cathode and anode plates was minimized as compared to previous conventional methods [3].
3.1 Differential Scanning Calorimetry Analysis From Fig. 3, the thermal loading for anode plate sample is increased slowly and corresponding glass transition temperature is measured. During the initial thermal loading, material exhibits exothermic nature up to 150 °C. Further increasing the temperature up to 375 °C, the material changes in steady state but once the temperature reach 400 °C again, material starts changing into endothermic peak. The complex peak is observed at 424 °C as an onset value. However, the DSC measurement takes only about 10 min. While charging and discharging, there may be chance for the material to react with atmospheric gases (CO2 ); as a result, the material gets turned into carbonated. When the percentage of the carbonates is increased, the battery performance is reduced for every 15%. The cathode material (Fig. 4) exhibits very sharp endothermic peak due to crystalline behavior. Like anode analysis, cathode plate also initially shows exothermic nature then turns into endothermic behavior. Beyond onset temperature, the material
Fig. 3 Thermal loading influence on anode plate at terminal point portion
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Fig. 4 Thermal loading influence on cathode plate at terminal point portion
continuously reduces the exothermic condition. It happens due to melting of material at this particular temperature. The peak values are obtained at 327 °C, and onset value is measured at 324 °C. The endothermic peaks at 250 and 260 °C correspond to hydrocarbonate dehydration, and at 370 °C, it is related to its decomposition. The curve for the carbonated cured paste features, 3BS peaks at 240 and 270 °C, two new peaks at 340 and 390 °C are analyzed from the degradation of the hydrocarbonate formed in the paste particularly under deep discharge rate from the onset temperature. The positive plate must be robust and sufficient enough to withstand the mechanical stresses which produced due to different molar volumes of the active material. From the obtained result, there exists a clear evidence which shows that the active material structure transforms as per charging/discharging cycles. Next DSC analysis is taken at bottom of the plate to identify the material behavior. From Figs. 5 and 6, it was observed that anode and cathode plate experiences less influence on temperature as compared with Figs. 3 and 4. This influence is because of temperature and chemical reactions. The less hydrogen content and uniform structure in failed battery are equated to the characteristics of chemically influenced material. This have attracted hydrogen-loss model to greater extent.
3.2 Thermogravimetric Analysis Result TGA analysis is used to identify the material mass loss with respect to temperature influence. The analysis is carried out in two different areas of both anode and cathode
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Fig. 5 Thermal loading influence on anode analysis, the test portion is taken at bottom of the plate
Fig. 6 Thermal loading influence on cathode, the test portion is taken at bottom of the plate
materials. This will help us to understand and identify the nature of degradation of material due to temperature loading. Figures 7 and 8 show the mass loss of anode and cathode plate at the terminal lead position. The losses are observed at 450 °C due to chemical corrosion and ion exchange, which leads to the material loss. Another phenomenon for material loss is growth of grids and corrosion which also degrades the life time of battery. The results depict that the rate of grid corrosion is influenced by
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Fig. 7 Degradation of anode material at near terminal point
Fig. 8 Degradation of cathode material at near terminal point
electrode potential, electrolyte composition and temperature, grid alloy composition and microstructure. The creep properties estimate the growth and elongation of grid members and growth during battery service. The results of the sample taken from bottom of the plate are illustrated in Figs. 9 and 10. The results exhibit the reduced mass loss compared to Figs. 7 and 8. The amount of reactions influences the material orientation which in turn influence the
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Fig. 9 Degradation of anode material at bottom of the plate
Fig. 10 Degradation of cathode material at bottom of the plate
material structural properties. The internal resistance of the material is also influenced by the temperature.
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4 Conclusion The best efficient thermal analyzing methods to find the basics of material nature are DSC and TGA. Thermal analytical methods give a new information that the different temperature influence is observed at single plate. This nature will lead to changes in material composition and physical properties in a single plate. TGA results clearly show the material degradation behavior. As like previous analysis, this also achieves different material loss value at single plate.
References 1. Yang S, Li R, Cai X, Xue K, Yang B, Hu X, Dai C (2018) Enhanced cycle performance and lifetime estimation of lead-acid batteries. New J Chem 42(11):8900–8904 2. Spanos C, Berlinger SA, Jayan A, West AC (2016) Inverse charging techniques for sulfation reversal in flooded lead-acid batteries. J Electrochem Soc 163(8):A1612–A1618 3. Rand DAJ (1996) Energy storage in lead-acid batteries: the Faraday way to sustainability. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 354(1712):1529–1544 4. Catherino HA (2006) Complexity in battery systems: Thermal runaway in VRLA batteries. J Power Sources 158(2):977–986 5. Catherino HA (2007) Secondary batteries: lead acid battery thermal runaway (No. TARDEC18509-RC). Tacom Research Development and Engineering Center, Warren, MI 6. Zhu H, Tan JJ, Xu ZL, Xu JS (2009) Experimental research on charging characteristics of a pressure-controlled VRLA battery in high-temperature environments. J Zhejiang Univ-Sci A 10(3):418–422 7. Matrakova M, Pavlov D (2006) Thermal analysis of lead-acid battery pastes and active materials. J Power Sources 158(2):1004–1011
Distance Operated Manipulator: A Case Study for Rose Plucking Utkarsha K. Mehta, Srushti R. Hippargi, Bhagyesh B. Deshmukh, and Roohshad Mistry
Abstract Few applications such as cutting shrubs and trees with prickles lead to damage to the hands while cutting it conventionally. The research has been undertaken for ease of rose plucking. A distance operated manipulator is designed and manufactured to overcome the issue of cutting roses by conventional way. The working of the tool is controlled by a lever operated mechanism. By a Bowden cable, the lever is connected to a brake caliper. The force applied by fingers at the lever is amplified at the point of application due to mechanical advantage. The endeffector consists of a gripper-cum-cutter. When the lever is operated, the stem of the rose is first gripped and then shearing of stem takes place. This unique feature of mechanism results into safe harvesting. Due to its simplest construction, it is a beneficial method. Trials have demonstrated the effective use of manipulator for rose plucking. Keywords Roses · Harvesting · Gripper · Cutter
1 Introduction Floriculture means flower cultivation is being practiced in India since times immemorial, and floriculture has blossomed into a viable business only in recent years. Considering the potential, this sector has helped in generating income and employment opportunities. Commercial cultivation of cut flowers such as rose, orchids, gladiolus, carnation, anthurium, gerbera and lilies has also become popular. Rose flower is one of the most important flowers commercially. Roses are dicot herbaceous often cultivated as a garden or potted annual flower. The required working hours to produce a rose are 3.770–5.240 h/ha, and cutting production accounts for 11.8–19.3% of this time. Therefore, the mechanization of cutting production has been strongly desired with rose industries. The acceptability of roses in trading particularly for exporting to European countries depends on adequate post-harvest longevity [1]. Harvesting of flowers is an important operation in floricultural practice. Economics as related U. K. Mehta (B) · S. R. Hippargi · B. B. Deshmukh · R. Mistry Walchand Institute of Technology, Solapur, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_100
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to timeliness, competition for labor and good market prices have pushed for mechanization of almost all agricultural operations. Mechanization of small farms is uneconomical as per Indian scenario. In small farms, harvesting of roses is carried out by conventional methods, such as traditional tool-scissors, pruner, lopper, etc., where labors work either with bare hands or wearing glows which may hamper human body them due to prickles on rose. To avoid damage, drudgery and to increase the output, a simple mechanism is needed with precise shearing action. This paper presents a simple mechanism for precise shearing and gripping action.
2 Literature Review Many conventional methods are available for plucking and pruning the roses. Considering the most traditional tool-scissor, this can pluck the branches by shearing. The pruner, more advanced over scissors, is having the ability of plucking with quite ease [2]. Again for distance actuation, the tools like lopper can be used which may have telescopic rod which will increase its reaching area [3]. Loppers are a type of scissors used for pruning twigs and small branches, like secateurs with very long handles. Figure 1 shows the harvesting of roses by lopper and scissor. Loppers are the largest type of manual garden cutting tool. This special kind of tool is operated by both the hands [4]. The tools mentioned above should not be used with bare hands, as pickles can harm the hands of worker, and hence it becomes necessary to wear gloves. As both the hands are engaged while operating the tools like loppers, etc., the problem may be faced for handling the plucked roses which may fallen off the ground. It will cause damage to the roses reducing value added to it. We have observed some issues of human factors (ergonomically discomfort) associated with all these conventional hand tools. The angle made between thumb and first finger
Fig. 1 Harvesting of roses by conventional method [5]
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while using conventional scissors is 30°, which develops stresses on hand. The manipulator described in this paper will be cost effective and able to resolve problems faced during plucking the roses. The simpler construction and cost-effectiveness make it affordable to small-scale farmers.
3 Novelty of the Device As the roses have prickles, it is quite harmful to cut the roses manually. Our design makes it easier and user friendly. The design uses links and a gripper with extended attachment of cutter. The device is small and portable.
4 General Specifications Design is capable of shearing and gripping simultaneously. The force exerted by fingers is around 30 N. Using a lever, this force is amplified to about 100 N. Flexible cable transfers the force and motion in order to ensure gripping and cutting of a rose. (Gripping takes place followed by cutting) A pipe is used to mount grippercum cutter at remote end of the pipe. Actuation force is provided by using a flexible cable (driven by a lever) to the caliper. Gripper and cutter assembly (end-effector) is installed on the inner jaws of caliper. Actuation of lever first grips the rose and then cuts the stem. Gripping and cutting action is obtained by single actuation. No damage to rose petals as no contact with petals.
5 Design Details and Methodology 5.1 Problem Identification Currently, in most small-scale farms, harvesting of roses is done by conventional methods which either damage the workers or flower itself, reducing value to it. Solution to this problem is mechanization, which is very costly.
5.2 Possible Solutions A robot with pneumatic actuator, sensors and picking mechanism can be used to automate the process of rose harvesting. But for small-scale rose harvesting, it is not feasible due to high installation cost of a robot. To provide an alternative for
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Fig. 2 CATIA model
small-scale rose harvesting, a feasible, economic device is proposed in this paper. Figure 2 shows the CATIA model of the device.
5.3 Design Considerations The basic criteria considered in the design of this device are included: • • • • •
Easy to use and handle Lightweight Height adjustable and flexible Cheap and simple design Fabrication with locally available materials.
5.4 Selection of Materials Brake lever: Carbon fiber (CFRP) or aluminum is commonly used for brake lever material. Aluminum is used where high strength and toughness are required which considerably contributes to increase in weight of brake lever. Thus, carbon fiber (CFRP) is used in this design due to its good rigidity, resistance to corrosion and high strength to weight ratio. Telescopic rod: Carbon fiber or aluminum alloy is used for telescopic rod due to its lightweight, intermediate modulus which often provides high resistance to bending. Grippers: Single-pivot side pull brake caliper is used. EPE foam sheet is attached to the brake shoes, which grips the rose stem.
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Blades: Rectangular blades of carbon steel 75Cr1 are used to shear rose stem as it allows high hardness and durability than the stainless steels. Bowden Cable: It is a flexible type of cable, which is used to transmit mechanical force. Handle grip: Foam, plastic, or matte rubber-based anti-skid grip.
5.5 Dimensions • • • • • •
Total weight of device: less than 1 kg Length of telescopic rod (extended): 800-1000 mm Gripping clearance: 8 mm Outer diameter of rod: 20 mm Handle/grip width: 25 mm Thickness: 1.5 mm.
5.6 Calculation of Forces The pull force at the point of application is calculated by the principle of moment. Figure 3 shows a schematic diagram of the device. P ×L = F ×b
(1)
where F = Force at point of application (fulcrum). b = Distance between wire holes to nut and bolts contact point (fulcrum). P = Hand force acting to the lever surface. L = Distance between wire holes to the lever surface. The induced bending stresses are calculated for both materials (aluminum and carbon fiber) using primary equations and stresses induced are within permissible limits. Induced bending stress (aluminum material): 28.51 N/mm2 . Induced bending stress (carbon fiber material): 26.85 N/mm2 .
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Fig. 3 Schematic diagram of the device
5.7 Anatomical Analysis While designing hand tools, it is essential to take into account the hand and finger force capabilities. Hand tools require diverse and sometimes extreme levels of force exertion depending on the movements involved; hence, in order to minimize discomfort and injuries of the upper extremities, finger force capabilities needs to be considered. Movements and exertions of the upper extremities, such as reaching, gripping and pinching, combined with repetition in a forceful and/or awkward manner are known contributing factors to the precipitation and aggravation of CTDs [6]. The range of motion (ROM) of the hand is the most commonly used functional measurement variable. Investigations concerning the strength capabilities of the hand began with explorations of grip strength. Average grip strength for males is stated as 504.2 N and that of females as 311.0 N [7]. However, strength measurements can vary depending on the instructions provided to the individual and the postures maintained during the force exertions. Positioning of the arm can significantly affect the maximum force exertions produced during grip. The Altezza level angle should be 100°-150° while operating the manipulator for comfortable holding. In general, a hand holding a manipulator could be abstracted as a ‘first’ type of lever as shown in Fig. 4a, and in this case, the fulcrum is located between the load and effort. The effort arm is usually kept more than the load arm to get mechanical advantage.
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Fig. 4 Type of Lever
If we extend our view to the elbow, the whole system could be abstracted as a ‘third’ type of lever as shown in Fig. 4b, the effort is located between the load and the fulcrum. In this case, the load arm is always greater than the effort arm and the mechanical advantage is less than 1. Longer arm and a heavier mass could induce fatigue on the biceps. As a result, using a heavy handle for better handling could further increase arm fatigue, so the weight of handle is kept as low as possible [8].
6 Conclusion A novel design of distance operated manipulator is carried out and manufactured. Since carbon fiber is 28% lighter than aluminum, it is selected for the design. Although both materials, carbon fiber and aluminum are suitable for design. It is tested that the device is ergonomically safe to operate. This manipulator is a modular design, and it can be used various applications such as on the basic level replacing manual scavenging. Due to simple construction and vast applicability, distance operated manipulator can be manufactured on a large scale.
References 1. Fard SHH et al (2008) Determining the shear strength and picking force of rose flower (Rosa hybrida L.). In: 10th international congress on mechanization and energy in agriculture, 14–17 Oct 2008, Antalya-Turkiye 2. https://en.wikipedia.org/wiki/Pruning_shears 3. https://growerstools.com/blog/everything-need-know-about-loppers 4. https://en.wikipedia.org/wiki/Loppers 5. https://www.google.com/search?q=rose+harvesting+images&oq=rose+harvesting+images& aqs=chrome..69i57.8760j0j7&sourceid=chrome&ie=UTF-8 6. Palanisami et al (1994) The effect of sitting on peak pinch strength. In: Aghazadeh F (ed) Advances in industrial ergonomics and safety VI. Taylor & Francis, New York, pp 587–594 7. Schmidt et al (1970) Grip strength as measured by the Jamar dynamometer. Arch Phys Med Rehabil 321–327 8. Bhandari VB (2010) Design of machine elements, 4th edn., pp 121–122
Topology Structure Design of Fish-Based Propulsive Mechanisms Gaikwad Pankaj Manik and Pankaj Dorlikar
Abstract The research presented here has its centre point on imitation of fishes and the kinematics of propulsive mechanisms to be used in efficient and versatile autonomous underwater vehicles (AUVs). Today’s AUVs are based on submarine. The paper started with the analysis of topological structural attributes of 4–11 link mechanisms with 1–6 DOFs. Using Hong Sen Yan’s creative design theory for mechanical devices, all the possible combinations of propulsive mechanism were synthesized to generate an atlas of 1149 new propulsive mechanisms subject to isomorphism, which will provide more inputs in the design and fabrication of the AUV models. Keywords Autonomous underwater vehicle · Mechanisms · Kinematic chains
1 Introduction Biomimetics is the study of shape of naturally existing biological living beings and their locomotion techniques employed especially for synthesizing similar things by the use of artificial mechanisms which can imitate actual mechanisms Sfakiotakis [1] and Salazar [2]. Fish along with other species like birds are ideal species to mimic. The different types of propulsion methods employed by fishes (see Fig. 1). The fish swimming propulsion types mentioned above is being used by varieties of fishes in their daily locomotion. Available fish-based swimming propulsion is classified into different types, such as Thunniform, Anguilliform, Carangiform, Ostraciform, Labiform, Amiiform, Diodontiform, Gymnotiform, Rajiform, Balistiform and Tetradontiform. These locomotion techniques are employed by fishes which depend upon their body shape and also in accordance with the habitat in which they
G. P. Manik (B) · P. Dorlikar Department of Mechanical Engineering, Army Institute of Technology, Pune 411015, India e-mail: [email protected] P. Dorlikar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_101
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Fig. 1 Taxonomy of propulsion methods employed by fishes
are found. Some of the species of fishes also make use of their appendages specifically fins (dorsal, anal, pectoral, caudal) alone and also in combination of two or more type of fins in order to achieve a near perfect propulsive mechanism which will help the fishes to swim swiftly in water.
2 Creative Design Procedure Based on the measures used in the design of existing mechanisms, Dr. Hong Sen Yan presented creative design procedures for the creation of all the feasible topological layouts of mechanisms by Yan [3]. Flowchart (see Fig. 2) shows the proposed followed for the creative design of mechanisms. The mains steps involved here can be inferred from the book of creative design methodology by Yan [3]. The creative design theory proposed by Dr. Hong Sen Yan has to be followed successively, viz. Generalization, Number Synthesis, Specialization, Particularization and finally Atlas of new propulsive mechanisms. The main steps that are there in creative design methodology are as follows: 1.
2.
Identifying existing designs with required design enumeration which the designers would like to have and achieve the topological features of the said propulsive mechanism designs so developed. Selecting existing designs arbitrarily and transforming it into equivalent kinematic chain.
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Fig. 2 Flowchart of Design Methodology [4]
3.
4.
5.
6.
Synthesizing or realizing the collection of generalized kinematic chains that have the same number of links and joints as the realized kinematic chains as obtained in Step 2, based on this procedure of number synthesis, simply select from the earlier chains. Assign types of links and joints to each generalized chain as obtained in step 3, to have the collection of realistic specialized kinematic chains so as to have the desired design prerequisites and constraints. Particularization or count of each possible specialized kinematic chain as obtained in step 4 into its equivalent symbolic format, to have the collection of mechanisms for the mechanical devices as per the needs. Looking for the existing available designs from the collection of existing designs, to have the database of new designs that are capable of realizing the actual fish motion.
3 Procedure Followed In this paper, the study started with exploring different types of propulsive mechanisms that are already there in different patents and research papers, which have
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Fig. 3 A propulsive mechanism designed using Epicyclic Gear Train by Jiangsu University of Science and Technology [5]
the ability to simulate the fin-based motion of swimming beings. Some of equivalent kinematic diagrams (see Figs. 3, 4, 5 and 6). After going by the existing mechanisms, the next step is to make equivalent mechanisms using different kinematics links and thereafter by number synthesis technique for the number of links, joints whether the said kinematic mechanism follows kutzbach–grubler criteria (mobility equation) for the DOF’s available in mechanism so as to find out the number of inputs required, when actually realizing the mechanisms. Fig. 4 A propulsive mechanisms designed using simple links by Dumas [6]
Fig. 5 A propulsive mechanisms designed using simple joints by Pablo Valdivia y Alvarado [7]
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Fig. 6 Bionic fish movement mechanism by Xiang [8]
4 Equations 1.
The basic equation used in the synthesis of mechanism is the mobility equation by Grubler as follows F = (3N − 1) − 2L
2.
The no. of loops given by Bedi et al. [9] as L=
3.
(N + 1) − F = J −N +2 2
(4.2)
Maximum no. of loops are given by N L = k C1 + k C2 + k C3 + · · · + k Cn
4.
(4.1)
(4.3)
where k = L − 1 (i.e. Number of Independent loops) and is ‘n’ number of loops. Joint-Loop Adjacency Matrix (JLAM) is a matrix of dimension J × L, elements are 0 or 1 defined by J L AMi, j (J × L) = 1 if ith joint is a constituent of jth = 0 otherwise
5.
(4.4)
Joint-Joint Adjacency Matrix (JJAM) is a symmetric matrix of dimension J, elements are 0 or 1 is defined as [6] J J AMi, j (J × L) = 1 if ith joint is a constituent of jth = 0 otherwise
6.
(4.5)
Count of kinematic chains is given in Torgal et al. [10, 11] as L=
3N − (F + 3) 2
(4.6)
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Link of highest degree in the chains of given N and F is N when F is odd 2
(4.7)
N +1 when F is even 2
(4.8)
K = K = 8.
In the synthesis of generalized kinematic chains, the selection of links is a crucial step is to generate chains. It is generally expressed as A L = N L2 /N L3 /N L4 /N L5 . . .
(4.9)
where N L2 means the no. of binary links, N L3 means no. of tertiary links and so on. The link selection of chains with N L links and N J joints can be obtained by solving the below two equations: N L2 + N L3 + N L4 + · · · + N Lm = N L
(4.10)
2N L2 + 3N L3 + 4N L4 + · · · + m N Lm = 2N L
(4.11)
where m is the maximum no. of joints incident to a link and is constrained by the below equation given by Yong Sen Yan et al. [3]. m = N J − N L + 2 when N L ≤ N J ≤ 2N L − 3
(4.12)
m = N L − 1 when 2N L − 3 ≤ N L (N L − 1)/2
(4.13)
And the number of joints N J is constrained by the following equation N L ≤ N J ≤ N L (N L − 1)/2
(4.14)
By solving Eqs. (4.10)–(4.14), all possible link assortments of generalized kinematic chains can be obtained which are summarized in Table 1. From the above table, it is clear that for a particular number of links at various degrees of freedom a total of 1149 unique mechanisms can be generated subject to isomorphism which means removing the mechanisms that generate similar type of Table 1 Comparison of results with earlier research No. of links
4
5
6
7
8
9
11
Total
DOF
1
1
1
2
1
3
2
4
2
4
6
Kinematic chain
9
9
12
4
16
7
40
10
839
184
19
1149
Topology Structure Design of Fish-Based Propulsive Mechanisms Fig. 7 JJAM of 8 Link 1 DOF Mechanisms
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propulsive mechanism and only taking the mechanisms that have unique propulsive mechanisms. Also from the computer program used by the research scholars mentioned in the acknowledgement and for the purpose of justification of the results, kinematic chains are drawn for mechanism with 9 links and having 2 degrees of freedom. Similarly, the same procedure is followed to realize other mechanisms with different number of links at various degrees of freedom. For drawing the chains, Joint-Loop Adjacency Matrix (JLAM) and Joint-Joint Adjacency Matrix (JJAM) as given in Eqs. (4.4) and (4.5) play a vital role. In case of the JLAM and JJAM matrices, the order of the matrices is decided by the no. of links and the matrix is a square and the elements of the matrix are determined by the if a link or a particular joint is connected to a particular joint and loop, then in that particular row and column number 1 is assigned and if the said pair is not connected, then the entry in that particular row and column in 0. Depending upon the above-mentioned theory matrix for a specific arrangement (see Figs. 7 and 8).
5 Results After taking into consideration, the data compiled in the link assortment table above following result table is generated and the results obtained for 9 links and 2 degrees of freedom mechanisms in order to compare with the results obtained in the previous research conducted are as shown in Table 2.
6 Conclusion This paper analyses the attributes of topological structures of earlier propulsive mechanisms. By using the creative design procedures for mechanism generation, 18 possible new propulsive mechanisms are proposed here after doing away with the identical mechanisms using isomorphism technique and then unique propulsive mechanisms are realized using commercial 2D mechanism synthesis software SAM 7.0™, and by this software, displacement, velocity and forces on mechanism
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Fig. 8 Chains of 8 links and 1 DOF
Table 2 Comparison of results with earlier research No. of links
DOF
Chains generated using algorithm by Hsieh et al. [10]
Results of Mruthunjaya et al. [12–14]
Results of Hwang et al. [13]
Results of Rao and Deshmukh [14]
9
2
40
40
40
40
could be found easily out along with the coupler curves as shown in the software to envisage the actual mechanism when in motion. Thereafter, the best suitable mechanism which perfectly imitates the actual fish motion is selected for usage in the fish inspired propulsive device. The mechanisms presented in (see Fig. 9) examples of fin undulation and oscillation mechanisms. Note: that the mechanism presented below represent the propulsive mechanism of fishes and the actual length of the mechanism rely upon the real size of the fish and the prototype size into which the mechanism is to be installed.
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(i)
(i i)
(iii )
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
Fig. 9 Unique mechanisms realized using commercially available SAM 7.0™ software
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(xvi)
(xvii)
(xviii)
Fig. 9 (continued)
Acknowledgements We beholden towards all those who had helped us in gratification of this paper. We do oblige to the following research scholars who have helped us while completing the kinematic section of the paper. Dr. Gurusharan Singh Bedi (Director Technical Education, Chhattisgarh). Dr. Shubhashis Sanyal (Professor NIT Raipur, Chhattisgarh). Dr. Hong Sen Yan (National Cheng Kung University, Taiwan, China). Dr. Yii Wen Hwang (National Chung-Cheng University, Taiwan, China).
References 1. Sfakiotakis M, Lane DM, Bruce J, Davies C (1999) Review of fish based swimming modes for aquatic locomotion. IEEE J Oceanic Eng 24(2) 2. Salazar R, Fuentes V, Abdelkefi A (2018) Classification of biological and bio-inspired aquatic systems: a review. IEEE J Oceanic Eng 148:75–114 3. Yan HS (2008) Creative design of mechanical devices. Springer-Verilog, Singapore 4. Zhang T, Zhou C, Wang C, Zhang X (2011) Flapping wing mechanism design based on mechanical creative design theory. In: International conference of mechatronics science, electric engineering and computer 5. Shuzheng W, Zhu WX (2014) Tailfin imitating propelling device. China Patent, CN104260864A 6. Dumas G (2007) Mechanism for control of an oscillating wing. WIPO Patent, WO2008144938A1 7. Alvardo PV, Toumi KY (2004) Mechanised fish Robot exploiting vibration modes for locomotion. U.S. Patent, US7865268 8. Weidong S (2016) Bionic fish movement mechanism. China Patent, CN106275536A 9. Bedi GS, Sanyal S (2013) Loop based algorithm for autonomous sketching of planar kinematic chains. In: Proceedings of 1st and 16th national conference on mechanism and machines (iNaCOMM 2013), IIT Roorkee, India 18–20 Dec 2013 10. Hsieh CF, Hwang WY, Yan HS (1998) Generation and sketching of generalized kinematic chains. In: Proceedings of the ASME 2008 international design engineering technological conference and computers and information in engineering conference, 3–6 Aug, New York 11. Torgal S (2016) Application of maxcode algorithm for the enumeration of kinematic chains of 9 link and 2 Degree of Freedom. SmartCom 2016, CCIS 628, pp 581–589 12. Mruthunjaya TS (1984) A computerized methodology for structural synthesis of kinematic chains: Part 1 formulation. Mech Mach Theory 19(6):487–495
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13. Mruthunjaya TS (1979) Structural synthesis by transformation of binary chains. Mech Mach Theory 14:221–231 14. Mruthunjaya TS (1984) A computerized methodology for structural synthesis of kinematic chains: Part 2-Application to several fully or partially known cases. Mech Mach Theory 19(6):497–505 15. Hwang WM, Hwang YW (1992) Computer aided structural synthesis of planar kinematic chains with simple joints. Mech Mach Theory 27(2):189–199 16. Rao AC, Deshmukh PB (2001) Computer aided structural synthesis of planar kinematic chains obviating the test for isomorphism. Mech Mach Theory 36:489–506
Impact of SOC Estimation on Primary Frequency Regulation for Sustainable Grid Energy Storage System J. Dhanaselvam
and V. Rukkumani
Abstract In the present scenario, it is very important to concentrate on grid energy storage to maximize renewable energy utilization. Nowadays, commercially feasible projects have been developed to store energy in large level by the development of battery storage technology (BST). In power system, when grid power is lost, battery storage system can deal with the renewable intermittency. Because of intermittent nature, the renewable system provides imbalanced frequency in the grid. So energy storage systems are required to balance the frequency variations by the way of charging and discharging and keep the frequency level at desired limit. Underfrequency events are to be seriously considered and when it happens. State-of-charge estimation is a crucial part in energy management system because SOC estimation involves in modeling and optimizing battery performance in terms of extension of life cycle, cost reduction, and safe operation of batteries for various applications including smart grid application. So estimation of SOC provides a key factor for optimized energy management and control system design. In this paper, equivalent circuit model-based SOC estimation is analyzed based on, mapping of the % of SOC directly with the circuit parameters such as open circuit voltage (OCV) and impedance (Z), and the availability the SOC (tk) is calculated by integrating ampere hour integral method with this one. And also, how the estimation of SOC regulates the grid frequency is also analyzed. Keywords Grid energy storage · Battery storage technology (BST) · Imbalanced frequency · Grid frequency regulation · SOC estimation
J. Dhanaselvam (B) Srikrishna College of Technology, Coimbatore, Tamil Nadu 641025, India V. Rukkumani Sri Ramakrishna Engineering College, Coimbatore, Tamil Nadu 641022, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_102
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1 Introduction 1.1 Introduction The unpredictable characteristic during increased wind penetration is highly challengeable in terms of stability of the system. The first and foremost problem arises during this state is oscillatory frequency. Due to the development of control strategies of wind turbine, it is possible to produce less fluctuation in the power output to make the frequency constant [1]. Besides all the techniques, the energy storage system provides a regulatory output power to the external disturbances. But some of the energy storage technologies have been eradicated because of the disadvantages such as low energy density, specific geographic requirement, and so on. To overcome this, battery technologies are analyzed in many researches, and comparatively, it produces better frequency regulation or better primary frequency control to the grid-connected wind system [2]. Generally, the frequency compensation provided by battery technologies is used for smoothening wind power with fast response. But the capacity of the batteries decreases with increase in discharge rate. So, state of charge must be considered for continuous monitoring of wind voltage fluctuations, and it must be kept within the specific limit (Fig. 1).. State-of-charge estimation is a crucial part in energy management system. Because SOC estimation involves in modeling and optimizing battery performance in terms of extension of life cycle, cost reduction, and safe operation of batteries for various applications including smart grid application. So estimation of SOC provides a key factor for optimized energy management and control system design [3]. Despite SOC is a very important parameter in BESS, it cannot be measured directly. In this paper, state-of-charge estimation is determined by equivalent circuit model-based estimation method where lookup table-based SOC estimation is incorporated in it for estimating open circuit voltage by considering the parameters base voltage, slope, internal resistance, and temperature is also taken for the consideration. AC BAR PV ARRAY
Wind System
Lithium- Ion Battery Energy Storage System
DC to AC Converter
Electric Grid
AC to AC Converter Voltage controlled DC to AC Inverter Circuit
Fig. 1 Block Diagram of proposed energy storage system [1]
Smart Homes and other loads
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Fig. 2 Droop characteristics of battery at 25 °C [4]
2 State of Charge 2.1 Equivalent Circuit Model Unlike conventional generating units, grid-connected battery energy storage systems can offer upward and downward frequency regulations during uncertainty conditions by discharging and charging respectively. Generally, the primary frequency regulation is needed to be supplied uninterruptedly based on the droop characteristics [4]. Normally, the frequency deviation lies between ± 20 MHz and ± 200 MHz from the reference grid frequency of 50 Hz (Fig. 2).. Equivalent circuit model-based estimation describes about the internal state of the battery which is described by the state equations. The open circuit voltage model of Li-ion battery is done by the following expression. OC V sOC = V o + β(S O C − 0.5) − M/S O C where • • • • •
OCV SOC —Open-circuit voltage at specific SOC point Vo —Base voltage β—Slope SOC—State of charge with variation from 0 to 1 M—SOC Correction factor (Fig. 3).
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Fig. 3 Droop characteristics of battery at 25 °C
In this model, the current at a particular time is calculated based on the available capacity of the battery. The battery SOC is calculated by ampere hour integral method which is fused into the model-based method to calculate available SOC. t k S O C(K ) = Y 0 −
τ I L (t)d t/ Q t0
where • • • •
SOC(K)—SOC att K Y 0 —SOC att 0 τ —Coulomb efficiency Q—maximum available capacity.
The advantage of model-based method is that the internal states such as the chemical composition concentration, over-potential, and degradation of the battery life can be easily estimated [5]. Like conventional generation units, battery energy storage system provides frequency regulation as upwards and downwards regulation by discharging and charging, respectively (Fig. 4)..
3 State-of-Charge Estimation for PFR It is very important to maintain the smart grid network state within the operational limits by balancing or re-balancing the supply and demand. Without initiating instabilities and operational constraints, how to integrate load participation for many applications is not clear. So, battery energy storage system plays an important role here to compensate and regulate the frequency in terms of primary frequency control [6, 7] (Fig. 5)..
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Fig. 4 Up–downregulation characteristics
Fig. 5 Proposed system model
The following flowchart illustrates how the frequency is maintained during upregulation and down-regulation by charging and discharging. BESS maintains it consistently in this regard (Fig. 6). Lithium-ion battery energy storage system is re-established every time when the frequency reaches ±2 Hz. In smart grid applications, the PFR must be established
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START
Measurement of PFR
49.8
NO
NO
49.98 HZ
50.02
50.2 Hz
YES
YES UP- REGULATION BESS Discharging
SOC Establishment 50%
SOC Establishment 10%
YES
NO
SOC Reestablishment 50%
NO
UP- REGULATION BESS Discharging YES SOC Establishment 90%
YES
NO
END
Fig. 6 SOC estimation flow diagram [2]
within 15 min, and when it is not, the PFR service will be completely interrupted [8–10]. At this time, 50% of the SOC will be re-established.
4 Results and Discussion By the analysis, the percentage of primary frequency regulation establishment is obtained. By this method, it is possible to achieve maximum success rate of PFR by maintaining SOC of Li-ion battery pack. By comparing with calendar aging percentage, the comparison is given in Table 1 (Figs. 7, 8 and 9).
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Table 1 Lion battery decomposition by analyzing SOC SOC re-establishment
Charging–discharging cycle (%)
During 50 Hz
12.5
Upregulation
11.8
Downregulation
Fig. 7 SOC % during all the regulation period
Calendar aging (%) 6.291 8.189
5.6
14.32
20
10
Chargingdischarging cycle %
5
Calendar Aging %
15
0 During 50Hz
Up Down regulaƟon regulaƟon
Fig. 8 SOC estimation for various temperature effects
5 Conclusion and Future Work In this paper, an analysis is done for an impact of state-of-charge estimation for primary frequency regulation services for smart grid applications. It is seen that the various temperature and charging–discharging cycles will have an influence on primary frequency regulation. And also, it is seen that when temperature increases says about 35 °C, the average life cycle is about 60 months. At the same time when the temperature is maintaining at an average value or low temperature value, the life
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Fig. 9 SOC impact analysis for various cycle depth
time of the battery will be increased. Finally, to have a proper primary frequency regulation an SOC must be maintained and correctly established. In future, a further analysis can be made to reduce time duration for establishing an SOC below to the minimum.
References 1. Stroe D-I, Knap V, Swierczynski M, Stroe A-I, Teodorescu R, Operation of grid-connected lithium-ion battery energy storage system for primary frequency regulation: a battery lifetime perspective. IEEE Trans Ind Appl. https://doi.org/10.1109/TIA.2016.2616319 2. Conto J (2012) Grid challenges on high penetration levels of wind power. In: Power and energy society general meeting, 2012 IEEE, July 2012, pp 1–3 3. Denholm P, Ela E, Kirby B, Milligan M (2010) The role of energy storage with renewable electricity generation. Technical Report NREL/TP-6A2- 47187, National Renewable Energy Laboratory, Jan 2010 4. Scrosati B, Garche J (2010) Lithium batteries: status, prospects and future. J Power Sources 195:2419–2430 5. Pérez G (2015) Enhanced closed loop State of Charge estimator for lithium-ion batteries based on Extended Kalman Filter. Appl Energy 155:834–845 6. Yang X, Jia C, Zhang P, Yi F, Sun Z, Zhang J (2015) Fuzzy control strategy of energy storage for wind-storage system. ICEMS, 25–28 Oct 2015, Pattaya, Thailand 7. Tesfahunegn SG et al (2011) Optimal shifting of Photovoltaic and load fluctuations from fuel cell and electrolyzer to lead acid battery in a Photovoltaic/hydrogen standalone power system for improved performance and life time. J Power Sources 196:10401–10414 8. Soylu E, Soylu T, Bayir R (2017) Design and implementation of SOC prediction for a li-ion battery pack in an electric car with an embedded system. Entropy 19:146. https://doi.org/10. 3390/e19040146
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9. Saravanan S, Thangavel S (2013) Fuzzy logic controller-based power management for a standalone solar/wind/fuel cell fed hybrid system. J Renew Sustain Energy 5:053147 10. Bañosa R, Manzano-Agugliarob F (2011) Optimization methods applied to renewable and sustainable energy: a review. Renew Sustain Energy Rev 15:1753–1766
Kinematic Modelling of UR5 Cobot Using Dual Quaternion Approach Mohsin Dalvi , Shital S. Chiddarwar , M. R. Rahul , and Saumya Ranjan Sahoo
Abstract This paper presents a framework based on dual quaternions for obtaining the forward and inverse kinematic models for a 6-dof (degrees of freedom) UR5 cobot. A forward kinematics model based on dual quaternions is developed for the cobot. Dual quaternion differential kinematics involving Jacobian transpose and damped least squares methods along with pose error feedback are used for determining the inverse kinematics model. Implementation of inverse kinematics methods for a given trajectory shows that Jacobian transpose method is faster, but gives more jerky motions and is less immune to singularity compared to damped least squares. Keywords Cobot · Dual quaternions · Differential kinematics
1 Introduction When robots are programmed to automate a manual task, one requirement is to have the end-effector change its pose (position and orientation) as smoothly as a human hand while moving over a trajectory. Euler angles remain the popular technique for representing orientations, but suffer from gimbal lock and ambiguities [1]. Quaternions are a more robust technique than Euler angles and rotation matrices due to compact representation, computational efficiency, and immunity from kinematic and mathematical singularities [2]. However, the available literature on modeling with quaternions shows that there are ambiguities in representation (Hamiltonian, NASA-JPL), handedness (right, left) and frames (global or body) [3]. In addition, using vectors for translations and quaternions for rotations when both transformations occur simultaneously leads to inconsistencies [4]. A dual quaternion (DQ) uses the concept of dual numbers to unify rotations and translations into a single state instead of defining separate vectors for them [5], and
M. Dalvi (B) · S. S. Chiddarwar · M. R. Rahul · S. R. Sahoo Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra 440010, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_103
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lends well to screw transforms [6]. Dual quaternions have been used for kinematic modeling and pose control of serial manipulators [7, 8]. Differential kinematics is a widely-used approach for inverse kinematics (IK) modelling. DQs and quaternion exponential maps have been used to solve IK problems by considering joint limits [6]. Dealing with Jacobian matrix is well discussed in literature, unlike the dual quaternion differential operator. The flow of the paper is as follows: Dual quaternion preliminaries are introduced in Sect. 2. The forward and inverse kinematic models for a serial manipulator are developed in Sect. 3 and applied to UR5 cobot in Sect. 4. In Sect. 5, the results are discussed and the paper is concluded in Sect. 6.
2 Mathematical Preliminaries Quaternions, invented by Hamilton, are an extension of complex numbers to three dimensions and are represented as a = a0 + a1 i + a2 j + a3 k = [a0 , a T ]T
a0 , a1 , a2 , a3 ∈ R, a ∈ R3 , a ∈ H (1)
where a0 is a scalar, a is a vector, and i, j, k are orthogonal unit vectors in 3D space that satisfy i2 = j2 = k2 = ijk = −1. The addition operation is element-wise, given by a + b = [a0 + b0 , a T + bT ]T . The multiplication operation is non-commutative and is given by b a0 b0 − a · b = H (a) 0 (2) ab = b a0 b + b0 a + a × b ⎡ ⎤ 0 −a3 a2 T −a a , I = diag (1, 1, 1) and [a]× = ⎣ a3 0 −a1 ⎦. where, H (a) = 0 a a0 I +[a]× −a2 a1 0 √ 2 The quaternion norm is given by a = aa ∗ = a0 + a.a, where a ∗ is the conjugate, defined as a ∗ = [a0 , −a T ]T . A unit quaternion is given by aˆ = a / a and has unit norm. The identity quaternion 1 = [1, 0T ]T is obtained when a quaternion a = [0, 0T ]T is multiplied by its inverse, which is given by a −1 = a ∗ / a2 . For a unit quaternion, a −1 = a ∗ . Dual numbers were developed by Clifford and are given by a = ar + ad where a ∈ D, ar ∈ R is the real part, ad ∈ R is the dual part, and is the dual unit that satisfies = 0, 2 = 0. A real function is extended to dual space by using i = 0 ∀ i ≥ 2 in Taylor series expansion of f (a + b) so as to give f (a + b) = f (a) + b f (a). Therefore, relations such as ea+b = ea + bea , cos (a + b) = cos a − b sin a and √ √ b a + b = a − √ are obtained. 2 a Dual quaternions (DQs), developed by Clifford, consist of a dual scalar and a dual vector as a = a 0 + a. It is rewritten to give a = ar + ad , where ar , ad ∈ H. Addi-
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tion is element-wise, given by a + b = (ar + br ) + (ad + bd ), while multiplication is given by br H (ar ) 0 a b = ar br + (ar bd + ad br ) = H (ad ) H (ar ) bd
(3)
Three exist, namely, dual conjugate a¯ = ar − ad , quaternion conjugate
∗ conjugates a = ar∗ + ad∗ , and combined conjugate a¯ ∗ = ar∗ − ad∗ . The norm of a DQ √ is obtained from a = a a ∗ = ar 2 + 2 (ar 0 ad0 +ar ·ad ). If ar = 1 and ar 0 ad0 + ar ·ad = 0, then a = 1. The identity DQ is 1 = [1, 0T ]T + [0, 0T ]T and the inverse is given by a −1 = ar−1 − ar−1 ad ar−1 . If ar = 0, then a −1 does not exist.
3 Kinematic Modelling of Serial Manipulator The unit quaternion rˆ = [cos θ2 , uˆ T sin θ2 ]T represents rotation by angle θ about unit ˆ Given a rotational quaternion rˆ = [r0 , r T ]T , the rotation parameters θ and uˆ vector u. are obtained as θ = 2 cos−1 (r0 ) and uˆ = r/ sin θ2 . In DQ form, rotation is represented as r = rˆ + 0. Let translation t = [tx , t y , tx ]T be depicted in quaternion form as t = [0, t T ]T and in DQ form as t = 1 + t/2. Then, the combined transformation q consisting of rotation r followed by translation t is given in DQ form as q = t r = rˆ + 21 t rˆ ˆ cos θ2 t T + sin θ2 (t× u) ˆ T ]T = [cos θ2 , sin θ2 uˆ T ]T + 21 [−sin θ2 (t· u),
(4)
Given a transformation DQ q = qr + qd , the rotation and translation components are respectively obtained using rˆ = qr and t = 2qd qr∗ . When a vector p0 , represented by DQ p0 = [1, 0T ]T + [0, p0 T ]T , is subjected to transformation q, the new vector p1 is obtained in DQ form as p1 = [1, 0T ]T + [0, p1 T ]T = q p0 q¯ ∗ . This DQ is used to obtain the forward kinematics (FK) model of a serial manipulator with n lower pair joints (revolute or prismatic). Figure 1 shows link i between joints i and i + 1 of a serial manipulator. Coordinate
frames {i − 1} and {i} are attached to these joints, and an intermediate frame i is placed at intersection of Z i−1 and X i axes. Classical Denavit-Hartenberg (or DH) conventions [9] are used to obtain the link and joint parameters. In order to coincide frame {i − 1} with frame {i}, rotation and translation motions are carried out, first about Z i−1 axis, and then about X i axis. These are screw transformations q iZ and q iX respectively and are carried out using using Eq. 4, as shown in Table 1. Let Cθ = cos θ2i , Sθ = sin θ2i , Cα = cos α2i , Sα = sin α2i , A = a2i and D = d2i . Also represent the DQs by 8 × 1 vectors as [ar T , ad T ]T . Then, the overall DQ that maps frame {i} to frame {i − 1} is given by
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Fig. 1 Screw motions for frame transformation
Link i Link i-1
i’
Screw(Z , , )
Table 1 Screw transformations carried out using Eq. 4
{i − 1} to i Frame transformation Rotation angle θ Rotation axis uˆ Translation direction t uˆ · t uˆ × t Transformation DQ
i−1
θi (0, 0, 1) (0, 0, di ) di 0 q Z = [Cθ , 0, 0, Sθ , −DSθ , 0, 0, DCθ ]T
i to {i} αi (1, 0, 0) (ai , 0, 0) ai 0 q X = [Cα , Sα , 0, 0, −ASα , ACα , 0, 0]T
q i = q iZ q iX = [η1 , η2 , η3 , η4 , −Dη4 − Aη2 , −Dη3 + Aη1 , Dη2 + Aη4 , Dη1 − Aη3 ]T
(5)
where, η1 = Cθ Cα , η2 = Cθ Sα , η3 = Sθ Sα and η4 = Sθ Cα . For an n-degree serial manipulator, the transformation DQ 0 q n mapping end-effector frame {n} to base frame {0} is given by n i−1 0 (6) qn = qi i=1
Now, given an end-effector pose 0 q n in DQ space, differential kinematics is used for mapping a differential change in joint parameters (θ˙ ) to a differential change in pose (q) ˙ through the Jacobian matrix J. ˙ q˙ = J θ,
wherei-th column of Jis J i =
∂ 0q n ∂θi
(7)
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For obtaining the inverse kinematics (IK) model of the serial manipulator, inverse differential kinematics is used. However, for a non-square Jacobian matrix, the pseudo-inverse is found by pre-multiplying both sides of Eq. 7 by J T and rearranging the terms so as to obtain
−1 T J q˙ (8) θ˙ = J T J During singularity condition, when two joint axes get coincident such that actuating either joint produces the same differential movement of the end-effector, then, the two corresponding columns of J become equal, and matrices J as well as ( J T J) become singular. One way to deal with this is to use the Jacobian transpose (J T ) method [10], where ( J T J)−1 is approximated to α , given by α=
q˙ T β βT β
,
where, β = J J T q˙
(9)
To further reduce the chances of J J T losing rank, the damped least squares method (DLS) is used [10]. Here, a damping constant δ ≈ 0.001 in the diagonal elements modifies Eq. 8 to
−1 T J q˙ θ˙ = J T J − δ 2 I
(10)
where I is identity matrix. Computational efficiency is attained if, instead of finding T
−1 J J − δ I , Eq. 10 is solved for θ˙ through iterative techniques such as GaussSiedel method. The differential operator q˙ in Eq. 7 is obtained from the dual number extension of the quaternion differential operator q˙ = 21 ω q [11] and is given by q˙ =
1 1 ξq = [0, ω T ]T + [0, (v + t × ω)T ]T q 2 2
(11)
where ξ is twist in world frame, ω ∈ R3 is angular velocity vector, v ∈ R3 is linear velocity vector, and t is obtained from t = [0, t] = 2qd qr∗ where qr and qd are real and dual parts of the pose q. Now, let end-effector pose be q k and corresponding joint parameters be θ k at some time interval k. Then, during next time interval k + 1, for some differential change in pose q˙ k , the IK model gives differential change in joint parameters θ˙ k . Then, θ k gets updated to θ k+1 = θ k + θ˙ k and corresponding pose obtained from FK model is Q F (θ k+1 ). However, due to linearisation, this pose does not match the updated pose q k+1 = q k q˙ k . The pose error ( q)k+1 is given by ( q)k+1 = Q F (θ k+1 )∗ q k+1 This error is fed back to the next differential change in pose q˙ k+1 .
(12)
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4 Application to UR5 Cobot The UR5 cobot (or collaborative robot) from Universal Robots, as seen in Fig. 2a, is a 6R serial manipulator having 0.85 m reach and 5 kg payload capacity. The cobot architecture is derived using classical DH conventions [9] as seen in Fig. 2b and Table 2. For implementing the framework, a library for dual quaternions is developed in Python 3.7, and run on a Windows 8 workstation with Intel i5-4570 3.2 GHz processor and 8 GB DDR3 RAM. The DQ outputs of the FK model are tallied with results from RoboAnalyzer software. Two IK models, one using J T method, and other using DLS method, are built, and are applied to the trajectory shown in Fig. 3. The red, green, and blue arrows correspond to the end-effector roll, pitch, and yaw axes, respectively. X2 J3
Z2 X4
J4
Z3
J5
X1
Z1 J2
Z0
X3
X6
Z5 Z6
X5
Z4 J6
TCP
J1
X0
(b) Frame assignment
(a) UR5 cobot Fig. 2 UR5 cobot (a) with its frame assignment (b) Table 2 DH parameters of UR5 cobot Joint i ai (m) αi (rad.) 1 2 3 4 5 6
0 0.425 0.392 0 0 0
π/2 0 0 π/2 −π/2 0
di (m)
θi (rad.)
0.0892 0 0 0.1093 0.09475 0.0825
θ1 θ2 + π/2 θ3 − π/2 θ4 + π/2 θ5 θ6
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Fig. 3 Closed trajectory showing end-effector frame markers
5 Results and Discussion The results of the DQ FK model matched perfectly with those obtained from RoboAnalyzer software as well as the FK model derived using homogeneous transformation matrices (HTM). Compared to 234 additions and 356 multiplications needed in HTM FK model, the DQ FK model required 224 additions and 312 multiplications. While DQs represented end-effector pose using 8 parameters, HTMs required 12 parameters for the same task. The pose specified by DQs is not intuitive to humans. However, using DQs over HTMs led to reduction in computational time by up to half in some cases. Applying the two IK models on the trajectory for 50 times each showed that J T method gave solutions faster, but also gave more jerky motions. For J T , mean solution time ranged from 12ms to 39ms, whereas for DLS, the same ranged from 33ms to 87ms. The graphs of quaternion components in Fig. 4 show that fluctuations in DLS method are lesser than in J T method. It is seen that the trajectory loop did not get closed in either method, which means pose error did not get removed even when simple feedback was used. This makes a case for exploring more feedback control methods.
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Fig. 4 Using end-effector DQ pose elements qd0 and qd1 to compare curves traced by J T and DLS IK models
6 Conclusion In this work, dual quaternions (DQs) were used to develop the forward kinematics (FK) and inverse kinematics (IK) models for a serial manipulator. The DQ differential operator as well as the J T and DLS methods of solving inverse differential kinematics in DQ space were discussed. This approach was implemented for the UR5 cobot and errors obtained during simulation over a closed trajectory were studied. Though DQs seem unintuitive to humans, they are computationally efficient and lend themselves well to working with elaborate trajectories. Future works include DQ implementation of fast IK algorithms such as FABRIK [10] and DQ spline interpolation for pathplanning problems.
References 1. Corke P (2017) Robotics, vision and control: fundamental algorithms In MATLAB® , Springer Tracts in Advanced Robotics, vol 118. Springer, 2nd edn 2. Shoemake K (1985) Animating rotation with quaternion curves. In: ACM SIGGRAPH computer graphics, vol 19. ACM, pp 245–254 3. Sommer H, Gilitschenski I, Bloesch M, Weiss S, Siegwart R, Nieto JI (2018) Why and how to avoid the flipped quaternion multiplication. CoRR 4. Allmendinger F, Eddine SC, Corves B (2018) Coordinate-invariant rigid-body interpolation on a parametric C1 dual quaternion curve. Mech Mach Theor 121:731–744 5. Jia YB (2013) Dual quaternions. Iowa State University, Tech. rep 6. Kenwright B (2013) Inverse kinematics with dual-quaternions, exponential-maps, and joint limits. Int J Adv Intelligent Syst 6(1 & 2):53–65 7. Radavelli L, Simoni R, De Pieri E, Martins D (2012) A comparative study of the kinematics of robots manipulators by Denavit-Hartenberg and dual quaternion. Mecánica Computacional Multi-Body Syst 31(15):2833–2848 8. Pham HL, Perdereau V, Adorno BV, Fraisse P (2010) Position and orientation control of robot manipulators using dual quaternion feedback. In: 2010 IEEE/RSJ international conference on intelligent robots and systems, pp 658–663 9. Siciliano B, Sciavicco L, Villani L, Oriolo G (2009) Robotics: modelling, planning and control. Springer, London
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10. Aristidou A, Lasenby J (2009) Inverse kinematics: a review of existing techniques and introduction of a new fast iterative solver. University of Cambridge, Tech. rep 11. Wang X, Han D, Yu C, Zheng Z (2012) The geometric structure of unit dual quaternion with application in kinematic control. J Math Anal Appl 389(2):1352–1364
Design of XY Air Bearing Stage for Ultra-Precision Rajesh Kumar and Jitendra Prasad Khatait
Abstract Precision motion system is used in many industries including electronics, optics, and automotive. In silicon wafer lithography process, the resolution has gone up as high as 10 nm and still going higher. Hence, ultra-precision machines have emerged with nano-level accuracy to cater to this requirement. This paper proposes a novel ultra-precision motion stage design with long range motion. The proposed machine has H-type configuration, supported on air bearings. Ironless direct drive motor is applied for actuation with motion feedback by optical linear encoders. The workspace is 300 mm2 with maximum velocity of 1 m/s and maximum acceleration of 10 m/s2 . Designed static stiffness of the machine is 398 N/μm and 156 N/μm in X and Y-axis, respectively. Dominant vibration mode is predicted at 46 Hz in vertical mode of the gantry. A PID control is implemented, allowing higher positioning performance. The maximum tracking error predicted in plane for low speed is 2 μm. Keywords Precision · Air bearing · XY
1 Introduction In recent years, demand of precision components in semiconductor, biomedical, storage media, optoelectronics, aerospace, and automotive industries have increased many times. Applications like micro or nano-machining, lithography, metrology, etc., require precision motion system. For example, the most advanced lithography system of today known as deep ultraviolet (DUV) lithography uses 193 nm wavelength lasers which can resolve features up to 38 nm while extreme ultraviolet (EUV) lithography uses 13.5 nm radiation wavelength reducing resolution to less than 10 nm. Hence, the positioning resolutions of less than 10 nm are now an industry standard [1]. Not only accuracy but high speed and long range is also required. Fast scanning is necessary to maintain viable production and cost offset. For example, flexible electronics are subjected to roll to roll manufacturing which has a resolution of R. Kumar · J. P. Khatait (B) Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_104
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10 μm with alignment precision of ±2.5 μm in a workspace of 300 mm × 300 mm at top scan speed of 400 mm/s [2]. Hence, in response to these demands, new ultra-precision stages are required having nanometer level accuracy, long range as well as high speed.
2 Literature Review Precision stages being an important field from industry perspective, there have been lots of work in this direction. Various kinds of configurations have been designed involving guideways, preloads, actuators, architecture, sensor and control. Zschaeck et al. [3] developed a double-H-type XY positioning system supported by two linear v-groove high precision guideways and driven by two ironless linear motors. Shinno et al. [4] developed voice coil motors actuated aerostatic XY planar stage for nanomachining. Tomita et al. [5] developed XY positioning stage composed of four linear motors while guiding the stage by linear motion ball bearings. Chung et al. [6] developed an XY positioning stage supported by magnetically preloaded air bearings. Shinno et al. [7] built XY-theta table system supported with four porous vacuum preloaded bearings. Gorniak [8] developed a long range air bearing stage with a T-type configuration. Lu and Usman [9] developed a novel 6D direct drive technology for planar stages having long range with positioning resolution of 1um. It uses magnetic levitation as support and actuation from linear magnet arrays and stereo-vision as position sensor. The earlier trend can be seen as stages with linear guide driven by ball screw which had inherent limitation of hysteresis and backlash. Then, hybrid drive stages were developed which used ball screw for coarse and long range positioning while using flexure or PZT or VCM for fine positioning. These structures had, however, problems due to unpredictable friction in coarse stage and mutual interdependencies among stages. After this linear motor-driven stages with air bearings reduced their complexity while allowing long motion range. Then, a new approach like magnetic levitation further improved the performance of these precision stages but currently limited by short strokes and higher complexity. There are very few ultra-precision table systems which enable simultaneous realization of high-speed feed drive as well as nanometer positioning accuracy for over long motion range with simple mechanical structure and control. Hence, the proposed motion stage design is directed to meet these requirements.
3 XY Stage Design The XY stage has modular design with series architecture shown in Fig. 1. Three identical modules of linear stages are assembled into planar stage using flexural
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Fig. 1 Design of XY stage
joints. This makes design general in nature, applicable for single to multi-degree of freedom applications. The positioning stage has an H-type configuration with two linear aerostatic guideways in Y-axis and one in X-axis. Unlike stacked linear stages design in “plus” shape which has inherent dynamics imbalance, it has dual y-axis H-type design, giving it better dynamics. Ten air bearings on each moving carriage have been used to provide linear guidance. These porous air bearings are preloaded by opposite air bearings which provide bidirectional load capacity and increases bearing stiffness. Three bearings making a plane and two defining a line constrains total of 5 degree of freedom. Since each air bearing is preloaded by another air bearing, total of 10 air bearing configuration provides higher stiffness though being over constrained and non-kinematic. The given configuration of air bearings is based on recommendations by one of air bearings manufacturers’ design guide [10].
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The x-axis carriage is connected with flexural joints such that it constrains six DoFs of the x-axis carriage and also allows yaw motion which is possible due to non-synchronous behavior of two y-axis actuators. The yaw motion can then be controlled from two position sensors on y-axis. Two actuators on y-axis provide high acceleration and velocity to the stage improving its turnaround performance. It is also tried to make it a coplanar stage having location of center of force and center of mass in the same plane to reduce moments, improving the dynamics of the system. The carriages are end supported over a precision ground rectangular section granite beam. Whole stage is based on granite base to minimize error due to thermal expansion. Positioning feedback is provided by high resolution optical encoder attached on support beam of each axis, especially dual encoders on y-axis which provides better orthogonality/yaw control. Each encoder is mounted close to the worktable plane to reduce Abbe error. Air compressor and filters are used to provide clean and dry air to air bearings. Further positioning error is reduced by motion controllers using closed loop feedback mechanism. Payload is mounted on top of moving carriage.
3.1 Stiffness Analysis The maximum carriage load capacity is determined by the maximum load capacity of bearings (which is a function of air gap between bearing and the surface), the preload applied (higher preload reduces the carrying capacity), size of the air bearing (larger size, higher capacity) and air pressure supply. Major factor to be considered is that in working condition carriage should maintain minimum allowable gap between bearing and the surface. Recommended fly height is around 5–6 μm. For correct determination of load carrying capacity, load-lift chart supplied by the bearing vendor is used. Each bearing is modeled as spring of constant stiffness. The stiffness values of each air bearing are obtained from manufacturer’s catalogue. A stiffness matrix can be calculated for the stage as given in Table 1. Natural frequency can then be calculated as follows (includes compliance due to air bearing only) (Table 2). Table 1 Stiffness due to air bearing in XY stage
Air bearing stiffness X
464
N/μm
Y
232
N/μm
Z
112
N/μm
θx
0.29
N m/μrad
θy
0.79
N m/μrad
θz
1.2
N m/μrad
Design of XY Air Bearing Stage for Ultra-Precision Table 2 Natural frequency of XY stage
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Natural frequency (Hz) X
542
Y
766
Z
266
θx
347
θy
524
θz
127
3.2 Error Budget Major sources of error are quantified and presented in the form of error budget predicting the overall accuracy of the machine. Critical areas can then be determined and solutions can be developed to minimize individual errors [11, 12]. The linear positional error for single axis consists of encoder scale error, the resolution of the encoders and the uncertainty of the encoder signal. The straightness error is due to the geometry of the guideways. Angular error are also called yaw, pitch and roll errors. These errors are created from the contours of the guideways. Dynamic errors occur from stiffness of machine components during acceleration and from external loading. Thermal error consists of the expansion of encoder scale. An error budget is prepared including above factors. Uncompensated and compensated positioning accuracy of 6.41 μm and 1.33 μm has been predicted, respectively.
3.3 Flexure Design Many mechanisms require small displacements or rotations. In such cases, elastically deforming parts (also called flexures) can be used to create points of rotation or parallel guides with a relatively high stiffness in the constrained directions. These parts are also free of backlash and friction which makes them suitable for use in precision applications. A typical XY stage is “rigid” in nature having 12 DoFs constrained, six on each ends. This causes undesired stress and control issues. Ideal requirement is to have only six DoFs constrained for the bridge axis so that the two actuators will be like “moving world.” Soemers [13] describes how it can be exactly constrained using wire flexures as shown in Fig. 2. Two wires y1 , y2 in Y-axis, thus constraining Y translation and yaw rotation (controlled using two actuators). Three wires in Z-axis z1 , z2 , z3 constraining three DoFs (z translation and roll and pitch rotation). Lastly, one wire in x-axis x 1 constrains x translation. The physical realization through wire flexures is quite difficult, and there is lack of robustness. So sheet flexures can also be used to serve similar purpose.
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Fig. 2 Kinematic connection of gantry stage [13]
Apart from the bridge axis being exactly constrained, it requires yaw compliance to accommodate small non-synchronous behavior of Y-axis actuators and deviation in parallelism between two Y axes. Flexural couplings attached to crossbeam can provide yaw compliance thus allowing active yaw and straightness compensation and improving orthogonality control. A combination of flexural couplings is added to the XY stage design as shown in Fig. 3. In XY assembly, cross flexure (Fig. 3a) provides yaw compliance at one end constraining 5 DoFs while parallel flexure (Fig. 3b) constrains 3 DoFs (two in-plane translation and one in-plane rotation). Thus, the total of 8 DoFs are constrained, overconstraining once in roll direction (by cross flexure) and twice in pitch direction (by parallel flexure). Though this arrangement is over constrained, it will contribute to higher stiffness. Monolithic flexures made from single metal plate are also viable options. Though such flexures appear simple, they have comparatively higher stiffness and bending stress (occurring at connections) but can be good alternatives. Two such flexures are proposed in Fig. 4.
4 FEA 4.1 FE Model Figure 5 shows the meshed model of XY stage. The system contains a total of 346,426 elements with element size of 5 mm. The simplest element for the modeling of the stage is the hexahedron element.
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a)
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b)
c) Fig. 3 a Cross flexure, b parallel flexure, c assembly
Fig. 4 Monolithic flexural couplings
Fig. 5 Meshed model
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The structure of the bolt connections are modeled using rigid element since their local deformation has no impact on the overall performance of the machine. Further bracket, beam and magnet tracks are connected using 1D weld elements as they are assumed to be rigidly connected members. Air bearings are modeled as linear spring element of constant stiffness. Multiple-node end connection with the spring using 1D rigid elements have been used to avoid stress concentration.
4.2 Analysis and Results Modal analysis has been performed for three cases of assembly. In the first case, structure has been assumed of very high stiffness to capture stiffness due to air bearing only. So that it can be correlated with analytical calculation. In the second case, crossbeam is mounted rigidly with gantry structure as shown in Fig. 6a. 1D rigid elements have been used for connection on both ends simulating bolted connections. In the third case, assembly is done using flexural couplings as shown in Fig. 6b. In last two cases, structural compliance has been taken into account. Tables 3 and 4 show results. In Table 3, FEA results are in good correlation with analytical ones In Table 4, the first two modes are translational modes in X and Y direction whose frequency will be determined from the servo bandwidth. Since, here, it is not taken into account, it appears as rigid modes. The dominant mode in the above simulation is the yaw mode which is 100 Hz for bolted assembly. In case of flexural coupling assembly, yaw mode has very low frequency showing yaw compliance which can be controlled by Y-axis actuators. 46 Hz is next dominant frequency (Fig. 7).
Fig. 6 Two types of connections of crossbeam a bolted (top), b flexural (bottom)
Design of XY Air Bearing Stage for Ultra-Precision Table 3 Natural frequency correlation between FEA and analytical
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Calculated frequency (Hz)
FEA frequency (Hz)
Air bearing only
Air bearing only
X
542
522
Y
766
787
Z
266
366
θx
347
471
θy
524
561
θz
127
137
Table 4 Modal frequencies Mode
Modal frequency (Hz) (air bearing only)
Modal frequency (Hz) (bolted assembly)
Modal frequency (Hz) (flexure coupling assembly)
1
–
–
–
2
–
–
–
3
137
100
2.6
4
246
138
46
5
366
201
53
6
446
294
74
7
471
306
109
8
522
335
144
9
561
369
229
10
614
393
255
Fig. 7 Yaw mode
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5 Control Dynamic modeling and prediction of the motion system forms a major portion of the design process. In this section, dynamic modeling of XY air bearing stage is presented. Similar analysis is done by García-Herreros et al. for rail guide-based dual drive gantry stage [14]. The XY stage can be considered as three-degree of freedom mechanism which can be described by the schematic in Fig. 8. Main physical parameters are given in Table 5. x 1 , x 2 and Y are measured quantities using linear encoders in each stage module. These quantities can be further mapped to X, θ and Y which can give global location of center of mass of work table. A Lagrangian-based model is derived. The equation of motion of the system can be written as M q¨ + H q˙ + C q˙ + K q = f
(1)
where M, C and K are the inertia, viscous damping and stiffness matrices, H is the coriolis and centripetal acceleration matrix, f is the vector of forces, and q is the vector of coordinates (X, θ, Y ). The equation of motion is further simplified based on assumptions given very small yaw angle θ, negligible centripetal and coriolis forces as compared to inertia, damping and spring forces and negligible influence over the motion of worktable due to linear and angular accelerations of the cross-arm and vice versa except due to angular acceleration of worktable on cross-arm.
Fig. 8 Model schematic
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Table 5 Parameters of XY stage for dynamic modeling Parameters of the XY stage
Unit
Symbol
Description
Value
m1
Mass of carriage X 1
10
m2
Mass of carriage X 2
10
kg
m3
Mass of crossbeam
20
kg
m4
Mass of carriage Y (including payload of 10 kg)
20
kg
J3
Rotational inertia of crossbeam
0.55
kg-m2
J4
Rotational inertia of carriage Y with payload
0.07
kg-m2
cv1
Viscous friction coeff. of actuator 20 X1
N/(m/s)
cv2
Viscous friction coeff. of actuator 20 X2
N/(m/s)
cy
Viscous friction coeff. of actuator 20 Y
N/(m/s)
c1
Viscous friction coeff. flexural coupling 1
10
Nm/(rad/s)
c2
Viscous friction coeff. flexural coupling 2
8
Nm/(rad/s)
k1
Torsional stiffness of flexural coupling 1
2000
Nm/rad
k2
Torsional stiffness of flexural coupling 2
1800
Nm/rad
L
Length of crossbeam
0.57
m
d
Offset of center of mass from crossbeam center line
0.1
m
kg
Simplified Model Ms q¨s + Cs q˙s + K s qs = f s
(2)
where
(m 1 − m 2 ) L2 − m 4 Y m1 + m2 + m3 + m4 Ms = 2 (m 1 − m 2 ) L2 − m 4 Y J3 + J4 + (m 1 + m 2 ) L4 + m 4 d 2 + m 4 Y 2 (cv1 − cv2 ) L2 cv1 + cv2 Cs = 2 (cv1 − cv2 ) L2 cv1 + cv2 + (cv1 + cv2 ) L4 0 0 Ks = 0 k1 + k2
(3)
(4)
(5)
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fs =
FX Fθ + m 4 d Y¨
(6)
T qs = X θ
(7)
m 4 Y¨ + c y Y˙ = Fy
(8)
where M S , C S and K S are the simplified inertia (3), damping (4) and stiffness (5) matrices, f S is the vector of simplified forces (6), and qS is the vector of coordinates X and θ (7).
5.1 Results A Simulink model in MATLAB is set up using the simplified equation of motion. A PID-based controller has been designed. Three-independent axis control has been used namely X 1 , X 2 and Y based on three actuators. Here, independent axis control assuming identical dynamics for each axis, all the three PID controllers have been tuned as K p = 1000, K i = 800 and K d = 500. A step input test is performed, followed by sinusoidal input and circle test of radius 150 mm to check for the positioning and tracking performance. From Fig. 9, it can be seen that Y response is fast as compared to X 1 and X 2 due to dynamic coupling between Y slider and cross-arm motion. Sinusoidal response in Fig. 11 shows that the maximum individual axis error is 20 μm for x 1 , x 2 -axis, while for Y-axis its 15 μm. Inter-axis offset error between x 1 and x 2 is 0.1 μm. Maximum tracking error (Fig. 13) in circle test of 150 mm radius at 0.1 m/s was found to be below 2 μm (Figs. 10 and 12).
6 Conclusion In this paper, a novel mechanical design of an XY air bearing stage for ultra-precision has been proposed. The serial architecture with H-type configuration of the stage provides better dynamics for high-speed applications. The design being modular makes it possible to assemble many such linear stages into higher DoF stages. Using air bearing pads for support over granite guideway eliminates error due to friction like hysteresis and backlash. Non-contact and direct drive features of the motion stage give better positioning performance than traditional ball screw and sliding support stage. An error budget is prepared to check theoretical performance of the system. It is predicted to be maximum uncompensated positioning error of 6.4 μm. A flexural coupling is designed to provide yaw compliance to relax stringent criteria of parallel
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Fig. 9 Step input response
dual axis stage and compensate for dynamic differences between parallel actuators. Design performance of XY stage is studied using FEA. Fundamental frequency of vibration for XY stage is observed to be vertical motion at 46 Hz. A PID controller is applied; tracking error is found to be below 2 μm at low speed.
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Fig. 10 Input sinusoidal trajectory
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Fig. 11 Error response of sinusoidal input
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Fig. 13 Tracking error plot in circle test
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References 1. Wagner C, Harned N (2010) EUV lithography: lithography gets extreme. Nat Photonics 4(1):24–26 2. Jain K, Klosner MARC, Zemel MARC, Raghunandan S (2005) Flexible electronics and displays: high-resolution, roll-to-roll, projection lithography and photoablation processing technologies for high-throughput production. Proc IEEE 93(8):1500–1510 3. Zschaeck S, Amthor A, Ament C (2011) Decentralized high precision motion control for nanopositioning and nanomeasuring machines. In: IECON 2011—37th annual conference on IEEE industrial electronics society. IEEE, pp 546–551 4. Shinno H, Yoshioka H, Taniguchi K (2007) A newly developed linear motor-driven aerostatic XY planar motion table system for nano-machining. CIRP Ann Manuf Technol 56(1):369–372 5. Tomita Y, Makino K, Sugimine M, Taniguchi N (1996) High-response XY stage system driven by in-parallel linear motors. CIRP Ann Manuf Technol 45(1):359–362 6. Chung TT, Chu CH, Chian HF, Huang C, Fan KC, Yen JY, Szu KI (2011) Structural design and analysis of a nano-positioning planar motion stage. IEEE, pp 833–838 7. Shinno H, Hashizume H, Yoshioka H, Komatsu K, Shinshi T, Sato K (2004) XY-θ nanopositioning table system for a mother machine. CIRP Ann Manuf Technol 53(1):337–340 8. Gorniak JM (2010) Design and metrology of a precision XY planar stage. Master’s thesis, University of Waterloo 9. Lu X, Usman I (2012) 6D direct-drive technology for planar motion stages. CIRP Ann Manuf Technol 61(1):359–362 10. Bearings NWA (2006) Air bearing application and design guide 11. Donaldson RR (1980) Error budgets. Technology of machine tools 5 12. Slocum AH (1992) Precision machine design. Society of Manufacturing Engineers, Dearborn MI 13. Soemers HMJR (2001) The design of high performance manipulators. In: 2001 IEEE/ASME international conference on advanced intelligent mechatronics, proceedings, vol 1. IEEE 14. García-Herreros I, Kestelyn X, Gomand J, Coleman R, Barre PJ (2013) Model-based decoupling control method for dual-drive gantry stages: a case study with experimental validations. Control Eng Pract 21(3):298–307
Design and Fabrication of a Bio-inspired Soft Robotic Gripper Ayush Agarwal, Ankit Baranwal, G. Stephen Sugun, and Prabhat K. Agnihotri
Abstract Compliant gripping is a promising way to protect delicate objects from the damage caused by contact pressure. Present work proposes the design of a soft gripper inspired by human fingers. The experimental fabrication of gripper is realized by means of a 3D printed hard skeleton made of acrylonitrile butadiene styrene (ABS) which is actuated by PDMS-based soft dielectric elastomer actuator. Different actuator configurations are explored to demonstrate the design flexibility of the present approach. Experimental results show that both closing and opening mode of gripper actuations are possible by suitably placing the elastomeric layer. Finally, it is shown that lightweight objects can be precisely handled using three-fingered gripper having claw type configuration. Keywords Soft actuator · Dielectric elastomer · Three-fingered gripper · Closing-opening mode actuation
1 Introduction Design and fabrication of soft gripper are required for robots used in health care and food industry. Bio-inspired soft gripper has several advantages over existing rigid and complex robotic grippers such as better control of gripping force, adaptability toward the shape of the object, and feel of human touch. [1, 2]. Human fingers are the best example of such gripper made of soft muscle and hard skeleton. While muscles act as actuator, skeleton provides strength and act as a passive support defining the degree of freedom of the finger. Currently, there are many actuation mechanism such as pneumatic structure [3], shape memory alloys (SMA) [4], fluidic elastomeric actuator (FEA) [5], shape morphing polymers (SMP) [6], and dielectric elastomer actuators (DEAs) [7–9] which are being studied extensively for soft robotics. DEAs are very promising due to their lightweight, large deformation, fast response time, and costeffective characteristics. DEAs are soft and highly stretchable resembling mechanical A. Agarwal · A. Baranwal (B) · G. Stephen Sugun · P. K. Agnihotri MAdMatLab, Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_105
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responses of muscles [10]. Basic structure of DEA consists of a dielectric membrane sandwiched between two compliant electrodes. On the application of electric field across the two electrodes, thickness of DE membrane decreases and surface area increases [11]. The deformation of membrane is later converted in useful actuation through various DEA configurations [12]. Among the existing possibilities, bend type minimum energy structure is more suited for gripper application [13, 14]. A minimum energy structure composed of a pre-stretched DEA bonded to a stiffer frame. This arrangement attains equilibrium configuration by bending in-plane on the removal of constraint. Application of electric field changes the state of stress in DE membrane. This results in unfolding of whole structure to reach new equilibrium configuration. In this paper, design and development of a soft three-fingered gripper are proposed based on the application of minimum energy structure of bend type actuator. The skeleton of multi-segmented finger is produced by 3D printing, and PDMS sheet is used as DE membrane. Different fabrication strategies have been proposed to improve the performance of actuator without changing DE membrane material.
2 Design of Gripper Figure 1 shows the proposed design of three-fingered gripper developed in this work. Acrylonitrile butadiene styrene (ABS) was used as the hard core for 3D printed fingers. These fingers were fixed at 120° from each other onto a rigid fixture forming the gripper hand as shown in Fig. 1a. Each finger consists of three segments which were connected by pin joints. This gives a rotational degree of freedom to segments with respect to each other. The tip of finger has smallest length of 18 mm and was coated by a thin film of PDMS layer to provide soft contact to the gripped objects. In order to ensure higher rotation of the tip, the length of subsequent segments increases as we move away from the fingertip. Moreover, the segments of a finger were designed in such a way that their rotation relative to each other would be constrained in 0 < θ < 90 limit. The backward rotation of fingers is constrained by providing an extended feature near the pin joint as shown in the side view of Fig. 1b. The actuation of skeleton was enabled by bonding a 30% pre-stretched PDMS film to each finger (Fig. 1c). After removing the constrained, the pre-stretched film deforms in-plane forming a curved finger configuration of Fig. 1d. In order to optimize the gripper design, different configurations were tested and summarized in Table 1. In design A, carbon grease coated pre-stretched PDMS sheet was glued only on one side of the finger (Fig. 1c). In a rather different arrangement of design B, while one side is bonded with pre-stretched soft layer, an unstretched layer is glued on the other side of skeleton (Fig. 1d). In this configuration, the equilibrium is achieved between two opposing forces of stretched and unstretched PDMS sheets. The actuation of both, design A and design B, is in opening mode. By changing the actuating layer of design B from side 1 to side 2, the same configuration is extended
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Fig. 1 Design of a gripper hand and b single finger. The skeleton configuration having PDMS sheet on c one side and d both sides in design A and design B
Table 1 Design possibilities of fingers explored in present work Name
Side 1
Side 2
Voltage applied membrane of side
Weight (g)
Design A
Pre-stretched PDMS sheet
No sheet bonded
Side 1
5
Design B
Pre-stretched PDMS sheet (125 µm thick)
Unstretched PDMS sheet (125 µm thick)
Side 1
5.4
Design C
Pre-stretched PDMS sheet (125 µm thick)
Unstretched PDMS sheet (125 µm thick)
Side 2
5.4
Design D
Pre-stretched PDMS sheet (250 µm thick)
Unstretched PDMS sheet (125 µm thick)
Side 1
5.7
to achieve closing mode actuation in design C. Later on, the thickness of PDMS layer used in design B is doubled to get design D. Fabrication of PDMS-based DE layer is the next step to process the soft gripper as shown in Fig. 1. To this end, a sacrificial layer of PVA was coated on glass substrate
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Fig. 2 a Fabrication process of PDMS thin film and b removal of PDMS thin sheet from the glass substrate
and a cavity of desired dimensions is formed with the help of acrylic tape. PDMS was poured into this cavity, and excess mixture was scrapped with the help of a sharp blade as shown in Fig. 2a. The uncured PDMS layer was subjected to vacuum for 30 min to remove any air bubble trapped before curing it at 80 °C for 1 h. The glass substrate was then dipped in water to dissolve the PVA layer and collect the PDMS sheet (Fig. 2b).
3 Results and Discussion The actuators were characterized by applying voltage to the PDMS membrane through the carbon grease using a high voltage power supply (IONICS 10 kV, 1 mA). The connections were made using copper tape. The voltage was increased in step of 0.5 kV till the breakdown of PDMS sheet. The actuation was recorded through a video camera and image sequences of video were analyzed by Image J software for actuation displacement of fingertip. Figure 3a compares the tip displacement as a function of applied voltage for various configurations of Table 1. Figure 3a shows that the actuation displacement varies non-linearly with the applied voltage. While design A records largest actuation of 7.8 mm at 3.5 kV, a maximum actuation of 3.8 mm at 5 kV and 2.8 mm at 6.5 kV is observed for design B and D, respectively. This observation reveals that use of PDMS sheet on both sides of skeleton in design B helps to achieve higher breakdown voltage in comparison to design A. Moreover, the increase in thickness of PDMS layer from design B to design D leads to higher breakdown voltage but much lower tip displacement. Design C is the voltage is applied on side 2 of the actuator. It is observed that actuator closes instead of opening in contrast to other three cases. Based on these observations, design B is preferred for further analysis in comparison with other tested configurations. Figure 3b shows the setup used to measure blocking force applied by the finger in design B as a function of the applied voltage. When voltage is increased, the finger opens and tension in string increases which reflects on the weighing scale.
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Fig. 3 a Actuation state of on application of voltage, b plot of actuator displacement against applied voltage, c arrangement for blocking force measurement, d plot of blocking force against applied voltage
Maximum blocking force achieved for design B is 0.87 mN at 4 kV (Fig. 3c) which is comparable to the reported data for similar configuration [15]. As the fingers of design B opens on the application of voltage, no voltage is required during the time when object is gripped. Consequently, lesser energy is required for operation of this type of actuator. To quantify the opening of claw type arrangement on application of voltage, we can assume a circle passing through three points of fingertips. At V = 0, the diameter of this circle is 25 mm which increases to 32 mm at 4 kV. A lightweight object is clamped between the three fingers. When voltage is applied, due to the opening motion of fingers, the object drops down at 3 kV as shown in Fig. 3d. Using the data shown in Fig. 3a, c, the efficiency η, which is defined as the ratio of mechanical energy stored to electrical energy supplied
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(U m /U c ), of each finger is calculated as follows. Mechanical energy U m stored in each finger (design B) is given by the relation [15], Um =
m
Fi (V ) · x
i=0
where F i is the blocking force at the tip as function of applied voltage V at every 0.5 kV step, and x is displacement at same voltage step. The number of steps is represented by m. The values of F i and Δx are given in Fig. 3a, c. From this, the calculated mechanical energy is 3.3 × 10−6 J. On the other hand, the electrical energy U c supplied to griper is calculated as, Uc =
1 CV 2 2
where C is the capacitance of DEA and V is maximum voltage applied. The capacitance is obtained from C = ε0 ε (A/d) where ε0 is the permittivity of the free space, ε is dielectric constant of PDMS, A is the area of electrode and d is the thickness of PDMS sheet. While ε0 = 8.85 × 10−12 F m−1 , the dielectric constant (ε) of PDMS is given in supplier data sheet as 2.6 [16]. The area (A) and thickness (d) of PDMS sheet are continuously changing throughout experiment so dimensions of stretched PDMS sheet A = 80 × 14 mm2 and d = 0.125 mm are used. Putting these values with maximum applied voltage of V = 4 kV from Fig. 3c, the electrical energy supplied to actuator comes out to be U c = 1.4 × 10−3 J. Using the calculated values of U m and U c , the electromechanical efficiency for each finger found to be around 0.23%. The obtained efficiency is relatively lower in comparison with reported values [15] which may be due to the rigid skeleton used to represent bones in human fingers. Improving the actuator efficiency using some modified designs and better actuator material will be the subject of further study.
4 Conclusion In summary, we have developed an actuator having hard skeleton-like human finger and soft skin like properties on surface due to the use of PDMS. It is shown that both opening and closing mode actuation is possible by changing the location of actuating DE layer on skeleton. Maximum displacement in opening mode was 3.8 and 4.7 mm in closing mode. The combined actuation of these two modes will be 8.5 mm. This actuation can be achieved at about 4 kV. This is a very beneficial because we are getting more actuation using same material by using smart design and fabrication method. This study may guide to fabrication of human-like hands which can be applied in the field of soft robotics.
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References 1. Suzumori K (1996) Elastic materials producing compliant robots. Rob Auton Syst 18:135–140 2. Manti M, Hassan T, Passetti G, D’Elia N, Lasachi C, Cianchetti M (2015) A bioinspired soft robotic gripper for adaptable and effective grasping. Soft Rob 2(3):107–116 3. Guoliang Z, Yangdong H, Weiqiang D (2019) A soft pneumatic dexterous gripper with convertible grasping modes. Int J Mech Sci 153–154:445–456 4. Riad A, Alhamany A, Benzohra M (2017) The shape memory alloy actuator controlled by the Sun’s radiation. Mater Res Express 4 5. Barreiros J et al (2019) Fluidic elastomer actuators for haptic interactions in virtual reality. IEEE Robot Autom Lett 4:277–284 6. Bowman CN (2014) Smart shape changing and shape morphing polymeric materials. Polymer 55(23) 7. Kofod G, Paajanen M, Bauer S (2006) Self-organized minimum-energy structures for dielectric elastomer actuators. Appl Phys A 85:141–143 8. Ahmadi S, Mattos AC, Barbazza A, Soleimani M, Boscariol P, Menon C (2012) Fabrication and performance analysis of a DEA cuff designed for dry-suit applications. Smart Mater Struct 22 9. Qiu Y, Zhang E, Plamthottam R, Pei Q (2019) Dielectric elastomer artificial muscle: materials innovations and device explorations. Acc Chem Res 52:316–325 10. Gu G, Zhu J, Zhu L, Zhu X (2017) A survey on dielectric elastomer actuators for soft robots. Bioinspiration Biomimetics 12 11. Suo Z (2010) Theory of dielectric elastomers. Acta Mech Solida Sin 23(6):549–578 12. Romasanta LJ, Lopez-Manchado MA, Verdejo R (2015) Increasing the performance of dielectric elastomer actuators: a review from the materials perspective. Prog Polym Sci 8 13. Kofod G, Paajanen M, Bauer S (2006) Self-organized minimum-energy structures for dielectric elastomer actuators. Appl Phys A Mater Sci Process 85(2):141–143 14. Petralia MT, Wood RJ (2010) Fabrication and analysis of dielectric-elastomer minimumenergy structures for highly-deformable soft robotic systems. In: Proceedings of IEEE/RSJ international conference on intelligent robots and systems, pp 2357–2363 15. Shintake J, Rosset S, Floreano D, Shea HR (2013) Effect of mechanical parameters on dielectric elastomer minimum energy structures. In: Proceedings of SPIE electroactive polymer actuators and devices (EAPAD) 16. Technical data sheet SYLGARD™ 184 silicone elastomer
Experimental and Simulation Study of Haptically Enabled Robotic Teleoperation for NOTES Sarvesh Saini
and Pushparaj Mani Pathak
Abstract Natural Orifice Transluminal Endoscopic Surgery (NOTES) involves the surgical treatment of a patient by reaching the location of surgery through the natural orifices of the human body. In the Robotic-NOTES, there is a bilateral teleoperation system between master and slave robots. Here, the Phantom Omni haptic device that gives motion trajectory and gets the force feedback is used as a master. The surgical manipulators, i.e., a miniature robot that follows the trajectory given by the master and interacts with the environment is used as a slave. In this work, firstly, the study of kinematic relation for the tip position and joint angles of the master and slave is carried out. Then the joint angles for a tip position of the slave robot are calculated for the trajectory planning. After that, a virtual environment is created to get the force feedback for the master from the slave while performing tissue manipulation for the virtual stomach model. Force feedback that we get in a virtual haptic environment will help in training a naive surgeon. Variation in the forces applied to the stomach virtual model and the force feedback in the master device is negligible. Keywords In-Vivo robot · NOTES · Haptic feedback · Teleopration
1 Introduction Inclusion of Robots in the medical field has the potential for every step of treatment like diagnosis, therapeutics, pre-operative planning, post-operative care, hospital carriage and scheduling, quality check, and rehabilitation, etc. [1]. Specifically, the inclusion of robots in surgery has turned out to be a great advantage. For each medical procedure, there is a requirement of some specific structure, intelligent and user friendly interface [2]. Robot-assisted laparoscopy has significantly reduced recovery time and hospital stay [3]. But laparoscopy has its drawback say operating tools with fewer degrees of freedom. The first robotic surgery was introduced in 1985 to S. Saini (B) · P. M. Pathak Mechanical and Industrial Engineering Department, Indian Institute of Technology, Roorkee, Uttarakhand 247667, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_106
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improve accuracy in neurosurgical biopsy [4]. A significant achievement came in the form of an Automated Endoscopic System (AES) for Optimal Positioning (AESOP). The camera was operated by a surgeon using a foot pedal and later using sound recognition. Teleoperation to robotic surgery involves the surgical tool robot as a slave and another robot as a master. Da Vinci has taken open surgery to minimally invasive surgery (MIS) with lesser scars, less blood loss, and reduction in recovery time, thus improving the surgical outcomes. Presently, Single-Port of Access (SPA) and Natural Orifice Transluminal Endoscope Surgery (NOTES) is mostly being pursued by robotic surgery techniques. These involve either a single incision or natural body orifice to introduce the surgical tool into the human body. The development of a robotic platform for NOTES is reported in literature [5]. For NOTES environment, there are three requirements viz. (i) effective force transmission at the tool/endoscope tip, (ii) sufficient DOFs for the desired workspace, for precise positioning of surgical tools, (iii) adequate visualization of the surgical field. Mihir et al., explained the forward kinematic analysis of in-vivo robot for biopsy and surgeries [6], and the technique for improvement in tool tip accuracy is suggested in the work [7]. The introduction of the haptic system to the medical field has become a great advantage. Haptic refers to the science of touch and force feedback during human-computer/robot interaction with two-way communication. Creating a virtual environment (with force feedback) for the surgical training of junior surgeons can be very useful [8]. Bondgraph modeling for hybrid trajectory and force control is presented in literature [9]. It helps them in better understanding of the real situation during surgeries. McDemott et al. explained the assembly of a virtual simulation environment [10]. Salamanca et al. has demonstrated a haptically enabled surgical simulation environment by comparing two platforms [11]. After consideration of the available literature, an attempt was made to build a bilateral teleoperation system for NOTES. The slave is a miniature robot and master is a Phantom Omni haptic device. For this, a miniature robot has 4 DOFs, and it is mounted on the tool channel of an endoscope. The endoscope also having 4 DOFs. There will be an increase in a significant amount of maneuverability in endoscopic-based tissue manipulation. Further, haptically enabled teleoperation will help the surgeon in sensing the amount of force applying in tissue manipulation. Here, in Sect. 2 the kinematic relation between tip position and joint angles is shown for the master haptic device and slave miniature robot. In Sect. 3 experimental description is presented. (a) To find the joint angles for the slave robot for any tip position in the workspace. The joint angles for a tip position are calculated using inverse kinematics for the trajectory planning. (b) Simulation in a haptic environment to get haptic feedback while performing tissue manipulation. The virtual environment is created in Microsoft visual studio using Phantom Omni as master robot and SolidWorks model of miniature robot model as a slave. The haptically enabled virtual model of the stomach has been introduced to get the force feedback for the master from slave while performing tissue manipulation. The variation of forces applied to the stomach model and the master get haptic feedback is negligible. In Sect. 4, the results are discussed, and in Sect. 5, future scope and conclusion are presented.
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2 Method and Meterial 2.1 The Relation Between Joint Angle and Tip Position of Master Device Phantom Omni Forward kinematic analysis is obtained for the master device (Phantom Omni haptic device), as shown in Fig. 1. Phantom Omni is an electromechanical device that can get kinesthetic feedback. It has six DOF stylus with three revolute and three gimbal joints. The encoder placed in the base of the robot with three joints are active joints and three joints are passive joints. Its D-H (Denavit-Hartenberg) parameters are given in Table 1, are obtained from literature [12]. Where a is link length,α is twist angle, d shows link offset and θ is joint angle. Using D-H parameter notation method, a 4 × 4 homogenous transformation matrix was used to relate (i − 1)th coordinate frame to (i)th coordinate frame as given in (1).
Fig. 1 Link lengths and frames for Phantom Omni [13] Table 1 D-H parameters of phantom omni master device Joint no. Parameters αi−1 ai di 1 2 3 4 5 6
π /2 0 −π /2 π /2 −π /2 π /2
0 L1 0 0 0 0
0 0 0 L2 0 0
θi −60∼60 0∼105 −100∼100 −145∼145 −70∼70 −145∼145
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⎤ −sθi 0 ai−1 cθi ⎢ sθi cαi−1 cθi cαi−1 −sαi−1 −sαi−1 di ⎥ i−1 ⎥ ⎢ i T = ⎣ sθ sα cαi−1 di ⎦ i i−1 cθi sαi−1 cαi−1 0 0 0 1 ⎡
(1)
Tip position of master (Phantom Omni) obtained using forward kinematics is given by (2)–(4). (2) xm = − sin θ1 (L 2 sin θ3 + L 1 cos θ2 ) ym = −L 2 cos θ3 + L 1 sin θ2 + L 3
(3)
z m = −L 2 cos θ1 sin θ3 + L 1 cos θ1 cos θ2 − L 4
(4)
where, L 1 = L 2 = 13.335 cm are link lengths and L 3 = 2.335 cm and L 4 = 16.835 cm are workspace transformation offsets between end effector’s origin and the first joint [13].
2.2 The Relation Between Joint Angle and Tip Position of the Slave Miniature Robot In the virtual environment, the position of the Clipper of SolidWorks model of a miniature robot corresponds to the tip of the phantom device in the master workspace. This miniature robot is also termed as in-vivo robot because it is developed to be inserted entirely into the peritoneal cavity for NOTES and Laparoscopic surgery. The miniature in-vivo robot is a four DOF robot consisting of three links and a clipper. A flexible shaft is a connected in-vivo robot using a coupler. A linear actuator placed outside the body provides translatory motion as 1st DOF and helps to cover intricate locations inside the abdominal cavity. Rotation about flexible shaft provides 2nd DOF. The planer motion by articulated links provides 3rd DOF and Clipper provide 4th DOF, which is used for tissue manipulation. To locate the position of the Clipper of miniature in-vivo robot in a virtual environment concerning origin where the coupler is assumed to have an origin, the joint angles for the slave robot are calculated through inverse kinematics algorithm. The D-H parameters for finding the tip position of the slave robot are given in Table 2. Real fabricated and SolidWorks model of in-vivo slave robot is shown in Fig. 2a and b respectively. The tip of the slave/in-vivo robot is given by (5)–(7). Where L represents the link length of the robot. Using (1) homogenous transformation matrix of each frame related to previous frame is 0 0 1 2 3 4 5 6T = 1T 2T 3T 4T 5T 6T ;
Experimental and Simulation Study of Haptically … Table 2 D-H parameters of slave robot Joint no. Parameters αi−1 ai−1 1 2 3 4 5 6
0 −90◦ 0 0 0 0
0 0 L1 L2 L3 L4
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di
θi
d 0 0 0 0 0
φ θ1 θ2 θ3 θ4 0
Fig. 2 a Prototype of in-vivo robot. b SolidWorks model of in-vivo robot [6]
⎡
cφc1234 −cφs1234 ⎢ sφc1234 −sφs1234 0 ⎢ 6 T = ⎣ −s 1234 −c1234 0 0
⎤ −sφ cφ(L 1 c1 + L 2 c12 + L 3 c123 + L 4 c1234 ) cφ sφ(L 1 c1 + L 2 c12 + Ls3 c123 + L 4 c1234 ) ⎥ ⎥ 0 d − (L 1 s1 + L 2 s12 + L 3 s123 + L 4 s1234 ) ⎦ 0 1
The final tip position is as, xti p = cφ(L 1 c1 + L 2 c12 + L 3 c123 + L 4 c1234 )
(5)
yti p = sφ(L 1 c1 + L 2 c12 + L 3 c123 + L 4 c1234 )
(6)
z ti p = d − (L 1 s1 + L 2 s12 + L 3 s123 + L 4 s1234 )
(7)
3 Experiement 3.1 Finding Joint Angles for the Tip Position of the Miniature Robot For the fabricated model, link lengths are equal to 1.1 cm. Figure 3a, b respectively shows the initial and final configuration of the links and the desired position. We take, l1 = L 1 + L 2 , l2 = L 3 + L 4 , d = 0 (no linear motion) and φ=0 (planar motion case)
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Fig. 3 a Initial configuration of the links. b Final configuration of links
to simplify the analysis. Joint angles for the slave robot are calculated using inverse kinematic algorithm for the in-vivo robot as obtained in [14]. Equations (5)–(7) can be solved as the case of a two-link manipulator as given by (8) and (9). x = l1 c1 + l2 c12
(8)
y = l1 s1 + l2 s12
(9)
Here c12 = cos(δ1 + δ2 ) and s12 = sin(δ1 + δ2 ). δ1 and δ2 are the joint angles for the two links of the manipulator. The values of joint angles are obtained using analytical method as mentioned below [15]. Using (8) and (9), we can write, x 2 + y 2 = l12 + l22 + 2l1l2 cos δ2 cos δ2 =
x 2 + y 2 − l12 − l22 2l1l2
If cos δ2 > 1, then point is not in workspace, sin δ2 = ± 1 − cos2 δ2
Now to find δ1 , we can write
(10) (11)
(12)
δ2 = atan(s2 , c2 )
(13)
x = k 1 s1 − k 2 s1
(14)
y = k1 s1 − k2 c1
(15)
where, k1 = l1 + l2 c2 and k2 = l2 s2 consider k1 and k2 k1 = rcosα
(16)
k2 = rsinα
(17)
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Using Eqs. (14) and (15) can be written as x = rcosα. cos δ1 − rsinα.sinδ1
(18)
y = rcosα. sin δ1 + rsinα.cosδ1
(19)
3.2 Simulation Experiment for Haptic Feedback • Force required for the tissue manipulation is depends upon the material properties of the stomach model, i.e., pring constant and damping factor in our model. • Force applied in the y-direction is more than the forces used in x and z-direction as tissue manipulation take place is in the y-direction. • Variation in the force exerted on the stomach model and the force felt in the master device as haptic feedback is negligible. Now, tan(α + δ1 ) =
(y/r ) (x/r )
δ1 = atan2(y, x) − atan2(k1 , k2 )
(20) (21)
The plot given in Fig. 4 shows the joint angles of the slave robot for different tip position of slave robot corresponding to the tip positions of the master device during a typical interaction. Here, dotted line shows the δ1 and solid line shows δ2 for achieving different tip positions during an interaction. It can also be inferred from Fig. 3b. Variation in δ1 is more compare to δ2 . It shows δ1 is more responsible for reach compared to δ2 and δ2 is more responsible for orientation compared to δ1 .
3.3 Simulation in Haptic Environment to Get the Forces For creating a virtual environment, a 3D model of the stomach and the four DOF slave tools were developed in SolidWorks commercial software. Then these models are converted into any of 3DS or PLY or OBJ for compatibility with Visual Studio C++ 2010. The QHHeaderWin32 is the main header file. It includes sub-libraries like global, device space and QHwin32class, that are utilized in the programming. Steps for developing the haptic experiment of the stomach and slave cursor tool (concerning origin) are shown in Fig. 5. The slave robot and stomach model concerning origin Fig. 6a shows that of virtual space, and deformations in the stomach model are shown in Fig. 6b. Force exerted on the stomach model and haptic/master device stylus will follow the mathematical model F = kx − c x˙ (spring-mass-damper) the system with appropriate spring and damping constants [16]. In our experiment, the considered values of mass, spring stiffness and damping constant are 0.1, 0.2 and 0.05
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Fig. 4 Joint angles for different tip positions of slave robot
respectively. Corresponding forces in the stomach model and stylus of the master device are shown in Fig. 7a and b, respectively. The haptic feedback force is slightly less than the interaction force between the stomach and stylus. This difference in the force may be due to friction and signal attenuation in teleoperation. It can be improved by using a better processor and by minimizing joint friction in the haptic device. Overall it is seen that the force in the virtual environment and the one felt in the stylus match each other to a large extent, thus indicating a fair implementation of haptic feedback in the virtual environment.
4 Result and Discussion From the above experiments following results are obtained.
4.1 Joint Angle for the Tip Position of Miniature Robot • Variation in δ1 is more compare to δ2 . δ2 is responsible for reach compare and δ2 is responsible for orientation.
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Fig. 5 Flowchart for creation of virtual environment for haptic experiment
Fig. 6 a Virtual haptic environment showing in-vivo cursor and stomach model with respect to origin. b Deformation of stomach on pressing with slave robot tip/cursor
4.2 Simulation Experiment for Haptic Feedback • Force required for the tissue manipulation is depends upon the material properties of stomach model, i.e., spring constant and damping factor in our model. • Force applied in y-direction is more than the forces applied in x and z-direction as tissue manipulation take place is in y-direction. • Variation in interaction force and stylus haptic feedback force is is negligible.
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Fig. 7 a Variation of absolute force and forces in different direction forces by virtual cursor exerted on stomach model. b Variation of absolute force and its components in master/haptic stylus
5 Conclusion and Future Scope • Teleoperation of Miniature robot can be useful in different fields such as tissue manipulation in in-vivo surgery, hazardous site exploration, etc. • Hyper redundant miniature robot gives different orientation for reaching the same point in the workspace. It will be helpful for the surgeon in tissue manipulation inside the intrinsic workspace of the stomach. • With more realistic properties of the stomach model, this system can be used for surgical simulation for NOTES. • The system can further be developed as real-time bilateral teleoperation system for NOTES. Acknowledgements The financial assistance to Mr. Sarvesh Saini in the form of a Research Fellowship (SRF)(sanction file no. is 09/143(0895)/2017-EMR-I) of the Council of Scientific and Industrial Research (CSIR), New Delhi is greatly acknowledged.
References 1. Taylor RH (2006) A perspective on medical robotics. Proc IEEE 94(9):1652–1664. https://doi. org/10.1109/JPROC.2006.880669 2. Casals A (1999) Medical robotics at UPC. Microproc Microsys 23(2):69–74 (1999). https:// doi.org/10.10007/1234567890 3. Farghaly SA (2013) Current surgical treatment option, utilizing robotassisted laparoscopic surgery in obese women with endometrial cancer: Farghaly’s technique. J Egypt Nat Cancer Inst 25(2):57–61. https://doi.org/10.1016/j.jnci.2013.03.002 4. Ballantyne GH (2002) Robotic surgery, telerobotic surgery, telepresence, and telementoring: review of early clinical results. Surg Endosc interv Tech 16(10):1389–1402. https://doi.org/10. 1016/j.jnci.2013.03.002
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5. Wang Z, Phee SJ, Wong J, Ho KY (2012) Development of a robotic platform for natural orifice transluminal endoscopic surgery. Gastrointest Interv 1(1):40–42. https://doi.org/10.1016/j.gii. 2012.08.010 6. Sutar MK, Pathak PM, Sharma A, Mehta N, Gupta V (2013) Forward kinematic analysis of in-vivo robot for stomach biopsy. J Rob Surg 7(3):281–287. https://doi.org/10.1007/s11701012-0375-y 7. Reiter A, Allen PK (2010) An online learning approach to in-vivo tracking using synergistic features. In: 2010 IEEE/RSJ international conference 2010 intelligent robots and systems (IROS), IEEE (2010), pp 3441–3446. https://doi.org/10.1109/IROS.2010.5650852 8. Maniya JB, Pathak PM, Mishra BK (2010) Design and development of virtual objects to be used with haptic device for motor rehabilitation. J Softw Eng Appl 3(10):990–997. https://doi. org/10.1109/IROS.2010.5650852 9. Saini S, Pathak PM, Orlando MF (2019) Bondgraph modelling for the master-slave robotic teleoperation system. In: 2019 28th IEEE international conference on robot and human interactive communication (RO-MAN), New Delhi, India, pp 1–6 (2019). https://doi.org/10.1109/ RO-MAN46459.2019.8956409 10. McDermott SD (1999) A haptic assembly and disassembly simulation environment and associated computational load optimization techniques. J. Comput Inf Sci Eng. 1(1):113–122. https:// doi.org/10.1115/1.1389085 11. Salamanca MLP, Navarro JMS, Esmeral JS (2011) Analysing collision detection in a virtual environment for haptic applications in surgery. Ingeniera e Investigacin 31(1):204–212. https:// doi.org/10.1115/1.1389085 12. Silva AJ, Ramirez OAD, Vega VP, Oliver JPO (2009) Phantom omni haptic device: kinematic and manipulability. In: Electronics, robotics and automotive mechanics conference. IEEE (2009), pp 193–198. https://doi.org/10.1109/CERMA.2009.55 13. Sansanayuth T, Nilkhamhang I, Tungpimolrat K (2012) Teleoperation with inverse dynamics control for phantom omni haptic device. In: SICE annual conference (SICE), 2012. IEEE, pp 2121–2126 14. Ahmmad SM, Khan M, Rahman M, Billah M (2013) Position control of a four link hyper redundant robotic manipulator. Asian J Scient Res 6(1):67–77 15. Craig JJ (2005) Introduction to robotics: mechanics and control. Pearson Prentice Hall Upper Saddle River (2005) 16. Toolkit O (2005) The SenSable technologies Inc., NJ, USA
Design of Robust Backstepping Controller for Four-Wheeled Mecanum Mobile Robot Zeeshan Ul Islam, Shital S. Chiddarwar, and Saumya Ranjan Sahoo
Abstract The use of mobile and portable robots is increasing in structured and unstructured environments. A robot with high dexterity and maneuverability in confined spaces, such as omniwheeled robots, is required for such tasks. This paper presents the trajectory tracking capability of four-wheeled mecanum mobile robot in the presence of external uncertainties and disturbances. Firstly, the dynamic equation of the robot is derived; then, the efficacy of a robust backstepping (RBS) controller was tested on a reference trajectory. Simulation results prove that the proposed controller tracks trajectory with greater accuracy than other standard controllers. Keywords Mobile robotics · Trajectory tracking · Robust controller · Mecanum wheels · Dynamic modeling
1 Introduction Omnidirectional mobile robots have the added advantage that they can perform translatory and rotatory motion independently and synchronously. This provides high maneuverability well suited for factory floors, warehouses and rescue operations. One such kind of mobile robot is four-wheeled mecanum mobile robot (FWMMR). The wheels have rollers aligned at an angle of ±45◦ with the axis of the wheel, helping it slide in the lateral direction. Our platform has one pair of diagonal wheels having rollers at an angle of 45◦ and the other pair at −45◦ The kinematic [1] modeling of mecanum-wheeled mobile robot is a difficult task which becomes more challenging with increased number of unactuated and actuated joints of the wheels. The dynamic [2] modeling of a FWMMR becomes difficult with
Z. U. Islam (B) · S. S. Chiddarwar · S. R. Sahoo Visvesvaraya National Institute of Technology, Nagpur, India e-mail: [email protected] S. S. Chiddarwar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_107
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increasing number of independently actuated wheels. These modeling for FWMMR have been done before [3], which were majorly torque-based approaches. Here, the modeling is done using input voltage to motors. Uncertainties and external disturbances that will come in real world are also incorporated in the dynamic equation derived by Newton–Euler approach [4]. In the situation of bounded, unknown uncertainty backstepping controller can effectively control a nonlinear dynamics for desired performance. A switching control law is further added to it to make it robust to higher perturbations and effectively and quickly converge the system to required trajectory thus making it a robust backstepping controller (RBS). This paper proposes the above methodology for the control of the FWMMR, and results are compared against well-tuned PID controller and backstepping controller. The main difference between this work and the previous works [5, 6] is as follows: (1) To ensure realistic performance, uncertainty and disturbance are taken into account, (2) the backstepping controller with and without robustness is compared to PID controller in situations having varying uncertainty, (3) motor dynamics is formulated using control input voltage and (4) the controller is designed with Lyapunov stability.
2 Kinematic Modeling The FWMMR is shown here in Fig. 1 in the top view, and there are a total of six frames in the system. The frames associated with the wheels are Owi Xwi Ywi , where i represents the wheel number. The robot frame of motion is Om Xm Ym Zm , and the world frame is On Xn Yn Zn . γi represents the angle of roller of the it h wheel, which is +45◦ for wheels first and third and −45◦ for the second and fourth wheel. Let the position vector Pm = [xm ym φm ]T represents the position of the robot in the Om Xm Ym Zm frame, where xm and ym are the positions of the mobile robot, and φm is the orientation of the mobile robot with respect to the Xm axis. In terms of the wheel’s rotational velocity around the wheel’s axis, i.e., θ˙i and wheel’s radius R the value of velocity vector, P˙ m is given as [˙xm y˙ m φ˙ m ]. ⎤ ⎡ −1 x˙ m ⎣ y˙ m ⎦ = R ⎣ 1 4 1 φ˙ m ⎡
d1 +d2
1 1
−1 1
−1 −1 d1 +d2 d1 +d2
⎡ ⎤ ⎤ θ˙1 1 ⎢θ˙2 ⎥ ⎥ 1 ⎦⎢ ⎣θ˙3 ⎦ 1 d1 +d2 θ˙4
(1)
Taking the rotation matrix of transformation that relates the frame Om Xm Ym Zm and On Xn Yn Zn , the velocity vector P˙ n = [˙xn y˙ n φ˙ n ] in the world reference frame is given as
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Fig. 1 Schematical top view of mobile robot
⎡ ⎤ ⎤ θ˙ ⎡ −C(φ) − S(φ) C(φ) − S(φ) −C(φ) − S(φ) C(φ) − S(φ) ⎢ ˙1 x˙ n ⎣ y˙ n ⎦ = R ⎣−S(φ) + C(φ) S(φ) + C(φ) −S(φ) + C(φ) S(φ) + C(φ)⎦ ⎢θ2 ⎣θ˙3 4 1 −1 −1 1 φ˙ n d1 +d2 d1 +d2 d1 +d2 d1 +d2 θ˙4 ⎡
⎤ ⎥ ⎥ ⎦ (2)
where C(φ) = cos(φ) and S(φ) = sin(φ)
3 Dynamic Modeling In Fig. 2, the forces acting on the mobile robot platform have been shown. The forces due to actuator of the ith wheel in X and Y direction are given as Fxi and Fyi , respectively. The net torque acting on the entire platform by the driving forces is represented by τ . The external force acting on the system is represented by Fe acting at an angle of δ with Ym and at a distance of b from the front wheel’s center. Now, applying the Newton’s second law of translation and rotation, i.e., F = ma, and τ = I α in the frame Om Xm Ym Zm and using rotation transformation matrix, we have P˙ n = Rnm (φ)P˙ m and Fn = Rnm (φ)Fm . Upon solving and rearranging [7], we have x¨ (t)3X 1 = f (˙x)3X 1 + g(x)3X 4 u(t)4X 1 + h(Fe , φ, δ)3X 1 + ξ(t)3X 1
(3)
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Fig. 2 Free body diagram of the FWMMR
where ⎡ f (˙x) =
1 (βx S(2φ)˙yn − βy S(2φ)˙yn + 2βy x˙ n + 2C 2 (φ)(βx − βy )˙xn + 4K1 x˙ n 2M 1 ⎣ (−βx S(2φ)˙xn + βy S(2φ)˙xn + 2βx y˙ n + 2C 2 (φ)(βx − βy )˙yn + 4K1 y˙ n 2M 1 ˙ (−2(d1 + d2 )2 K1 − βz )φ) 2In
⎡ 1 − (Sφ + Cφ) M1 (−Sφ + Cφ) − M1 (Sφ + Cφ) K2 ⎣ 1 M (−Sφ + Cφ) M1 (Sφ + Cφ) M1 (−Sφ + Cφ) g(x) = √ M 1 2 (d + d2 ) − I1n (d1 + d2 ) − I1n (d1 + d2 ) In 1 ⎡ h(Fe , φ, δ) =
1 (2Fe S(φ)S(δ) + 2Fe C(φ)C(δ)) 2M 1 ⎣ (−2Fe S(φ)C(δ) + 2Fe C(φ)S(δ)) 2M 1 (d F C(δ) − d2 Fe S(δ) + bFe S(δ)) 2In 1 e
⎤ ⎦
⎤ 1 (−Sφ + Cφ) M 1 (Sφ + Cφ) ⎦ M 1 (d + d2 ) In 1
⎤ ⎦ ξ(t) = [ξx ξy ξφ ]T
where C(φ) = cos(φ), S(φ) = sin(φ), C(δ) = cos(δ) and S(δ) = sin(δ) and K1 and K2 are motor coefficients in the driving force by motor given as Fdr = K1 Rθ˙ + K2 u [8] and ξ(t) is uncertainty in the environment.
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4 Controller Design In backstepping(BS) control design [9, 10], the system is firstly broken down into smaller subsystems, and input of each subsystem is considered as virtual input. All the virtual control laws are integrated by stepping back to generate the actual control law, and its stability is taken care of by Lyapunov function. Let the position vector be defined by x1 = [xn yn φn ]T , velocity vector by x2 = [˙xn y˙ n φ˙ n ] and control inputs by u(t) = [u1 u2 u3 u4 ]T . The two subsystems are given as x˙ 1 = x2
(4)
x˙ 2 = f (x) + g(x)u(t)
(5)
where x2 acts as the virtual control input for subsystem 1. The errors are defined as e1 = [x1 − x1d ] = [xn − xnd yn − ynd φn − φnd ]T
(6)
e2 = [x2 − x2d ] = [˙xn − x˙ nd y˙ n − y˙ nd φ˙ n − φ˙ nd ]T
(7)
where x1d and x2d represent desirable position and velocity, respectively. Taking the Lyapunov function as V (e1 , e2 ) =
1 1 [e1 ][e1 ]T + [e2 ][e2 ]T 2 2
V˙ (e1 , e2 ) = [˙e1 ][e1 ]T + [˙e2 ][e2 ]T
(8) (9)
To have V˙ (e1 , e2 ) ≤ 0, we take [˙e1 ] = −Ka [e1 ] and [˙e2 ] = −Kb [e2 ] where Ka and Kb are 3X 3 positive definite matrix. Thus, we have [˙x1 ] = [˙x1d ] − [Ka ][e1 ]
(10)
[˙e2 ] = [˙x2 ] − [¨x1d ] − [Ka ]2 [e1 ]
(11)
From Eqs. (4, 5) and (10, 11) and the Lyapunov design conditions, the backstepping controller is given as in Eq. (12), and the discontinuous switching controller is given as in Eq. (13). u(t)bs = g −1 (x)[Ka2 [e1 ] + [¨x1d ] − f (x) − h(Fe , φ, δ) − Kb [e2 ]]
(12)
u(t)sw = g −1 (x)[Kc sat(s/ψ)]
(13)
where s = λ1 e1 + e˙ 1 and is called weighted average error, and sat is defined as sat(x) = x, if |x| < δ and sat(x) = Sign(x), if |x| ≥ δ, and Sign(x) = 1, ifx ≥ 0 and
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Sign(x) = −1, ifx < 0. Here, ξ(t)max ≤ |Kc | and λ are 3X 3 matrix The robust backstepping controller is finally given as u(t) = g −1 (x)[Ka2 [e1 ] + [¨x1d ] − f (x) − h(Fe , φ, δ) − Kb [e2 ]] + g −1 (x)[Kc sat(s/ψ)] (14)
5 Simulation Results To verify the effectiveness of the proposed control method, a reference nonlinear trajectory tracking is done with external uncertainties and disturbances. The values of the parameters of the robot [7] are taken as M = 6 kg, In = 0.0945 kgm2 , d1 = 11 cm d2 = 18 cm b = 10 cm βx = βy = βz = 0.02 K1 = 0.087 N/V K2 = −11.4 kg/s, and the boundary layer thickness is chosen as δx = 0.5 m δy = 1 m and δφ = 0.25 rad. Results of error in trajectory tracing have been calculated in the form of integral square error (ISE), integral average error(IAE) and integral time average error (ITAE). The equation of trajectory ∀ t in (0 s, 30 s) xd =
3 sin(t) 9 sin(t) cos(t) yd = φd = 0 2 1 + sin (t) 1 + sin2 (t)
(15)
and the uncertainty is given as ξx (t) = [15 sin(t) ξy (t) = 17 sin(t) ξφ (t) = 13 sin(t)]T
(16)
∀ t in (0 s, 30 s) The external disturbing force is given as Fe = 10 N for 4 ≤ t ≤ 6 and δ = 1 rad for 4 ≤ t ≤ 6 The parameters of the RBS controllers are obtained after over 250 iterations of parameter estimation toolbox in MATLAB R2016a. ⎡
⎤ ⎡ ⎤ ⎡ ⎤ 64.06 0 0 128.08 0 0 5.34 0 0 Ka = ⎣ 0 70.29 0 ⎦ Kb = ⎣ 0 140.04 0 ⎦ λ = ⎣ 0 17.14 0 ⎦ 0 0 60.04 0 0 140.63 0 0 2.6 and Kc is a diagonal 3X 3 matrix with each value = 20 The initial position and orientation of the FWMMR is [xn yn φn ] = [3 0 0]T . From Fig. 3, it can be observed that the RBS controller tracks the trajectory with accuracy greater than PID and backstepping. PID undergoes a lot of chatter in the last few seconds. However, RBS follows the reference path smoothly. Figure 4 shows the control input voltage supplied to actuators, and RBS follows a sinusoidal curve smoothly but PID undergoes rapid fluctuation for the entire interval. In practice, this will damage the actuator. Hence, for practical purposes, RBS is a better choice. The errors of RBS, PID and backstepping are compared in Table 1. Figures 5 6 and 7 show the magnitude of weighted error in all the three degrees of freedom with respect
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4 Desired PID RBS Backstepping
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Fig. 3 Trajectory tracing plot
to the boundary layer for the RBS controller. It can be clearly seen that the system is constrained within the boundary layer, except for certain time instants in Y axes, which maybe perhaps due to slightly higher uncertainty in Y direction.
6 Conclusion and Future Scope In this paper, a robust backstepping controller for FWMMR is developed. The robot platform is given external disturbances and nonlinear uncertainties. The dynamic equation of the FWMMR is developed in input affine form. Using the backstepping technique of integrating all the virtual inputs of the subsystem, the robust backstepping control law is developed. The stability of the control law is verified using the Lyapunov method. Lastly, simulations were performed to check the efficacy of the proposed controller on a reference path, and it keeps the system safely within boundary limits. The proposed controller outperforms PID and backstepping in terms of trajectory tracking, error magnitude and practical usability. In the future, the controller would be developed in a way to have a dynamically adjusting boundary layer that would provide two-fold benefits of reducing the controller bandwidth and also to improve accuracy of the trajectory tracing. The control policy would be implemented on the hardware system in order to determine the practical feasibility and efficacy of the controller.
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-4
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Fig. 4 Plot of control input voltage versus time
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Table 1 Tracking performance comparison Controller ISE(m) IAE(m) RBS
[0.14 319.22 0.054]T 10−6 [8.09 5.09 9.971]T 10−4 [4.34 × 10−8 0.003 0.002]T
PID BS
ITAE(m)
[0.0053 2.5
0.003]T 10−3
[4.00 3.20 1.40]T 10−3 [2.94X10−5 0.007 0.006]T
[0.080 3.792 0.049]T [6.03 4.79 2.11]T [0.04 11.5 9.65]T
1 Weighted Error Limits of Boundary Layer
Weighted Error in X(m)
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
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Fig. 5 Plot of weighted error in X and boundary layer 2 Weighted Error Limits of Boundary Layer
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Fig. 6 Plot of weighted error in Y and boundary layer
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Weighted Error in phi(rad)
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Fig. 7 Plot of weighted error in phi and boundary layer
References 1. Tzafestas SG Introduction to mobile robot control. Elseiver Publication 2. Tlale N, De Villiers M (2008) Kinematics and dynamics modeling of a mecanum wheeled mobile platform. In: 15th International conference on mechatronics and machine vision in practice. IEEE, pp 657–662 (2008) 3. Abdel Rahman M et al (2014) A description of the dynamics of a four-wheel Mecanum mobile system as a basis for a platform concept for special purpose vehicles for disabled persons. In: 58th Ilmeanau Scientific Colloquium, Technische Universität Ilmenau, 08–12 2014 4. Mittal RK, Nagrath IJ Robotics and control. Tata Macgraw Hill Publication 5. Villers M, Bright G (2010) Development of a control model for a four wheel mecanum vehicle. In: 25th International conference of CAD/CAM, robotics and factories of the future conference, 13–16 2010 6. Hamid KR (2014) Adaptive tracking with external force disturbance rejection for uncertain robotic systems. Int J Control, Automat Syst 12(1):169–176 7. Alakshendra V, Chiddarwar S (2016) Adaptive robust control of mecanum wheeled mobile robot with uncertainties. Springer, Non Linear Dyamics 8. Viet TD, Doan PT, Hung N, Kim HK, Kim SB (2012) Tracking control of a three-wheeled omnidirectional mobile manipulator system with disturbance and friction. J Mech Sci Technol 26(7):2197–2211 9. Behera L, Kar I (2010) Intelligent systems and control: principles and applications, 2nd edn. Oxford University Press 10. Sahoo RS, Chiddarwar S, Alakshendra V (2017) Intuitive dynamic modeling and flatness-based nonlinear control of a mobile robot. Sage Publication: Simulation: Transactions of the Society for Modeling and Simulation International (2017)
Dynamic Analysis of a Magnetohydrodynamic Journal Bearing of Circular Cross Section in a Rotating Coordinate Frame Debasish Tripathy and Kingshook Bhattacharyya
Abstract An infinite journal bearing of circular cross section lubricated with an electrically conducting fluid has been considered for this study. Unlike previous studies involving such bearings encountered by the authors, a rotating coordinate system has been used for deriving the dynamic parameters of the bearing. In addition to the above, the electric field and flow rate relations have been perturbed by use of electromagnetic boundary conditions to eliminate the same from the Reynolds equation. The basic Reynolds equation and its perturbed forms have been solved analytically to derive the stiffness and damping coefficients in the rotating coordinate frame. Keywords Magnetohydrodynamic · Bearing · Dynamic stiffness · Damping
1 Introduction Magneto hydrodynamic bearings first evoked interest at the beginning of the second half of the last century. Initial studies by Kuzma [1, 2], and Sasada et al. [3] considered a fully flooded bearing. Kamiyama [4] introduced Reynolds boundary condition thus considering a partially flooded bearing. Similar studies were also carried out by Shvarts [5, 6] and Dudzinsky et al. [7]. Malik and Singh [8] extended such analyses to the case of a finite bearing. Stability behavior of finite magnetohydrodynamic journal bearings was reported by Kulkarni and Rao [9]. In the literature studied, there were no instances of studies of dynamic parameters of such bearings or their stability in a rotating coordinate frame. The present study is an effort to fill up this void. In the process of attempting the same perturbed forms of the electromagnetic boundary conditions and the consequent flow rate and electric field, relationships have been derived and introduced in the formulations which will be discussed next.
D. Tripathy · K. Bhattacharyya (B) Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_108
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Fig. 1 Schematic diagram of MHD journal bearing
2 Formulations 2.1 Assumptions The bearing is assumed to be infinitely long and of circular cross section as shown in Fig. 1. The radius of the shaft is R, the center-to-center distance between the bearing and the shaft is e, with the maximum being the clearance C, while the attitude angle is φ. W is the weight of the shaft. The x, y coordinate system is as described in the figure. The film thickness h is a function of x. The shaft rotates with a speed of ω and processes with a speed of ωp . The external magnetic field is radial and is given by the relation B = −B0 R/r , where r is the radial distance of a point within the bearing from the origin. The viscosity of the lubricant is μ, the electrical conductivity is σ, the pressure of the fluid film is p, and ez is the electric field in the z-direction.
2.2 Non-Dimensionalization and Function Definitions The non-dimensional variables are defined as follows y h u pC 2 x ,P = ,M = θ = ,η = , H = ,U = R h C ωR μω R 2
σ B0 C, E = μ
σ C ez μ ωR
∂ We define a variable = (θc + ε0 sin θc )−1 and operators Dη = ∂η , Dθ = ∂θ∂ , Dτ = ∂τ∂ . Table 1 provides a list of functions used in the derivations that follow.
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Table 1 Functions related to pressure profile calculations f i = f (M Hi ) = tanh(0.5M Hi )
gi = g(M Hi ) = 2 tanh(0.5M Hi ) − M Hi
f i = f (M Hi ) = Msech2 (0.5M Hi ) θ F01 (θ) = 0 g0−1 dθ θ F1Im1 (θ) = 0 M 2 Mg0−1 + θc dθ
gi = g (M Hi ) = −M tanh2 (0.5M Hi ) θ F02 (θ) = 0 f 0 g0−1 dθ θ F1Re1 (θ) = 0 (M E 0 − Dθ P0 )g0−1 g0 cos θdθ
F2Im1 (θ) = F1Re2 (θ) = F2Re2 (θ) =
θ 0
θ 0
M 2 Mg0−1 + θc dθ
F2Re1 (θ) =
0.5M 2 g0−1 f 0 cos θdθ
F1Im2 (θ) =
θ
2 −1 0 0.5M g0 f 0
sin θdθ
F2Im2 (θ) =
θ 0
θ 0
θ 0
(M E 0 − Dθ P0 )g0−1 g0 sin θdθ M 2 Mg0−1 sin θ + (1 − cos θc ) dθ M 3 g0−1 cos θ + sin θc − 2 f 0 ε0−1 dθ
2.3 System Equations In keeping with Ref. [1], but with the following velocity, boundary conditions commensurate with Couette flow of a rotating coordinate frame, namely U (η = 0) = 1 −
ωp ωp = 1 − Dτ φ, U (η = 1) = − = −Dτ φ ω ω
(1)
The complete solution for the velocity profile is in any radial section of the fluid film which is given in non-dimensional form by
U = −M −2 (Dθ P − M E) + cosh(M H η) M −2 (Dθ P − M E) − Dτ φ + 1
M H −2 M (Dθ P − M E) − Dτ φ + coth(M H ) − sinh(M H η) tanh 2 (2) The non-dimensional flow rate is given as 1 Q=
U H dη = M −3 (Dθ P − M E)g + M −1 (1 − 0.5Dτ φ) f
(3)
0
The conservation of mass equation and the consequent non-dimensional Reynolds equation in rotating frame is Dθ Q + Dτ H = Dθ M −3 (Dθ P − M E)g + M −1 (1 − 0.5Dτ φ) f + Dτ H = 0 (4) Following Ref. [1], for an insulated bearing in open circuit condition, we have
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D. Tripathy and K. Bhattacharyya
θc
θc H (θ )dθ = M
E 0
Qdθ θc : cavitation boundary
(5)
0
2.4 Perturbed Equations and Solutions We introduce the following perturbations which are standard in stability analysis of hydrodynamic bearings. Additionally, the electric field is also perturbed. H0 = 1 + ε0 cos θ, H = H0 + cos θ ε1 eiλ R τ + sin θ ε0 φ1 eiλ R τ , ωp ε = ε0 + ε1 eiλ R τ , λ R = ω iλ R τ φ = φ0 + φ1 e , P = P0 + P1 ε1 eiλ R τ + P2 ε0 φ1 eiλ R τ , E = E 0 + E 1 ε1 eiλ R τ + E 2 ε0 φ1 eiλ R τ
(6)
A new form of perturbation of the flow rate is introduced in order to handle the electric field variables and the boundary conditions in a convenient fashion, namely Q = Q 0 + (Q 1Re + i Q 1Im0 + i Q 1Im sin θ )ε1 eiλ R τ + (Q 2Re + i Q 2Im0 + i Q 2Im cos θ )ε0 φ1 eiλ R τ
(7)
The perturbed flow rates and corresponding equations are obtained from (3) and (4) Q 0 = M −3 (Dθ P0 − M E 0 )g0 + M −1 f 0 , Dθ Q 0 = 0
(8)
Q 1Re = M −3 (Dθ P1Re − M E 1Re )g0 + 0.5M 2 f 0 + (Dθ P0 − M E 0 )g0 cos θ , Dθ Q 1Re = 0 (9) Q 1Im0 + Q 1Im sin θ = M −3 (Dθ P1Im − M E 1Im )g0 , Dθ [Q 1Im0 + Q 1Im sin θ ] + λ R cos θ = 0 (10)
Q 2Re = M −3 (Dθ P2Re − M E 2Re )g0 + 0.5M 2 f 0 + (Dθ P0 − M E 0 )g0 sin θ , Dθ Q 2Re = 0 (11) Q 2Im0 + Q 2Im sin θ = M −3 (Dθ P2Im − M E 2Im )g0 λ R f0 , Dθ [Q 2Im0 + Q 2Im sin θ ] + λ R sin θ = 0 −2 ε0 M
(12)
Dynamic Analysis of a Magnetohydrodynamic Journal …
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The perturbed electric fields are obtained from (5) = (θc + ε0 sin θc )−1 , E 0 = Mθc Q 0 E 1Re = Mθc Q 1Re − Mθc sin θc 2 Q 0 , E 1Im = Mθc Q 1Im0 + M(1 − cos θc )Q 1Im E 2Re = Mθc Q 2Re − Mθc (1 − cos θc )2 Q 0 , E 2Im = Mθc Q 2Im0 + M sin θc Q 2Im Applying Reynolds boundary conditions P0 (0) = P0 (θc ) = Q0 =
(13) ∂ P0 ∂θ (θc )
= 0, we get
M f 0 (θc ) − E 0 g0 (θc ) , P0 (θ ) = M 3 Q 0 F01 (θ ) − M 2 F02 (θ ) + M E 0 θ M2
(14)
Applying boundary conditions Pi (0) = Pi (θc ) = 0 for the perturbed pressures, we get M 2 θc2 2 sin θc Q 0 + F1Re2 (θc ) − F1Re1 (θc ) , F1Re3 (θc ) + M 2 θc2 F1Im2 (θc ) , Q 1Im2 = −λ R Q 1Im1 = λ R F1Im1 (θc ) M 2 θc 2 (1 − cos θc )Q 0 + F2Re2 (θc ) − F2Re1 (θc ) , Q 2Re = F2Re3 (θc ) + M 2 θc2 F2Im2 (θc ) , Q 2Im2 = −λ R Q 2Im1 = λ R F2Im1 (θc )
P1Re (θ ) = F1Re1 (θ ) − F1Re2 (θ ) + Q 1Re M 3 F01 (θ ) + M 2 θc θ Q 1Re =
(15)
− M 2 θc sin θc Q 0 2 θ
P2Re (θ ) = F2Re1 (θ ) − F2Re2 (θ ) + M 3 F01 (θ ) + M 2 θc θ Q 2Re − M 2 θc (1 − cos θc )2 Q 0
(16)
F1Im2 (θc ) P1Im (θ ) = λ R F1Im1 (θ ) − F1Im2 (θ ) , F1Im1 (θc ) F2Im2 (θc ) P2Im (θ ) = λ R f 2Im1 (θ ) − λ R f 2Im2 (θ ) F2Im1 (θc )
The stiffness and damping coefficients are evaluated following [9] as θc S R R = −2
θc P1Re (θ ) cos θ dθ , Sφ R = −2
0
P1Re (θ ) sin θ dθ , 0
(17)
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D. Tripathy and K. Bhattacharyya
θc S Rφ = −2
θc P2Re (θ ) cos θ dθ , Sφφ = −2
0
DR R = −
D Rφ
2 λR
2 =− λR
P2Re (θ ) sin θ dθ
(18)
0
θc P1Im (θ ) cos θ dθ , Dφ R = − 0
θc P2Im (θ ) cos θ dθ , Dφφ 0
2 λR
2 =− λR
θc P1Im (θ ) sin θ dθ 0
θc P2Im (θ ) sin θ dθ
(19)
0
3 Results and Discussion The pressure profiles for a bearing operating at ε = 0.8 with variation in Hartmann number (M) are shown in Fig. 2 The corresponding variation of non-dimensional stiffnesses and dampings (for λR = 1) are shown in Fig. 3. The pressure profiles corroborate results of [4] and indicate the effectiveness of the magnetic field in bolstering the load-bearing capabilities of a fluid film bearing (Fig. 2). However, the fact that a strong magnetic field is a mixed blessing is brought out by the variation of stiffness and damping characteristics with increasing M. While S RR and S Rφ show a decreasing trend with M, S φφ increases with M, and S φ R increases initially followed by a decrease. Similarly DRR and DRφ show an increasing trend with M, while Dφφ and Dφ R decrease. Hence, we may conclude that simply increasing M can result in instability of the bearing as the effect of M on the different stiffness and damping coefficients is not always favorable and a stability analysis is needed to design a bearing with optimal M. Fig. 2 Variation of P0 with M
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Fig. 3 Variation of non-dimensional stiffness and damping coefficients with M
References 1. Kuzma DC (1963) The magnetohydrodynamic journal bearing. J Basic Eng Trans ASME Ser D 85(3):424–427 2. Kuzma DC (1964) The finite magnetohydrodynamic journal bearing. J Basic Eng Trans ASME Ser D 86(4):445–448 3. Sasada T, Kurosaki Y, Honda K, Kamijo K (1974) MHD journal bearing in a magnetic field perpendicular to its axis 1st report, analysis of an infinitely long bearing. Bull JSME 17(114):1645–1651 4. Kamiyama S (1969) Magnetohydrodynamic journal bearing (report 1). Trans ASME J Lubr Technol 91:380–386 5. Shvarts IA (1966) Magnetohydrodynamic lubrication theory for a cylindrical bearing. Mekhanika Zhidkosti i Gaza 1(1):9–15 6. Shvarts IA (1966) Investigation of the basic characteristics of the MHD bearing, Izv. AN SS13R. Meklaanika Zhidkosti i Gaza 1(4):189–191 7. Dudzinsky SJ, Young FJ, Hughes WF (1968) On the load capacity of the MHD journal bearing. Trans ASME J Lubr Technol 90(1):139–144 8. Malik M, Singh DV (1980) Analysis of finite magnetohydrodynamic journal bearings. Wear 64:273–280 9. Kulkarni PA, Rao BVA (1977) Stability behaviour of finite MHD journal bearings. Mech Mach Theory 12:293–302
Synergistic Effect of Pocket and Bionic Texture on the Performance Behaviours of Thrust Pad Bearing J. C. Atwal and R. K. Pandey
Abstract Performance behaviours of a sector-shaped hydrodynamic thrust bearing have been presented herein, incorporating the combined geometry of a pocket and a fish (Rohita Labeo) scale texture on the pad surface. The governing Reynolds equation incorporating the mass conservation algorithm is discretized employing FEM and then the solution of the sets of the equations by applying FBNS method. Based on the numerical investigation, it is found that the textured pad enhances the film load capacity and reduces the friction coefficient than the conventional pad case. Keywords Thrust pad bearing · Textured pad · Fish scale texture
Nomenclature a b h h h1 h2 hd hT Mr n N Nf N fp Nr Nθ p
Fish texture width (m) Spacing of fish textures (m) Film thickness (m) = h/ h 2 Inlet film thickness (m) Minimum (exit) film thickness (m) Pocket/Texture depth (m) = (h + h d )/ h 2 = (2r z + b)/(a + b) Unit normal vector Shape function = θ f /θc = (θ f − θ p )/θc Total elements in radial direction Total elements in sliding direction Gauge pressure (N/m2 )
J. C. Atwal (B) · R. K. Pandey Department of Mechanical Engineering, I.I.T. Delhi, New Delhi 110016, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_109
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p r r r initial rz R1 R2 R (or B) θf θ0 θp μ0 φ
J. C. Atwal and R. K. Pandey
= ph 22 /(μ0 R12 ) Radial coordinate (m) = r/R1 = (0.5 × R − r z ) Width of pocket in r direction (m) Inner radius of bearing (m) Bearing outer radius (m) Bearing width (m) Circumferential extents of pocket/texture Pad angle (degree) Circumferential extents of pocket in case of pocketed-fish textured pad Viscosity 40 °C temperature (N-s/m2 ) Speed (s− 1 ) Cavity or void fraction (1 − ρ/ρ0 )
1 Introduction Due to growing awareness towards the energy and resource conservations, worldwide efforts are being made by the investigators to build up energy-efficient pad thrust bearings. A hydrodynamic thrust pad bearing is used for guiding and supporting the axially (thrust) loaded rotors efficiently. It is essential to mention here that fluid film pad thrust bearing involves smearing of the oil at the pad and runner interface, which produces significant frictional losses. In addition to this, it has been found that 25– 50% of total fluid film bearing losses occur in feeding the oil to the bearings. This loss arises due to power accounted in accelerating the oil in feed ports and grooves and against the friction drag on the wetted shaft and thrust collar. This demonstrates that it is an important task to decrease the power losses in bearings. Presently different surface textures (involving different shapes of nano and micro geometries) and macropockets are being employed by the researchers for exploring and improving the tribo-performances of the different bearings [1–4]. It has been demonstrated that for low film thickness ratios (in the range of 1.0001–1.5), the textures provided on the sliders/pads towards the oil entry side yield the substantial-high loadcarrying capacity and reduced friction coefficient [5–9]. Employing microgrooves of the square, trapezoidal, circular and triangular cross-sectional shapes on pads’ surfaces, numerical investigations have been conducted by the authors to find the best performing cross-sectional shape of groove among these [6, 7]. This investigation revealed the best results with the microgrooves of square cross-sectional shape positioned towards the entry region on the pad. From the presented literature review, it is noticed that the design of surface textures with tiny geometries and its positioning on the pad surface are very important from the performance perspective of fluid film bearing. Moreover, it is also observed that conceiving the new textures for exploring to improve further the performance
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behaviour of thrust pad bearings is an important research task. Hence, the main objective of this paper is set to conceive new textures on the thrust pad for numerical investigation and, accordingly, to achieve the best performing texture ever reported in the literature.
2 Governing Equations and Computational Procedure The partial differential equation employed in the numerical simulation and the computational approach adopted is presented in this section. Figure 1a, b shows the CAD model of sector-shaped thrust pads and coordinate system used in this paper. However, Fig. 1c–e illustrate through the CAD model the geometries of the pocket, the pad having fish texture and pocket with fish textured pads for visualization point of view. Two views of these pockets/textures are shown in Fig. 1f–h with major variable symbols.
2.1 Film Thickness Expressions The film thickness variation between thrust collar and the conventional pad is taken in the numerical model using the following relation: h = h 1 − (h 1 − h 2 )(θ/θ0 )
(1)
In the presence of pocket/texture on the surface of the pad, the film thickness relation is modified as follows:
Fig. 1 a CAD model of sector-shaped pad thrust bearing; b employed coordinate system; c CAD model of pocketed pad d CAD model of the fish textured pad; e CAD model of the pocketed-fish textured pad; f two views of the pocketed pad; g two views of the fish textured pad; h two views of the pocketed-fish textured pad
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J. C. Atwal and R. K. Pandey
h T = h + hd
(2)
where h d = h, h is depth of pocket/texture. 1.
Expression of h d in the presence of pocket The equations for defining pocket, which is shown in Fig. 1f, are written as below: θ = 0; θ = θ f ; r = (R/2 − r z ); r = (R/2 + r z )
(3)
Using Eq. (3), the film thickness expressions for pocketed pad are as follows: hd = 2.
h; if θ ≤ θ f ; (R/2 − r z ) ≤ r ≤ (R/2 + r z ) 0; else
(4)
Expression of h d in the presence of fish texture The equations for boundaries are: ri = (R − 2r z )/2; rl = ri + (b + a)(k − 1) + θ ; ru = rl + (b + a)(−1 + k) + a; Mr = (b + 2r z )/(b + a); ru = a + (b + a)(−1 + k) + rl ; rl1 = ri + 2θc + (b + a)(−1 + k) − θ ; ru1 = rl1 + a + (k − 1)(a + b) (5) where k = 1, 2, . . . , Mr . If ‘N f ’ is an even number, taking all values of ii for a value of k yields the following expressions: ⎛
⎞ h ; h d = ⎝ h ⎠ ; 0 ;
⎫ if(ii − 1)θc ≤ θ ≤ iiθc ; rl ≤ r ≤ ru ; ii = 1, 3, 5, . . . , (N f − 1) ⎬ if(−1 + ii)θc ≤ θ ≤ θc ii; rl1 ≤ r ≤ ru1 ; ii = 2, 4, 6, . . . , N f ⎭ else
(6) If ‘N f ’ is an odd number, taking all value of ii for a value of j results, the following expressions: ⎛
⎞ h ; h d = ⎝ h ⎠ ; 0 ;
⎫ if(ii − 1)θc ≤ θ ≤ iiθc ; rl ≤ r ≤ ru ; ii = 1, 3, 5...N f ⎬ if(θc ii − θc ) ≤ θ ≤ iiθc ; rl1 ≤ r ≤ ru1 ; ii = 2, 4, 6...(N f − 1) ⎭ else
(7) 3.
Expression of h d in the presence of pocketed-fish texture Equations (3)–(7) are employed synergistically for modelling of film thickness for the pocketed-fish textured pad.
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2.2 Reynolds Equation for Computation of Pressure in the Film Reynolds equation incorporating the mass-conserving cavitation model is expressed as follows [10]:
3
3 1 ∂ h ∂p ∂((1 − φ)h) 1 ∂ h ∂p r = + 2 r ∂r 6μ ∂r r ∂θ 6μ ∂θ ∂θ
(8)
where the lubricant’s density ‘ρ’ is constant and is equal to ‘ρ 0 ’ in the full film zone; φ is cavity fraction, which is equal to zero in the full film zone, and it varies between 0 and 1 in the cavitation zone. The solution of Eq. (8) provides the values of p and φ. The constraint pertaining to mass conservation is modelled using Fischer–Burmeister–Newton–Schur (FBNS) algorithm [11]. FBNS approach leads to the following equation: p+φ−
p2 + φ 2 = 0
(9)
Equation (8) has been discretized employing the FEM. In this method, Eq. (8) is multiplied with weight function N, which is assumed to be differentiable once for r and θ, followed by integration of the resulting equation over the element domain yields:
e
where F1 =
3
1 ∂ ∂h ∂ ∂φh + N r dr dθ = 0 (F2 ) − (F1 ) + r ∂r ∂θ ∂θ ∂θ
h r ∂p, 6μ ∂r
F2 =
3
1 h ∂p . r 2 6μ ∂θ
∂N ∂ ∂ ∂ F1 ∂ ∂N F1 + N F1 or − N = F1 − (N F1 ) (N F1 ) = ∂r ∂r ∂r ∂r ∂r ∂r ∂ ∂θ
(F1 N ) = N
(10)
∂ F1 ∂θ
+ F1
∂N ∂θ
or − N
∂ F2 ∂θ
=
∂N ∂θ
F2 −
∂ ∂θ
(N F2 )
(11) (12)
Using Eqs. (11) and (12) in Eq. (10) results:
∂φh ∂N ∂N 1 ∂ ∂ ∂h + F2 − N + N r dr dθ = 0 (F N1 ) − F1 (F2 N ) − r ∂r ∂r ∂θ ∂θ ∂θ ∂θ
e
Using the component form of gradient theorem in Eq. (13) yields:
(13)
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J. C. Atwal and R. K. Pandey
∂ (N F1 )dr dθ = ∂r
e
e
N F1 n r ds
(14)
r N F2 n θ ds
(15)
e
∂ ∂θ
(N F2 )r dr dθ = e
Using Eqs. (14) and (15), the following relation is obtained: 3 3 1 h ∂N ∂ p h ∂N ∂ p ∂φh ∂h + 2 +N −N r dr dθ 6μ ∂r ∂r r 6μ ∂θ ∂θ ∂θ ∂θ e 3 3 h ∂p 1 h ∂p + − N nr r n ds = 0 6μ ∂r r 6μ ∂θ θ
(16)
e
3
h r ∂∂rp + Assuming qn = n r 6μ
e
3
1 h ∂p n , r 6μ ∂θ θ
Eq. (16) becomes:
3 3 1 h ∂N ∂ p h ∂N ∂ p ∂φh ∂h + 2 +N −N r dr dθ − N qn ds = 0 (17) 6μ ∂r ∂r r 6μ ∂θ ∂θ ∂θ ∂θ e
Now p and φh are approximated in an element using the following expressions: p e (r , θ ) =
n
p ej N j , φ e h(r , θ ) =
j=1
n
φ ej h j N j
(18)
j=1
Using Eq. (18), Eq. (17) is written as: ⎡
⎤ 3 3 n n n ∂Nj ∂Nj ∂Nj h h ∂ N 1 ∂ N ∂h e e e ⎣ pj pj φjh j + 2 +N − N ⎦r dr dθ 6μ ∂r ∂r ∂θ ∂θ ∂θ r 6μ ∂θ j=1 j=1 j=1 e (19) − N qn ds = 0
e
The ‘2n’ independent equations are needed to find the values e e algebraic e p , p , . . . , p of 2n unknowns n . Therefore, n independent shape functions 1 2 e e N1 , N2 , . . . , Nne are used for generating n equations and other n equations are generated found complimentary constraints. Hence, the ith equation is obtained by substituting N = Nie in Eq. (19) as follows:
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⎫ ⎧
3 3 n ⎨ ⎬ h ∂ Ni ∂ N j 1 h ∂ Ni ∂ N j r dr dθ p ej + 2 ⎭ ⎩ 6μ ∂r ∂r r 6μ ∂θ ∂θ j=1 e ⎫ ⎧
n ⎨ ⎬ ∂Nj ∂h e Ni h j + r dr dθ φ j + Ni − r dr dθ − Ni qn ds = 0 ⎭ ⎩ ∂θ ∂θ j=1 e e e
(20) Equation (20) can be written as: n
K iej p ej +
j=1
n
Biej φ ej − f ie − Q ie = 0
(21)
j=1
where
3 3 . ∂Nj 1 h ∂ Ni ∂ N j h ∂ Ni ∂ N j Ni h j + 2 = r dr dθ, ..Biej = r dr dθ, 6μ ∂r ∂r r 6μ ∂θ ∂θ ∂θ e e
∂h r dr dθ , Q ie = Ni qn ds f ie = Ni ∂θ e e
K iej
[K ]{ p} + [B]{φ} = { f } + {Q}
(22)
The boundary pressure values are known; thus, the column vector {Q} is omitted in the calculation of pressure and fractional film content. Numerical results have been generated using elements 160(Nθ ) × 160(Nr ) in the computational domain. These numbers of elements are chosen based on mesh independent tests. The following convergence criteria have been employed during the numerical computation of variables: Nθ +1 Nr +1 i=1
j=1
Nθ +1 Nr +1 i=1
j=1
| p(i, j)new − p(i, j)old | ≤ 10−6 | p(i, j)old |
(23)
|φ(i, j)new − φ(i, j)old | ≤ 10−6 |φ(i, j)old |
(24)
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J. C. Atwal and R. K. Pandey
3 Results and Discussions The pressure result obtained based on the proposed model has been compared with the results of the authors [5] for developing confidence in the present investigation. Excellent matching between both the findings is found in Fig. 2; thus, the proposed model is considered validated. Input data of Table 1 are used for producing the results presented in this paper. Figure 3 shows the comparative pressure profiles at the mid-width of the pad for the data mentioned in Table 1 and given in the caption. It can be seen in this figure that for constant minimum film thickness and film thickness ratio, the pressure values are maximum with the pocketed-textured pad. High-pressure values are obtained with a pocketed-textured pad due to the largest restriction of flow in the circumferential direction. Variations of enhancement of film load capacity and reduction in coefficient of friction as compared to the conventional pad are presented in Fig. 4. The maximum increase in the load capacity occurs at 50 µm of texture and pocket depths.
Fig. 2 Comparison of pressure values
Table 1 Input data
Parameter
Value
Lubricant viscosity (μ0 ), Pa-s
0.139
Pad inner radius (R1 ), m
0.030
Pad outer radius (R2 ), m
0.055
Sector angle of pad (θ 0 ), degree
45
Number of pad (Z n )
6
Cavitation pressure (gauge), Pa
0
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Fig. 3 Variation of pressure profiles with pocket/textured pads (U = 11 ms−1 , sh = 16.20 µm, h2 = 30.0 µm, θ f /θ 0 = 0.7, θ p /θ 0 = 0.6, b/a = 0.5, b/R = 0.01, hd = 50.0 µm, 2r z /R = 0.9)
Fig. 4 Performance parameters variation with depth of the pocket and texture (U = 11 m/s, sh = 16.20 µm, h2 = 30.0 µm, θ f /θ 0 = 0.7, θ p /θ 0 = 0.6, b/a = 0.50, b/R = 0.01, hd = 50.0 µm, 2r z /R = 0.90)
Figure 5 illustrates the variation of film load capacity and friction coefficient with fish texture spacing. The maximum load capacity and minimum coefficient of friction with fish textured pad occur at a dimensionless spacing value of 0.01. However, Fig. 6 represents the performance parameters of the bearing with a dimensionless circumferential extent for pocketed with a fish texture pad. The dimensionless circumferential extent of 0.6 for pocket then followed by a dimensionless circumferential extent of 0.1 for fish texture yield the highest film load capacity and minimum friction coefficient.
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Fig. 5 Variation of performance parameters with b/R for fish texture pad (U = 11 m/s, sh = 16.2 µm, h2 = 30 µm, θ f /θ 0 = 0.7, hd = 50 µm, 2r z /R = 0.9)
Fig. 6 Performance parameters variation with θp/θ 0 for pocket with fish textured pad (U = 11 m/s, sh = 16.20 µm, h2 = 30.0 µm, θ f /θ 0 = 0.7, b/a = 0.50, hd = 50.0 µm, 2r z /R = 0.90)
From Fig. 7, it can be noted that with a rise in dimensionless circumferential extent, the load-carrying capacity first increases up to a point (dimensionless circumferential extent = 0.7), and thereafter, it diminishes. Moreover, the pad having pockets shows greater load capacity compared to a fish textured pad. Enhanced pressure magnitude in the pocket pad gives a higher value of load-carrying capacity than the fish textured pad. Figure 8 depicts the variation of film load capacity and friction coefficient of the fish textured pad with the parameter ‘b/a’. It can be seen that as the value of ‘b/a’ increases, the film load capacity decreases and friction coefficient increases. The film load capacity is maximum at ‘b/a = 0.5’, and it is minimum at ‘b/a = 2’.
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Fig. 7 Performance parameters variation with θ f /θ 0 for pocket and fish textured pad (U = 11 m/s, sh = 16.20 µm, h2 = 30.0 µm, b/a = 0.50, hd = 50.0 µm, 2r z /R = 0.90)
Fig. 8 Performance parameters variation with b/a for fish textured pad (U = 11 m/s, sh = 16.20 µm, h2 = 30.0 µm, θ f /θ 0 = 0.70, hd = 50.0 µm, 2r z /R = 0.90)
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4 Conclusions Based on the investigations presented herein, the following points are concluded: 1.
2.
3.
4. 5.
The pocket with fish texture, pocket and fish texture pads have yielded performance improvement in the bearings in comparison with the conventional thrust pad. The pocket with fish texture, pocket and fish texture pads have yielded enhancement in the film load capacity in the range of 85–93%, 83–89% and 62–71%, respectively, in comparison to the conventional pad. Friction coefficient has reduced with pocket/texture pads. The pocketed-fish textured pad, pocketed pad and fish textured pad have yielded a reduction in the friction coefficient falling in the range of 45–47%, 43–46% and 39–41%, respectively, in comparison to the conventional pad. Texture depth played a vital role in improving the performance behaviour of thrust pads. Texture depth varies with operating parameters. Pocketed-fish textured pad possessing the pocket and texture up to the dimensionless circumferential length of 0.7 produced optimal performance behaviours.
Acknowledgements and Declaration of Conflict of Interests The authors would like to acknowledge the usage of IIT Delhi’s Baadal and HPC facilities. Moreover, the authors declare that there are no conflicts of interest in publishing this paper.
References 1. Gropper D, Wang L, Harvey TJ (2016) Hydrodynamic lubrication of textured surfaces: a review of modelling techniques and key findings. Tribol Int 94:509–529 2. Sudeep U, Tandon N, Pandey RK (2015) Performance of lubricated rolling/sliding concentrated contacts with surface textures: a review. J Tribol 137:031501 3. Rahmani F, Pandey RK, Dutt JK (2018) Performance studies of powder lubricated journal bearing having different pocket shapes at cylindrical bore surface. J Tribol 140:031704 4. Kharbanda JK, Pandey RK (2014) Application of tribology for enhancing the life of sugar mill roll bearing and journal. Int Sugar J 116:490–495 5. Gherca AR, Maspeyrot P, Hajjam M, Fatu A (2013) Influence of texture geometry on the hydrodynamic performances of parallel bearings. Tribol Trans 56:321–332 6. Aggarwal S, Pandey RK (2017) Frictional and load-carrying behaviours of micro-textured sector shape pad thrust bearing incorporating the cavitation and thermal effects. Lubr Sci 29:255–277 7. Aggarwal S, Pandey RK (2018) Performance investigation of micro-pocketed textured pad thrust bearing. Ind Lubr Tribol 70:1388–1395 8. Atwal JC, Pandey RK (2020) Performance analysis of thrust pad bearing using microrectangular pocket and bionic texture. Proc Instit Mech Eng Part J J Eng Tribol. https://doi. org/10.1177/1350650120940076
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9. Atwal JC, Pandey RK (2020) Film thickness and friction investigations in a fluid film thrust bearing employing new conceived micro-texture on pads. J Tribol. https://doi.org/10.1115/1. 4048500 10. Giacopini M, Fowell MT, Dini D, Strozzi A (2010) A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. J Tribol 132:041702 11. Woloszynski T, Podsiadlo P, Stachowiak GW (2015) Efficient solution to the cavitation problem in hydrodynamic lubrication. Tribol Lett 58:18
Analysis of a Soil-Moisture Sensor for Potential Failure Modes and Mass Manufacturing Mihir Mogra, Rajesh Aouti, N. S. Rakesh, Alishan Ahmed, R. Ashwin, Jose Joseph, and G. K. Ananthasuresh
Abstract We present techniques for mass manufacturing of a soil-moisture sensor that includes a heat probe and a temperature probe. The requirement of mass manufacturing necessitated a few changes in design of the heat probe. We also present an improved design using the standard method of reliability analysis. Potential failure modes were identified for the sensor, and the modifications to the design were made accordingly. The approach to design was aimed at improving the accuracy and repeatability of the sensor as well as making it suitable for mass manufacturing. Automating the process of manufacturing the sensor as a mechatronic system is outlined. By doing so, we aim to bring down the cost of the sensor considerably, thus enabling the use of sensors in large quantities for deployment in a field. Keywords Reliability · DFMEA · DPHP
1 Introduction Soil-moisture sensors are beneficial for controlled irrigation practices in agriculture. Since the moisture level might vary from point to point in a field, enough sensors need to be deployed over the entire field. Hence, the sensor must be made affordable, reliable, and accurate. This can be achieved through automated mass-manufacturing processes while ensuring reliability. In our earlier work [1, 2], we developed, prototyped, analyzed, and tested a soil-moisture sensor. It takes, on an average, 10 h to make one sensor manually. It is a laborious process and manual method affects the consistency of sensor performance. In this paper, we undertake translational research aspects to make the sensor suitable for producing in large numbers with consistent performance for extensive field studies. The sensor under consideration is called a dual-probe heat-pulse (DPHP) sensor (Fig. 1). It consists of two probes: a heater and a thermocouple. The heater comprises a nichrome wire (50 µm diameter) wound around a relatively thick copper wire M. Mogra (B) · R. Aouti · N. S. Rakesh · A. Ahmed · R. Ashwin · J. Joseph · G. K. Ananthasuresh Indian Institute of Science, Bengaluru, Bengaluru, Karnataka 560012, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_110
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Fig. 1 a Prototype of the DPHP soil-moisture sensor and b its elements
(0.5 mm diameter) and enclosed in a steel tube (0.8 mm ID and 1.5 mm OD). It works on the principle of Joule heating by applying a certain voltage pulse (3.3 V) across the nichrome wire for predetermined time. This heat travels through the soil and peak temperature is detected by the thermocouple which is at a distance from the heater. Based on different moisture levels of the soil, the volumetric specific heat of the moist soil varies, which is computed using the temperature rise recorded by the thermocouple. Thus, the moisture level is detected using the relation between volumetric specific heat and temperature rise [3]: Tmax =
q eπr 2 C
(1)
where Tmax is the maximum rise in temperature at a radial distance r from the heater, q the heat input, and C the volumetric heat capacity of the soil.
2 Reliability Analysis A design for failure mode and effect analysis (DFMEA) [4] was performed on the envisioned sensor by listing the potential failure modes and prioritizing them to mitigate the undesired effects sequentially. The design changes suggested by DFMEA were checked for automated manufacturing and overall reliability before finalizing the design and process. By considering the repeatability and precision of sensors, changes in design were made. Hence, a reliability study was carried out to check for the long-term effect of the design changes. There are many tools and techniques available for improving the
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reliability of the system; we have used design for failure modes and effect analysis (DFMEA), which involves identifying the potential failure modes and improving the design accordingly, thus making the sensor robust. DFMEA involves assigning an individual number (out of 10) to a failure mode based on its severity (SEV), occurrence (OCC), and detectability (DET). These scores are multiplied to get the risk priority number (RPN). The RPN is used to prioritize the failure modes so that corrective actions can be taken to reduce the frequency, severity, and/or improve the detectability of the failure mode. The objective is to design the sensor such that RPN value is minimized to the target value (or less). DFMEA can be conducted at a component, subsystem, or system level. At the subsystem level, the heater probe is a critical component of the sensor, and DFMEA for the same was carried out as can be seen in Table 1. Based on RPN of the initial design, the following modifications were made: 1.
2.
3.
4.
5.
Thin-film Parylene coating is replaced with powder-coating technique. Powder coating has better adhesive properties as compared to Parylene and does not affect the heating of the nichrome. Hence, it is capable of serving the primary function of electrically insulating the nichrome wire in a much more efficient manner as compared to Parylene. A single thicker copper wire is used instead of folding a thin copper wire. The thick wire could be aligned in the center of the steel tube with the support from casing, thus ensuring uniform distribution of heat in all directions. This is in contrast to folded copper wire, which could not be aligned centrally and had the disadvantage of localized heating at certain points where the wire could touch the steel tube (Fig. 2). Pulse-arc welding is used to seal the tube from one end. But due to excessive heat released during welding, the surface of steel can oxidize, thus exposing the free iron ions to moisture, which can corrode the steel surface. This was overcome using a standard process of passivation, which removes the oxide layer and forms a thin and strong corrosion-resistive chromium oxide layer [5, 6]. Parylene coating could be damaged easily due to the friction between the nichrome wire and steel tube. Although thermal grease did not provide a significant contribution to heat conduction, due to its electrically insulating and thermally conducting nature, it was used to prevent shorting of the circuit between the nichrome and steel tube. Due to the good adhesive strength of powder coating, thermal grease can be eliminated. This led to an elimination of many problems from thermal grease (e.g., uniformly filling in steel tube) while bringing down the cost of the sensor. The voltage capacity of the nichrome wire was checked by increasing the voltage at a fixed rate for a certain time period. It was observed that powder coated nichrome can sustain voltage up to 13 V, which suits our application since the maximum operating voltage for the sensor is 5 V.
The DFMEA chart was updated as shown in Table 2. It can be observed that the RPN values have been reduced significantly. Additional electrical testing will be carried out to ensure the reliability of the heater element in extreme conditions as
Bending/buckling of SS tube
Nichrome wire breaks
Can come off easily
Can touch the SS tube leading to localized heating
Non-uniform filling of SS tube
Wire breaks from overheating
SS strength
Nichrome tensile properties
Parylene coating of nichrome wire
Folded copper wire
Thermal grease (TG)
Voltage capacity of nichrome
Breaks the circuit
Uneven conduction
Parylene can scrape off due to friction
Shorts the circuit
Breaks the circuit
8
6
7
8
8
Damages the heating 8 element
Corrosion of SS Damages the heating 8 tube in presence of element moisture
SS corrosion resistance
SEV
Potential failure modes
Functional parameter/design parameter
Potential effects of failure
Subsystem
Component
Soil-moisture sensor
Heater probe
System
Table 1 DFMEA chart for the initial design of the heater probe
8
4
3
5
OCC
Surge in voltage
High viscosity of TG
3
5
SS tube is 8 electrically conducting in nature
Weak adhesive coating
Weak material properties
Material properties
Welding of one end of SS tube leads to corrosion
Potential causes/mechanism of failure
Sensor might stop working
Erroneous reading
Erroneous reading
Erroneous reading
Sensor might stop working
Current design/process controls
3
6
5
7
4
3
4
DET
72
150
280
448
128
72
160
RPN
Check failure modes of nichrome with coating for working voltage
Passivation of steel
Alternate coating required
Check with automation prototype
Standard tests to be conducted
Passivation of steel
Recommended actions
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Fig. 2 a Initial design of heater element of sensor with nichrome wire wounded around folded copper wire. b Modified design of heater element of sensor with nichrome wire wound and re-wound around thick copper wire
well. Similarly, DFMEA will be constructed for other subsystems and components to ensure the overall reliability of the sensor.
3 Mass Manufacturing of the Sensor Figure 3 depicts a step-by-step procedure to manufacture a DPHP sensor. Keeping the affordability of the sensor in mind, a mechanism as shown in Fig. 4 will be implemented. Since the sensor works on the principle of Joule heating, the most important part of the sensor is the heater. Hence, the consistency in the manufacturing of heater is of primary importance. As shown in Fig. 3, the processes involved in making the heater are winding of the nichrome wire around the copper wire and inserting the heater element (nichrome + copper) in the steel tube. To achieve uniform winding in the making of the heater element, the mechanism depicted in Fig. 4 will be used. The mechanism is actuated by a single motor. It comprises of two arms rotating in opposite directions; the motion of arms is enabled with the help of bevel gears. The nichrome wire is fed to the gripper that is hinged to the arms. The copper wire is fed through the hollow shaft, along the axis, with a constant predetermined feed rate. The arms are at an unequal distance from the axis so that the arms can complete full revolutions. At the hinges between the arm and gripper, a spool (containing torsion spring) of wire with a relatively high strength is connected. The wires from the spool run along the gripper through it and meet at the axis of the mechanism, where they
subsystem
Bending/buckling of Damages the SS tube heating element
Nichrome wire breaks
Some spots on wire without coating
Wire breaks from overheating
SS strength
Nichrome tensile properties
Powder coating of nichrome wire
Voltage capacity of nichrome
8
8
8
8
SEV
Breaks the circuit 8
Shorts the circuit
Shorts the circuit
Damages the heating element
Corrosion of SS tube in presence of moisture
SS corrosion resistance
Potential effects of failure
Potential failure modes
Functional parameter/design parameter
component
Soil-moisture sensor
Heater probe
System
Table 2 DFMEA chart for the modified design of the heater probe
Surge in voltage
Improper spraying
Material properties
Material properties
Welding of one end of SS tube leads to corrosion
1
3
4
3
2
Potential OCC causes/mechanism of failure
Sensor might stop working
Erroneous reading
Erroneous reading
Current design/process controls
3
3
4
3
4
DET
24
72
128
72
64
RPN
Conduct experiments to check adhesive strength
Check with automation prototype
Standard strength tests to be conducted
Recommended actions
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Fig. 3 Stepwise making of the sensor
are connected. The intersection of the wires is glued to the hollow shaft connected to the driven bevel gear. The initial distance between the gripper arms is equal to the length of nichrome to be wound around copper. The nichrome wire is fed to the gripper arms and is cut after the gripper has held the two ends firmly. The nichrome wire is glued to the copper wire at a single point before the process starts. When the arms start rotating, the copper is fed along the axial direction. Since the arms rotate in the opposite direction
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Fig. 4 Mechanism for automating winding process
around the copper wire, nichrome starts winding from both sides of the copper wire. As the nichrome starts to wind, the distance between the gripper must decrease since the length of the nichrome wire between the grippers is constant. Since the nichrome does not have enough strength to pull the gripper arms, the stronger wire (which also starts winding around the hollow shaft simultaneously as nichrome starts to wind around copper) pulls the arm inwards. As the stronger wire winds around the hollow shaft, it will start drawing wire from the respective spools connected to the hinges. This tension generated in the stronger wire while drawing from spool is responsible for pulling the arms inwards toward the axis. Thus, the required length is wound around the copper. Once the winding process completes, the torsion spring inside the spool pulls the strong wire back, which rotates the hollow shaft in the opposite direction, causing the arms moves back to their original position, thus making it ready for the next iteration of winding.
3.1 Calculations for Feed Rate of the Copper Wire For the mechanism described in the previous section to work properly, the feed rate of copper wire and the rotational speed of the arms have to be worked out. As seen from Fig. 5a, the nichrome wire of length ln (87 mm) has to be wound around copper of length lc (15 mm). The diameter of the copper wire is dc , x is the length set aside on either side of the nichrome wire for soldering onto the connecting board. Hence, the length that needs to be wound is (ln − 2x). We have considered x
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Fig. 5 Figure depicting critical dimensions of nichrome and copper wire
to be 10 mm. Using this data, we can calculate the number of turns in winding and spacing between each winding. From Fig. 5b, it can be inferred that the length of single winding is (π dc )2 + s 2 , where s is the spacing between each winding. Since winding takes place from both sides of the copper wire simultaneously, the number of turns in winding (n) and the spacing between each winding (s) can be found using ln − 2x n (π dc )2 + s 2 = 2
(2)
ns = lc
(3)
Assume that the time taken to wound nichrome around copper is t. Then, in this time t, the copper wire needs to be fed by lc . Therefore, the feed rate of copper = lc /t. Since the lengths of arm 1 and arm 2 (r1 and r2 ) are different, the rotational velocities of each arm shall be different. However, each arm has to wound the same length in time t so, the tangential velocity of each arm will be equal. Thus, ω1r1 = ω2 r2 =
ln − 2x 2t
3.2 Assembly of the Heater Probe Nichrome wire wound around copper forms the heater element. To form the complete heater, the heater element is placed inside a stainless steel (SS) tube. The length of the copper wire is equal to the length of the SS tube, and the inner diameter of the SS tube has a value close to the diameter of copper wire (with the nichrome wound around it). Hence, the heater element fits perfectly inside the SS tube. To ensure it is placed centrally inside the SS tube and to prevent any loose assembly, the heater
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Fig. 6 Heater assembly process
element is secured inside the SS tube by applying a drop of glue on the top of the heater element after it is assembled inside the tube. To obtain the desired assembly, the process shown in Fig. 6 is proposed. When the construction of heater element from the winding process is completed, it is sent to the inclined plane, which funnels it into the nozzle. A plunger is then actuated to push the heater element into the steel tube beneath it, which is carried by the conveyer belt. The speed of the conveyor can be determined based on the supply rate of the heater element from the winding process. While pushing the heater element into the steel tube, the plunger also deposits a drop of glue on top of it (similar to the working of a syringe). Hence, it becomes a continuous process and the heater is assembled at regular intervals. This completes the assembly of the heater which is then sent to other processes as depicted in Fig. 3.
4 Closure The automation setup will bring down the cost of making a sensor by cutting down the labor costs involved in making the critical parts. The setup improves the reliability of the system by bringing uniformity in manufacturing. A reliability study was carried out for the DPHP soil-moisture sensor, which resulted in systematic improvement in the performance of the sensor. Further, assembly processes can also be automated similarly. This would enable affordable and quick mass manufacturing of sensors, thus enabling the objective of deployment of DPHP soil-moisture sensors in large quantities in the field.
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Acknowledgements This work was supported by the NNeTRA project of the Indian Ministry of Electronics and Information Technology.
References 1. Jorapur N, Palaparthy VS, Sarik S, John J, Shojaei M, Ananthasuresh GK (2015) A low-power, low-cost soil-moisture sensor using dual-probe heat-pulse technique. Sens Actuators A Phys 233(1):108–117 2. Palaparthy VS, Mondal S, Singh DN, Bhagini MS, Ananthasuresh GK (2018) Effect of spatial variations and desiccation cracks on the DPHP and MPHP sensors. Sens Actuators A Phys A 279:638–648 3. Campbell GS, Calissendorff C, Williams JH (1991) Probe for measuring soil specific heat using a heat-pulse method. Soil Sci Soc Am J 55:291–293 4. Raytheon six sigma DFMEA, “DFMEA with suppliers”. https://www.raytheon.com/sites/def ault/files/connections/rtnwcm/groups/public/documents/content/rtn_connect_dfmea_pdf.pdf 5. Passivation of stainless steel. https://wine.appstate.edu/sites/wine.appstate.edu/files/Diversey_ PassivationofStainlessSteel.pdf 6. Pickling and passivating stainless steel. https://www.worldstainless.org/Files/issf/non-imagefiles/PDF/Euro_Inox/Passivating_Pickling_EN.pdf
Evaluation and Validation of Weld Joint Fatigue in Vibration Using Notch Stress Approach Ashish Patil and Swapnil Bhende
Abstract Welds are most susceptible to the vibration loading. When subjected to loading, weld failures are observed at the throat or at the toe. Vibration fatigue evaluation is different from that of static fatigue evaluation as dynamic characteristics of the considered system play important part in stress generation. Weld fatigue under vibration needs different methodologies and one of these is explained in this paper. The paper contains details of the application of notch stress approach and its comparison with the two other approaches, namely nominal and structural stress approach. Keywords Welds · Structural stress · FEA · Fatigue · Vibration
1 Introduction Welded joints are permanent type of joints which have wide variety of applications. Lightweight design is the advantage, welded joints provide over bolted and riveted joints. With the large-scale applications, there comes the necessity to design and analyze these joints with accuracy. Due to inherent complex damage mechanisms which include material irregularities, microcracks in the weld bid, and the local notch at the weld toe, simulating weld joint is a difficult task [1]. Weld failures are usually observed at the weld root and toe. There are multiple theories which are proposed to evaluate weld fatigue: Nominal stress theory, Structural stress theory, Notch stress theory, and Linear Elastic Fracture Mechanics (LEFM) [2]. These theories are classified into two categories as global and local theories. Local theories are more precise, but they require more computational time as compared to the global theories. Hence, it is advisable to choose the theory based on the accuracy needed and the amount of time available for the computation. It has been observed that various theories have various limitations too. Structural stress A. Patil (B) · S. Bhende Eaton Technologies Pvt. Ltd., Eon Free Zone, Kharadi, Pune 411028, India e-mail: [email protected] S. Bhende e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_111
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theory is useful for the failures originating from the toe while notch stress approach is useful for the both type of failures: originating from toe and root [3]. Nominal stress approach can be used for evaluating welds in a very quick time but with limited accuracy [4]. Nominal stress approach uses the S–N curves to represent the standard weld configurations using FAT numbers. International Institute of Welding (IIW) has provided set of S–N curves which provide S–N curve for each configuration [5]. Though most of the common configurations are covered under nominal stress approach, there are product design-specific welds that are not available under this approach. Structural stress theory can be utilized when observed failure is originating from the toe of the weld. This theory works for hand in hand with Finite Element Analysis (FEA) to identify the high-stress area which are also called Hot Spots. Hence this theory is also called as Hot spot stress theory [6]. In this approach, stress at the weld is identified using the extrapolation method. Stresses are obtained at the distance 0.4t and 1.0t and extrapolated using the following formula [7]. σ hs = 1.67 ∗ σ @0.4t − 0.67 ∗ σ @1.0t where σ hs—Hot spot stress at the weld toe. Though effective, this method is only useful to address the toe failures. The accuracy-complexity correlation for the four approaches show that with the increased accuracy, amount of time for the modeling and complexity in modeling increases. Hence the selection of each approach should be based on the design phase and available resources.
2 Notch Stress Approach According to IIW, notch stress is the total stress at the root of the notch. The decrement in the stress at the notch, for a certain depth, can be replicated using the enlarged notch radius at the weld—parent metal interface. This method was developed by Radaj. Figure 1 shows the difference between structural stress and the notch stress at the similar location for the same loading conditions. It is evident from the figure that structural stress does contain only membrane and the bending stresses but not the nonlinear stresses due to local stress raisers. Hence, notch stress includes both the type of stresses: due to component geometry and due to local stress raisers. Notch stress for the complex geometries can be found out with FEM modeling. Care must be taken to mesh the notched area of the weld properly. IIW recommends element size less than 1/6th of the notch radius for the linear elements and element size less than 1/4th of the notch radius for higher order elements [9]. Notch needs to be meshed with proper sized elements and to be connected with other parts with smooth transition mesh. For the huge models, sub modeling needs to be done as this
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Fig. 1 Difference between notch and structural stress at the weld-parent material interface
method is mesh sensitive. Due to high mesh density, this approach requires higher computational time as compared to other approaches. Based on the desired higher accuracy of the results and availability of computational resources, this approach can be implemented.
3 Details of Analysis and Validation 3.1 Modeling Notch stress approach has been implemented on the tube to plate weld to predict the structural integrity of the structure under vibration loading in FEA. The structure consists of a tube which carries a load at the one end while the other end is fixed to the plate via welding. This whole structure is subjected to the vibration loading at the base. Vibration test is performed in two steps: 1. 2.
Sweep from 0 to 30 Hz to identify the natural frequency of the structure. Dwell at the obtained frequency in the sweep for the output response of 3G.
For qualification, the required number of test cycles the structure should withstand is 1.0 × 105 . Based on the frequency identified in the sweep, the test duration in seconds is calculated by multiplying frequency with test cycles, for which structure should sustain the vibration. The detailed simulation procedure and test procedure is as below: The geometrical representation of the structure assessed and the details about the welded geometry are as shown in Fig. 2. According to Notch stress approach, the notch details/geometry needs to be introduced in the toe of definite dimensions. For the tube thickness more than 5 mm, notch radius should be of 1mm while for the tube thickness less than 5 mm, notch radius reduces to 0.05 mm [1]. In this application, 0.05 mm of notch radius has been used.
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Fig. 2 Details about the assembly understudy and position of externally introduced notch
Fig. 3 Mesh details
All different parts of the assembly are connected through bonded contacts at the required locations. The meshing was performed in accordance with IIW recommendations mentioned in the introduction section. The precaution was taken to mesh the notch and area around notch with high density mesh to capture stresses appropriately. Figure 3 shows the meshing used in the modelling for the current assembly. Input acceleration is applied at the base (at the constraint node locations shown in Fig. 4) in such a way that 3G response is recorded at the Accelerometer mounted near center of gravity.
3.2 Simulation Results Frequencies obtained from the frequency analysis with the above boundary conditions are 7 Hz (Dominant in Horizontal plane), 10 Hz (Dominant in Vertical plane), 59 Hz and 60 Hz for the first four modes. Out of these four, 7 and 10 Hz, being within the excitation frequency range till 30 Hz, are of interest. The mode shapes for these frequencies can be interpreted from Fig. 5.
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Fig. 4 Boundary conditions
Fig. 5 Mode shape results after modal analysis
After the frequency analysis, harmonic analysis was performed to predict the stresses. It was found that 10 Hz mode was most damaging and resultant stresses obtained are as shown in Fig. 6.
Fig. 6 Maximum principal stress plot
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Table 1 Results summary of the analysis results Sr. No.
Material
S–N curve
Max principal stress
Allowable stress from S–N curve
Required no. of cycles
Prediction
1
Aluminum
FAT 71
43.1 ksi
20.4 ksi
1.0 × 105
Failure
From the obtained stresses, the next step was to obtain life by utilizing an appropriate S–N curve. According to IIW standards, the required S–N curve for Aluminum is FAT 71. Based on the universal curve mentioned above, calculated life is less than 1.0 × 105 cycles and hence, analysis predicted failure for the structure in the test (Table 1).
3.3 Test Validation Test was performed with the help of vibration shaker to obtain 3G response at the CG. Sine sweep performed on the structure showed a resonance frequency at 9.15 Hz in a vertical plane (Fig. 7). The obtained frequency from the analysis is 10 Hz which shows that test validates the simulation. When structure was vibrated (Dwell test) at the obtained frequency from the sweep, crack was observed at the weld toe which further validates the theory as well as the failure predictions in the considered case. Figure 8 shows the cracked region of the tube.
Fig. 7 Frequency response from the test
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Fig. 8 Observed crack ad its location in the assembly under consideration
Table 2 Comparison table for the approaches used for the evaluation of weld fatigue in vibration Approach
Applicability to the current problem
Meshing
Geometry modification
S–N curve
Nominal stress approach
NA
Mesh insensitive
NA
NA
Structural stress approach
Applicable
Mesh sensitive
NA
FAT 40
Notch stress approach
Applicable
Mesh sensitive
Introduction of FAT 71 notch
4 Conclusions Notch stress approach can be effectively used to evaluate weld fatigue through simulations. Notch stress approach is sensitive to mesh density near the notch. Hence, sufficient solid 3D mesh needs to be modeled near the entire area of interest which increases the time for solution. Nominal stress approach is applicable for standard set of weld configuration while structural stress approach is applicable for weld toe failures. Notch stress can be used to evaluate weld failures of all kinds. Availability of Simulation time and amount of required accuracy are the deciding factors while employing this approach. Table 2 depicts the actual comparison between all the three approaches [7].
References 1. Fricke W (2012) IIW guideline for the assessment of weld root fatigue. IIW Guideline for the Assessment of Weld Root Fatigue, Revision 3 2. Zamzami I, Susmel L (2016) On the accuracy of nominal, structural, and local stress-based approaches in designing aluminium welded joints against fatigue. Int J Fatigue. ISSN 0142-1123
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3. Malikoutsakis M, Savaidis G (2009) An approach to the effective notch stress concept to complex geometry welds focusing on the Fe modeling of weld ends. In: 3rd ANSA & μETA international conference 4. Wægter J (2009) Fatigue design based on S-N data 5. Hobbacher A (2016) Recommendations for fatigue design of welded joints and components. IIW Document IIW-2259-15 6. Fuštar B, Lukacevi´c I, Dujmovi D (2018) Review of fatigue assessment methods for welded steel structures 7. Patil A, Bhende S (2018) Evaluation and comparison of frequency domain fatigue life approaches for weld joints. In: F2018-NVB-019, SAE FISITA conference
Trajectory Control and Force Control of Biomimetic Fingers by Tendon-Based Actuation System Using Bond Graph Vijay Saini, Simran Pal Singh, Neeraj Mishra, and Anand Vaz
Abstract Tendon-based actuation in natural fingers gives dexterity to execute daily life complex motions with ease. It is important to understand the biomechanical structure responsible for these motions. In the proposed work, a single link tendonbased actuation system with two biomimetic fingers has been designed and developed for trajectory and force control. This setup clearly demonstrates the concept of actuation of prosthetic finger of a partially impaired hand by using the motion of remaining natural fingers. The microcontroller plays the role of the human central nervous system (CNS) and provides pulse width modulation (PWM) signals generated through a proportional, integral and derivative (PID) control algorithm to the motor driver for actuation of the motor. The rotatory motion of the motor is converted into translatory motion of the nut by the lead-screw and slider-nut mechanism. Translation of the nut causes joint rotation of the active finger through string-tube mechanism. For joint angle feedback, optical potentiometer is used. For force feedback, force sensor based on strain gauges is designed and developed. The system is modelled using the graphical technique of bond graph, which is very useful for modelling these complex multi-energy domain biomechanical systems. In modelling the system, inertias both rotational and linear, frictional losses, string-tube mechanism modelling, losses in the motor inductance, etc., are all taken into account. The setup serves as a low-cost open-architecture-based controller for investigation of finger tendon mechanics and control. Keywords Prosthetic fingers · Bond graph · Force sensor · Processing
V. Saini (B) · S. P. Singh · N. Mishra · A. Vaz Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology Jalandhar, Jalandhar, Punjab 144011, India A. Vaz e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_112
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1 Introduction The human hand is a vital organ in our human body. It can perform a variety of tasks very easily and efficiently due to its complex internal structure and arrangement. Humans easily learn how to move and control the movements of the hands and the fingers to grasp and hold objects. Yet how are these movements controlled [1]? It is easy for humans to perform prehensile tasks. What is the control mechanism involved in performing these tasks? A need is therefore felt to develop an experimental setup to answer these questions. In addition, if there is partial hand impairment, it is not easy to perform the above tasks. Partial human hand impairment considered here is when the hand has lost one or more fingers but retains the ability of the remaining finger. The experimental setup should be such that it cannot only explain the complex actuation system using muscles and tendons but also the concept of the prosthesis and how to actuate prosthetic fingers using the remaining natural fingers [2, 3]. The proposed experimental setup additionally addresses the issues of trajectory control and force control of fingers using a tendon-based actuation system. It also demonstrates the implementation of the concept of ‘opposition space’ [4]. An experimental setup has been designed and developed to address these issues. It consists of various essential components such as two prosthetic single link fingers (developed through rapid prototyping), slider-nut and lead-screw mechanism, fishing wires, string-tube mechanism, 12 V DC motor, Motor Driver (L298N), Arduino UNO microcontroller (ATmega328P), Optical potentiometer (COPAL JT22-320500), force sensor, two power supplies, etc. Fishing wires act like tendons, slider-nut and lead-screw mechanism acts like muscles, Arduino microcontroller acts as the brain of the human, which generates the PWM signals. These control signals, which are based on the PID algorithm, are used to drive the motor. The motor is coupled with the slider-nut and lead-screw mechanism. The optical potentiometer is used as angle feedback at the joint of the finger, and force sensor is used for feedback of force at the tip of the finger. The experimental setup has been modelled and simulated through the bond graph approach [5].
2 Methodology The designing and development of the experiment setup are explained in detail. An experimental setup is prepared for replicating the complex mechanism of the index finger and the thumb and their actuation through tendons and muscles.
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2.1 Prosthetic Fingers Mechanism In this work, the two prosthetic fingers each contain a single link. Each link acts as a distal part of the respective fingers. It is sufficient to perform the objective of implementing the trajectory control and force control of biomimetic finger using tendon-based actuation system. The setup is composed of two single link fingers: link 1 of finger 1 and link 2 of finger 2. Finger 1 acts like an index finger, and finger 2 acts like a thumb. These two links are hinged on a third fixed link as shown in Fig. 1. In the actual human finger, finger joints are generally spherical. However, in this model, the natural spherical joints are replaced by the hinged joints containing a pulley type of arrangement [6]. Each joint contains two ball bearings to reduce the friction. This pulley arrangement is the part of distal link. It is designed in such a way that tendons could be wrapped easily around the pulleys and provides actuation torque to the joint. The torque helps in the rotation of the links with respect to the fixed link. The holes are used for fixing one end of the tendons so that the links get the rotational effect. There is a potentiometer holder to hold the optical potentiometer. The potentiometer holder is bolted on to the fixed link. Designing and drafting of the different part of 3D printed prosthetic fingers are carried out in SOLIDWORKS, and their assembly is as shown in Fig. 1.
2.2 Lead-Screw and Slider-Nut Mechanism Lead-screw and slider-nut mechanism consists of a lead-screw, a flexible coupling, a nut, a bearing and a structure to support all these components. The actual setup of the lead-screw mechanism is as shown in Fig. 2.
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When the motor rotates, power is transmitted to the screw through the coupling. This rotation of the lead-screw results in the translational movement of the nut. As the nut slides on a frame, to reduce its friction, a lubricant is applied underneath the nut. The direction of the nut movement is decided by the direction of rotation of motor either in a clockwise or anticlockwise direction. One end of the tendons is connected to the nut so it could generate a force in the tendon likewise in the actual muscle–tendon arrangement.
2.3 String-Tube Mechanism The string-tube mechanism is a great mechanism for transforming the motion from one joint to the other joint. A major benefit of the string-tube mechanism is that the joints need not be parallel and need not be fixed. This mechanism is very helpful in the case where the distance between the two joints is changing with time: Active connection. In the setup, Link 1 is an active link as a drive from the motor through the tendons is given to this link. The link acts like an index finger which is the remaining working finger of the patient. A string-tube mechanism is required to transmit the motion from the nut of the lead-screw and slider-nut mechanism to the active joint of link 1. The string-tube mechanism is used for the actuation of the link as shown in Fig. 3. Two tendons per link joint are needed to actuate the link as tendons work only in tension. One tendon is for flexion, and the other is for the extension. One end of the tube is fixed on a fixed link while the corresponding string end is connected to the next link 1 of the same joint. Each string winds around the pulley of distal link. The other ends of the tubes are fixed on the slider-nut frame while the corresponding string ends are connected to the common slider-nut in an alternate configuration. In this configuration, one string is connected at the head and the other on the tail side of the nut. This configuration is used to reduce the number of actuators as both the strings of a link are actuated using a common slider-nut.
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Passive connection. In the setup, link 2 is a passive link as no direct drive is given to the link 2. As it is coupled to the finger of link 1 in an unlike configuration through strings, the motion of link 1 will drive link 2 through passive connection. The passive finger acts like a prosthetic finger say thumb which will get the drive from the active finger. In an unlike configuration, the prosthetic thumb will rotate in the same sense as that of the active finger but in opposite direction. In the proposed setup, for the passive connection as shown in Fig. 4, it is not required to use the string and tube mechanism as both the fingers joints of link 1 and link 2 are parallel and fixed. So, only strings are used to actuate the passive finger. Two strings are required for passive connections as well. One end of the first string is connected to the link 1 and then winds around the pulley of link 1 and to the pulley of link 2 and then the other end is Fig. 4 Schematic arrangement of passive connection using strings only (unlike configuration)
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connected to the link 2. Likewise, the second string is required as string works only in tension. Rotation of link 1 will exert a torque in the active joint. This torque will exert a force in the strings. This force will transmit tangential to the joint of link 2 and then generates torque in the joint of link 2. This torque will result in the rotation of link 2. During this rotation, the second string does not play any role as it is in a loose state. Its role will come into the picture when link 1 rotates in the opposite direction. At that time, the first string will come into a loose state.
2.4 Experimental Setup A fully assembled experimental setup for trajectory and force control of tendons of a biomimetic finger is shown in Fig. 5. In this figure, the interconnection between different components and workspace of the setup is as shown.
2.5 Force Sensor Design and Development The human hand has the capability of applying different forces depending upon the shape, size and weight of the object to grasp and lift the object. Therefore, there is a need to control the fingertip forces so that the prosthetic finger can apply the exact force which the patient should require to handle an object with precision. So, a force sensor is required to get the feedback from the fingertip. In the proposed work, a force sensor using strain gauges has designed and developed as shown in Fig. 6, which can even measure the minute forces even the touch on the fingertip of the prosthetic finger.
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Fig. 6 Force sensor a CAD model b actual
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Schematic of the actual force sensor circuit for the force sensor. Actual circuit diagram for the force sensor is as shown in the Fig. 7. Calibration of force sensor. At the known values of weights, the values of voltages are recorded and then a graph is plotted between the voltage and the force (in Newton). Then a linear equation is obtained which represents the force at the fingertip as a function of voltage coming from the force sensor circuit.
2.6 Bond Graph Modelling The bond graph approach is addressed to model the experimental setup. Bond graph model for single joint actuation system for trajectory control. Trajectory control is a crucial aspect in terms of the variation of the fingers position to handle and grip any object. Figure 8 shows the modelling of a single joint actuation system for trajectory control. In this, different parts are modelled such as motor, lead-screw,
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Fig. 8 Bond graph model for single joint actuation system for trajectory control
slider-nut, string-tube mechanism, active joint of the link and PID algorithm with the motor driver. The desired angular position of the link is tracked using a PID control algorithm as shown in the PID WBGO in Figs. 8 and 9. The concept of Word Bond Graph Objects (WBGO) is implemented to represent the components subsystems as Objects. PID stands for proportional, integral and derivative control. It is a closed-loop feedback control system. The input to the PID algorithm is a flow, in terms of joint angle error. The information of actual joint velocity is taken using a signal bond from the link angular velocity junction. The output of the PID algorithm is torque corresponding to the design error. The information of torque is sent to the motor controller, and it
Fig. 9 Bond graph model for single joint actuation system for force control
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produces the required current. The motor controller is modelled as a source of flow for the motor. Bond graph model for single joint actuation system for force control. Modelling of the force control is also necessary for gripping the object precisely or for applying the power of the fingers. So, it plays a requisite role in grasping the finger. Figure 9 shows the modelling of a single joint actuation system for force control of the fingertip. Modelling of the actuation for the force control is the same up to the revolute joint of the finger as that in trajectory control after then it needs some modification. As the force is applied at the fingertip, so it is necessary to convert the torque at the finger joint to the force at the tip of the finger using a transformer. Linear inertia is taken into account. As the finger is assumed to apply a force on the fixed surface, then it is modelled with a source of flow zero with a coupling representing compliance and resistance of force sensor. Force signal is taken from the ‘0’ junction with a negative sign and on computing error by comparing it with the desired force. The error on differentiating routes on to the PID algorithm as a source of flow signal for computing the required torque. The information of the torque is sent to the motor controller, and it produces the required current, which is given to the motor as a source of a flow.
2.7 Software Development For controlling purpose, program codes are written in Arduino IDE and processing software with reference to PID front end code by Brett Beauregard (2009). The Arduino code runs the main code which includes the PID algorithm and implementation of a timer interrupts using TimerOne.h library. Processing software is used to plot the graph of the processed variables from Arduino in real time. Finally, the design of the graphic user interface and the serial communication functions/helpers between Arduino and processing have been done [7].
3 Results and Discussions The proper interpretation of the results for step input, sinusoidal input is being done. Various system responses are perceived and some of them are presented for trajectory control and force control of the biomimetic finger.
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Fig. 10 System response at K p = 80, K i = 0, K d = 8 and sampling time = 0.045 s
3.1 Results for Trajectory Control For step input. A tuned response for step input should be necessary to get the best response. By changing the values of gains by trial and error method, the tuned response is obtained as shown in Fig. 10.
3.2 Results for Force Control For step input. As controlling of force is done by the force sensor, which is very sensitive to small pressure changes at the fingertip. So, it is troublesome to control the fluctuation generated during the force control in the system response. Figure 11 shows the tuned response for force control at sampling time is equal to 0.010 s.
4 Conclusion In this work, the study of human fingers and their different motions have been analysed from the prosthesis point of view. With the help of this educational experimental setup, the biomechanics of the finger actuation is clearly explained. Instead
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Fig. 11 System response at K p = 0.10, K i = 0, K d = 0 and sampling time = 0.01 s
of going through the conventional machining for manufacturing, additive manufacturing is successfully used for 3D printing the biomimetic fingers, because this is less time-consuming. A force sensor using strain gauges which is being used for force measurement at the tip of the biomimetic finger can measure even small forces due to a touch at the fingertip. String-tube or Bowden cable-based actuation system is demonstrated and implemented. The concept of ‘opposition space’ is implemented successfully for passive connection. The bond graph methodology used to model the complete experimental setup has taken into account almost every component including string-tube mechanism, rotational and translational coupling, inertias of the moving bodies, control algorithm, losses, etc. The graphical representation of real-time results in GUI through processing (open-source platform) by doing serial communication with Arduino UNO microcontroller has given proper explanation about the variations in the desired and actual values. From the controlling aspect, the trajectory control and force control have been successfully achieved by using the feedback sensors where a potentiometer is used for angular position feedback and force sensor for force feedback and by PWM signals generated using the PID algorithm. The experimental setup serves as a very low-cost open-architecture-based controller for the study of various concepts related to finger tendon mechanics and control.
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References 1. Fowler NK, Nicol AC, Condon B, Hadley D (2001) Method of determination of three dimensional index finger moment arms and tendon lines of action using high resolution MRI scans. J Biomech 34(6):791–797 2. Vaz A, Hirai S (2007) A bond graph approach to the analysis of prosthesis for a partially impaired hand. J Dyn Syst Meas Contr 129(1):105 3. Vaz A, Hirai S (2003) Actuation of a thumb prosthesis using remaining natural fingers. In: Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS 2003), Las Vegas, USA, 1998–2003 Oct 4. Iberall T (1997) Human prehension and dexterous robot hands. Int J Robot. Res. 16 5. Kalra S, Vaz J, Mishra N (2013) Development of a test rig for the study of musculoskeletal actuation of human finger. In: ACM International conference proceeding series 6. Armstrong TJ, Chaffin DB (1978) An investigation of the relationship between displacements of the finger and wrist joints and the extrinsic finger flexor tendons. J Biomech 11(3):119–128 7. Libraries homepage. https://processing.org/reference/libraries/. Last accessed 18 June 2019
Design of a Two Degrees of Freedom Actuator for Rehabilitation Robotic Applications Saurav Kumar Dutta, B. Sandeep Reddy, and Santosha Kumar Dwivedy
Abstract Rehabilitation robots and many other bio-inspired applications deal with multiple degrees of freedom joints. Generally, each degree of freedom of such a joint is actuated by “an equal” number of single degree of freedom actuators which occupy a lot of space surrounding the joint and is a major hurdle in the development of orthotics and exoskeletons. This paper attempts to model a robotic system where the number of actuators used is less than the degrees of freedom of the robot, while maintaining full actuation. The model of a two degrees of freedom robotic system using only a single actuator is considered. Pneumatic artificial muscles (PAMs) are used as these actuators are flexible and safe, have high power-to-weight ratio and have resemblance with human muscles. This model would be greatly beneficial in the development of rehabilitation robots as less number of PAMs would be used for actuating the degrees of freedom about a joint, while maintaining full actuation. Keywords Soft actuator · PAM
1 Introduction One of the major challenges behind the development of any prosthetic arm or leg is to incorporate in it as many degrees of freedom (like extension, flexion, adduction, abduction, pronation, supination, etc.) as is found in a human arm or leg. Also, many joints in a human body like shoulder, elbow and wrist have multiple degrees of freedom. One of the common ways of developing such a joint is to cascade as many single degree of freedom actuators as is the number of degrees of freedom in the joint. For example, Salazar et al. [1] have developed a human-inspired mechatronic finger. They used four pairs of pneumatic muscles in antagonistic arrangement to actuate the four degrees of freedom of the mechatronic finger. Therefore, the development of a prosthetic hand or arm with 27 and 32 degrees of freedom, respectively, Saurav Kumar Dutta (B) · B. Sandeep Reddy · Santosha Kumar Dwivedy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_113
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would be a difficult task considering the space constraint around the joints in a hand. Moreover, due to the excellent power-to-weight ratio and flexibility as compared to the conventional actuators, the pneumatic muscles serve as soft actuators and hence a natural choice as substitutes for human muscles in prosthetics, exoskeletons and human–machine interactions as in haptics [2–5]. Due to these features, PAMs are used not just as actuators but also for flapping of wings or flaps in aerospace applications [6]. A detailed information on PAMs and its different types can be found in the work of Daerden and Lefeber [7]. In the literature, the researchers have mostly used a pair of PAMs in antagonistic arrangement to actuate a single degree of freedom joint in rehabilitative devices like in the works of Tsagarakis and Cladwell [3] and Ferris et al. [8]. Due to these difficulties, the present paper comes up with the conceptual design of a two degrees of freedom joint actuation using only a single PAM. The paper has six sections. Section 2 discusses a single degree of freedom actuator with a pair of PAMs and a single PAM, Sect. 3 presents the model of the two-degrees-of-freedom actuator using a single PAM, Sect. 4 presents the model of the PAM in the proposed actuator, Sect. 5 presents the mathematical modelling of the proposed actuator, and Sect. 6 concludes the paper.
2 A Single Degree of Freedom Actuator The proposed model of the two-degrees-of-freedom actuator can be more easily understood by first going through the model of a single degree of freedom actuator. Figure 1a shows the antagonistic arrangement of a pair of PAMs, actuating a link. Both the muscles work antagonistically to get a bidirectional motion. Figure 1b shows the actuation of a single degree of freedom joint or link using just one PAM. In this model, the pulley is not used. The link is attached to a platform above the ground by
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Fig. 1 A single degree of freedom actuator using a the antagonistic arrangement of a pair of PAMs and b a single PAM
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a hinge. A small solid block with a hole in it is attached to the casing of the hinged joint. The solid block is fixed at its place and does not move. The thread at the upper end of the muscle, through the hole, is attached to the link directly. When the muscle contracts, the link moves from its initial position and makes an angular displacement of θ . When the muscle relaxes, the link goes back to its initial position due to gravity. In Fig. 1b, the PAM with the solid line indicates the contraction of the PAM and the PAM with the dashed line indicates the same PAM under relaxation. Therefore, it can be concluded that the model of Fig. 1b also works exactly similar to the model of Fig. 1a. But the model of Fig. 1b has an edge over the model of Fig. 1a as the link is executing the same bidirectional rotation with a single PAM. It is to be noted that in the absence of the gravitational force, the link would only execute a unidirectional motion. The reason being when the muscle relaxes, the wire becomes loose and the link will not be able to go back to its initial position by its own.
3 A Two-Degrees-of-Freedom Actuator A two-degrees-of-freedom actuator with traditional approach would require two pairs of PAMs, each pair of PAM working antagonistically for actuating each degree of freedom, as can be seen in Fig. 2a. It is evident from Fig. 2a that the traditional model is very clumsy and congested. The two pulleys are assembled in a way that the planes of the two pulleys are perpendicular to each other. The out-of-the-plane normal for the blue pulley is e3 -axis, and the out-of-the-plane normal for the green pulley is e2 axis. This particular arrangement of PAMs and pulleys takes lot of space and plays a major hindrance for the development of two and three degrees of freedom prosthetic joints.
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In the last section, it is observed that by removing the pulley and attaching the thread directly to the link helped in doing away with one out of the two PAMs. Therefore, the proposed model of the two-degrees-of-freedom actuator would also have no pulleys. Figure 2b shows the model of the two-degrees-of-freedom actuator using one PAM. In this case, the link is attached to the platform by a spherical joint. A solid block with two holes is attached to the casing of the spherical joint. The axes of these two holes are perpendicular to each other, as can be seen in Fig. 2b. Further, in this model, the PAM has one inlet for the compressed air and has two outlets, which are sealed. Since the PAM has two outlets, two threads can be attached to the PAM. The pink-coloured thread, through the hole, is attached at the top of the link. The pink-coloured thread is responsible for the actuation of the link about e3 -axis. The orange-coloured thread, through the other hole, is attached to the side of the link. The orange-coloured thread is responsible for the actuation of the link about e2 -axis. When the muscle contracts, the link moves in a three-dimensional space with its actuation both about the e2 - and e3 -axes. When the muscle relaxes, the wires become loose and the link is free to move in the three-dimensional space subjected to any arbitrary external force. If the system is in vacuum, the link would continue to be in its displaced position and will not return to its initial position by its own. The muscle in Fig. 2b can also be actuated selectively such that when the pink-coloured wire is pulled, the orange-coloured wire remains loose and vice versa. This ensures that the link has independent motions both about e2 - and e3 -axes.
4 Model of PAM in the Proposed Actuators The traditional McKibben muscles work in a way that when the muscle contracts, its length decreases and when the muscle relaxes, its length increases. There is little deformation along the lateral directions during both contraction and relaxation of the muscles. The models of the actuators shown in Figs. 1 and 2a use PAMs of the McKibben type. Also, the McKibben-type PAMs can carry load only when they contract. These PAMs are unable to carry load when they relax. Hence, these PAMs are often used antagonistically. The single degree of freedom actuator with a single McKibben PAM shown in Fig. 1b can only have unidirectional motion in the absence of gravitational force. However, if an extensor contractor pneumatic artificial muscle (ECPAM) [9] is directly connected to the link, as shown in Fig. 5, the single degree of freedom actuator with a single ECPAM would exhibit bidirectional motion and its motion would be similar to that of the actuator shown in Fig. 1a. In Fig. 5, the ECPAM drawn by solid line indicates the ECPAM under contraction, the same ECPAM drawn by dotted line indicates the ECPAM under extension and the same ECPAM drawn by dashed line indicates the ECPAM under relaxation. In the usual McKibben-type PAMs, the tube or the bladder has one inlet and one outlet for the flow of compressed air. The bladder gets inflated when the outlet is
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sealed. The sealed outlet of the tube provides the attachment for joining the wires or threads to the joint or the link, as shown in Fig. 3a. In the PAM for the proposed two-degrees-of-freedom actuator, two bladders are taken inside a braided mesh. As can be seen in Fig. 3b, the inlet of the two bladders is same but their outlets are different. In other words, it can be said that the PAM has one inlet and two attachments for connection of wires or threads coming out of the two outlets of the two tubes. The PAM used in the actuator of Fig. 2b is basically the PAM of Fig. 3b. When air gets filled into both the bladders of the PAM, it results in actuation of the link about both e2 - and e3 -axes simultaneously. However, by using a 5/2-way directional control valve, as shown in Fig. 4, both the bladders of the PAM, for the proposed two-degrees-of-freedom actuator, can be selectively filled which would lead to the independent actuation of the link about e2 - and e3 -axes, as shown in Fig. 5b.
Fig. 3 a McKibben type of a PAM and b the proposed model of a PAM
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5 Mathematical Modelling of the Proposed Two-Degrees-of-Freedom Actuator The equations of motion of the link driven by the proposed two-degrees-of-freedom actuator of Fig. 5b can be derived by following the convention of Fu et al. [10] as ¨ 2×1 + [h(q, q)] ˙ 2×1 + [c(q)]2×1 = [(q)]2×1 [M(q)]2×1 [q]
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where q is the vector of joint variables (α, β), q˙ and q¨ represent the joint velocities and accelerations, respectively, and [M(q)], [h(q, q)] ˙ and [c(q)] represent the mass matrix, coriolis vector and gravity vector, respectively. It may be noted that in Eq. 1, the force vector, [(q)]2×1 , is also a function of the joint variables. This is because the torque exerted at the joint is a cross product of the force exerted by the proposed PAM and the distance between the PAM and the point on the link at which the wire is attached. Also, the force–contraction characteristics of most of the PAMs generally exhibit a nonlinear relationship [11]. Hence, empirical force–contraction relations obtained from experimental data points are also used to characterize a PAM [12].
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In this work, for simplicity, it is assumed that the force–contraction characteristics of the proposed PAM exhibit a linear relationship. Also, it is assumed that the torque remains constant throughout the operation of the PAM. The torque is taken as the product of the force exerted by the PAM and the distance between the PAM and the point of attachment of the wire on the link when both α and β are equal to zero. The properties of the link are: mass=1kg, length=0.6m, distance of centre of gravity is 0.3m, measured longitudinally from any of the ends of the link, and radius of the circular cross section is 0.05m. The distance between the PAM and the point of attachment of the pink-coloured wire on the link is 0.25 m, and for orange-coloured wire, this distance is 0.2 m. It is assumed that each of the PAMs, on contracting, executes a force of 1 N. On putting these values in matrix Eq. 1, two nonlinear equations are obtained in α and β. Using ode45 in MATLAB, the equations can be solved for α(t) and β(t). Figure 6 shows the time response plots of α(t) and β(t).
6 Conclusion In this paper, the model and working of a single degree of freedom and two-degreesof-freedom actuators using a single PAM are presented. The single degree of freedom actuator uses an extensor contractor PAM, whereas the proposed two-degrees-offreedom actuator uses a PAM with two bladders inside a braided mesh. A simplified mathematical model of the proposed two-degrees-of-freedom actuator is used to obtain the time response plots of the two degrees of freedom. One of the limitations of the proposed two-degrees-of-freedom actuator is that it can have only unidirectional motion and not bidirectional motion. Future work will be to fabricate the PAM of the proposed actuator and also to improve the mathematical model considering all the nonlinearities relating to PAM.
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References 1. Oliver-Salazar M, Szwedowicz-Wasik D, Blanco-Ortega A, Aguilar-Acevedo F, Ruiz-González R (2017) Characterization of pneumatic muscles and their use for the position control of a mechatronic finger. Mechatronics 42:25–40 2. Ferris DP, Czerniecki JM, Hannaford B (2005) An ankle-foot orthosis powered by artificial pneumatic muscles. J Appl Biomech 21(2):189–197 3. Tsagarakis NG, Caldwell DG (2003) Development and control of a soft-actuated exoskeleton for use in physiotherapy and training. Autonomous Robots 15(1):21–33 4. Reynolds D, Repperger D, Phillips C, Bandry G (2003) Modeling the dynamic characteristics of pneumatic muscle. Ann Biomed Eng 31(3):310–317 5. Wu Y-C, Chen F-W, Liao T-T, Chen C-T (2019) Force reflection in a pneumatic artificial muscle actuated haptic system. Mechatronics 61:37–48 6. Woods BK, Gentry MF, Kothera CS, Wereley NM (2012) Fatigue life testing of swaged pneumatic artificial muscles as actuators for aerospace applications. J Intelligent Mater Syst Struct 23(3):327–343 7. Daerden F, Lefeber D (2002) Pneumatic artificial muscles: actuators for robotics and automation. Eur J Mech Environ Eng 47(1):11–21 8. Ferris DP, Gordon KE, Sawicki GS, Peethambaran A (2006) An improved powered ankle-foot orthosis using proportional myoelectric control. Gait Posture 23(4):425–428 9. Al-Fahaam H, Nefti-Meziani S, Theodoridis T, Davis S (2018) The design and mathematical model of a novel variable stiffness extensor-contractor pneumatic artificial muscle. Soft Robot 5(5):576–591 10. Fu KS, Gonzalez R, Lee CG (1987) Robotics: Control Sensing. Vis. Tata McGraw-Hill Education, New York 11. Kalita B, Dwivedy S (2019) Dynamic analysis of pneumatic artificial muscle (PAM) actuator for rehabilitation with principal parametric resonance condition. Nonlinear Dyn 97(4):2271–2289 12. Li H, Ganguly S, Nakano S, Tadano K, Kawashima K (2010) Development of a light-weight forceps manipulator using pneumatic artificial rubber muscle for sensor-free haptic feedback. In: 2010 International Conference on Applied Bionics and Biomechanics
Kinematics/Dynamics Analysis with ADAMS/MATLAB Co-simulation of a SolidWorks Designed Spatial Robot Arm with Control and Validation of Results Vikash Kumar Abstract This paper presents kinematic and dynamic analysis of a 3-R robot arm with ADAMS/MATLAB co-simulation, and its control with PID and PID-based fuzzy logic controller; finally, simulated results are compared with numerical method in MATLAB, for kinematics/dynamic analysis, a three degree of freedom robot arm with all revolute joint is designed in SolidWorks, designed model is exported in ADAMS, and exported plant motion is controlled with the MATLAB/Simulink software. ADAMS software is now widely applied to most of the areas such as robotics, automobile, mechanism and so on for an automatic dynamic analysis of the mechanical system. The virtual model plant created by ADAMS software is approaching the real model dynamics analysis in MATLAB with numerical methods, and it can provide a credible result for the real model simulation with ode45 numerical method test and analysis. Keywords Spatial arm · Co-simulation · Fuzzy control · PID control · Dynamic analysis
1 Introduction Robotics a field that is grooving day by day, and many researchers still trying to develop an effective algorithm (position and force control algorithms) to develop a robot that is cost effective and utilized in most of the field for the maximum task. Nowadays, robots used in industries have a wide range of applications, mostly for welding and painting robots in car plants [1], electronic board assembly in electronics and computer fields, repairing nuclear installations in nuclear plant [2], etc. Even though all these tasks can a robot do accurately and precisely and also having more advantages, development of a robot is required a lot of time and amount that involves for testing and validation of robot before installation is done for the final task which it is made for. With the help of ADAMS /MATLAB simulation robot can be analysed V. Kumar (B) Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_114
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Fig. 1 SolidWorks designed three-links robot manipulator, and same designed model is exported ADAMS
kinematically and dynamically [3]. The advantage of co-simulation is that it provides a virtual environment for analysis purpose, and it does do not require any type of dynamics equation or modelling of system [4]. That’s makes it easy and effective for dynamics analysis, and its results are comparable with any numerical and analytical method for validation. Co-simulation of a SolidWorks designed robotic arm and its control with PID [5] and PID-based fuzzy controller is the aim of the paper. To test the simulation results and method for analysis of robot manipulator, a robot manipulator is designed in SolidWorks with 3-DOF as shown in Fig. 1, and all links dimensions, taken for analysis, are shown in Table 1. With taking Table 1, parameters, a three-link manipulator is designed in SolidWorks, it is exported in ADAMS for dynamics analysis, and the designed and exported model is shown in Fig. 1. To export the designed model into ADAMS, the initial step is first assembled each of the three parts of the robot arm in SolidWorks from “Assembly command” in SolidWorks software. Then, assembled arm is saved as “Parasolid” file format, because ADAMS can only recognize some specific file format of SolidWorks. At next, this saved model is imported into ADAMS, the imported model of manipulator does not contain mass and inertia as per material, so in ADAMS, we defined the type of material for each link, and according to type of material, all link parameters are automatically assigned to model. All assigned parameters as shown in Table 2 are taken to analyses the dynamics. Table 1 Manipulator parameter used for robot links design in SolidWorks Links
Length in mm
Width in mm
Height in mm
Link 1
500
Diameter of shaft 50 mm
Link 2
370.2460679378
44.0492135876 mm
27.0246067938
Link 3
237.7796579879
27.7779657988
18.8889828994
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Table 2 Manipulator links mass and inertia parameters data taken from ADAMS Links
Mass
I xx (kg mm2 )
I yy (kg mm2 )
I zz (kg mm2 )
Link 1
7.173866 kg
1.3414708741E+05
1.34149E+05
2231.75143
Link 2
3.754276 kg
5.1823273E+04
5.145724E+04
Link 3
1.063915 kg
6020.2525946
5985.052737
822.1331641 98.2997074
2 Kinematic Analysis, Dynamic Modelling and Control of the Manipulator 2.1 Kinematic Analysis Before exporting ADAMS model of manipulator into MATLAB software, here, discussion on forward kinematic equations and analysis of them is done. For these coordinates, frames of each joint and Denavit-Hartenberg (D-H) parameter [3] are taken for analysis of kinematics as shown in Fig. 2 and Table 3. The transformation matrices for each frame are computed with the help of Table 3. D-H parameter and transformation matrix [3] as given below are used. Fig. 2 Attached coordinates of robot joints
Table 3 Denavit-Hartenberg parameters Frame
ai−1
α i−1
di
θi
0–1
0
0
L1
θ1
1–2
0
90
0
θ2
2–3
L2
0
0
θ3
3–4
L3
0
0
0
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⎤ −sθi 0 ai−1 cθi ⎢ sθi · cαi−1 cθi · cαi−1 −sαi−1 −sαi−1 · di ⎥ i−1 ⎥ ⎢ 1 T =⎣ sθi · sαi−1 cθi · sαi−1 cαi−1 cai−1 · di ⎦ 0 0 0 1 ⎡
(1)
After using the D-H parameter table, we get a transformation matrix for each frame, like putting first frame D-H parameters in the transformation table, we get the transformation of frame 1 to frame 0. It is represented as 0 T1 and given in the following way ⎤ cθ1 −sθ1 0 0 ⎢ sθ1 cθ1 0 0 ⎥ 0 ⎥ T1 = ⎢ ⎣ 0 0 1 L1 ⎦ 0 0 0 1 ⎡
(2)
Similarly, we get the transformation matrix for each frame, from frame 4 to frame 0 and multiply all transformation matrices to get the relation between third link end point P (X 4 , Y 4 , Z 4 ) and base frame B0 and get 0 T4 transformation matrix as given below 0
T4 = 0 T1 1 T2 2 T3 3 T4
(3)
The last column of the 0 T4 matrix gives the relation between the X, Y and Z coordinate of the end point P with respect to base frame B0 , which are ⎡
P(X 4 ,Y4 ,Z 4 )
⎤ ⎡ ⎤ px cos θ1 [L 2 cos θ1 + L 3 cos(θ2 + θ3 )] = 0 P4 = ⎣ p y ⎦ = ⎣ sin θ1 [L 2 cos θ1 + L 3 cos(θ2 + θ3 )] ⎦ pz L 1 + L 2 sin θ1 + L 3 sin(θ2 + θ3 )
(4)
2.2 MATLAB/ADAMS Co-simulation Robot kinematics, dynamics and control analysis, with the help of ADAMS and MATLAB, make easy and accurate, so it is the best tool for applying control command to examine system behaviour. With co-simulation of MATLAB/ADAMS that makes a combined program, it is a more efficient package to simulate and analyses a system like robot manipulator. In order to make co-simulation workable, a common format of input and output is needed that helps to make changes in one program automatically which is made by user in other programs. To recognize MATLAB exported data by ADAMS, ADAMS requires the activation of its control plug-in manager that exports ADAMS control plant model to MATLAB/Simulink software.
Kinematics/Dynamics Analysis with ADAMS/MATLAB Co-simulation … Step:1 Build the solid works
Step:6 Analysis of results in Matlab model
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Step:2
Step:3
SolidWorks Model Exported in ADAMS
Identify the ADAMS inputs and outputs
Step:5 Simulate the model
Step:4 Build the control system in Matlab Simulation using block diagram
Fig. 3 ADAMS/MATLAB co-simulation process steps
ADAMS and MATLAB/Simulink co-simulation mean that to make a multi-body system in ADAMS software, define input–output parameters related to system equations and then export system control plant created by ADAMS to MATLAB/Simulink and set control scheme [6]. During the simulation, the data exchange process between ADAMS’s virtual prototype and MATLAB control program is happening. Where ADAMS works on the mechanical system equations solution, and MATLAB works to resolve the equations of control system. They together complete the whole system simulation and control process. The flow chart of co-simulation process is displayed in Fig. 4. ADAMS has complex seven matrices for the dynamic model of the robot and its motion that is sent to MATLAB during simulation process. The first four are state-space representation matrices A, B, C and D. The fifth and sixth matrices are related to inputs and outputs that are predefined. And the last one has information about plant state variables [7, 8] (Fig. 3).
2.3 Dynamics and Control System of the Manipulator Dynamics of Manipulator For analysing robot arm dynamics, here, we used transformation matrix 0 T 4, which is fourth column as given in Eq. (4), and these X, Y and Z coordinates give a relationship between endpoint and base of the manipulator and set dynamic parameters (Fig. 2), where the base coordinate system coincides with one end of the link 1. Basic dynamic parameters of links are defined as follows: m1, m2 and m3 are, respectively, masses of the links 1, 2 and 3; I 1 , I 2 and I 3 are, respectively, rotational inertia around Z-axis; θ 1 , θ 2 and θ 3 are, respectively, angle variable; L 1 , L 2 and L 3 are, respectively, lengths
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Fig. 4 PID controller for co-simulation
of the three links, and L c1 , L c2 and L c3 are, respectively, lengths between centre of mass of the links and joint axes. Euler–Lagrange method: Total kinetic energy of robot manipulator is given as: 1 I1 + I2 + I3 + m 2 L 22 cos θ2 + m 3 {L 22 cos θ2 + L 23 cos(θ2 + θ3 ) 2 1 +2L 2 L 3 cos θ2 cos(θ2 + θ3 )}]θ˙12 + [I2 + I3 + m 2 L 22 ]θ˙22 2 1 1 2 ˙2 2 ˙ + [I3 + m 3 L 2 ]θ3 + m 3 [L 3 (θ2 + θ˙3 )2 + 2L 2 L 3 θ˙2 (θ˙2 + θ˙3 ) cos θ3 ] 2 2
TK =
(5)
Total potential energy is given as: U P = g[m 1 L c1 + m 2 (L 1 + L c2 sin θ2 ) + m 3 (L 1 + L 2 sin θ2 + L c3 sin(θ2 + θ3 ))] (6) Lagrange function is given by L = TK − U P
(7)
And with above Lagrange, we get the system dynamic equation as: d dt
∂L ∂ θ˙i
−
∂L ∂θi
= τi
(8)
With the use of Eq. (8), we can calculate each joint torque as τ = M(q)q¨ + C(q)q˙ + g(q) + f (q) ˙
(9)
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where τ is torque on each joint actuator, M is inertia matrix, q is joint coordinate vector, C is Coriolis and centrifugal acceleration matrix, g is gravity vector, and f is residual dynamics matrix. For this example, the manipulator has the following mass matrix in the dynamic model [3]: m 11 m 12 m 13 M = m 21 m 22 m 23 m m m 31 32 33
(10)
where 2 m 11 = I1 + I2 + I3 + m 2 lc2 cos2 θ2 + m 3 (l2 cos θ2 + lc3 cos(θ2 + θ2 )]2 2 m 33 = I3 + m 3lc3
(11) (12)
Similarly, centrifugal force, Coriolis force and gravity force and with obtained torque equations we can analysis robot manipulator system numerically with different numerical method. Control System In robotics, control of a robot manipulator is considered as a spine of the system; with the help of controller, a robot can do the assigned task automatically and precisely. So, the controller is very important part in today’s robotics; a controller is nothing but it is a form of equation and written is in form of algorithms that can be understood by the computer programs. The objective of controller implementation in robotics is that the end point of a robot called end effector follows the desired path and orientation accurately and complete the assigned task automatic with higher precision [9]. The motion control of a manipulator either serial or parallel manipulator is done with wide control scheme like feed-forward [2], optimal feedback linearization [10], PID, adaptive backstepping tracking, PID with sliding mode control [11], inverse dynamics control [9, 10, 12], kinematics and dynamics-based PID controllers [13] sliding mode control [5], output integral sliding mode controllers [12] and smooth integral sliding mode controllers [14]. Most of these control schemes require mathematical models of the system to implement a control law, so these controllers need the kinematic and dynamic analysis of the model before implementing controller; however, in ADAMS/MATLAB cosimulation, ADAMS takes care of kinematic and dynamic part and MATLAB sees control part of the system. In below-given two-section, PID and PID-based fuzzy controller implementation on manipulator is discussed. PID Controller Today most of the automatic system used a combination of proportional, integral and derivative (PID) control. This combination is widely applied in the automatic system
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because of its robustness and ease of implementation; in PID controller, we specified some gain to each error signal (proportional error signal, derivative of proportional error signal and integration of proportional error signal from 0 to current time), and the control low for PID controller is given as below u = K P e(t) + K D
de(t) + KI dt
e(t)d(t)
(13)
If we ignore dynamic properties of joint driver, then according to PID control law, the driven torque becomes
τ = K P (θdesired − θ ) + K V θ˙desired − θ˙ + K I
(θdesired − θ )
(14)
θ desired and θ˙desired are denote desired signal of joint angle and angular joint velocity respectively. K P , K V and K I = Respectively related gain coefficients. In the above equation, K P , K D and K I are gain of the error signal in the control loop, and their value remains fix after assigning according to desired response pattern of the output results of the system. These gains modified the main system equation, and according to that damping coefficient and frequency of the system are changed. For any system to see its performance, its rise time, settling time, percentage overshoot and steady-state error for the input signal are used. Each gain (K P , K D and K I ) value has different effect on these performance parameters. For getting desired performance of the system, a very fine-tuning of the gain parameter is required, and it is an iterative process and done in a way like if we want to make system response fast, we increase proportional element, and if we want to make system slow, we increase damp that reduces system overshoot, and with increment of differential element, we can analyse each element effect on system behaviour. With this process, we get PID position control. MATLAB also has its inbuilt PID controller block in MATLAB control toolbox (MCT); however, for this paper, we have used our own calculated gain value that is computed by analysing transfer function output on different gain values with step input signal. In this co-simulation work, a simple PID control system was implemented for each individual angle outputs. Implementation of PID controller is shown in Fig. 4, and its tuned gain parameter and corresponding gains output results are discussed in results section. PID-Based Fuzzy Logic Controller As explained above section, the PID controller is used in the industry because of their reliability, robustness and ease of implementation. Although these conventional controllers are very easy to implement and gives effective results only with the linear system. However, they are not highly applicable for complex, time variant, nonlinear due to variations in parameter because of variations in load, friction and external disturbances [15–17] and also systems with delay. For nonlinear system, fuzzy controller is efficient tools. So a combination of conventional [18, 19] PID controller with fuzzy logic controller is better option that maintains easiness and
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Fig. 5 Fuzzy system basic configuration
robustness of PID controller, and its nonlinear effect of the arm is handle by fuzzy logic controller. It improves the system performance in steady and transient state, in this paper, a PID controller is tuned with fuzzy logic controller, and for this work, MATLAB control toolbox fuzzy controller is used. The fuzzy system basic configuration is shown in Fig. 5, and it has four components [11, 16, 18–22]: 1. 2. 3. 4.
Fuzzifier: Maps crisp points into fuzzy sets. Fuzzy inference engine: Mapping between the fuzzy sets in the input space U ⊂ Rn to the fuzzy sets in the output space V ⊂ Rm . Fuzzy rule-based system: It is the heart of the whole system in the sense which comprises fuzzy rules describing how the fuzzy system performs. Defuzzifier: Maps fuzzy sets of output space into crisp points.
There are various forms for fuzzy controller P, PD, PI, controller and PD-type fuzzy controller. In the present work, a PID-type fuzzy controller is applied. In the fuzzy PID controller, the control variables are the error (e) and the change of the error (e). The rule base is given as: If error (e) is negative large (NL) and derivative of error (e) is positive large (PL), then u is zero. If error (e) is zero and derivative of error (e) is negative large (NL), then u is negative large. If error is zero (ZE) and derivative of error (e) is negative small (NS), then u is negative small. If error is zero and derivative of error (e) is zero, then u is zero. If error is zero and derivative of error (e) is positive small (PS), then u is positive small. If error is zero and derivative of error (e) is positive large, then u is positive large. If error is positive large and derivative of error (e) is negative large, then u is zero. If error is positive large and derivative of error (e) is zero, then u is positive large.
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In the present work, the product inference rule and based on the above rule, the membership function used is given in Fig. 6 and Table 4. Membership functions for error input and output gains. Implementation of PID-based fuzzy controller is represented by block diagram as shown in Fig. 7, in this gain of the controller signal is decided by fuzzy controller, it’s not remain fixed as like PID controller. In this gain is varying according to some defined limit and controller output signal vary according to gains. Selected controller gains and its output results is discussed in results section.
Fig. 6 Membership functions for inputs (e) and (e). and control signal memberships function for outputs (Kp, Ki and Kd)
Table 4 Rule base for fuzzy control e/e
NL
NS
ZE
PS
PL
NL
NL
NS
NS
PS
PL
PL
NL
NS
PS
PS
PL
Fig. 7 Fuzzy-based PID controller for co-simulation
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3 Simulation and Results To run the simulation, ADAMS exported plant name assigned during plant export time. In MATLAB, command window is called by entering “Adams sys” command, and a control plant is created that contains dynamic model information. For simulation, MATLAB Simulink tool is used in which a complete simulation control model is created, and output results are analysed with different gains value. Gains are selected as shown in Table 5. After some trail end error method for PID controller and for fuzzy-based controller, limits of the gains are extended that vary according to the input error signal, and output results are shown in Fig. 8. It shows joint angle in degree versus time graph for each joint with PID and PID-based fuzzy controller, and its output is compared with the desired signal. The end point of the link three taken as an end effector and its coordinate X, Y and Z variation with input signal is shown below in Fig. 9. Table 5 Controller gains for PID and PID fuzzy controller
Constants
PID controller gains
PD-based fuzzy controller
KP
5
2–5
KI
1
1–2
KD
0.3
0.1–0 0.9
Fig. 8 Joint 1 and joint 2 angle variation with time for square and sine were taken as an input
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Fig. 9 Variation in X, Y and Z coordinate of the endpoint with joints inputs
4 Conclusions In this work, an alternative approach with the use of ADAMS/MATLAB software for the kinematics, dynamics and control of the virtual developed three DOF robot has been presented and proposed. With the proposed approach, we are able to analyses inverse kinematics, forward kinematic, the forward dynamic of the manipulator without the generation of complex dynamics equations and without hard and complex analytical methods. In this paper, kinematic, dynamic, control and variation of each joint angle with time are shown, and output of each joint is compared with desired input. With ADAMS/MATLAB software, variation of controller gains effect on joints is also shown in joints angle graphs. Further, by ADAMS post-processer, we can cross-check all the data, like inertia, centrifugal acceleration, gravity, Coriolis acceleration and instantaneous power of the drive motors with the results that were obtained from MATLAB calculation. As future work, with the help of this approach, several robotic systems behaviour will be tested. Also, trajectory generation behaviour of the manipulator will be analyzed in 3D space to study its velocity, acceleration and force on each manipulator’s joints. Some other control algorithms like optimal control can be also tested.
References 1. Park KT, Shin YJ, Park CH, Mo YS, Jeong DC (2008)Robot application for the assembly process of an engine part. ICCAS 2. Ray DD, Singh M (2010) Development of a force reflecting tele-robot for remote handling in nuclear installations. CARPI 3. Craig JJ (2003) Introduction to robotics: mechanics and control, 3rd edn. Prentice Hall, United Kingdom 4. Sosa-Méndez D, Lugo-González E, Arias-Montiel M, García-García RA (2017) ADAMSMATLAB co-simulation for kinematics, dynamics, and control of the Stewart–Gough platform. Int J Adv Robot Syst 1–10
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5. Kisir S, Bingul Z (2012) Position control and trajectory tracking of the Stewart platform. In: Kucuk S (ed) Serial and parallel robot manipulators—kinematics, dynamics, control and optimization. InTech, Croatia, Rijeka, pp 179–202 6. Zhu DL, Qin JY, Zhang Y (2010) Research on co-simulation using ADAMS and MATLAB for active vibration isolation system. IEEE ICICTA 2:1126–1129 7. Affi Z, Romdhane L (2006) ADAMS/SlMULINK interface for dynamic modeling and control of closed-loop mechanisms. In: Proceedings of the 7th WSEAS international conference on automatic control, modelling and simulation, Prague, Czech Republic, pp 353–356 8. Naing ZZ, AI-Mamun MP (2009) Integrated ADAMS+MATLAB environment for the design of an autonomous single wheel robot. In: Industrial electronics, IECON 35th annual conference of IEEE 2009, pp 2253–2258 9. Taghirad HD (2013) Parallel robots: mechanics and control, 1st edn. CRC Press, Boca Raton, FL, USA 10. Tourajizadeh H, Yousefzadeh M, Tajik A (2016) Closed-loop optimal control of a Stewart platform using an optimal feedback linearization method. Int J Adv Robot Syst 13(144):1–9 11. Mathworks. https://in.mathworks.com/help/fuzzy/what-is-sugeno-type-fuzzy-inference.html 12. Fraguela SH, Fridman L, Alexandrov VV (2012) Output integral sliding mode control to stabilize the position of a Stewart platform. J Franklin Inst 349:1526–1542 13. Ding B, Cazzolato BS, Stanley RM et al (2014) Stiffness analysis and control of a Stewart platform-based manipulator with decoupled sensor-actuator locations for ultrahigh accuracy positioning under large external loads. J Dyn Sys Meas Control 136(6):1–12 14. Kumar P, Chalamga A, Bandyopadhyay B (2015) Smooth integral sliding mode controller for the position control of Stewart platform. ISA Trans 58:543–551 15. Huang SJ, Lee JS (2000) A stable self-organizing fuzzy controller for robotic motion control. IEEE Trans Ind Electron 47(2):421–428 16. Yoo BK, Ham WC (2000) Adaptive control of robot manipulator using fuzzy compensator. IEEE Trans Fuzzy Syst. 8(2):186–199 17. Lewis FL, Yesildirek A, Liu K (1996) Multilayer neural-net robot controller guaranteed tracking performance. IEEE Trans Neural Networks 7(2):388–398 18. Li W (1998) Design of a hybrid fuzzy logic proportional plus conventional integral derivative controller. IEEE Trans Fuzzy Syst 6(4):449–463 19. Li W et al (2001) Tracking control of a manipulator under uncertainty by fuzzy P + ID controller. Fuzzy Sets Syst 122:125–137 20. Chen G, Joo YH (2001) Introduction to fuzzy control systems. Department of Electrical and Computer Engineering, University of Houston, Houston, Texas 77204-4793, USA 21. Fuzzy logic controllers and methodology, advantages and drawbacks. Antonio Sala, Pedro Albertos and Manuel Olivares 22. Mathworks. https://in.mathworks.com/help/fuzzy/types-of-fuzzy-inference-systems.html
Joining Aluminum Open Cell Sponge by Friction Stir Welding A. Chandru and S. V. Satish
Abstract Metallic sponges delivered from aluminum 6XXX series are a standout among the most as of late created ultra-lightweight cellular metals. Metallic sponge features exceptionally high-energy ingestion limits with low density and are consequently utilized for boundless applications in the production of ultra-lightweight cellular parts. The manufacture of cellular component parts, for example, hightemperature air or liquid filters and implants, is potential requirement zones relying upon the cell structure size and type of the metal sponge. Metal sponge of closed cell type is ordinarily appropriate for structural uses though open cell sponge; in general, it will be favored for utilitarian applications. Advancement of satisfactory joining innovations for these materials is a basic advance for their far-reaching modern use. The present paper portrays a strong solid-state welding technique that is fit for giving sound joints between aluminum 6XXX macro-porous sponge arrangements. The joining is accomplished by friction stir welding technique abbreviated as FSW through heat generation, softening of material and plastic deformation following the going of non-consumable rotating high-speed steel [HSS] cylindrical tool through the gap between the two solid edges of the sponge to be joined. Having optimized the welding conditions and without utilizing filler metal strong edges were created. The subsequent welds had comparable mechanical properties to that of solid FSW-welded aluminum 6XXX series. Keywords Joining · Metal sponge · Friction stir welding · Macro-porosity · Solid edges
A. Chandru (B) · S. V. Satish Department of Mechanical, PESIT-Bangalore South Campus Affiliated to Visvesvaraya Technological University, Belagavi, Bangalore, Karnataka 560100, India e-mail: [email protected] S. V. Satish e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_115
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1 Introduction Lightweight structural and auxiliary parts have been utilized progressively to lessen the heaviness of different vehicles, railcars, defense vehicles, marine cruisers, and airplane. Amid these cellular segments, multiple layer composite, high vitality retention limits, great damping impacts, and high strength-to-weight proportions are exceptionally compelling for mobility businesses and models. Metal sponge is classified as innovative metals for assembling multiple layers composite, preferred in lightweight boards, basic platform structures, and energy absorption [1]. Metal sponge applications are explored to manufacture vitality safeguards, heat exchanger, sound and magnetic shield, high-temperature filters, fuel cells, and restorative medical implants. Metal sponge as of now is not generally utilized on a modern industrial sector due to different aspects [2]. An expansion of information in their mechanical properties and advancement in cellular welding methods will urge researchers for investigation of these metals for modernized application along with the potential outcomes for their business exploitation. Open cell metallic sponge casted from aluminum termed as ‘aluminum sponge’ in the present paper can be viewed as one of the late created metal sponges equipped for use in load-bearing basic applications. To explore the maximum capacity of sponge metals, a need is to create techniques to weld sponge to sponge or solid metals. Recent investigations suggesting novel methods were explored for manufacturing metal sponge, prevalently aluminum sponge [1–3]. Some attempts have been made to the joining of these aluminum sponges just as aluminum sponge in vehicle applications as sandwich part; however, past experimentation on joining or welding of open cell macro-porous sponge has been found in the literature [4, 5]. Two late articles have investigated the weld ability and mechanical properties of aluminum sandwich foams of closed cell type [5]. Due to the cellular structure of the sponge, the utilization of customary welding procedures—for example, tungsten inert gas (TIG) welding—leads to crumbling of the cellular structure. Few encouraging outcomes were obtained during laser welding of aluminum sponge [6]. Aluminum precursor bonding was adapted for welding aluminum sponge as the temperature needed for bonding is around 500–580 °C of the melting temperature of the metals being welded. The FSW of aluminum sponge would be practical in light of the fact that the bonding pressure required building up joints having sound qualities is well over the Table 1 Data of density, porosity, and relative density of casted aluminum sponge Sponge specimen
Sponge density (kg/m3 )-ρ f
Porosity (%)
Relative density-ρ
Specimen 1
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35.887
0.64
Specimen 2
1492
43.698
0.55
Specimen 3
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53.698
0.45
Specimen 4
1117
57.894
0.41
Specimen 5
1300
50.943
0.48
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Fig. 1 Schematic representation of FSW. a Aluminum. b Aluminum sponge with solid edge
compressive strength of aluminum sponge during joining temperature. In this way, all practical techniques to join aluminum sponge should ideally encounter two primary requirements. To start with, the temperature for bonding ought to be underneath the melting point of aluminum sponge to counteract basic crumbling of cellular structure to be welded. Secondly, process of bonding ought to be related with development of semi-solid phase at weld interface, to remunerate the need to high pressure application and strength to the sponge amid the welding procedure. These conditions are met by friction stir welding [FSW] as indictaed in Fig. 1a; a capable welding method for aluminum sponge. This procedure is likewise equipped for delivering joints having great quality. The present paper investigates a FSW method with solid edge retention as shown in Fig. 1b, to deliver high-quality similar/dissimilar joints of aluminum 6XXX series foam.
2 Experimental Procedure The aluminum metal selected for the current work is commercial aluminum 6XXX series. The solid aluminum 6XXX series ingots were melted in electric induction furnace around 750–800 °C in graphite crucible. The molten aluminum is poured into metal mold containing porous die [7]. Sponge casting can be made from sand or metal mold. The gating elements such as runner and riser are separated by using band saw. Solidified sponge structure is subjected to air blast to remove the sand residue. Aluminum casted sponge billet was later machined using vertical milling machining as FSW needs the welding edges to be straight and has certain dimension constraints for clamping the workpiece in welding fixtures. The dimensions of the sample were 160 × 110 × 8 mm. The solid edges of the casted sponge are the faying
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Fig. 2 Macro-structure of a aluminum sponge, b aluminum sponge with solid edge
surfaces and are cleaned by emery paper to prepare the surfaces for welding. The thickness of the edge is 8 mm as the pin length of the tool is 5.7 mm. The specimens are placed in such manner such that solid edges are adjacent. Aluminum sponge which has been casted and machined as shown in Fig. 2, was joined together using FSW, a solid-state metal welding process that utilizes the heat generated due to friction between the non-consumable rotating tool and the workpieces along with the compressive force due to the tool to join the components together. To retain the position of the tool at or below the metal surface, a vertical downward force is needed. To prevent breakdown associated machinery, minimize excessive wear and tear on the tool, and prevent tool fracture, the FSW cycle should be regulated so that the forces on the tool are reduced and sudden modifications are avoided. As observed, the load acting during FSW ranged between 0.3 and 0.5 tons. The design of the tool and the pin profile is major element as a sound tool improvises weld quality and enables to operate at the maximum possible welding speed. It is desirable that the material used in tool to be sufficiently tough, strong, and hard wearing, at the joining temperature. Hence, high-speed steel with 10% cobalt having a shoulder length of 20 mm and pin length of 5.7 mm was selected. The weld quality depends majorly on the tool rotational speed and feed rate of the tool. The optimum tool rotational speeds for welding of aluminum foams are 700, 800, and 900 rpm. And as per literature review, the feed rate was fixed at 30 mm/min. The mechanical properties and microstructure will fluctuate a lot depending upon these factors. Via optical microscopy, microstructures were examined and were used to explore the composition at the weld interface. Mechanical tests were conducted on few welded samples to investigate the weld strength.
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3 Results and Discussion Initial attempts targeted at welding the aluminum sponge, by FSW, proved ineffective. Due to porosity content at the higher end and compressive load acting on the foam structure leads to collapsibility of weld foam region. Therefore, the solid edge retention on casted sponge was considered toward the use of welding techniques, like FSW, to eradicate or minimize any cellular structure damage to the sponge being welded. Figure 3a shows the surface of FSW welded aluminum foam at 900 rpm, in which a smooth bond is obtained at the faying surfaces without any structural damage. The weld joint can be obtained without the aid of filler metal and free from defects such as no kissing bonds, cavity, or groove-like defects as observed from the X-ray image shown in Fig. 3b. Porosity refers to the voids, i.e., the empty spaces on or within what should be a solid metal casting. It is a defect, but in case of metal sponge, it is an important characteristic and relative density (ρ) which is ratio of foam density to solid density (ρ f /ρ s ) which is deciding factor for exploring macroscopic properties like energy absorption, stiffness, thermal, electrical, and acoustic properties. Relative density influences the mechanical properties much stronger and complex compared to microstructure features. Figure 4 shows typical images of the microstructure of aluminum 6XXX sponge metal. Figure 4a represents parent metal section featuring fine but long needs of unmodified eutectic silicon in an aluminum alloy matrix. Angular particles of primary silicon are also seen. Figure 4b shows thermomechanical affected zone (TMAZ) features’ dendritic structure. Fine eutectic silicon particles are seen at the interdendritic regions. A few undissolved primary silicon particles are also seen. Figure 4c
Fig. 3 a Macro-structure of aluminum sponge welded by FSW. b X-ray image of aluminum sponge welded by FSW
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Fig. 4 Images of microstructure of FSW sample at 900 rpm
welded section shows metal flow. Fine eutectic silicon particles in the matrix of aluminum solid solution and fine particles of eutectic silicon are also seen. Compressive tests of uniaxial type according to ASTM E9 standards were conducted on the aluminum sponge with different porosity and relative densities at similar strain rate as indicated in Table 1. The compressive loads were applied along directions parallel to the direction of pouring of molten metal. Figure 5 shows the stress–strain curves of the aluminum sponge with varying levels of porosity and relative density. The aluminum sponge reveals stress–strain behavior compared with other metal sponge [8]. The stress–strain curve for metal sponge is characterized by linear increase of stress with strain at lower stress inputs (elastic deformation), succeeded with a long deformation plateau as the cell walls buckle and collapse, and finally cell walls get interconnected to each other resulting in flow stress increase causing a densification regime. Tensile tests were performed on the samples at room temperature on universal testing machine based on ASTM E8M-15a. Tensile samples were cut according to standards using electric discharge machining (EDM). Tensile tests showed that
Fig. 5 Compressive stress–strain curve for different porosity samples
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Fig. 6 Tensile stress–strain curve for different tool rpm
all welded tensile specimens were fractured near the welded region. However, the tensile specimens with low relative density were fractured at the base metal due to the presence of voids. Figure 6 shows stress–strain variation of friction stir welded samples at different tool rotation speeds. The results are equivalent to the strength of friction stir welded solid aluminum metal [9] because of the solid edge which results in mechanical properties of the weld to be unaffected by sponge porosity.
4 Conclusions The use FSW method for joining aluminum metal sponge with retention of solid edge to commercial aluminum 6XXX series proved effective. By FSW approach, mechanical strength of the aluminum sponge joints which are equivalent to that of the solid aluminum were reliably produced, without resulting in any cellular integrity damage or modification to the sponge during the welding. It is concluded that weld joints of the sponge with the solid edge have enhanced mechanical properties, due to absence of any precursor metal at the faying surface. Hence, FSW process may be considered for welding aluminum similar/dissimilar sponge. The novel method of solid edge retention by gravity casting technique for sponge manufacturing and welding of sponge by FSW technique proved very successful and defect free. Furthermore, the weld ability of various metals/alloys sponges could be explored by FSW. Acknowledgements The technical support extended to this research work from PESIT Bangalore South Campus affiliated to Visvesvaraya Technological University, Belagavi, is gratefully acknowledged by authors.
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References 1. Peng P et al (2018) High-performance aluminium foam sandwich prepared through friction stir welding. Mater Lett 236 2. Banhart J (2000) Manufacturing routes for metallic foams. JOM 52(12):22–27 3. Lefebvre L-P et al (2008) Porous metals and metallic foams: current status and recent developments. Special issue on metal foams, 2000, September, 2008, pp 775–787 4. Pogibenko AG et al (2015) The weldability of aluminium based foam materials. Weld Int 15 5. Kramer I et al (2006) Friction stir welding of foamable materials & foam core sandwiches. In: Conference on materials, processes, friction and wear MATRIB 6. Haferkamp H et al (2004) Laser based welding of cellular aluminium. Sci Technol Weld Joining 9(1) 7. Chandru A et al (2019) Casting of macro porous aluminum sponge with controlled cellular structure. Mater Today Proc 19:715–720 8. Koza E et al (2003) Compressive strength of aluminium foams. Mater Lett 58:132–135 9. Chandru A et al (2017) Evaluation of joint properties of aluminium alloy after friction stir welding. Int J Sci Res 88–93
Harmonic Response Analysis of Photovoltaic Module Using Finite Element Method Chaitanya V. Bhore , Atul B. Andhare , Pramod M. Padole , Chinmay R. Chavan, Vishal S. Gawande, V. Surya Prashanth, and R. Balagopal Chary
Abstract Nowadays, photovoltaic modules are installed superjacent to cars, ships, trains, metro stations, high rise buildings, etc. where they are subjected to dynamic loads, most significantly through wind and base excitation. Due to these dynamic conditions, vibrations are induced in photovoltaic modules. These vibrations affect the performance and life cycle of photovoltaic module in long run. Therefore, vibration analysis is necessary to understand nature and intensity of vibrations induced in the photovoltaic modules to ensure that they are in acceptable limit. In this paper, the effect of forced vibration on PV module is studied. Harmonic response analysis is performed using finite element method to understand the effect of forced vibrations. Finite element analysis is done for a particular wind speed and different mounting conditions of PV module. The response spectrum is found for excitation frequency ranging from 0 to 100 Hz in sweep generation mode. Vibration responses are plotted in terms of displacement spectrum. This study will provide basis for designing of structures or mountings, which will help in reduction of induced vibration, hence improving life and performance of PV module. Keywords Dynamic loading · Photovoltaic · Vibration · Harmonic
1 Introduction To match with the exponential growth of energy demand in today’s world, renewable energy sources are becoming a popular choice. Photovoltaic systems are one of the most famous renewable energy sources. They can be installed where solar radiation are available. Earlier, they were installed in static conditions such as on roof tops, on land, etc. But nowadays, PV modules are installed above cars, trains, metros, metro stations, ships, etc. where they are subjected to dynamic conditions. These dynamic C. V. Bhore (B) · A. B. Andhare · P. M. Padole · C. R. Chavan · V. S. Gawande · V. S. Prashanth · R. B. Chary Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India URL: http://vnit.ac.in/ © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_116
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Fig. 1 Schematic PV module in the air flow (black) with negative pressure (Blue) and positive pressure (red) and flow separation
conditions occur generally due to base excitation and wind loads [1, 2]. Vibration gets induced into such modules and affect its performance as well as life. PV modules are affected by its surrounding environment which may affect its effectiveness and will indirectly lead to decrease in productivity. From fluid engineering point of view, the PV panel acts as a barrier to wind flow when it is in inclined position; hence, wind load is an important aspect of analysis. As illustrated in Fig. 1 [3] due to flow of wind, low pressure and high pressure regions are formed, below and above the surface of PV module, respectively. Oscillations occur due to the variation of pressure on both sides of PV module, hence inducing vibration [3–5]. Most important factors while analysing operational efficiency and life cycle of photovoltaic cell or a complete system are wind and temperature fluctuation [6, 7]. Karl-Anders Weiss et al. [8] studied correlation between PV module deflection corresponding to wind velocity acting on its surface. Outdoor experimental measurements show the relation between deflection and wind speed and hence the dynamic behaviour of PV module. Correlation between pressure p acting on module surface and deflection y at the centre of surface is shown in Eq. 1. To analyse dynamic loading, indoor facility can be set up in which this model will be very helpful. p = −(62.689y + 0.21895y 2 + 0.1127y 3 )
(1)
Rain, snow, winds, etc. are some of the external mechanical loads to which PV modules are exposed. Strong dynamic conditions occur when rain or wind loads induce vibrations or otherwise only static conditions are created by steady wind and deposited snow. Connecting wires and PV cells are more likely to fail under dynamic loading conditions as fatigue failure. Cracks and micro-cracks may be formed due to induced vibration decreasing performance of photovoltaic module [9–11]. Due to
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such micro-cracks, efficiency of conversion from light to electrical energy decreases and rate of degradation of PV cell increases. Numerical simulation can be performed to predict such effect [12]. As seen from literature, dynamic loading can cause fatigue failures, cracks and micro-cracks due to induced vibration. This affects the life and performance of PV module. Therefore, it is important to study vibration response of PV module. In this paper, the effect of forced vibration on PV module is studied. Harmonic response analysis is performed using finite element method to understand the effect of forced vibrations. Finite element analysis is done for a particular wind speed and different mounting conditions of PV module. The response spectrum is found for excitation frequency ranging from 0 to 100 Hz in sweep generation mode. Vibration responses are plotted in terms of displacement spectrum.
2 Finite Element Analysis Finite element analysis is performed using ANSYS software. Before performing finite element analysis, it is necessary to specify all the assumptions and detailed specifications relevant in this study. This helps to distinguish as well as compare results with existing analysis. Assumptions made for finite element analysis are: • • • • • •
Wind speed is 50 km/h and flowing horizontal to ground surface. All four corners elements of PV module are fixed. Wind pressure is uniformly distributed over the surface of PV module. Wind load acting on PV module is harmonic in nature. Only first six natural frequencies and their modes are considered. Photovoltaic cell layer is ignored [8].
Table 1 shows the specification of photovoltaic module used for analysis. Figure 2 shows dimension of PV module under consideration. Four layers in PV module were glass, EVA, EVA, Tedlar in this sequence and their thicknesses are 3.2 mm, 400 µm, 400 µm, 350 µm, respectively [3]. Table 2 shows different material properties used in finite element analysis. Figure 3 shows four corners fixed coarse meshed finite element model with 685 elements and 5362 nodes. Eight-node solid element is used for modelling. From wind speed and mounting angle, dynamic load amplitudes were calculated. These were used as input harmonic loads. Frequency range of 0 to 100 Hz was used for harmonic response analysis. Before this study, natural frequencies of PV module were determined both by FEM and experiment. For validation of FEM model, experimental results were compared with that of FEM results and good correlation was found. Using FEM model, natural frequencies were determined for upto six modes of vibration. Six natural frequencies
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Table 1 Specifications of Multi/Poly—photovoltaic module Specifications Cell size Number of cells Front side glass Weight Dimensions (L × W × H ) Junction Box Frame Rated power (Pmax) Short circuit current (Isc) Open circuit voltage (Voc)
156.75 mm × 156.75 mm 72 Nos. 3.2 mm Thick low iron tempered 21.5 kg 1958 mm × 987 mm × 40 mm IP67 with 4 rails Anodized alimunium frame 315 W 8.92 A 46.15 V
Fig. 2 Photovoltaic module dimensions Table 2 Specifications of multi/poly–crystalline photovoltaic module Material Young’s modulus Poisson’s ratio (ν) (GPa) Glass EVA PV Cell Tedlar Aluminium
66.1 0.0668 112.4 2.076 70
0.23 0.33 0.28 0.33 0.33
Density (kg/m3 ) 2600 1030 2328 1370 2700
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Fig. 3 Finite element model of PV module
found were 17.47, 35.15, 39.18, 51.03, 53.56 and 63.32 Hz. This validated FEM model was used for determination of harmonic response of PV module to externally induced forced vibration.
3 Results Taking into consideration the weight of PV module and wind speed, load acting on the module was calculated. Four responses are found: two angular position of PV module at 21◦ and 90◦ , and two load condition considering pure wind load and base excitation of same magnitude as wind load. Figure 4 shows finite element model of wind load acting on PV module when module was kept at 21◦ angle from horizontal. Figure 5 shows the harmonic response spectrum of PV module when purely wind load is acting and module is inclined at 21◦ with respect to horizontal.
Fig. 4 Finite element model of PV module showing wind load
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Fig. 5 Frequency response with wind load and module at 21◦
Fig. 6 Frequency response with wind load and module at 90◦
Figure 6 shows the case of PV module for 90◦ inclination with horizontal, which makes the incident wind velocity direction perpendicular to module front surface. Maximum amplitudes are observed when excitation frequencies match with various natural frequencies of Pv module. Figure 7 shows the harmonic response spectrum of PV module when base excitation loads are acting and module is inclined at 21◦ with respect to horizontal. Figure 8 shows the case of PV module for 90◦ inclination with horizontal. Maximum amplitudes are observed when excitation frequencies match with various natural frequencies of PV module in case of base excitation loads also. Table 3 shows peak amplitudes and corresponding natural frequencies for different cases of inclination angles and load type as shown in above harmonic responses (Figs. 5, 6, 7 and 8).
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Fig. 7 Frequency response with base excitation and module at 21◦
Fig. 8 Frequency response with wind load and module at 90◦ Table 3 Peak amplitude and its corresponding frequency for different cases of response spectrum Case Frequency (Hz) Amplitude (mm) WL (21◦ ) WL (90◦ ) BE (21◦ ) BE (90◦ )
63.32 17.47 62.78 17.16
3.49e−5 8.30e−6 4.49e−6 2.91e−9
Wl wind load, BE base excitation)
4 Conclusions The effect of forced vibration on PV module is studied in this paper. Harmonic response analysis is done for different inclination in wind load dynamic condition. In case of dynamic conditions created by wind load and reaction force, it was observed that for inclination of 90◦ with horizontal peak amplitude of vibration occurs for first modal/natural frequency of module.
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But, when the inclination was decreased to 21◦ with horizontal, it was observed that peak amplitudes occurred at sixth mode, that is highest natural frequency of PV module. From above observation, we can conclude that as we go on increasing the inclination angle with horizontal in dynamic conditions created by wind load and base excitation, peak deflection occurrence moves from highest to lowest natural frequency modes. This harmonic response analysis enables us to quantify nature of vibration response for different dynamic conditions. With the help of dynamic response results, one can design suitable structures and mounting conditions which can prevent unwanted effects of externally induced vibration. Further study can be performed for different wind speeds at different inclination and with different support conditions.
References 1. Danyel R, Mischa B (2004) Policy differences in the promotion of renewable energies in the eu member states. Energy Policy 32(7):843–849 2. Parida V, Iniyan S, Goic R (2011) A review of solar photovoltaic technologies. Renew Sustain Energy Rev 15(3):1625–1636 ˇ Kristina K (2016) Analysis of external dynamic loads influence to 3. Art¯uras K, Audrius C, photovoltaic module structural performance. Eng Fail Anal 66:445–454 4. Menoufi K, Chemisana D, Rosell JI (2013) Life cycle assessment of a building integrated concentrated photovoltaic scheme. Appl Energy 111:505–514 5. Sanghoon Y, Sehyun T, Jinsoo K, Yongseok J, Kisuk K, Jiyoung P (2011) Application of transparent dye-sensitized solar cells to building integrated photovoltaic systems. Build Environ 46(10):1899–1904 6. Skoplaki E, Palyvos JA (2009) On the temperature dependence of photovoltaic module electrical performance: a review of efficiency/power correlations. Solar Energy 83(5):614–624 7. Dirk G, Emmanuel VK (1999) Aeolian dust deposition on photovoltaic solar cells: the effects of wind velocity and airborne dust concentration on cell performance. Solar Energy 66(4):277– 289 8. Weiss K-A, Assmus M, Jack S, Koehl M (2009) Measurement and simulation of dynamic mechanical loads on pv-modules. In: Reliability of photovoltaic cells, modules, components, and systems II, vol 7412. International Society for Optics and Photonics, pp 741203 9. Yih-Chih C, Jian-Zong L, Yu-Teng L (2011) Micro crack detection of multi-crystalline silicon solar wafer using machine vision techniques. Sens Rev 31(2):154–165 10. Köntges M, Kunze I, Kajari-Schröder S, Breitenmoser X, Bjørneklett B (2011) The risk of power loss in crystalline silicon based photovoltaic modules due to micro-cracks. Solar Energy Mater Solar Cells 95(4):1131–1137 11. Dallas W, Polupan O, Ostapenko S (2007) Resonance ultrasonic vibrations for crack detection in photovoltaic silicon wafers. Meas Sci Technol 18(3):852 12. Jørgensen M, Norrman K, Krebs FC (2008) Stability/degradation of polymer solar cells. Solar Energy Mater Solar Cells 92(7):686–714
Development of a Micro-forming System for Micro-extrusion Process of Micro-pin in AZ80 Alloy D. Rajenthirakumar, N. Srinivasan, and R. Sridhar
Abstract Manufacturing of components for micro-systems is a key enabling technology for a new generation of miniaturized devices. The micro-forming is to manufacture the parts or part features with the dimensions in sub-millimeter scale. The process has great potential to become a promising micro-manufacturing method. Micro-forming technology poses much higher demands on accuracy, velocity and mass production, and the common forming machines cannot satisfy these requirements. Micro-forming equipment with high speed and high precision has become an important research field for industrial application. In this work, micro-extrusion ability of bio-absorbable AZ80 alloy is examined using a novel micro-extrusion equipment consisting of forward extrusion assembly and a loading setup. This work aims at discussing the size effect-related deformation behaviors which will help to understand the mechanisms and fundamentals of the size effects in microextrusion process. The realization of such a productive forward extrusion assembly poses significant advantages when compared to the conventional manufacturing technologies in the production of nano- and micro-parts. Keywords Micro-forming · Micro-extrusion · Aluminium
1 Introduction With the development of micro-electro-mechanical systems, micro-forming, defined to be the production of parts and structures with at least two dimensions in the submillimeter range, becomes an important technology. Reviews of micro-forming by Engel and Eckstein [1] and Vollertsen et al. [2] highlight the key technologies and indicate that micro-forming is a promising way to fabricate micro-parts because of its significant advantages in mass production, providing controlled quality and low cost. Further, the usage of metal, ceramics or metallic glasses in micro-system technology D. Rajenthirakumar (B) · N. Srinivasan · R. Sridhar Department of Mechanical Engineering, PSG College of Technology, Coimbatore, Tamil Nadu 641004, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_117
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becomes important in future technology because these materials have higher strength and wear resistance than the polymers that are being used most widely. However, micro-forming technology is still at laboratory research status, and few industrial micro-forming process chains have been realized due to the size effects in micro-forming, when the component feature sizes are reduced to a few hundreds or tens of microns. Machines, forming tools and handling of micro-parts are critical elements which significantly determine industrial applications of micro-forming. The development of micro-parts by micro-forming, however, cannot totally be based on the traditional macro-forming knowledge. In micro-forming, the material deformation behavior is characterized by a few grains in the deformation zone. Different properties of grains make the deformation behavior inhomogeneous and difficult to predict. Further, the size effect-related deformation phenomena further affect the performance and product quality in terms of defect formation, dimensional accuracy and surface finish of the micro-formed parts. In this work, a novel micro-forming system (Fig. 1) is developed and equipped with a tool. Micro-extrusion process is investigated with the micro-forming system, and the optimum forming parameters are obtained. Based on the results, micro-pins are successfully fabricated with highdimensional accuracy. This work also aims at discussing the size effect-related deformation behaviors which will help to understand the mechanisms and fundamentals of the size effects in micro-forming process.
Fig. 1 Forward extrusion machine
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2 Development of Micro-forming System and Micro-extrusion Process In this work, micro-extrusion ability of bio-absorbable magnesium (AZ80) alloy is examined using a novel equipment consisting of forward extrusion assembly and a loading setup. The shape and dimensions of the split die used in the experiment are obtained by machining with a wire electrical discharge machine. The effects of miniaturization on micro-components and the material behavior during forward extrusion are investigated. By using a forming equipment in conjunction with a loading setup, the authors are able to investigate the force–displacement response for AZ80 alloy micro-extrusion with different grain size. Size effect on flow stress is also investigated using different size scaled specimens with different grain sizes. Extrusion tests performed on the samples of different grain size demonstrated that decreasing grain size caused an increase of flow stress. It is also determined that the flow stress decreases with the reduction of specimen size. Further, the present results demonstrate that high-strength micro-components could be fabricated by controlling grain refinement.
3 Summary The realization of such a productive forward extrusion machine poses significant advantages when compared to the conventional manufacturing technologies in the production of nano- and micro-parts. Understanding of the size effect-related deformation behaviors and phenomena and size effect mechanisms is crucial in micro-part development via micro-forming.
References 1. Engel U, Eckstein R (2002) Microforming—from basic research to its realization. J Mater Process Technol 125–126:35–44 2. Vollertsen F, Niehoff HS, Hu Z (2006) State of the art in micro forming. Int J Mach Tools Manuf 46:1172–1179
Topological Analysis of Epicyclic Gear Trains—Symmetry and Redundancy V. R. Shanmukhasundaram , Y. V. D. Rao , and S. P. Regalla
Abstract This proposal is for carrying out topological analysis of epicyclic gear trains (EGTs) starting from the graph theory based model. With respect to a given EGT (graph), the topological analysis is a phase of conceptual design where the designer determines all the possible combinations of input, output, and fixed links in an EGT. In this phase, symmetry analysis of an EGT is vital, as it becomes easy to discard kinematically equivalent input-output-fixed link combinations. Furthermore, it is essential to evaluate those input-output combinations causing one or more redundant links. Addressing these twin objectives, graph theory based methods are presented and elucidated with examples. Keywords Symmetry · Redundant gears · Topological analysis · Clutching sequences · Operating mode
1 Introduction Epicyclic gear trains (EGTs) represent a class of geared kinematic chains that are widely used as a major part of mechanical transmission systems, finding applications in automobiles, machine tools, robotic manipulators, etc. During the conceptual stage of design, it is common to represent the kinematic structure of an EGT as a simple graph where vertices and edges denote links and joints respectively [1]. The graph representation enables development of graph theory based algorithms which when applied upon a set of parent graphs, enumerates EGT chains of a certain complexity in the form of candidate graphs. Once all the non-isomorphic graphs (i.e. concepts) are enumerated, the next stage of conceptual design is topological analysis involving determination of all the distinct V. R. Shanmukhasundaram (B) Madanapalle Institute of Technology & Science, Madanapalle, Andhra Pradesh, India Y. V. D. Rao · S. P. Regalla Department of Mechanical Engineering, BITS PILANI, Hyderabad Campus, Secunderabad, Telangana 500078, India
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ways of assigning fixed, input, and output links, for a given EGT [2]. For a set of combinations of fixed, input and output links, the following need to be evaluated: (1) (2)
Topologically identical possibilities owing to the existence of symmetry. Those possibilities that render certain link/s as redundant.
As first effort in this regard, Olson et al. [2] analyzed all the distinct 5-link EGTs for combinations of an input, output and a fixed link and discarded those containing kinematically superfluous link/s. Hsu and Lin [3] devised a method based on the fundamental circuits of the EGT, for identifying redundant gears. Liu and Chen [4] introduced the concept of kinematic fractionation and related rules following which redundant links can be prevented while assigning different input and output links of the EGT chain. Del Castillo [5] derived functional constraints to be satisfied by an EGT graph to avoid idle links. These constraints when applied result in specific combinations of sun, planet, and carrier links of an EGT. All such EGT graphs with these combinations are then enumerated. However, Del Castillo [5] also mentions that the conditions imposed in the enumeration are necessary but not sufficient for guaranteeing all the inversions with driver and driven link combinations, to be free from idle links. Rao and Rao [6] and more recently Rajasri et al. [7] have used the Hamming number matrix in a procedure for partitioning the links of EGT into different classes based on symmetry. Their endeavor [6, 7] was to determine symmetrical links corresponding to topologically identical vertices in the EGT graph. The topological analysis of an EGT chain, being a basic phase of conceptual design, ultimately paves way for analyzing various kinematic configurations for speeds, torques, and in turn efficiency. A similar problem to topological analysis is that of enumerating the clutching sequences for an EGT graph to be used in multi-speed transmissions with sometimes multiple operating DOFs [8–12]. From the literature survey, it is evident that different works on EGTs were aimed at obtaining kinematic configurations of EGTs and understand the kinematic structure of an EGT for any symmetry as well as driver-driven link combinations that result in certain redundant link/s. Though graph theory based synthesis methods are well established for catering to the enumeration of distinct possibilities of EGTs, methods for topological analysis in the literature are not comprehensive enough and are very much needed for an EGT designer. The present work aims to address this aspect and is organized in the following manner. In Sect. 2, graph representation of an EGT is briefly dealt with. Determination of symmetry properties of an EGT are explained in Sect. 3 with suitable examples. Information of a given rotation graph is input to Nauty and Traces package [13, 14] and orbits and automorphism group are obtained as output, which are in turn used to characterize the symmetry inherent in the rotation graph [6, 7] and in the EGT. Section 4 introduces a flow graph model for visualizing the internal torques of the EGT. This flow graph can be constructed using the information from the rotation graph. Using connectivity properties of the flow graph, it is shown that, condition for the presence of redundant link/s can be expressed [15].
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2 Graph Representation of Kinematic Structure of EGTs In the graph of an EGT, vertices, and edges respectively imply the links and joints (pair connections). In this manner, through the topology existent in the graph, the kinematic structure inherent in the EGT is revealed. There exist two kinds of pairs in any EGT, namely, turning pairs and gear pairs. In the EGT graph, the edges are colored in order to differentiate turning and gear pairs. The graph of an N-link and F-DOF EGT consists of N vertices, (N − 1) turning pairs and (N − 1 − F) gear pairs [1]. In the graph of N-vertex EGT, all the (N − 1) turning pairs are connected (i.e. not disjoint). Therefore, the set of turning pair edges (TPE) constitutes a spanning tree and as a consequence, every gear pair edge along with path of corresponding TPEs gives rise to a fundamental circuit (FC). The TPEs are labeled with levels specifying the position of axis in space. Further, all those edges having same label are connected and constitute a tree subgraph with respect to the spanning tree graph of all TPEs [16]. In every FC, each TPE can possess one of the two different edge labels prevailing in the circuit. Recalling that, since the same labeled TPEs are not disjoint, therefore as a consequence, in every FC there is a transfer vertex (TV) that separates TPEs possessing one common label from those TPEs having the other common label. A rotation graph has the information of all the gear pair edges (GPEs) and its TVs (i.e. as in the FCs). Displacement graph/s can be generated by labeling the TPEs in different ways such that TVs of all FCs remain the same as given in the rotation graph. In Fig. 1a, a 5-link EGT is shown in its closed rotation graph representation [17] with TV marked as vertex-1. The three possible displacement graphs with respect to the rotation graph are shown in Fig. 1b–d. Traditionally, the structural synthesis of EGTs, with a given number of links and DOF, focuses on enumerating all the non-isomorphic rotation graphs and all the nonisomorphic displacement graphs with respect to every rotation graph [18]. Using a displacement graph, an EGT schematic can be sketched, thereby giving ideas to an EGT designer during the conceptual stage.
3 Determination of Symmetry Properties of EGTs 3.1 Application of Group Theory to Study Structure of EGT Graph Identical location of identical links or joints in an EGT is defined as symmetry in an EGT [6]. The topological symmetry of a simple graph can be studied using the theory of finite groups [19]. Owing to the correspondence between the kinematic structure of an EGT and its simple graph model, the symmetry inherent in the graph is also the symmetry of EGT chain. For an EGT graph, group elements refer to its vertices or edges. A group action refers to a particular transformation achieved by
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(a)
(b)
(c)
(d)
Fig. 1 a Rotation graph of a 5-link EGT; (b–e) Its 3 displacement graphs
permuting the group elements. All such exhaustive permutations (mappings) belong to symmetric group whose size given by:
group elements
Group size of symmetric group =
i.
(1)
i=1
group elements i. denotes factorial function. For the purpose of symmetry analwhere, i=1 ysis of a graph, permutations of group elements of the graph can be used to convey those compositions (group actions) that preserve the topology of the graph and therefore that of the EGT chain. All those structure-preserving permutations constitute the automorphism group, which is also a subgroup of the symmetric group. The group elements can be partitioned into equivalent classes or orbits. Those group elements belonging to the same equivalent class (vertex orbit or edge orbit), occupy similar topological positions in the corresponding graph. For a given group object (EGT graph), equivalence classes and automorphism group are related by the Burnside lemma [18]. Nauty and Traces program [14] can be used to obtain the orbits and automorphism group with respect to a rotation graph of EGT, as input. In our previous work [20] symmetry analysis results with up to 6 links were tabulated.
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3.2 Topologically Similar Possibilities Owing to Symmetry—Examples The knowledge of vertex orbits with respect to an EGT graph can be applied to the topological analysis of same. The EGT schematic of a displacement graph in Fig. 1c is given in Fig. 2 which is commonly known as “Ferguson paradox”. It is basically a compound EGT with link-1 as carrier link along with its associated compound planet being link-2 which meshes with suns: link-3 and 4 (external gear) and link-5 (internal gear). Generally, planet carrier link-1 is given an input. One among the three sun links is fixed and output power is derived from the remaining two suns. The symmetry analysis with respect to the EGT graph of Fig. 1 revealed the orbits as: {(1); (2); (3 4 5)}. Therefore, the sun links (3, 4, and 5) are all topologically similar since they all fall in the same vertex orbit (indicated in blue color in Table 1). As a result, the 3 suns can be interchanged for the choice as fixed link thereby leading to different inversions which are all kinematically equivalent as mentioned in Table 1. An EGT graph can be decomposed into kinematic units (KUs) by splitting them at the TVs [16]. In the previous example of a 5-link EGT, it was seen that similar links were associated with FCs belonging to a single carrier link (TV) and therefore only a single KU exists. In the case of multi-carrier EGT graphs (more than one KU), symmetrical links can exist across KUs if and only if the TV of respective KUs is topologically similar. As an example, a 6-link EGT is shown in Fig. 3a along with
Fig. 2 A schematic of EGT with respect to the displacement graph of Fig. 1c
Table 1 The kinematic equivalence of different operating modes owing to the graph symmetry Operating Input link Output links Fixed link Comments Mode Mode-1 1 4, 5 3 All the three modes are Mode-2 1 3, 5 4 kinematically equivalent. Mode-3 1 3, 4 5
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Fig. 3 a A 6-link EGT; b its 2 kinematic units belonging to TV-1 and TV-2 respectively
(a)
(b) the KUs in Fig. 3b. The vertex orbits are found to be: {(1 2); (3 4); (5 6)}. This 6-link EGT has TVs link-1 and link-2 being topologically similar and as a consequence all associated links from the respective KUs, are also similar. Link-3 in KU of TV-1 is similar to link-4 in KU of TV-2. Likewise, link-5 and link-6 are also similar. A possible schematic representation of this EGT is given in Fig. 4. There exist four central links for this 6-link EGT excluding the two planet links: the carrier links 1 and 2 as well as sun links 5 and 6. If we consider giving input to a link, fixing a link and deriving outputs from the remaining two links then there are many such possibilities. Owing to symmetry as apparent from vertex orbits, a link corresponding to a vertex orbit can be interchanged for another link falling in the same orbit thereby giving rise to similar possibilities as listed in Table 2. For example, Mode-1 and Mode-2 are similar because they are mutually convertible by
Fig. 4 A schematic representation with respect to the EGT graph of Fig. 3a
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Table 2 The sets of topologically similar possibilities for input-output-fixed links for Fig. 4 Operating Mode Input link Output links Fixed link Comments Mode-1 5, 6 1 2 Mode-1 and Mode-2 are topologically identical Mode-2 5, 6 2 1 Mode-3 1, 2 6 5 Mode-3 and Mode-4 are topologically identical Mode-4 1, 2 5 6 Mode-5 1, 6 2 5 Mode-5, Mode-6, Mode-7 Mode-6 1, 5 2 6 & Mode-8 are topologically Mode-7 2, 5 1 6 identical Mode-8 2, 6 1 5
interchanging link-1 and link-2 since they belong to same vertex orbit (marked in blue color); the other vertex orbit consists of links 5 and 6 (marked in green color) and they are outputs.
4 Identification of Redundant Links in an EGT Graph—Application of Connectivity Concept in Graph Theory 4.1 Background In the quest for kinematic configurations corresponding to different operating modes, there is a necessity to detect and avoid those, which present redundant link/s. This is because: a redundant link in an EGT with respect to a certain operating mode (i.e. given input and output links) can be taken out of the EGT without affecting the kinematic characteristics [2, 5]. Olson et al. [2] presented an approach that involves writing the angular velocity equations for all the gear pairs (FCs of the graph) of the EGT, followed by determining the expression for transmission ratio with respect to the input and output links. If the transmission ratio is not a function of angular velocity of any link of the EGT then that particular link is declared redundant. Hsu and Lin [3] came up with an interesting logic for detecting redundant links for an EGT with G gear pairs. It is based on generating combinations of FCs: 2, 3, 4, …, (G − 1), at a time. A general expression for existence criterion for redundant gears was derived as a function of number of FC combinations, number of input and output links. Based on the knowledge of input and output links (as presented by a certain operating mode); the existence criterion for redundancy is applied to those combinations of FCs that contain the particular input and output links. Hsu and Lin [3] computerized the procedure and applied it to common automotive EGT transmissions. Liu and Chen [4] used the concept of kinematic units (KUs) and proposed that in order to avert redundant link/s, the choice of input and output links must be spread
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across KUs of the EGT, such that there is not any KU which does not contain at least one actuating link or an output link.
4.2 Present Work A flow graph model is introduced, whose vertex set consists of all the links and gear pair connections. The edges of the flow graph represent internal torques. All the links in an EGT have rotary motion and their shafts naturally possess external torques. A rotary motion from one link to another is transmitted by means of a gear element (keyed to the respective shafts of links), resulting in an internal torque. The algebraic sum of all the internal torques at a link is equal to its external torque. Now, a distinction can be made regarding the nature of internal torques existing in an EGT namely: meshing torques and planet-Carrier torques. A meshing torque at a link is due to a gear pair connection with another link and it is numerically given by: Meshing torque at a link Gearing power at the pair connection = Relative angular velocity of the link in carrier frame of reference
(2)
A planet-carrier torque is responsible for the transfer of a rotating motion to a planet from a carrier or vice versa. It is numerically given by: Planet − Carrier torque = Arm length of carrier ∗ Meshing force of planet
(3)
A commonly used transmission in automotive industries, namely, the Simpson EGT is shown in Fig. 5. It is a 6-link EGT having link-1 and link-3 as suns. Link4 and Link-6 are the carriers whose corresponding planets are link-2 and link-5 respectively. The flow graph for the Simpson EGT is given in Fig. 6 whose vertex
Fig. 5 EGT schematic of the Simpson gear system
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Fig. 6 A flow graph model for visualizing internal torques of Simpson EGT of Fig. 5
set consists of the 6 links and the four gear pairs: G (1, 2), G (1, 5), G (2, 3) and G (4, 5). The edges of the flow graph along with their internal torques are tabulated in Table 3. Given the input and output links, the direction of internal torques is not obvious in a flow graph. This is because the direction of torques depends on the magnitudes of meshing powers which is determined by the values of angular velocities of the links. However, by employing the flow graph, it is possible to determine the presence of otherwise redundant link/s in the EGT with respect to the input and output links. The approach used here relies on the connectivity of the proposed flow graph model and it is explained with the help of an example given by Hsu and Lin [3]. With respect case-1 of Table 4, inputs are link-1 and link-3 and output link is 4. All those edges belonging to the possible paths from link-1 and link-3 leading to output link-4, are marked as red in Fig. 7a. It is evident that edge 5–6 is not marked red because whatever torque the planet link-5 transmits to the carrier link-6 does not contribute to the overall external torque at output link-4. Therefore, link-6 is redundant. In this operating mode, the torques transmitted to planet link-5 from link4 and link-1 is futile. This is because: under normal operating conditions (without any Table 3 Details of nature of internal torques with respect to flow graph of Fig. 6 Edge in the flow graph
Type of internal torque
2–4
Planet-carrier torque between links 2 and 4
5–6
Planet-carrier torque between links 5 and 6
2 − G(2, 3)
Meshing torque of link-2 owing to gear pair 2–3
3 − G(2, 3)
Meshing torque of link-3 owing to gear pair 2–3
1 − G(1, 2)
Meshing torque of link-1 owing to gear pair 1–2
2 − G(1, 2)
Meshing torque of link-2 owing to gear pair 1–2
5 − G(1, 5)
Meshing torque of link-5 owing to gear pair 1–5
1 − G(1, 5)
Meshing torque of link-1 owing to gear pair 1–5
4 − G(4, 5)
Meshing torque of link-4 owing to gear pair 4–5
5 − G(4, 5)
Meshing torque of link-5 owing to gear pair 4–5
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Table 4 Results of redundancy analysis to the Simpson EGT of Fig. 5 Operating mode
2 Inputs and 1 output link
Redundant gear pair/s
Redundant link/s
Case-1
1, 3, 4
None
5, 6
Case-2
1, 3, 6
None
None
Case-3
1, 4, 6
G (2, 3)
3
Case-4
3, 4, 6
None
None
(a)
(b)
Fig. 7 a Flow graph of case-1 with those edges marked red which constitute the paths between input links (1 and 3) and output link 4; b the two components of the flow graph obtained by splitting the graph at link-4 and link-1
circulating power), the torque supplied to link-5 takes away useful torque developed at link-4. Hence link-5 is also redundant. Another reasoning is also offered to justify that link-5 and link-6 are redundant: angular velocity of output link-4 can be calculated using the input angular velocities of link-1 and link-3 and by solving the Willis equations of FCs: 4(1 2) and 4(2 3). Both the FCs, which include the constituent links (1, 2, 3, and 4), belong to same kinematic unit. Another KU consists of FCs: 6(1 5) and 6(4 5). Link-1 and 4 are common to both the KUs (Fig. 7b). Since all the input and output links belong to same KU, they are kinematically fractionated with respect to the other KU [4]. Therefore, link-5 and link-6, being in a different KU than the input and output links, are redundant. Regarding the case-3 of Table 4; inputs are given to link-1 and link-4 and output is taken from link-6. As evident from the flow graph with respect to this mode in Fig. 8a, gear pair (2, 3) and link-3 are redundant. For cases 2 and 4, none of the links are redundant as apparent from Fig. 8b. This is because: all the edges in the flow graph are traversed by the possible paths between input and output links (all the edges belonging such paths are marked in red).
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(b)
Fig. 8 a Flow graph with respect to case-3; b flow graph for cases 2 and 4
5 Conclusions Given the wealth of numerous EGT graph solutions as design alternatives, there is always a need for rational guidelines to assist a designer in selecting a transmission for a particular application. The developed graph theoretic framework for topological analysis can help the EGT designer in evaluating various EGT transmission architectures and thereby synthesizing different operating modes, as per the requirements. If symmetry is present in the EGT graph, then some of the input-output link combinations (i.e. those concerning the topologically similar links), will turn out to be kinematically equivalent. The presence of redundant link/s in an operating mode will have a detrimental effect on mechanical efficiency. This is because: the meshing power due to the redundant link/s does not contribute to the power derivable at the output link. Using the connectivity properties of a flow graph, the presence or otherwise of redundant link/s can be found with respect to specified input and output links of the EGT. In summary, the topological analysis gives deeper insight about the kinematic structure and the kinematic interactions within the EGT. In order to answer the question as to which EGT concept is best for an intended application and what being the most efficient operating mode, it necessitates a detailed study of internal power flows. The topological analysis methods can be perhaps extended, in order to facilitate such a thorough power flow study; and it is planned as our future work in EGTs.
References 1. Buchsbaum F, Freudenstein F (1970) Synthesis of kinematic structure of geared kinematic chains and other mechanisms. J Mech 5(3):357–392 2. Olson DG, Erdman AG, Riley DR (1991) Topological analysis of single-degree-of-freedom planetary gear trains. J Mech Des 113(1):10–16
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3. Hsu CH, Lin YL (1994) Automatic identification of redundant gears in planetary gear trains. Math Comput Model 20(7):99–109 4. Liu CP, Chen DZ (2001) On the application of kinematic units to the topological analysis of geared mechanisms. J Mech Des 123(2):240–246 5. Del Castillo JM (2002) Enumeration of 1-DOF planetary gear train graphs based on functional constraints. J Mech Des 124(4):723–732 6. Rao AC, Rao YVD (2002) Symmetry in planetary gear trains. Indian J Eng Mater Sci 9:311–314 7. Rajasri I, Gupta AVSSKS, Rao YVD (2014) Symmetry and its effects on structures of planetary gear trains. J Inst Eng (India): Ser C 95(1):30–38 8. Hsieh HI, Tsai LW (1998) The selection of a most efficient clutching sequence associated with automatic transmission mechanisms. J Mech Des 120(4):514–519 9. Hwang WM, Huang YL (2011) Connecting clutch elements to planetary gear trains for automotive automatic transmissions via coded sketches. Mech Mach Theory 46(1):44–52 10. Peng ZX, Hu JB, Xie TL, Liu CW (2015) Design of multiple operating degrees-of-freedom planetary gear trains with variable structure. J Mech Des 137(9). https://doi.org/10.1115/1.403 0856 11. Gao MF, Hu JB (2018) Kinematic analysis of planetary gear trains based on topology. J Mech Des 140(1). https://doi.org/10.1115/1.4038072 12. Xu X, Sun H, Liu Y, Dong P (2019) Automatic enumeration of feasible configuration for the dedicated hybrid transmission with multi-Degree-of-Freedom and multiplanetary gear set. J Mech Des 141(9). https://doi.org/10.1115/1.4042846 13. Mckay BD, Piperno A (2018) Nauty and traces user’s guide (version 2.7). Technical report, Departimento di Informatica, Sapienza Universita di Roma 14. The nauty and Traces page. http://pallini.di.uniroma1.it/ 15. Harary F (2001) Graph theory. Narosa Publishing House 16. Shanmukhasundaram VR, Rao YVD, Regalla SP (2019): Enumeration of displacement graphs of epicyclic gear train from a given rotation graph using concept of building of kinematic units. Mech Mach Theory 134:393–424 17. Ravisankar R, Mruthyunjaya TS (1985) Computerized synthesis of the structure of geared kinematic chains. Mech Mach Theory 20(5):367–387 18. Yang W, Ding H (2018) The complete set of 1-DOF planetary gear trains with up to 9 links. J Mech Des. https://doi.org/10.1115/1.4041482 19. Duzhin SV, Chebotarevsky BD (2011) Transformation groups for beginners. Paperback. Universities Press, Hyderabad 20. Shanmukhasundaram VR, Rao YVD, Regalla SP (2019) Analysis of symmetry in epicyclic gear trains. In: Uhl T (eds) Advances in mechanism and machine science. IFToMM WC 2019, vol 73. Springer, Zurich, pp 1079–1090
Condition Monitoring and Identification of Misalignment with Initial Unbalance of Flexible Rotor-Bearing System Sankalp Singh, Hanmant P. Phadatare, and Barun Pratiher
Abstract Shaft misalignment, rotor unbalance, rubbing, bearing faults and cracks are the commonly observed faults in the rotor-bearing systems. Vibration signature has been considered to be one of the best ways to detect and diagnosis subsequent condition monitoring of rotor-bearing faults. A flexible shaft with a rigid disc supported by flexible bearings at both the ends has been considered for the present critical observations. The bearing is considered as a spring with linear stiffness and damping. Further, Timoshenko beam theory is considered to describe mathematically the model of rotor element. We used finite element analysis to obtain the critical whirling speeds of rotor-bearing system for different modes at various system parameters with the illustration of Campbell diagram and Fourier spectrum. Vibration spectrums in the transverse direction are illustrated for different modes of vibration using successive approaches of Hamilton principle, Galerkin’s method and Newmark’s integral method. Condition monitoring of vibration spectrums considering rotor unbalance and shaft misalignment as well as their identification has been analysed with FFT, time responses, phase portrait and Poincare’s map, numerically and experimentally. Keywords Modal analysis · Unbalance · Misalignment · Modal analysis · Condition monitoring · Identification
1 Introduction The most important configurations in rotor dynamics, i.e. shaft or shaft-disc assemblies are generally used to transmit power. Shaft misalignment, rotor unbalance, rubbing, bearing faults and crack are the commonly observed faults in the rotordynamic systems. Detecting vibration signature is one of most efficient ways to diagnosis the presence of faults in rotor-bearing system. Some researchers have worked on analysis of the rotating machines such as conditioning monitoring and S. Singh · H. P. Phadatare · B. Pratiher (B) Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur 342037, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_119
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diagnosis along with mathematical modelling of the system. Sekhar et al. [1] and Gibbons [2] studied the forces act on the coupling by parallel misalignment and angular misalignment, respectively. Mohanty et al. [3] studied vibration signature of shafting system to determine misalignments. In his book, Mahanty [4] described conditioning and health monitoring of the machineries. Sudhakar et al. [5] used equivalent loads minimization and vibration minimization method to determine the position and amount of unbalance in the system. Nelson et al. [6] formulated the rotor-bearing system considering the effect of rotatory inertia, gyroscopic moments, axial load, bearing stiffness and damping by finite element method in fixed coordinate and rotating coordinate system. Zorzi and Nelson [7] included all the parameters of [6] with the effect of viscous damping and hysteresis damping. Nelson [8] included all the parameter of [6] with the effect of shear deformation. He concluded that the accuracy of the finite element modal improves as the number of an element is increased. Isermann et al. [9] described evolution in the application of model-based fault detection and diagnosis. Xu and Marangoni [10] derived theoretical model and performed an experiment to the analysis of misalignment and unbalance. Patel and Darpe [11] conducted experiment for analysis of parallel and angular misalignment signature by different amount of misalignment and different running speed of the system. He explained the misalignment by orbit plot, time waveform and FFT. Very recently, Phadatare et al. [12, 13] studied nonlinear dynamics of flexible rotorbearing system using time response, fast Fourier transform, Campbell diagram and Poincare’s map. Karpenko et al. [14] modelled Jeffcott rotor model with preloaded snubber ring to analyse the interactions between the whirling rotor and the massless snubber ring. However, the experiment and the theory results are compared using bifurcation diagrams, Poincare’ maps and phase plane diagrams. Mohanty et al. [15] widely described in detail about the methods basically used in conditioning monitoring of machineries. Schwarz and Richardson [16] explained how to determine the modal parameter (modal frequency, damping ratio and modes shape) by shaker and hammer test and explained the parameter to determine them like about frequency response function, operating deflection shape, pretrigger delay, windows type, leakage and curve fitting methods, etc. Pratiher et al. [12, 13, 17] studied behaviour of the rotating system and used method of multiple scales to approximate solution of the mathematical model. He also presented free vibration analysis and behaviour of the rotating system under harmonic base excitation [14]. David Hutton [18] presented fundamental theory regarding finite element analysis. John Vance et al. [19] described analysis and principles of rotating machines. Piotrowski [20] has given that the basic steps required for alignment of the machine and types of maintenance. He explained about validity rule and different method to measure the amount of misalignment. Chandra et al. [21] used time frequency techniques to detect misalignment fault in rotating system and the detection performance were evaluated by comparing CWT, HHT and STFT. Most of rotating machines are subjected to major defects such as unbalance and misalignment as malfunctions experienced in either at manufacturing level or at assembly level which produce vibration. Thus, fault detection and diagnosis in
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rotating machine during operation are very important by both ways of economical and safety purpose. Hence, condition monitoring of the dynamic behaviour of the rotor-bearing system with initial unbalance and shaft misalignment is studied here. A mathematical model has been derived considering Hamilton’s principle and Galerkin’s method using finite element approach. Study of modal characteristics as well as detecting and diagnosis of the vibration caused by misalignment with a mass unbalance has been performed. The theoretical and experimental results have been verified. The results have been presented using Campbell diagram, time response, FFT, phase portrait plot and Poincare’ map. This also enables us to understand the system behaviour under mutual effect of unbalance and misalignment.
2 Mathematical Modelling 2.1 Disc Element Two translation motion u(s, t) = u b (s, t) and v(s, t) = vb (s, t) are considered for the disc. Two small rotations are α(s, t) and β(s, t) about x-axis and y-axis. The shear effect is neglected in the analysis of the disc. The frames of references for the disc mounted on a rotating shaft are shown as F(XYZ) in Fig. 1. It is an inertial frame and R(x yz) is local coordinate which is fixed to the disc. In order to find the orientation of the disc, following three consecutive rotations of the disc are considered: (1) β about Y axis defines x1 y1 z 1 (2) α about x1 axis defines x2 y2 z 2 (3) ϕ about z 2 axis defines x yz. The angular rate of relative to F(X Y Z ) can be obtained by the Euler angles as described in [6]. In rotor configuration, translation displacement has been considered since axial displacement is negligible as compared to the translation motion. The linear and rotational displacement of the element can be approximated by the relation [8].
Fig. 1 Eulerian angle and schematic diagram of the experimental set-up
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u(s, t) v(s, t)
α(s, t) = [N (s)]{q(t)}, β(s, t)
= [M(s)]{q(t)}.
(1)
T Here,{q(t)} = u 1 v1 α1 β1 u 2 v2 α2 β2 . The linear displacement can be u(s, t) = u b (s, t) + u s (s, t) and v(s, t) = vb (s, t) + vs (s, t).. The rotations of cross section associated with the bending are α(s, t) = −∂vb (s, t)/∂s; β(s, t) = ∂u b (s, t)/∂s. Therefore, the potential energy (U s ) of a shaft element of length (l) including the elastic bending and shear deformation can be expressed as 1 US = 2
l
E I α 2 + β 2 + k AG (u − β)2 + (v + α)2 ds.
(2)
0
The kinetic energy of a shaft element (T s ) and the disc (T d ) rotating at a constant speed (Ω), including the translation and rotational forms, are given by 1 TS = 2
l
2 1 ρ A u˙ + v˙ 2 + ρ I α˙ 2 + β˙ 2 − 2I p α β˙ ds + 2
0
Td =
l I p 2 ds.
(3)
0
1 1 1 1 m d u˙ 2 + m d v˙ 2 + Id (α˙ 2 + β˙ 2 ) + I p ϕ˙ 2 − Idp α β˙ ϕ. ˙ 2 2 2 2
(4)
Using Hamilton principle and then applying Galerkin’s method, the equation of motion can be expressed in following form as [1]. ( MTs + M Rs ) q¨es + (cv [Cvd ] − G s ) q˙es + K s qes = Q s .
(5)
Here, Qs is composed of the forcing terms like mass unbalance, interconnection forces and other external effects on the shaft. The unbalanced forces in both directions (x and y, respectively) are added at the node where the disc is located in the system, then these forces in the x and y direction can be expressed as Fx = m u e2 cos(t), Fy = m u e2 sin(t).
(6)
Here, mu is the mass unbalance and e is the position of eccentricity.
2.2 Shaft Misalignment This model considers the flexible coupling because it permits some amount of parallel or angular misalignment between two connecting shafts. Due to misalignment in rotating machinery, reaction forces and moments are developed in the coupling [1, 7]. These reaction forces are acting as periodic loads on the rotating shaft with a
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Fig. 2 Force diagram of parallel misalignment
periodic function of half-sinusoidal having a time period of π/. This model has considered 1 × and 2 × components of the reaction forces in finite element formulation. The Z1 is an axis of driving machine and Z2 is an axis of driven machine. The reaction forces and moment on both sides of the coupling are given below. For parallel misalignment:M X 1 = Tq sin θ1 + K b φ1 , M X 2 = Tq sin θ2 − K b φ2 , MY 1 = Tq sin φ1 − K b θ1 , MY 2 = Tq sin φ2 + K b θ2 ,
Fx1 = −M y1 − M y2 Z 3 ,
Fx2 = −Fx1 Fy1 = (Mx1 + Mx2 ) Z 3 , Fy1 = (Mx1 + Mx2 ) Z 3 ,
Fy2 = −Fy1 .
(7)
The nodal force vector Q 1c and Q 2c corresponds to the left and right nodes of the coupling element (Fig. 2). ⎧ ⎫ Fx1 sin(t) + Fx1 sin(2t) ⎪ ⎪ ⎪ ⎪ 1 ⎨ Fy1 cos(t) + Fy1 cos(2t) ⎬ Qc = , ⎪ ⎪ 0 ⎪ ⎪ ⎩ ⎭ 0 ⎧ ⎫ Fx2 sin(t) + Fx2 sin(2t) ⎪ ⎪ ⎪ ⎪ 2 ⎨ Fy2 cos(t) + Fy2 cos(2t) ⎬ Qc = ⎪ ⎪ 0 ⎪ ⎪ ⎩ ⎭ 0
(8)
Here, θ1 = sin−1 ( X 1 Z 3 ),θ2 = sin−1 ( X 2 Z 3 ),φ1 = sin−1 Y1 Z 3 and
φ2 =sin−1 Y2 Z 3 . The X and Y are the amount of misalignments in the horizontal and vertical directions, respectively. It can be obtained by reverse dial indicator method experimentally.
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3 Result and Discussion Effect of geometrical parameters like disc mass, disc position and shaft length on the critical whirling speed is analysed with help of Campbell diagram for different modes. Similarly, eigenspectrums are also portrayed for the different disc mass. As well as, effect of the misalignment with initial imbalance is also discussed in this section. Experiment Set-up and Procedures: Schematic diagram of the rotating system is prepared as shown in Fig. 1. The same setup is considered as a experimental setup. It comprises a flexible shaft with a rigid disk and the shaft is supported by bearings. One of the end of the shafts is coupled with a motor using the flexible coupling. The motor can be controlled to attain different speeds of the rotor using a control panel or computer interface. As well as the motor can be made accelerate/decelerate with desired rate till the specified speed using the control interface. Two proximity sensors are used to measure horizontal and vertical deflection of the shaft. The sensors are located using L-shaped holder as shown in Fig. 1. SpectraQuest DAQ system with a computer software interface is used to acquire and process the vibrations signals from the sensors for better interpretation of the vibrational behavior of the system. Here, specially manufactured papers with a thickness are used to vary the misalignment value by accommodating beneath the bearings attachment base. The misalignment value is measured near the coupling.
3.1 Modal Characteristics Modal analysis has been studied to evaluate the eigenfrequencies and eigenspectrums to understand critical whirling speeds, i.e. forward and backward natural frequencies with spin speed when the disc is located at L/3 distance for different disc mass by keeping other parameter constant. From Fig. 3, the increase and decrease in forward (N f )/backward (N b ) natural frequency of the rotating system can be observed with increase in the shaft spin speed (). This phenomenon of splitting the frequency in two branches is due to action of gyroscopic couple which is also spin speeddependent. For most of the applications, the rotating systems whirl with forward natural frequency (N f ). Therefore, it can be concluded that the critical speed of the shaft gets increased with the spin speed. It is also observed that the natural frequencies related to 3rd mode of vibration are greater than that of 2nd mode of vibration and 1st mode of vibration. It has been found that natural frequency in 2nd mode of vibration is nearly equal to four times as compared with 1st mode of vibration while natural frequency is find to be nearly eight times as compared with fundamental natural frequency. Similarly, the eigenspectrums corresponding 1st, 2nd and 3rd mode of forward natural frequency have been represented in Fig. 4, when the disc is at the mid-position. It helps to understand behavioural displacement patterns of the shaft
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Fig. 3 Campbell diagram and respective characteristic mode shapes for different disc mass
for different running speed. Based on the analysis of these eigenspectrums, proper positioning of the disc can be selected to vary the effect of gyroscopic couple on the system. It is expected from the shown eigenspectrums that larger amplitude of the eigenspectrums for a heavy disc can be observed as compared to that of smaller value of disc mass. Therefore, understanding the modal characteristics and its development over the various mode of vibration is required to monitor their condition of safe operating zone. Hammer test has been conducted to understand the modal characteristics of experimental set-up, and it is especially the eigenfrequency. Here, the experiment is performed at stand still position of the shaft. The calibrated hammer is used to apply initial excitation to the rotor system by striking at mid-point of the shaft. Resulting displacement behaviour of the system is captured and stored in the computer using data acquisition system. The data acquisition system includes proximity sensors, tachometer, Spectra-Pad signal processing unit, motor control and Spectra-Quest software module. The resulting data is stored in excel file and then converted into displacement form using calibration sheet of proximity sensors for further graphical presentation and analysis. The time response, fast Fourier, phase portrait and
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Fig. 4 Mode shapes for different disc mass
Poincare’s map are portrayed using this data as shown in Fig. 6. Free vibration analysis of the rotor-bearing system at zero spin speed has been demonstrated numerically and experimentally. Signal samples of 10 s duration are acquired using data acquisition system and better parts of the signals after hammer impact have been considered for further signal conditioning and analysis purpose. The corresponding time response has shown attenuation of the amplitude per oscillating cycle till the system come to rest position. This phenomenon is due to presence of inherent damping in the system that causes response to set at rest after some oscillations. This behaviour can be used to detect characteristics of the system at stand still position such as damping coefficient and natural frequency. A peak in FFT depicts the presence of frequency of free vibration. The system has lost energy per cycle of oscillations, and this is also observed from phase portrait and Poincare map. It has been depicted that the presence of the positive damping caused the equilibrium point to act as a stable attractor. Similar system parameters with the experimentally obtained damping coefficient have been used to verify proposed mathematical model. The numerical results are portrayed in
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Fig. 5 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare map with a disc located at L/3 distance from the bearing; h = 0.0161 m Theoretically
Fig. 5. Then, these results have been compared with experimental results (Fig. 6). In Figs. 5b and 6b, it has been observed that the error in prediction of the frequency obtained by mathematical model has been found less than 5%. As well as, Figs. 5a and 6a have depicted that the rate of attenuation of the signal amplitude observed to be same. So, the both results are in pretty much compliance and proposed model can be considered for further analysis. Slightly more amplitude of peak in FFT of the experimental result is observed compared to the theoretical; it may be result of more than one frequency because corresponding time response shows beat phenomenon. But it is neglected due to no detection of more than one in FFT plot.
3.2 Shaft Misalignment Alignment is very necessary for any system, if the system has misalignment; it produces vibration and results into loss of the power and suffer coupling wear, etc.
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Fig. 6 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare map with a disc located at L/3 distance from the bearing; h = 0.0161 m Experimentally
Any system like turbine, compressor and pump, etc., has work of power transmission. Those systems become vulnerable towards vibration due to the misalignment in the system. Results obtained numerically considering shaft misalignment have been depicted in Figs. 7 and 8 and corresponding experimental findings which have been obtained by considering parallel misalignment with initial unbalance are portrayed in Figs. 9 and 10. In Figs. 7 and 8, the rotating system behaviour has been portrayed by varying magnitude of the parallel misalignment (i.e. 1.08 and 2.16 mm). The corresponding time response history has depicted periodic behaviour with presence of more than one frequency (Figs. 7b and 8b). Increase in magnitude of the 2X frequency has been detected with increase of parallel misalignment magnitude from 1.08 to 2.16 mm. As well as, looping can be observed in the phase portrait diagram due to increasing magnitude of the 2*X component. To develop Poincare’s map, the response signal has been sampled with 70 Hz frequency. The resultant Poincare maps have shown two dots, and it has depicted double periodic behaviour of the system.
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Fig. 7 a. Time response b. Fast Fourier Transform plot c. Phase portrait d. Poincare’ map with a disc located at L/3 distance from the bearing; parallel misalignment 1.08 mm. Theroetically
Experimentally obtained results have been plotted in Figs. 9 and 8. In the time response, the beat phenomena have been observed. In FFT (Figs. 9b and 10b), the vibration has also shown peak at 1-X, 2-X, 3-X and 4-X and consequently it signifies presence of the misalignment in the experimental set-up. Same behaviour as described for that of theoretical results has been observed in these experimental results also. More no. of peaks have been observed in experimental results (Figs. 9b and 10b) but 3X and 4X have very small effect on the system behaviour. This can be depicted by comparing Figs. 7c and 8c with Figs. 9c and 10c, respectively, and the corresponding trajectories have similar behaviour on phase portrait. Parallel misalignment without an unbalance (m u ) It is always difficult to completely balance the rotating system. Defective manufacturing of the rotor or defective assembly are main cause of presence of an unbalance in the system. To analyse the misalignment effect, mathematical model of the rotating system with parallel misalignment by neglecting initial unbalance has been presented in Figs. 11 and 12. When there is no initial unbalance in the system, only the misalignment in the system at the 2*X frequency is dominated with respect to
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Fig. 8 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare’ map with a disc located at L/3 distance from the bearing; = 35 Hz, mu = 0.0006 kg and parallel misalignment 2.16 mm. Theoretically
1*X frequency, this has been observed from FFT plot. In the time wave form graph, two frequencies are seen clearly and looping of trajectory has been observed in phase portrait which is due to dominance of the 2*X component. Comparison of FFT plots of Figs. 8 and 12 depicts attenuation of the misalignment effect with presence of the unbalance mass. So, 1-X component is more dominated with respect to the 2-X component due to the unbalance mass. Thus, the misalignment effect gets attenuated comparatively but at the expenses of increased unbalance force. Same behaviour of the system also observed in Figs. 7 and 11. Thus, condition monitoring of the rotating system can be understood well with help of above analysis. These observations are helpful to detect the health condition of the rotating system prior to catastrophic failure of the system and as well as it will help to determine type, exact source and vibrational behaviour.
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Fig. 9 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare’ map with a disc located at L/3 distance from the bearing; = 35 Hz, mu =0.0006 kg and parallel misalignment 1.08 mm. Experimentally
4 Conclusion The free vibration analysis helps to identify the characteristics of the system, and the characteristics of the system are much in compliance for experimental and theoretical model. Experimentally and theoretically analysis is done for the mass unbalance of the system, and it is observed that when the system has only mass unbalance then only 1-X peak observed in FFT. When the system has only parallel misalignment then the 2-X peak gets dominated in the system. When the system has both initial unbalance and parallel misalignment but the initial unbalance amount is high, then the 1-X frequency is dominated and effect of 2-X frequency becomes less. This method may be applicable for the large systems like a turbine, compressor, gearbox, etc., to investigate the unbalance and misalignment in the system. The misalignment effect can be monitored or controlled at the expenses of increasing unbalance force as per applications need.
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Fig. 10 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare’ map with a disc located at L/3 distance from the bearing; = 35 Hz, mu = 0.0006 kg and parallel misalignment 2.16 mm. Experimentally
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Fig. 11 a. Time response b. Fast Fourier Transform plot c. Phase portrait d. Poincare’ map with a disc located at L/3 distance from the bearing; = 35 Hz and parallel misalignment 1.08 mm. (Theoretically)
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Fig. 12 a. Time response b. Fast Fourier transform plot c. Phase portrait d. Poincare’ map with a disc located at a L/3 distance from the bearing; = 35 Hz and parallel misalignment 2.16 mm. Theoretically
Acknowledgements This research paper is a part of project no “S/SERB/BP/20140012”, which is financially supported by Science and Engineering Research Board (SERB) under DST, Government of India. Authors thank Department of Science and Technology for providing the financial grant.
References 1. Sekhar AS, Prabhu BS (1994) Effects of coupling misalignment on vibration of rotating machinery. J. Sound and Vib. 185(4):655–671 2. Gibbons CB (1976) Coupling misalignment forces. In: Proceedings of the fifth turbo machinery symposium, gas turb. Lab., Texas, pp 1111–1116 3. Mohanty AR, Fatima S (2015) Shaft misalignment detection by thermal imaging of support bearings. In: IFAC conference paper 48-21, pp 554–559 (2015) 4. Mohanty AR, Machinery condition monitoring, principles and practices, CRC Press (2015) 5. Sudhakar AS, Sekhar AS (2011) Identification of unbalance in a rotor bearing system. J Sound Vib 330:2299–2313
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6. Nelson HD, McVaugh JM (1976) The dynamics of rotor-bearing systems using finite elements. J Eng Ind 19:75 7. Zorzi ES, Nelson HD (1977) Finite element simulation of rotor-bearings systems with internal damping. J Eng Power 8:76 8. Nelson HD (1980) A finite rotating shaft element using timoshenko beam theory. J Mech Desi 102:799 9. Isermann R, Balle P (1997) Trends in the application of model based fault detection and diagnosis of technical processes. Control Eng Pr 5(5):709–719 10. Xu M, Marangoni RD (1993) Vibration analysis of a motor-flexible coupling-rotor system subject to misalignment and unbalance, Part I: theoretical model and analysis. J Sound Vib 176(5):663–679 11. Patel T, Darpe A (2009) Experimental investigation on vibration response of misaligned rotors. Mech Syst Sig Proc 23:2236–2252 12. Phadatare HP, Maheshwari V, Vaidya KS, Pratiher B (2017) Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system. Int J Mech Sci 134:532–544 13. Phadatare HP, Pratiher B (2016) Nonlinear frequencies and unbalanced response analysis of high speed rotor-bearing systems. Procedia Eng 144:801–809 14. Phadatare H, Choudhary B, Pratiher B (2017) Evaluation of nonlinear responses and bifurcation of a rotor-bearing system mounted on moving platform. Non-lin Dyn 90:493–511 15. Karpenko EV, Wiercigroch M, Pavlovskaia EE, Neilson RD (2006) Experimental verification of Jeffcott rotor model with preloaded snubber ring. J Sound Vib 298:907–917 16. Schwarz BJ, Richardson MH (1999) Experimental modal analysis. CSI Reliability Week, Orlando, FL 17. Choudhary B, Pratiher B (2015) Numerical studies of a nonlinear flexible rotating system under harmonic ground motion. In: Proceedings of 9th IFToMM international conference rotor dynamics, vol 21, pp 1677–1687 (2015) 18. Hutton D (2004) Fundamental of finite element analysis. McGraw-Hill 19. Vance J, Zeidan F, Murphy B (2010) Machinery vibration and rotordynamics. Wiley (2010) 20. Piotrowski J (2007) Shaft alignment hand book. Third Edition by CRC Press 21. Chandra HN, Sekhar AS (2015) Fault detection in rotor bearing systems using time frequency techniques. Mech Syst Sig Proc 72(73):105–133
Effect of Unbalance with Bearing Flexibility on Vibration Phenomenon of Geometrically Nonlinear Rotating Shaft with Ball Bearing Hanmant P. Phadatare, Sankalp Singh, and Barun Pratiher
Abstract Free and forced vibration analysis of a geometrically rotating shaft supported on ball bearings has been studied using the numerical method and compared with the results obtained experimentally. This study is concerned with vibration analysis of geometrically nonlinear rotating model with a rigid disk. The shaft has been designed under the frame of Euler–Bernoulli beam theory with additional effects such as rotary inertia, gyroscopic effect, higher-order large deformations, and rotor mass unbalance in order to replicate an equivalent practical model of rotor-bearing system. The mathematical expressions have been derived to demonstrate the nonlinear free and forced vibrations of the rotating shaft coupled with rigid disk in two transverse planes. Solutions of the nonlinear equation are being obtained using method of multiple scales as well as numerical methods. Effects of rotor parameters such as bearing stiffness and damping coefficient are examined with help of this nonlinear mathematical model. The obtained results are portrayed for a better understanding free and forced vibration analysis with time response, FFT, phase portrait, and Poincare’s map. The present outcomes enable an understanding on how the system dynamics influenced with the variations in the values of different parameters. Keywords Flexible rotor · Bearing flexibility · Geometric nonlinearity · Modal characteristics · Unbalance responses · Nonlinear phenomena
1 Introduction Free vibration analysis has the most important role in designing vibrating systems, because high energy vibrations mostly occur at critical speed of the system, and it affects its regular functionality and sometime may likely to face possibility of catastrophic failure or weaken the system. There are number of applications of rotorbearing system such as turbomachinery, airplanes, pump, compressor, ships, and H. P. Phadatare · S. Singh · B. Pratiher (B) Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur 342037, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_120
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industrial machinery. Mostly found defect in rotating machines is the unbalanced mass. It may be due to poor manufacturing or assembly of the machine. Even wellbalanced rotor has small unbalanced mass under tolerable value. But at high speed applications, this small unbalanced mass applies large fluctuating or rotating force on the system. At the beginning of the machinery service period, this effect could be small but it may rise to a noticeable limit over the time and need attention. A brief history for understanding the unbalance responses of a flexible rotating machinery is presented here. Sturla et al. [1] investigated free vibration analysis of nonlinear slender rotating shaft with simply supported condition and analyzed effect gyroscopic couple, external damping, and rotational speed. Closed form of polynomial frequency equation and integral equation with forced condition are derived by Sturla et al. [1]. Karunendiren [2] presented an analytical method for the free vibration analysis of shaft with resilient bearing support. Melanson et al. [3] provides exact solution for the complex natural frequencies and complex normal modes, as well as investigated effect of the internal damping on the stability of the rotor-bearing system. Sheu et al. [4] studied dynamic behavior of rotating Rayleigh beam. Hosseini [5] investigated the effects of random properties on coefficient of variation (COV) of first mode eigenvalue. In [6], the free vibrations of an in-extensional rotating shaft with nonlinear curvature and inertia were considered. Shabaneh and Zu [7] investigated behavior of a rotating disk-shaft system with linear elastic bearings under free and forced vibration. Chang-Jian et al. [8] investigated the chaotic dynamics of a rotor-bearing system with nonlinear suspension using Poincare map, Liapunov exponents, and bifurcation diagrams. The effect of the shear deformation and rotary inertia of a rotor on its critical speeds was considered by Grybos [9]. Jei and Leh [10] analyzed uniform asymmetrical Rayleigh shaft with asymmetrical rigid disks and isotropic bearing. Kim et al. [11] investigated the free vibration of a rotating tapered composite Timoshenko shaft. Phadatare et al. [12, 15] studied dynamics of nonlinear phenomenon of rotating system using time response, fast Fourier transform, Campbell diagram ,and phase space. In another study, bifurcation analysis of nonlinear rotating system with excitation of moving platform has also been performed by Phadatare et al. [13]. Karpenko et al. [14] modeled Jeffcott rotor model with preloaded snubber ring to analyze interactions between the whirling rotor and the massless snubber ring. However, the experiment and the theory results are compared using bifurcation diagrams, Poincare´ maps, and phase plane diagrams. Shad et al. [16] studied nonlinear effect of higherorder deformation in analysis of a rotating system. Yamamotov et al. [17] have described detailed procedure for vibration measurement and imbalance fault detection in rotating machinery. Sudhakar and Sekhar [18] worked on identification of unbalance in a rotating system. In another study, Shekhar [19] investigated effects of coupling Misalignment on the rotating machines. Isermann and Balle [20] have considered basic fault detection and diagnosis methods. Amiya et al. [21] described methods of machine condition monitoring in his book. Nayfeh [22] discussed in detail the different types of perturbation methods with examples related to different fields in his book. Most of authors in the recent literature considered flexible bearing with the combination of rigid shaft and disk to analyzes the performances of rotor-bearing systems.
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Limited researchers studied the rotor dynamics considering flexible shaft with rigid disk supported by a flexible bearing. The characteristics of the bearings have a prominent role not only in the dynamics of modal parameters but also system performance characteristics. The primary concern of this paper is to understand the effect of bearings flexibility along with mechanical unbalance on the flexural vibration characteristics of the rotor-bearing system. The mutual contributions of different parameters under unbalance condition have also been studied. Similarly, the study the effect of damping and bearing stiffness to its behavior in forcing condition has been performed experimentally and theoretically. We here study the vibration aspects of geometrically nonlinear rotating shaft combined with rigid disk. The shaft has been modeled and Euler–Bernoulli beam element accounting of the effects of rotary inertia, gyroscopic effect, higher-order large deformations and rotor mass unbalance. In next section, a mathematical model of rotor-bearing system consisting of a flexible shaft with a disk simply supported by bearings has been formulated using Euler’s beam theory. As well as, stretching nonlinearity, gyroscopic effect, and rotary inertia effect have been taken into consideration.
2 Formulation of Mathematical Model The system is consisting of a flexible shaft, a rigid disk, and flexible bearings. The shaft has length L and the disk is located along the shaft length at a distance L d from an end of the shaft. The shaft is pinned at its both end and supported by the flexible bearings at same distance lb from both ends of the shaft. The bearing is characterized as a flexible linear spring. The bearing is characterized as a flexible linear spring. Expressions for kinetic energy of balanced rotor-bearing system can be expressed as [5] L Tt =
ρA 2 v˙ + w˙ 2 dx + 2
0
L
L ρI 2 2 α˙ + θ˙ dx + (2ρ I )α˙ θ dx 2
0
0
L 2 1 v˙ + w˙ 2 δ(x − L d ) dx + (m) 2 0
L 2 1 I α˙ + I2 θ˙ 2 + β˙ 2 δ(x − L d ) dx. + 2
(1)
0
Here, ρ, A, I, I d are the mass density, cross-sectional area, polar and diametrical moment of inertia of the shaft, respectively, and m is mass of the disk α, ˙ θ˙ and β˙ are the time derivatives of the angular displacement about the axes of Z, Y, and X.
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Dirac delta function has been incorporated in order to represent the disk effect in the distributed system. The kinetic energy due to unbalance mass has been expressed as [10, 11]. Tu = m u d(v˙ cos t − w˙ sin t) +
mu d2 . 2
(2)
Here, mu , and d are the unbalance mass and its location from geometric center of shaft. Strain energy of the shaft due to bending in rotating shaft is developed by neglecting the effect of term of time functioning in the consequence of displacement in x, y, z directions as the concept of u x = −zθ + yα, u y = v, u z = w [11]. Strain energy of the flexible shaft can be expressed as L U=
L 2 2
E I 2 2 E A 4 4 v + w v + w + 2 v w dx dx + 2 8
0
0
L L 2 2
2 2
v + w dx v + w dx + 0
(3)
0
Here, (·) and over the letters represent time and space derivatives, respectively. EA and EI are considered as axial rigidity and flexural rigidity, respectively. The bearings are considered to be 2D linear spring. The bearings are located at a distance L b from ends of the shaft. So, the strain energy stored in single bearing can be expressed as 1 Ub = 2
L [(K l v)v + (K l w)w]δ(x − L b ) dx
(4)
0
where K l is linear spring coefficients of the bearings in both directions. Dirac delta function has been incorporated in order to represent the bearing effect in the distributed system.
2.1 Hamilton’s Principle Hamilton’s principle and Galerkin’s method principle [16] with consideration of single mode vibration have been applied to derive equations of motion. Coupled nonlinear temporal equations for govern the dynamic forced vibration characteristics of systems have been developed using similar procedures [10, 16] and can be expressed as below.
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1 μ1 + μ2 V 3 + V W 2 + c V˙ = m u 2 dϕ(L d ) sin t 2 (5) 1 μ1 + μ2 W 3 + V 2 W + c W˙ = m u 2 dϕ(L d ) cos t. W¨ + κ1 V˙ + κ2 W + 2 (6) V¨ − κ1 W˙ + κ2 V +
The expressions for coefficients (κ1 , κ2 , μ1 , and μ2 ) can be illustrated as below. 2
= Md f (x) x=L d + Idy f (x) x=L d + ρ A
L
L f (x) dx + ρ I
0
Id x 2 2ρ A f (x) x=L d + κ1 =
κ2 =
EA μ1 =
L 0
f (x)
EI
4
L
f (x)
2
dx +
0
EA dx, μ2 =
l
L 0
f 2 (x) dx,
2
L
0
f (x)
2
dx,
0
2K l 2 f (x) x=L b ,
⎞ ⎛ L 4 mu ⎝ . f (x) dx ⎠ dx, m u =
0
(7) Equations (5) and (6) can be expressed as below for free vibration analysis of the system after neglecting effect of the unbalance terms, 1 μ1 + μ2 V 3 + V W 2 + c V˙ = 0 2 1 ¨ ˙ μ1 + μ2 W 3 + V 2 W + c W˙ = 0 W + κ1 V + κ2 W + 2 V¨ − κ1 W˙ + κ2 V +
(8) (9)
Since the governing equation of motion contains the nonlinear terms, the exact solutions are not available as obtaining closed form is somehow difficult. Thus, the perturbation approach (i.e., method of multiple scales) [12, 13] is used to get solution of the Eqs. (5), (6) and Eqs. (8), (9). One may also go for a numerical method for solution also. The MATLAB function ode45 is used to obtain numerical solution of these equations.
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3 Numerical and Experimental Investigation Experimental setup of the rotating system is prepared as shown in Fig. 1. The setup comprises a flexible shaft with a rigid disk and the shaft is supported by bearing at a distance L b from the both ends. One of the end of the shafts is coupled with a motor using the flexible coupling. The motor can be controlled to attain different speeds of the rotor using a control panel or computer interface. As well as the motor can be made accelerate/decelerate with desired rate till the specified speed using the control interface. Two proximity sensors are used to measure horizontal and vertical deflection of the shaft. The sensors are located using L-shaped holder as shown in Fig. 1. SpectraQuest DAQ system with a computer software interface is used to acquire and process the vibrations signals from the sensors for better interpretation of the vibrational behavior of the system. Here, a rotating flexible shaft has 0.93 m length (L) which is supported by the bearings at a distance of 0.05 m from its both ends. The shaft has 7800 kg/mm3 mass density (ρ), 200 GPa modulus of elasticity (E) and 0.0152 m diameter (d). A rigid disk is located along the shaft at a distance L d . The disk has 2340 kg/ mm3 mass density (ρ), 0.12 m diameter (D), and 0.03 m thickness. Equivalent damping coefficient is taken as 30 Ns/m. Numerical results are found by varying geometrical parameter of the system such as location of the disk, thickness of the disk, spinning speed of the shaft, bearing stiffness, and viscous damping.
3.1 Modal Characteristic Free vibration analysis of the rotor-bearing system has been performed at zero spinning speed by giving small initial excitation and received the output in the terms of time response. The results have been obtained by varying geometrical parameters of the system such as the location of the disk and the disk thickness. An experimental setup to represent a theoretical model of rotor bearing has been considered. The numerically obtained results have been compared with the experimental results in
Fig. 1 a Schematic diagram change it and b experimental setup of a rotor-bearing system
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terms of the system characteristic parameters (i.e., natural frequency and damping) to verify the proposed theoretical model. The fundamental natural frequency of the experimental setup having zero spin speed has been obtained by applying the small impact at the middle of the shaft using the calibrated hammer. The resultant responses of the system have been obtained in time domain then which has been converted into frequency domain with use of fast Fourier transform. The results of free vibration analysis in Figs. 2, 3, 4 and 5 are in terms of time responses and fast Fourier transforms. The numerical and experimental results have been presented in solid and dotted lines, respectively, in these figures for better comparison. Figures 2 and 3 have been portrayed to show change in characteristic behavior of the system due to change in disk position (L d = L/3 and L/6) with disk
Fig. 2 a Time response, b fast Fourier transform: with a disk located at L/3 distance from the bearing
Fig. 3 a Time response, b fast Fourier transform with a disk located at L/6 distance from the bearing
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Fig. 4 a Time response, b fast Fourier transform plot with a disk located at L/3 distance from the bearing. h = 0.032 m
Fig. 5 a Time response, b fast Fourier transform plot with a disk located at L/6 distance from the bearing; h = 0.032 m
thickness (h = 0.016 m). In Fig. 2b, frequency peaks can be observed at 57 and 57.21 Hz for numerical and experimental investigation, respectively. Such peaks can also be observed at 65 Hz and 66.9 Hz in Fig. 3b, 48.47 Hz and 49 Hz in Fig. 4b, 57.51 Hz and 60 Hz in Fig. 5b for the numerical and the experimental models, respectively. Figures 2 and 3 show the increase of the fundamental frequency of the system as the disk location changes from L/3 to L/6. From these observations, we can depict that the numerical results are in close compliance with experimental results as the error is less than 5%. Similar frequency change can be observed in Figs. 4 and 5 for another configuration of the system. As well as, the decrease in fundamental frequency of the system
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Fig. 6 a Phase portrait, b Poincare map corresponding to Fig. 2 with a disk located at L/3 distance from the bearing
with increase in the thickness of the disk can be also detected by observing Figs. 2, 3, 4 and 5. As the system has viscous damping, the initially excited amplitude of the system gets damped out and the system response comes to rest with time. For better understanding of this dynamic behavior the system, phase portrait and Poincare maps have also been plotted (Fig. 6). The fundamental natural frequency has been considered as a sampling frequency to plot the Poincare maps. We can observe in Fig. 6 that the trajectory has been attracted toward the origin and remains at the same position for infinite time. When the system was perturbed infinitesimally from the equilibrium point, it caused the system to attract toward the initial state. So, the equilibrium point behavior looks similar to an attractor (Fig. 6a). When Poincare plane cut the trajectory in phase space, the points have been observed approximately in a straight line and pointing toward the origin as well as the cluster of points at the origin has also been observed in Fig. 6b. Thus, it can be depicted that the motion would remain at rest after finite oscillations, which has shown periodical behavior. The corresponding damping coefficient has been obtained using time response data of the hammer test.
3.2 Unbalance Responses As the experimental system are difficult to fully balance, a small initial imbalance mass (mu = 0.0005 kg) in the system is determined using procedure as described in [22]. The experimental results have been obtained by varying system parameters such as spin speed and imbalance mass. These have been presented in Figs. 7, 8, 9 and 10 with consideration of steady-state part of the response. Due to presence of the
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Fig. 7 a Fast Fourier transform plot, b phase portrait, c Poincare’ map with a disk located at L/3 distance from the bearing; Ω = 35 Hz (Experimental)
Fig. 8 a Fast Fourier transform plot, b phase portrait, c Poincare’ map with a disk located at L/3 distance from the bearing: Ω = 45 Hz (Experimental)
Fig. 9 a Fast Fourier transform plot, b phase portrait, c Poincare’ map with a disk located at L/3 distance from the bearing; Ω = 35 Hz, mu = 0.0055 kg. (Experimental)
imbalance in the system, the unbalance force acts on the rotor with frequency of spinning speed (Ω). Figure 7a shows the peak at 35 Hz when spinning speed of the shaft has been set at 35 Hz and same type of behavior has also been observed for the spinning speed 45 Hz in Fig. 8.
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Fig. 10 a Fast Fourier transform plot, b phase portrait, c Poincare’ map with a disk located at L/3 distance from the bearing; Ω = 35 Hz, mu = 0.0095 kg. (Experimental)
From Figs. 7 and 8, it has been also observed that the magnitude of forcing frequency peak increased with increase in the spin speed (Ω). Further, it increases till critical speed and then may get decreased. The frequencies which are other than forcing frequency have comparably very small magnitude. This causes very small effect of these frequencies on vibration behavior of the system. Presence of damping in the system has led the transient term of the response to vanish at initial stage and resulted into steady-state response which has been dominated only by the forcing term. The phase portrait and Poincare’ maps have been plotted only for 1 s span of time response of the steady state. The spin speed (Ω) has been considered as sampling frequency to plot Poincare’ maps. From Figs. 7, 8 and 9, phase portrait diagrams have depicted presence of the two frequencies at the lower spinning speed and the low imbalance mass, as it shows looping of the trajectories with irregular closed loop. But at higher imbalance mass (Fig. 10), the phase portrait plot has shown closed loop with approximately oval shape. Consequently, the influence of frequencies other than that of the unbalance has been observed almost negligible, so, it has been depicted that the irregular behavior of the system has been eliminated due to increase of unbalance forcing term. The cluster of points at single location has been observed in Figs. 7, 8, 9, and 10 (Poincare’ map). It has been depicted that the system vibrated periodically with the sampling frequency (i.e., forcing frequency) and the system has dominance of the unbalance force. The theoretical results (Figs. 11 and 12) have not shown heart shape trajectory in phase portrait plots as compared to that of experimental results. This behavior is due to presence of more than one frequency and this can also be depicted from the FFT plots also. Source of the undefined frequency in the experimental model may be due to misalignment or lose assembly of the system, etc. But, Fig. 10 shows that this behavior has been attenuated by increasing unbalance magnitude in the system. This unbalance magnitude can be increased by varying the spin speed (Ω) or the unbalance mass (mu ). Theoretical analysis has shown dominance of the unbalance force in the response of the system as same as the experimental analysis.
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Fig. 11 a Fast Fourier transform plot, b phase portrait, c Poincare’ map with a disk located at L/3 distance from the bearing; Ω = 35 Hz, h = 0.016 m, mu = 0.0005 kg (Theoretical)
Fig. 12 a Fast Fourier transform plot, c phase portrait, d Poincare’ map with a disk located at L/3 distance from the bearing; Ω = 35 Hz, mu = 0.0095 kg. (Theoretical)
Further, the investigation of the rotor-bearing system has been extended theoretically to analyze effect of the damping and bearing stiffness with inclusion of the stretching nonlinearities. The obtained results have been portrayed in Figs. 13, 14. In Fig. 13, the results are related to different damping values such as c = 0.0, 2.5, and 30 Ns/m obviously, and the comparison of different damping values has shown early diminishing of transient term at high viscous damping value than the low damping value. With no damping (c = 0.0 Ns/m), the time response has shown beating behavior for infinite time due to presence of two frequencies as shown in Fig. 13. The presence of two frequencies in the vibration of the system has been depicted from corresponding fast Fourier transform plot (FFT) also. Initial five seconds data have been used to process the FFT plot with thousand samples per second sampling rate. The peaks have been detected at 40 and 72.4 Hz, which are spin speed of the shaft and natural frequency of the system, respectively. In the phase portrait plot, torus shape trajectory has been observed and it has also depicted presence of two frequencies in the vibration data. The Poincare’s map has been plotted by using plane cut the phase space at spin speed frequency. The Poincare’ map has shown intersection of plane as points distributed in an approximately circular closed-loop shape.
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Fig. 13 Time response, fast Fourier transform plot, phase portrait, and Poincare’ map: Ω = 35 Hz, a c = 0.0 Ns/m, b c = 2.5 Ns/m, c c = 30 Ns/m. (Theoretical)
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Fig. 14 Time response, fast Fourier transform plot, and phase portrait: Ω = 40 Hz, mu = 0.0095 kg a K l = 2.1151 × 103 N/m b K l = 2.1151 × 105 Ns/m c K l = 2.1151 × 107 N/m (Theoretical)
So, it can be concluded that these frequencies are not commensurate and the system has quasi-periodic behavior. There is possibility that the frequencies are commensurate at very long period. So, it is always hard to differentiate periodic and quasi-periodic behavior. This analysis has been limited to 30 s time history. Results with damping c = 2.5 Ns/m have been portrayed in Fig. 13b., time response has shown diminishing of transient term during initial 10–12 s and prevailing only steady-state response afterward. The FFT has been represented for initial five second samples. It has shown influence of natural frequency in the system behavior due to consideration of the transient part. The diagram of FFT for the steady state has shown one peak only and has depicted dominance of only one frequency, i.e., forcing frequency (Ω).
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As the phase portrait has been plotted by considering transient part of the response, it has shown torus type shape trajectory. But for the steady state, it has shown circular closed loop. One point has been observed on the Poincare map and it has been depicted that the steady-state behavior of the system is a periodical. For damping value c = 30 Ns/m, transient part of the response diminished in early stage so the system showed a periodical behavior of the steady state. Figure 14 has been plotted to analyze effect of change in stiffness of the bearings. Increase in the natural frequency of the system has been observed in the FFT plot due to increase in stiffness of the bearings, and some movement away from the forcing frequency has also been detected. Consequently, the decrease in amplitude of the system response has also been observed. The phase portrait plots have shown reduction of the closed loop almost to a point with increase in the stiffness of the bearing. At this configuration, decrease in transient part span in time domain has been observed with increase in the bearing stiffness and almost negligible at its high value.
4 Conclusion The nonlinear behavior of rotating shaft coupled with disk has been analyzed for the first mode of vibration accounting large elastic deformation. It has been observed that the significant influences of gyroscopic, rotary inertia, and mass unbalance effects on the dynamics of the rotor bearing. The present perspective of this work is to demonstrate vibration characteristics of a flexible rotor-bearing system with the experimental validation. We here illustrate the overall dynamics with time histories, FFT, Poincare’s section, and phase plane trajectories. The imbalance is already present in the system, thus 1× frequency can be seen in FFT and this 1× frequency dominates in response of the rotor-bearing system. As the transient response of the system damps out with time, only steady response can be observed in time response. The damping and stiffness of the bearings are also play an important role to decide the system qualitative behavior of the system. The damping s property of the bearing can be used to change the behavior of the system from the quasi-periodic to a periodic behavior. Moreover, a bearing with large stiffness value can be used to suppress the vibration amplitude. Acknowledgements Authors would like to thank “Science and Engineering Research Board (SERB)”, Government of India for providing the financial grant with project no “S/SERB/BP/20140012”. The results from this present work are part of this project.
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References 1. Sturla FA, Argento A (1996) Free and forced vibrations of a spinning viscoelastic beam. J Vib Acoust 118(3):463–468 2. Karunendiran S, Zu JW (1999) Free vibration analysis of shafts on resilient bearings using Timoshenko beam theory. J Vib Acoust 121(2):256–258 3. Melanson J, Zu JW (1998) Free vibration and stability analysis of internally damped rotating shafts with general boundary conditions. J Vib Acoust 120(3):776–783 4. Sheu GJ, Yang SM (2005) Dynamic analysis of a spinning Rayleigh beam. Int J Mech Sci 47(2):157–169 5. Hosseini SAA, Khadem SE (2005) Free vibration analysis of rotating beams with random properties. Struct Eng Mech 20:293–312 6. Hosseini SAA, Khadem SE (2009) Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia. Mech Mach Theo 44(1):272–288 7. Shabaneh NH, Zu JW (2000) Dynamic analysis of rotor–shaft systems with viscoelastically supported bearings. Mech Mach Theory 35(9):1313–1330 8. Chang-Jian CW, Chen CK (2006) Nonlinear dynamic analysis of a flexible rotor supported by micropolar fluid film journal bearings. Int J Eng Sci 44(15):1050–1070 9. Grybos R (1991) The effect of shear and rotary inertia of a rotor at its critical speeds. Arch Appl Mech 61(2):104–109 10. Jei YG, Lee CW (1992) Modal analysis of continuous asymmetrical rotor-bearing systems. J Sound Vib 152(2):245–262 11. Kim W, Argento A, Scott RA (1999) Free vibration of a rotating tapered composite Timoshenko shaft. J Sound Vib 226(1):125–147 12. Phadatare HP, Maheshwari V, Vaidya KS, Pratiher B (2017) Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system. Int J Mech Sci 134:532–544 13. Phadatare H, Choudhary B, Pratiher B (2017) Evaluation of nonlinear responses and bifurcation of a rotor-bearing system mounted on moving platform. Nonlinear Dyn 90:493–511 14. Karpenko EV, Wiercigroch M, Pavlovskaia EE, Neilson RD (2006) Experimental verification of Jeffcott rotor model with preloaded snubber ring. J Sound Vib 298:907–917 15. Phadatare HP, Pratiher B (2016) Nonlinear frequencies and unbalanced response analysis of high speed rotor-bearing systems. Procedia Eng 144:801–809 16. Shad MR, Michon G, Berlioz A (2011) Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending. Appl Math Modell 35:2145–2159 17. Yamamoto GK, Costa, C, Sousa JSS (2016) A smart experimental setup for vibration measurement and imbalance fault detection in rotating machinery. Case Studies in Mech Sys Sig Proc 4:8–18 (2016) 18. Sudhakar AS, Sekhar AS (2011) Identification of unbalance in a rotor bearing system. J Sound Vib 330:2299–2313 19. Shekhar AS, Prabhu BS (1994) Effects of coupling misalignment on vibration of rotating machinery. J Sound Vib 185(4):655–671 20. Isermann R, Balle P (1997) Trends in the application of model based fault detection and diagnosis of technical processes. Control Eng Pr 5(5):709–719 21. Amiya R (2015) Mohanty: machinery condition monitoring. CRC Press, Principles and Practices 22. Nayfeh AH (1981) Introduction to perturbation techniques. Wiley-Interscience, New York
Analysis of Parametric Influence on Control of a Two-Link Flexible Manipulator Incorporating a Payload Pravesh Kumar and Barun Pratiher
Abstract The control problem for the trajectory tracking of a flexible two-link manipulator incorporating a payload described by a nonlinear model is addressed in the present work. The influence of parametric variation of system attributes on the design of PD inversion-based nonlinear control of a two-link manipulator has been demonstrated. The kinetic energy expression for the system is obtained by using the position vector in generalized coordinate system. The Euler–Lagrange’s approach in conjunction with the assumed mode method is utilized to obtain the dynamic model of the system composed of four nonlinear ordinary differential equations which are simulated, and the results have been graphically illustrated. The effect of system parameters such as payload mass, joint mass, system inertia, and physical and geometric properties of the links are explored by comparing the simulation outcomes. Moreover, the presented results exhibit that the system parameters have a significant effect on the input–output characteristics and should be accounted for while designing such systems. Keywords Flexible manipulator · Euler–Bernoulli beam · Dynamic modeling · PD controller · Trajectory tracking
1 Introduction The traditional robot manipulators have been designed to minimize the vibration during and after the operation of manipulator and to achieve higher position accuracy by maximizing their stiffness. This design in turn has resulted in the higher power consumption and lower productivity due to their bulky design. Moreover, the payload carrying capacity of such manipulator is lower as compared to the manipulator system weight. On the contrary, the flexible manipulators have higher speed, higher payload capacity, larger maneuverability, smaller actuators, and lesser capital cost. The mechanical flexibilities have been inducted in the links as well as actuator to P. Kumar · B. Pratiher (B) Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_121
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exploit the advantages of flexible robot manipulators. However, the flexibilities have resulted in the challenging problem of increased vibrations and positioning inaccuracies of manipulators employed in precise applications in medical, space, military, and nuclear industries. Therefore, an appropriate manipulator dynamic model and control mechanism accounting of the rigid and flexural motions of the system is necessitated. Moreover, it is also important to understand the influence of system attributes on the control parameters which may help the design engineer in manipulating the system for accurate positioning. A brief literature regarding the flexible manipulators is presented in next to discern the recent developments in control and vibration suppression of robots. The vibration suppression by developing boundary and adaptive control laws of a three-dimensional Euler–Bernoulli beam with a point payload has been demonstrated by Al-Solihat et al. [1]. The estimation of variables representing the rigid and flexible motions of single link flexible manipulator by a robust nonlinear observer has been carried out by Chalhoub and Kfoury [2]. A smart structure including piezoelectric actuator being modeled as an Euler–Bernoulli beam having a point payload has been analyzed by Choi [3]. The active vibration control of the manipulator under actuator hysteresis and natural frequency variation through a robust control using quantitative feedback theory has been proposed. A control strategy for moving a torque-driven planar single-link flexible manipulator to a specified angle with simultaneous vibration suppression has been developed by Meng et al. [4]. A nonlinear state-dependent Riccati equation-based control strategy has been used for controlling the tip position of a single-link flexible manipulator by Shawky eta al. [5]. An attempt to reduce the lateral oscillatory vibrations due to longitudinal motion of a flexible link carrying a payload has been made by Shin and Rhim [6]. In order to reduce the lateral vibrations, a controller based on input-shaping technique has been developed with further discussion on the effects of payload mass and damping on the dynamic behavior of the system. An Euler–Bernoulli beam with a hub and tip mass payload undergoing rigid and elastic motions has been dynamically modeled by Tavasoli [7] using Hamilton’s principle. A model-based control law for asymptotic stabilization using Lyapunov-based boundary control for achieving the desired set point through rigid body motion with reduced vibrations has been demonstrated. A rigid link model with flexibility in the joint has been considered by Patel et al. [8] and developed a feedback linearization controller for the trajectory tracking. The results have been compared with those of the PD controller, and it has been illustrated that the proposed controller exhibits better performance in presence of joint flexibility. The trajectory tracking problem of a two-link manipulator with intermediatory lubricated revolute joint modeled as a short journal bearing for the purpose of reducing the friction and wear has been addressed by Sun [9]. A fuzzy self-tuning proportion-integration-derivative controller has been used to achieve the desired tip position and the influence of clearance and lubrication of journal on the precision of the manipulator has been demonstrated. A rigid-flexible two-link manipulator incorporating a point mass payload undergoing large deformation has been considered for vibration suppression using an optimal trajectory planning technique by Abe [10]. Abe and Hashimobo [11] considered the point-to-point motion of two flexible links
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driven by a single motor using PSO algorithm-based feed-forward control technique. The boundary controllers for the trajectory tracking of a flexible two-link manipulator under gravity exploiting singular perturbation approach has been demonstrated by Ashayeri et al. [12]. The flexible and rigid motions of the manipulator are decoupled using the two-time scale control theory in order to achieve the desired trajectory while suppressing the link vibrations. An adaptive state feedback control for the point-to-point motion and trajectory tracking of a two-link manipulator in planar motion has been used by Bai et al. [13]. The simulations illustrated small steady state and tracking errors, and it has been suggested to use harmonic gear reducer and direct actuators for the improved and efficient controller performance. Kumar and Pratiher [14] developed a nonlinear model of two-link flexible maniplator to compute the modal parameters and further to investigate the dynamic stability when the flexible joints are subjected to harmonic motions. Lochan and Roy [15] used the proportional-integral SMC and asymptotic SMC techniques to control the angular position and tip deflection of the two-link flexible manipulator with point payload. Oh and Kong [16] developed a two degree of freedom controller for a two-link manipulator based on rotating coordinate system. The controller has been designed based on the position and motion of the manipulator in the rotating coordinate system, and the results have been compared with the traditional controllers. The numerical investigation of semi-active parameter control technique for a flexible two-link robotic arm employed in turning process has been performed by Ozer et al. [17]. The links of the robotic manipulator are connected by revolute joints modeled as torsional springs representing the joint resilience of the arm. Perez et al. [18] addressed the trajectory tracking control of a two-link rigid manipulator using the adaptive neural network control. It is evident from the literature that various control schemes and methods have been developed for the single- and two-link flexible manipulators for vibration suppression and position accuracy. However, the studies regarding the influence of system attributes and configurations of the robot manipulator on the control parameters are trivial. As the robot manipulator of various capacities and configurations is being used in different industrial operations for different applications where the end point control is the major concern, it becomes inevitable to study the system performance under parametric variations. Thus, here the dynamic modeling of a planar two-link flexible manipulator where links are assumed to be Euler–Bernoulli beam element in generalized coordinate system is developed. The Euler–Lagrange’s principle is used to obtain the nonlinear flexible motion of the links and rigid motion of the hub joints which are further discretized using assumed mode method. Further, a modelbased inversion technique in conjunction with proportional-derivative controller is used to investigate the influence of the system variables such as link length, payload capacity, material of links, joint inertia and joint mass on the control parameters such as control torque and modal displacements for the desired angular position of the joint angles. The present study shall have a significant contribution in development of effective control strategies for the flexible manipulators involved in various industrial applications performing different operations.
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2 Dynamic Modeling and Governing Equations In this section, we establish a dynamic model which describes the dynamics of planar two-link flexible robot with revolute pair. Consider a planar flexible two-link manipulator incorporating payload being driven by two motors representing flexible the hub joints as shown in Fig. 1. A brief dynamic modeling has been presented considering the planar revolute motions of the joints along with links undergoing flexible motion. A global coordinate system represented by (X, Y ) with X , Y as the unit vectors is assumed; xˆ1 , yˆ1 and xˆ2 , yˆ2 are the orthogonal unit vectors of the moving coordinate system attached with first and second link, respectively. The relationships between the rotating and inertial frame of reference of first and second link are, respectively, given as:
xˆn yˆn
n ⎤ n
sin cos δ + v δ + v , t) , t) (L (L i i (i−1)x (i−1)x ⎥ Xˆ ⎢ ⎥ i=1 i=1n =⎢ n ⎦ Yˆ . (1) ⎣
− sin δi + v(i−1)x (L , t) cos δi + v(i−1)x (L , t) ⎡
i=1
i=1
The position vector of general point and end point on the link is given as: Ri =
x (vi + y)
(xi ,yi )
, and Ri+1 = Ri (L , t) +
x vi+1 (L , t)
(xi+1 ,yi+1 ) .
(2)
The kinetic energy (T ) and potential energy of the system (U) are expressed as:
Fig. 1 A planar two-link flexible robotic manipulator connected with revolute pair
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⎡ L ⎤ i n 2 ⎣ ρi RitT Rit dx (x,t) + m i RitT Rit (L ,t) + Ji δit + v(i−1)xt (L , t) ⎦. T = 0.5 i=1
0
(3) ⎤ ⎡ L i n ⎣ E i Ii (vi x x )2 dx ⎦. U = 0.5 i=1
(4)
0
at the joints and the non-conservative work The external torques τ1,2 applied done by structural damping c1,2 in the links are: 2 2 δWnc = 0.5 c1 v1t + c2 v2t
(5)
In order to obtain the governing equations of motion to adequately describe the dynamics of two-link flexible manipulator, the Euler–Lagrangian approach is applied. The Euler–Lagrangian equation is given by: d dt
∂L ∂ p˙ i
−
∂L ∂ Wc + = Qi ∂ pi ∂ pi
(6)
T Here, L = T − U, Q i = 0 0 τ1 τ2 , pi = u 1 u 2 ϕ1 ϕ2 . The governing nonlinear equations of motion along with the hub dynamics have been obtained by substituting Eqs. (3)−(5) in Eq. (6) and are expressed below. The governing equations of the joint dynamics are expressed as: L 1 0
ρ1 A1 xv1tt + x 2 δ1tt + v12 δ1tt + 2v1 v1t δ1t dx
+ (m 1 + ρ2 A2 L 2 ) L 1 v1tt + L 21 δ1tt + v12 δ1tt + 2v1L v1t δ1t (L , t)
L 2 +
ρ2 A 2 0
+ mP
2v2 v2t δ1t + 2v2 v2t δ2t + 2v2 v2t v1xt + v22 δ1tt + v22 δ2tt dx 2 2 2 2 +v2 δ1t v1xt + xv2tt + x δ1tt + x δ2tt + x v1xtt
2v1 v1t δ1t + v12 δ1tt + L 1 v1tt + L 21 δ1tt + 2v2 v2t δ1t + 2v2 v2t δ2t +2v2L v2t v1xt + v22 δ1tt + v22 δ2tt + u 22 v1xtt + L 2 v2tt + L 22 δ1tt + L 22 δ2tt + L 22 v1xtt
+ J1 δ1tt + J2 {δ1tt + δ2tt + v1xtt (L , t)} = τ1 .
2v2 v2t δ1t + 2v2 v2t δ2t + 2v2 v2t v1xt + v22 δ1tt + v22 δ2tt dx ρ2 A2 +v22 δ1t v1xt + xv2tt + x 2 δ1tt + x 2 δ2tt + x 2 v1xtt 2v2 v2t δ1t + 2v2 v2t δ2t + 2v2 v2t v1xt + v22 δ1tt + v22 δ2tt + mP +v22 v1xtt + L 2 v2tt + L 22 δ1tt + L 22 δ2tt + L 22 u 1xtt (L ,t)
L 2 0
(L ,t)
(7)
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+ J2 {δ1tt + δ2tt + v1xtt (L , t)} = τ2 .
(8)
The governing equations of motion for the links are expressed as: 2 + c1 v1t + E 1 I1 v1x x x x = 0. ρ1 A1 v1tt + xδ1tt − v1 δ1t ρ2 A2
(9)
2 2 2 v2tt + xv1xtt (L , t) + xδ1tt + xδ2tt − v2 δ1t − v2 δ2t − v2 v1xt (L , t) −2v2 δ1t δ2t − 2v2 δ2t v1xt (L , t) − 2v2 δ1t v1xt (L , t)
+ c1 v1t + E 2 I2 v2x x x x = 0.
(10)
3 Closed Form Equations of Motion In this section, the derivation of the output and state-equations composed of rigid body rotation of joints and flexible motion of the links of the manipulator is realized to demonstrate the influence of important system variables on the control parameters specifically on the angular tip positions, modal displacements, and input torques. Now, the deflection u 1,2 of a point located at a distance x along the links of the manipulator using assumed mode method is expressed here as: u 1 (x, t) = φ1 (x)q1 (t), and u 2 (x, t) = φ2 (x)q2 (t).
(11)
Here, q1 (t), and q2 (t) are the modal displacements of first and second link, respectively; φ1 (x), and φ2 (x) are the eigenfunction of first and second link for first mode of vibration, respectively, derived by Kumar and Pratiher [13] for rigid joint conditions and expressed here as: ¯ = G 11 cos γ¯ 1 x¯ − cosh γ¯ 1 x¯ + G 12 sin γ¯ 1 x¯ + G 14 sinh γ¯ 1 x¯ . φ1 (x)
(12)
φ2 (x) ¯ = H11 cos ξ γ¯ 1 x¯ + H21 sin ξ γ¯ 1 x¯ + H31 cosh ξ γ¯ 1 x¯ + H41 sinh ξ γ¯ 1 x¯ . (13) The first mode eigenfrequency γ¯ 1 can be obtained by numerically solving the determinant of the coefficient matrix resulting from the boundary conditions, and integration constants can be calculated as explained in Kumar and Pratiher [13] for the given system parameters. Substituting the expressions in Eq. (11) in Eqs. (6, 8, 10, and 12), the dynamic model of the two-link manipulator with extended payload can be expressed in matrix form as:
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⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ζ1 q1,2 , δ1,2t q1tt q1t 0 ⎢ q2t ⎥ ⎢ ζ2 q1,2 , δ1,2t ⎥ ⎢ 0 ⎥ ⎢ q2tt ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥. M q1,2 ⎢ ⎣ δ1tt ⎦ + η q1,2t , δ1,2t ⎣ δ1t ⎦ + ⎣ ⎦ ⎣ τ1 ⎦ 0 δ2tt δ2t τ2 0
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⎡
(14)
Here, M q1,2 is the mass matrix consisting of rigid and flexible motion components, η q1,2t , δ1,2t is the Coriolis and centrifugal component matrix, ζ q1,2 , δ1,2t is the stiffness matrix, and last matrix column in Eq. (14) represents the force matrix containing the torque exerted at the joints of the manipulator. Now, the open input loop torques are calculated, and those are capable of replicating the given trajectories for both the joints (δ1d , δ2d ) of the manipulator. Firstly for the required given trajectories, the resultant modal deflections (q1d , q2d ) are calculated by numerical integration of governing equations of flexible manipulator dynamics from of Eq. (14) for zero initial conditions. Now, the required input torques (τ1d , τ2d ) of the system are obtained for the rigid body motion of the manipulator from the inverse dynamics of Eq. (14) by using the desired trajectory (δ1d , δ2d ) and computed modal deflections (q1d , q2d ). In order to add robustness to the system, a linear feedback proportional-derivative (PD) controller on the joint trajectory error is used and expressed as: ˙ τinput = τd (qd , q˙d , ϕ˙d , ϕ¨d ) + 1 D K p (ϕd − ϕ) + K D (ϕ˙d − ϕ)
(15)
Here, K p and K D are the proportional and derivative gains, respectively, and can be selected suitably so that the poles of the linearized system of Eq. (14) are in left half of s plane. When hubs are driven by the torques τinput given in Eq. (15), the desired trajectories of the joints are obtained.
4 Results and Discussions The procedure of inverse dynamics of the system is exploited to design and determine the open-loop torque accounting of the rigid body motion of the joints for the accurate positioning of the links of manipulator. Here to conduct the simulations, the geometrical and physical characteristics for both the links have been considered as b1,2 = 0.05 m, h 1,2 = 0.0055 m, ρ1,2 = 7800 kg/m3 , E 1,2 = 2.1 × 1011 N/m2 , m 1 = 0.1 kg, m 2 = 0.1 kg, J1,2 = 0.02 kgm2 , and L 1,2 = 0.5 m. The K p and K D are taken as (100, 150) and (500, 600), respectively. A sinusoidal torque profile is used to position the first link at 60° and second link at 30°, respectively, in 4 s. The influence of payload on the angular position, input torque, and modal displacement is shown in Fig. 2. It can be observed that both the links achieve the desired positions and the error (of order 0.001) reduces with the increase of payload mass from no payload condition to 0.5 kg. The required input torque and the amplitude of the modal displacements also increase significantly with the increase in payload mass.
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Fig. 2 Effect of variation of payload mass on the input torques, angular positions, and modal displacements
It can also be observed that the tip have small amount of residual vibrations even after the required torque is removed. Hence, while lifting, an increased amount of payload will require increased amount of torque and the end point vibrations have to be suppressed for accurate positioning. A negligible effect on the input torque of the second hub is noticed with the increase in second joint mass as shown in Fig. 3; however, the torque input of the first joint increases due to the increase in overall inertia of the system driven by first joint. The increase joint inertias increase the power consumption of the actuators due to increased input torques which is shown in Fig. 4, and it also increases the amplitude of residual vibrations of the end effector which may result in increased error trajectory tracking of the manipulator. The end effector vibrates at significantly higher amplitudes for larger joint inertias after reaching the desired set angular positions. The increase in length of the first link increases the required amount of torque for both the joints; however, the rigid motion of the end point of the second link increases
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Fig. 3 Effect of variation of joint mass on the input torques
Fig. 4 Effect of variation of joint inertia on the input torques and modal deflections
significantly which is evident from Fig. 5. While the end point vibrations of the first link damp down after some time for the larger link length, the tip of the second link keeps on oscillating for a larger duration. It is observed from Fig. 6 that, as the material of the links is changed from steel to aluminum, the input torque reduces, but the tip of the manipulator links vibrates with large amplitudes. The residual vibrations are prominent in case of the aluminum. Thus, the aluminum shall reduce the power consumption of the manipulator system significantly, but the design engineer will have to reduce the end point vibrations for accurate trajectory tracking.
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Fig. 5 Effect of variation of first link length on the input torques and modal deflections
Fig. 6 Effect of variation of material of the links on the input torques and modal displacements
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5 Conclusions In the present work, mathematical model of a flexible two-link manipulator having revolute pairs undergoing small deformations incorporating the inertial nonlinearities under the influence of externally applied joint torques is presented. The generalized coordinate system in conjunction with the Lagrangian formulation is used to develop the dynamics of the manipulator accounting for the rigid joints and flexible links motions. The four temporal nonlinear governing equations of the motion of the links and joints are obtained using assumed mode method and the eigenfunctions satisfying the boundary conditions of the manipulator. An inverse dynamic procedure is perused to analyze the influence of system attributes on the control parameters such as input torques to the joints, the angular position, and end point vibrations of the links. The increase in mass of payload, mass of joints, inertia of the joints, and link length increases the required torque input significantly and hence increases the power consumption of the manipulator. The increase in link length and the joint inertia witness an adverse effect on the end point vibrations, the tip of the link vibrates at large amplitudes during the operation, and residual vibrations are observed even after the completion of duty cycle of the required torques. The change in material of the links from the steel to aluminum witness a decrease in required torques, but position accuracy also decreases simultaneously. The present study shall enhance the understanding of the performance of flexible manipulators under the parametric variations and will be effective in designing the appropriate control techniques for vibration suppression under such conditions.
References 1. Al-Solihat MK, Nahon M, Behdinan K (2018) Three-dimensional nonlinear coupled dynamic modeling of a tip-loaded rotating cantilever. J Vib Control 24(22):5366–5378 2. Chalhoub NG, Kfoury GA (2005) Development of a robust nonlinear observer for a single-link flexible manipulator. Nonlinear Dyn 39:217–233 3. Choi SB (2006) Vibration control of a smart beam structure subjected to actuator uncertainty: experimental verification. Acta Mech 181:19–30 4. Meng Q, Lai X, Wang Y, Wu M (2018) A fast stable control strategy based on system energy for a planar single-link flexible manipulator. Nonlinear Dyn Publ Online 94(1):615–626 5. Shawky A, Zydek D, Elhalwagy YZ, Ordys A (2013) Modeling and nonlinear control of a flexible-link manipulator. Appl Math Model 37:9591–9602 6. Shin H, Rhim S (2015) Modeling and control of lateral vibration of an axially translating flexible link. J Mech Sci Technol 29(1):191–198 7. Tavasoli A (2015) Dynamic modeling and nonlinear boundary control of hybrid Euler-Bernoulli beam system with a tip mass. Proc Inst Mech Eng Part K: J Multi-Body Dyn 229(1):3–15 8. Patel A, Neelgund R, Wathore A, Kolhe, JP, Kuber MM, Talole JP (2006) Robust control of flexible joint robot manipulator. In: 2006 IEEE international conference on industrial technology, Mumbai, pp 649–653 (2006) 9. Sun D (2106) Tracking accuracy analysis of a planar flexible manipulator with lubricated joint and interval uncertainty. ASME, J Comput Nonlinear Dyn 11(5)
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10. Abe A (2009) Trajectory planning for residual vibration suppression of a two-link rigid-flexible manipulator considering large deformation. Mech Mach Theo 44(9):1627–1639 11. Abe A, Hashimobo K (2015) A novel feedforward control technique for a flexible dual manipulator. Robot Comput-Integr Manuf 35:169–177 12. Ashayeri A, Eghtead M, Farid M (2008) Trajectory tracking for two-link flexible arm via two-time scale and boundary control methods. ASME. International mechanical engineering congress and exposition, October 31 to November 6 (2008) 13. Bai M, Zhou DH, Schwarz H (1998) Adaptive augmented state feedback control for an experimental planar two-link flexible manipulator. IEEE Trans Robot Autom 14(6) 14. Kumar P, Pratiher B (2018) Nonlinear modeling and vibration analysis of a two-link flexible manipulator coupled with harmonically driven flexible joints. Mech Mach Theo 131:278–299 15. Lochan K, Roy BK (2015) Position control of two-link flexible manipulator using low chattering SMC techniques. IJCTA 8(3):1137–1145 16. Oh S, Kong K (2015) Two-degree-of-freedom control of a two-link manipulator in the rotating coordinate system. IEEE Trans Ind Electron 62(9) 17. Ozer A, Semercigil SE, Kumar RP, Yowat P (2013) Delaying tool chatter in turning with a two-link robotic arm. J Sound Vib 332(6):1405–1417 18. Perez PJ, Perez JP, Soto R, Flores A, Rodriguez F, Meza JL (2012) Trajectory tracking error using PID control law for two-link robot manipulator via adaptive neural networks. Proced Technol 3:139–146
An Assistive Chair Using a Series-Elastic Actuator Ankur Kushwaha, Yash Agrawal, Sandeep Khandai, K. V. S. Hari, and G. K. Ananthasuresh
Abstract Series-elastic actuators are mechatronic devices used for force control. They consist of an actuator in series with a spring in conjunction with a sensor for measuring the actuator displacement and another for spring displacement. A controller helps deliver required force profile by driving the actuator in a feedback mode as per the measured displacements and the applied load. We present design, prototyping, and testing of a series-elastic actuator integrated with a slider-rocker linkage. This device is retrofitted to a chair such that the user is assisted while sitting and rising. Multi-body dynamics using Simscape and control system design are discussed. Keywords Arthritis · Rehabilitation · Elderly
1 Introduction The elderly often have difficulty in the routine activity of sitting in and rising from a chair. This could be attributed to their inability to bear excessive loads on the knees and hip. It is, therefore, beneficial to have an assistive appendage in the chair to provide support and actuation, which is otherwise provided by a caretaker. An allmechanical device of that kind was developed by our group [1, 2]. It had an inclined seat that can rotate about a virtual axis using a compliant hinge mechanism with the Y. Agrawal · G. K. Ananthasuresh Department of Mechanical Engineering, IISc Bengaluru, Bengaluru, Karnataka, India e-mail: [email protected] G. K. Ananthasuresh e-mail: [email protected] A. Kushwaha (B) · S. Khandai · K. V. S. Hari Department of Electrical and Communication Engineering, IISc Bengaluru, Bengaluru, Karnataka, India e-mail: [email protected] K. V. S. Hari e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_122
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Fig. 1 a Prototype of actuated assistive chair with series-elastic actuator (SEA), b Torque profiles to be customized for 60 kg user, and c SEA device built in this work
help of open-section shells that bend and twist simultaneously to provide required torque-angle profiles. A pair of cams, wherein one decides the torque-angle profile for sitting in and another for rising, needs to be customized for individual users. To enable such customization, we have designed an actuated chair and a multi-body dynamics simulation programme. The actuated chair uses a series-elastic actuator (SEA) in a slider-rocker linkage. This is the focus of this paper (Fig. 1). The slider-rocker linkage, which connects the SEA to the chair frame and the seat, is designed such that the SEA does not rotate much w.r.t. the frame. This ensures that the dynamics of SEA are decoupled from the dynamics of the linkage, by neglecting the rotations altogether. The stroke, dead length, and overall size of the SEA are designed accordingly. The torque-profile, is converted to a displacementprofile using relations pertaining to the linkage. The SEA comprises an actuator element (motor and lead screw) assembled in series with springs, employing two feedback sensors. This is used to provide a variable force output at each stroke value, which, via a single slider-rocker linkage, is translated to the chair seat. As a concept, the SEA has existed since 1995 [3], finding applications in various natural tasks of mechatronic systems as well as addressing the problems involving failure of power transmission components due to shocks. It is amenable to force control, as shown by [4–6]. The device works by reducing a force control problem, (where you move the actuator when force exceeds some value), to a position control problem, (where you move the actuator until some position is reached). The springs are assembled in series with the actuator, such that force felt at the end is commensurate with the deformation in the springs. We control this deformation by controlling the position of the actuator, such that at each stroke, this deformation is equal to a reference value generated from the requisite torque-profile. Feedback of the actuator stroke and device stroke is taken from a twin opticalincremental encoder set, positioned at the seat and the motor. The difference between reference value and measured value is fed as error to the PID controller, implemented in the micro-controller. The nominal model of the system is obtained from
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experiments on a manufactured prototype and used to find the PID parameters. This completes the design stage. The multi-body dynamic simulation programme models the linkage and the user limbs whose parameters (weight, lengths of limbs) are measured. Customization of our mechanical chair comprises three steps: (i) simulation with user parameters to see what torque-angle profile helps in suitable sitting and rising; (ii) real experience of the user with profile using the actuated chair and tuning of the profile based on that; and (iii) designing the pair of cams as per the profile that user likes (see [1] for details). The remainder of the paper is organized as follows. The mechanical design aspects of the device are discussed in Sect. 2. The relations needed for motor selection and dynamics equation for the device are presented in Sect. 3. Control system design is presented in Sect. 4. Electronics implementation is described in Sect. 5. Multi-body dynamic simulations are presented in Sect. 6. After completing tasks of all sections a protoype is operated for a test user and the results are analyzed in Sect. 7. Section 8 is closure.
2 Series-Elastic Actuator The primary idea behind a Series-Elastic Actuator (SEA) is a spring element in series with an actuator element such that the displacement of the device is the sum of the displacements of the actuator and the spring. This system works in conjunction with two feedback sensors, preferably one measuring the displacement of the device and one measuring the actuator displacement. The deformation in the spring can be related to the force felt at the actuator end by multiplying with a spring stiffness.
2.1 Design and Retrofitting on Chair Frame Amplification of force from the actuator is of essence because the device has to support users in wide range of weights. We choose to go with a power screw since they are easy to obtain and by nature provide force amplification. This is in conjunction with compression springs attached in series and two rotary encoders, one at the motor end and one at the seat. The device has one revolute joint with the seat and one with the chair frame in a slider-rocker mechanism. Ball screws are the ideal choice as power screws since they provide low frictional resistance to motion and low noise with minimal backlash. Because of the manner in which the SEA-device is mounted on the seat, the ball screw will be in compressive load at all times of operation. When the seat is being raised, the torque applied by the motor will be in the same direction as the motion, extracting power from the power supply. When the seat is lowered, i.e., when the occupant is sitting, the torque applied on the motor is in the direction of motion. This makes the motor run in a generative
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Fig. 2 SEA—Components
modes, with higher than ordinary currents which are not desirable. To counter this, we use a lead screw, which relies on the sliding contact between surfaces of the screw and the nut to transmit motion and forces, having a self-locking feature. This happens when the lead angle is lower than the friction angle, making it necessary to apply torque on the lead screw, in the direction of motion. Consequently, the generative mode of motor is never activated. This comes with a trade-off in the form of requirement of higher power from the motor, due to lower mechanical efficiency of the transmission of a lead screw as compared to a ball screw. Some relations and parameters that help in the selection of the motor are presented in the next section. Figure 2 shows the manufactured prototype with major components labeled. We have used a DC motor, which is coupled to an optical incremental rotary encoder. Using a 4:1 timing belt-pulley drive, the motion is transferred to a lead screw, with kinematical accuracy and high transmission efficiency. This lead screw is supported by angular contact ball bearings on both sides. A lead nut moves on the lead screw, guided on two guide rods on either side. This is connected on another part using connecting plates. The end-effector and compression springs are assembled on additional guide pins. This is all mounted appropriately on a base plate. Figure 3 shows the SEA device, mounted on the chair frame along with its geometry. We have two feedback encoders, one measuring the seat angle, from which the device stroke could be derived, and one measuring the stroke of the lead nut. The output from these can be processed to give the instantaneous compression in the spring, which is proportional to force felt at the seat, provided that we neglect
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Fig. 3 SEA—Assembly a Prototype, b Schematic
friction, backlash, and inertial forces. We convert the required torque profile to a required spring deformation profile as a specified function of angle of the seat. Using this force-control could be reduced to a position-control problem, i.e., maintaining the required spring compression at measured seat angle. This position-control can be achieved by implementing a proportional-integral-derivative (PID) control scheme.
3 Motor Selection and Governing Equations This section will serve as the derivation for open-loop dynamics governing equations. If not stated otherwise, the symbols have their usual meanings.
3.1 Kinematics Figure 3 shows the geometry at a general angle of the chair seat, θ . Here, s is the length of the device, l, h and r are the indicated lengths. Using geometrical relations we can write (1) s = l 2 + h 2 + r 2 + 2r (h sin θ − l cos θ ) s˙ =
˙ (h cos θ + l sin θ ) θr s
(2)
xr =
Tr eq s K r (h cos θ + l sin θ )
(3)
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We can find the torque exerted on the seat by a force equal to K x, acting along the device . Since we wish to control this torque, if we equate the two we can find the spring compression to be maintained at that θ , denoted by xr . In this expression, Tr eq is the requisite torque felt at the seat, which is a function of θ . x˙r =
Treq s θ˙ + Tr eq s˙ Tr eq s
xr +
˙ sin θ − l cos θ ) 2 K r θ(h xr Tr eq s
(4)
Here, () represents differentiation with respect to θ . Using these relations, we find the angular speed of the motor(N ) as, N =γ
|(x˙r + s˙ )| = φ˙ p
(5)
Preceding relations determine the motor speed as well as necessary variables for the calculation of errors.
3.2 Dynamics 3.2.1
Leadscrew
Lead screws have different impedance for motion in both the directions. Also, they could be designed to have a self-locking feature such that any amount of axial force is unable to move the lead nut, which is used in our application. The expressions for torque required to turn the lead screw during extension and retraction are: T = (K x + Meq (x¨ + s¨ )) β= β=
μ cos λ + sin λ cos λ − μ sin λ
μ cos λ − sin λ cos λ + μ sin λ
dm β 2
(6)
f or extension
(7)
f or r etraction
(8)
It is important to note that β > 0 for both the cases and greater for the extension case. At this stage, we have enough data to select the motor. These relations were simulated on an Excel sheet where the dimensions, torque and seat RPM, etc. were input parameters and motor torque, angular velocity, and power were the output parameters. The spring stiffness was taken such that 20 mm of compression produces 1000 N of force, enough to produce approximately 175 Nm of torque at the seat hinge. We take a safety factor of 1.3 to take into account inertial effects and unwanted friction
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at other locations. The motor selected and procured has a power output of 350 W at rated RPM of 2750 RPM and provides 1.1 Nm rated torque. This motor, with a 4:1 ratio timing pulley-belt transmission, drives the lead screw. The torque applied by the motor comes out to be T Text = (9) γη where η is the transmission efficiency, which is 95% to 98% for timing belt drive systems. Note that we do not consider the inertial forces due to the slow rotation (≈ 0.01 rad/s) of the device w.r.t. the pivot on the chair frame.
3.2.2
DC Motor
Using preceding relations, we can find the governing equations for open loop dynamics. We start with the model of the motor. Tmot − Beq N = Jeq N˙ + Text Jeq N˙ + Text + Beq N κ ˙ + Beq N˙ Jeq N¨ + Text Jeq N˙ + Text + Beq N +L + κN Vin = R κ κ i=
(10) (11)
(12)
The second term in the preceding equation needs elaboration. Since the value of L comes out to be very small, it is possible to completely neglect the term, hence we do ˙ . Further manipulation yields a second order equation not need an expression for Text in x, (13) α1 x¨ + α2 x˙ + α3 x = Vin − (α2 s˙ + α1 s¨ ) where α1 =
Rγ Meq dm β R Jeq γ + κp 2κη
α2 = α3 =
Beq R γ +κ κ p β Rγ dm K 2κη
(14)
(15)
(16)
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4 Control System The variable, Vin , is the input variable while x is our output variable and s˙ is a disturbance. We have an additional constraint that x has to follow a profile w.r.t. s, i.e., it is a function of s. The nature of s depends on physical properties of the user like mass, limb lengths, weight distribution, and the effort applied from joints, which change from person to person. The aim of this analysis is to provide a robust control system such that effects of s are filtered out using PID control.
4.1 Error Calculation The relation for reference signal xr for a selected profile is given by Eq. (3). Following relations are needed to calculate the actual values of the output variable and its derivative: xm =
pφm 2 − l + h 2 + r 2 + 2r (h sin θm − l cos θm ) 2π γ
(17)
x˙m =
θ˙m r (h cos θm + l sin θm ) pNm − 2π γ l 2 + h 2 + r 2 + 2r (h sin θm − l cos θm )
(18)
where the subscript m represents instantaneous measurement through the encoders. Subsequently, we subract these from reference values to get the instantaneous errors. The integral error is calculated by simple integration of the error in the source code by implementing Simpson’s 1/3rd rule.
4.2 Transfer Equation Form and System Block Diagram The equations for the transfer function model for the system are given by KC X (s) = X r (s)
α1
(τ s + 1) (s)
− M(s)
s2 +
α2 s α1
(s)
KC KD 2 α3 + K C (s) = s 3 + α2 + s + s+ α1 α1 α1 τ
(19)
(20)
where reference signal X r (s) is calculated from Eq. (3). The control system block diagram is shown in Fig. 4. We use a PID control scheme with inner loop rate feedback to filter out effects of sudden jumps in the reference signal. K C , K D , τ are the PID
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Fig. 4 Control system block diagram
parameters that need to be tuned to achieve a robust control. M(s) is the instantaneous velocity of the device, which is to be treated as an unknown disturbance. The characteristic polynomial ( (s)) dictates dynamics of the system along with many performance parameters such as steady-state performance, settling time, and peak overshoot. The aim of PID control is to mould the characteristic polynomial to the following form by obtaining the appropriate PID parameters from the nominal model of the system. (s) = (s 2 + 2ζ ωn s + ωn2 )(s + a)
(21)
For robotics operations, ζ is chosen as 1 and ωn is chosen as high as permitted by the hardware, instabilities, and saturation [7]. The pole a has to be placed such that the dominance condition is satisfied and it does not affect the dynamics setup by ζ and ωn , taken around five times the product of ζ and ωn . Once we fix these parameters, relations to calculate the PID parameters follow. K C = α1 (ωn2 + 2ζ ωn a) − α3
(22)
K D = α1 (a + 2ζ ωn ) − α2
(23)
τ=
KC α1 ωn2 a
(24)
Although we have an estimate for the nominal model of the system from the constants, we conducted experiments where the system was excited by step input of voltage and the angular velocity was measured. The system model was estimated from that to get experimental values of α1 and α2 . Next, we tested the system for various values of ωn to find the one with the best performance. The results are discussed in the following sections.
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5 Electronics and Integration The electronics sub-system is a critical part of the device and its architecture is shown in Fig. 5. It has (a) microcontroller, (b) encoders, (c) motor driver, (d) SMPS, (e) motor, and a (f) limit switch. The encoders are the most critical parts. The DC motors have commutator brushes that produce sparks everytime they lose contact and the environment is exuded with the electromagnetic interference caused by this phenomena. Encoders, via the wires that carry signals get affected by this noise. So we need to ensure their fidelity. The twin incremental optical encoder set are used to measure instantaneous stroke, compression, and velocities given by equations (1) and (2). They are procured from Autonics and have a resolution rating of 2500 and 360 pulse per revolution respectively. They are provided with four channels, A, A.B, B. Here A and B provide square-pulse form output. A and B are 90◦ out of phase from each other. A and B are the inverted forms of the respective signals, used for noise immunity. The four form a differential line driver signal with IC AM26C32CN used as the reciever at the microcontroller end. The microcontroller (TM4C123GH6PM, Texas Instruments, 80 MHz clock frequency, 32 KB RAM) interfaces with encoders, limit switches, and the motor, implementing the control algorithm. This microcontroller has a quadrature encoder module which is used to decode the signals from a differential line reciever. The direction of rotation is determined by current and previous states of channel A and B, done within the microcontroller. A counter is present to either increment or decrement on every rising and falling edge of each channel. In clockwise rotation the counter increments
Fig. 5 Electronics subsystem architecture
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and vice versa in counter-clockwise rotation. Since four edges are present per pulse, the resolution is four times that of the resolution rating. Thus, for the motor encoder, each rotation is quantised in 1440 parts and provides us with a 1.4 μm resolution. Similarly, for the seat encoder, we have 0.036◦ resolution. They provide instantaneous angular velocities calculated from the time duration between subsequent edges. We use a 350 W brushed-permanent magnet DC motor with rated current of 19.2 A, rated torque of 1.1 Nm, rated voltage of 24 V, and rated angular speed of 2750 RPM. This is a close fit for the motor required for the purpose of our device, as discussed in Sect. 3.2.1. H-bridge (Cytron) is used to control the motor with rating of 24 V and 20 A, which matches the output from the SMPS, ensuring that current remains limited to the continuous operating range of the motor. The motor is driven by 24 V Pulse Width Modulation (PWM) signals with frequency of 10 kHz while the voltage is varied by modulating the duty cycle.
6 Multi-Body Dynamics Simulations We have isolated the user part from the device such that the user interacts with torque profiles as a passive nonlinear spring. These torque profiles need to be optimized for classes of people such that user comfort is maximized. We can categorize people on the basis of their overall mass and their Body Mass Index and test different torque profiles to identify which profiles cater to different categories of users. Primarily, the effort spent by a person in sitting can be characterized by the torque at the knee joint. If the user’s weight is low, during sitting, some torque profiles would be too stiff and he/she would stop at the point of static equilibrium, unable to reach the sitting posture. Similar condition could occur for an over-weight person with a torque profile intended for lower weight class while rising. In this section, we present the use of multi-body dynamics simulations to ensure that appropriate profile is operated for a user. The multi-body dynamic (MBD) analysis is done using Simscape toolbox on Matlab. For simplicity, the human body is divided into three parts: shank with the foot, thigh, and torso including neck and head, all connected with revolute joints, in a 2-D model. The proportions of lengths and weights w.r.t. to the total height and mass are taken from [8], given in Table 1. In a three-link link model, the center of mass of each link is considered at its center. The seat is taken as a ternary link with dimensions identical with the actual one. The interaction between the seat and human, through the thigh, is defined by a sub-system consisting of transform sensor block, which uses a contact model. The foot is fixed to the ground. MBD analysis is the first step to ensure that torque profiles serve their intended purpose. In the simulation experiments, we perform a test to ensure that the person would be able to completely sit using the bespoke profile and similarly if the person is able to stand without using any of his/her own muscles. In the first case, while sitting, we allow the user to sit due to gravity alone, with torque-free conditions at the knee joint. While rising, the occupant rises in similar
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Table 1 Mean limb lengths and weight distribution. H is the total height and W is the total weight Limb Description Limb Length (H) Limb Weight (W) Lower Leg Thigh Upper Body
0.285 0.245 0.470
0.11 0.22 0.67
Fig. 6 Different configurations the user takes while operation. While sitting the sequence is forward and while rising the sequence is backward
conditions. In the second case, we specify a finite movement in both hip and knee joints while sitting and compare the results with and without external torque profiles. We wish to emulate the first case with real users (Fig. 6).
7 Device Operation We have implemented the discussed control scheme of the prototype. Using MBD analysis, we can get a suitable torque profile for a person. This is important since we have a limit of current and voltage the motor could be safely subjected to. At any θ , as the force applied exceeds the reference value, the motor adjusts the lead nut by applying some voltage according to the PID control law. Also, when a higher force is applied, the motor will run in a saturated state, the maximum voltage will be applied. Further, this device could be programmed to follow different force profiles for sitting in and rising. This device is tested on a person weighing 60 kg and 5 feet 10 inches tall. The reference torque profile versus actual torque profile obtained from the test is plotted in subsequent figures, one for rising and one for sitting in. As expected, during rising, actual value remains below the reference value and during sitting, the reference value is below actual value; it follows with acceptable accuracy. Now, we are able to provide the subjects with a variety of torque profiles to choose from and thereby identify the most comfortable one for her/him (Fig. 7).
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Fig. 7 System performance while sitting and rising
8 Closure An actuated assistive chair is designed to enable customization of an all-mechanical version presented in [1]. Simulations were performed to obtain an appropriate torque profile required for sitting and standing, using Simscape toolbox of MATLAB. However, the torque profile that is comfortable to a user depends on the relative posture of the body with respect to the chair frame and physical parameters. Hence, we created an actuated chair using which users can experience the torque profile to assess the level of comfort. A control system is successfully implemented using a 32-bit microcontroller interfacing with incremental optical encoders as sensors and brushed DC motor along with power screw as actuator, to track the torque profile as a function of seat angle. A series-elastic actuator is utilized to provide various torque profiles and get the user feedback in terms of comfort level. Acknowledgements This project was supported by the Technology Initiative for the Disabled and Elderly (TIDE) programme of the Department of Science and Technology, Government of India. We would like to acknowledge contributions from Dhananjay Yadav in making the Simscape model for MBD analysis and construction of a previous prototype which served as an inspiration for this idea. We also thank Navin Engineering, Bengaluru for assistance in manufacturing of the prototype.
References 1. Hampali S, Pai AS, Ananthasuresh GK (2019) An open-section shell designed for customized bending and twisting to ease sitting and rising in a chair. In: Badodkar D, Dwarakanath T (eds) iNaCoMM 2017, Machines, Mechanism and Robotics. Lecture Notes in Mechanical Engineering. Springer, Singapore, pp 427–439 2. Sarojini D, Lassche TJ, Herder JL, Ananthasuresh GK (2016) Statically balanced compliant two-port bistable mechanism. Mech Mach Theor 102:1–13 3. Pratt GA, Williamson MM (1995) Series elastic actuators. In: Proceedings 1995 IEEE/RSJ international conference on intelligent robots and systems. Human Robot Interaction and Cooperative Robots, IEEE
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4. Paine N, Oh S, Sentis L (2014) Design and control considerations for high-performance series elastic actuators. IEEE/ASME Trans Mechatronics 19(3):1080–1091 (IEEE) 5. Kong K, Bae J, Masayushi T (2009) Control of rotary series elastic actuator for ideal force-mode actuation in human–robot interaction applications. IEEE/ASME Trans Mechatronics 14(1):105– 118 (IEEE). https://doi.org/10.1109/TMECH.2008.2004561 6. Kong K, Bae J, Masayushi T (2011) A compact rotary series elastic actuator for human assistive systems. IEEE/ASME Trans Mechatronics 17(2):288–297 (IEEE). https://doi.org/10.1109/ TMECH.2010.2100046 7. Ogata K (2010) Modern control theory, 5th edn. Prentice Hall, One Lake Street, Upper Saddle River, New Jersey 07458 8. Anthropometric Data. https://www.ele.uri.edu/faculty/vetter/BME207/anthropometric-data. pdf. Last Accessed 19 Jul 2019
Natural Control of Virtual Models of Mechanisms Using Leap Motion for Interactive Learning Sachin Pullil
and Rajeevlochana G. Chittawadigi
Abstract The study of mechanisms is imperative in every mechanical engineering programme. Diagrams and videos are the most popular methods of teaching the kinematics of mechanisms. However, such approaches create a one-way communication environment, wherein students consume information without much practical understanding. Occasionally, physical models of mechanisms are used for instruction, but this approach is expensive and limited to simple mechanisms. This paper presents the use of Leap Motion sensor, which reads input from the user’s hand, to manipulate 3D models of mechanisms in a virtual environment. This allows the student to interact with the model of a mechanism to facilitate his/her understanding of it. The proposed method can also be used to control physical prototype of mechanisms. Keywords Mechanisms · Kinematics · Simulation · Leap Motion · Teaching
1 Introduction The study of planar and spatial mechanisms is essential in every mechanical engineering curriculum due to their extensive use in the industry. A prime example is the slider–crank mechanism that forms the basis for all reciprocating engines. Throughout the history of mechanical engineering, institutions have continually relied on diagrams in textbooks to teach complex mechanisms to students. However, this method of instruction is often ineffective in inculcating a thorough understanding of mechanisms [1]. Now, with the advent of technology, video lectures and tutorials have gained popularity in educating students in the workings of intricate mechanisms. Nonetheless, both of these methods create a one-way communication environment, wherein the students simply consume information presented to them, without a practical understanding. Research shows that interactive learning (i.e., based on two-way communication) environments bolster learning in students [2]. S. Pullil · R. G. Chittawadigi (B) Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_123
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A method to interactively teach the kinematics of mechanisms is seen in [3], where a 3D simulation software called MechAnalyzer was introduced, which allows the user to manipulate the link lengths and angles of multiple preloaded mechanisms. However, this software requires inputs to be keyed in and does not allow for natural control of mechanisms. In another instance, various models that use haptics have been employed to instruct students on mechanisms [4]. These make use of force feedback to imitate the inertia of real-world mechanisms. While effective, this method requires the instructors to create virtual and sometimes physical models of each mechanism to be taught, which can be time-consuming and costly. In this study, we propose a novel way of educating students, where they can naturally control the input to a 3D model of a mechanism and study its output. Using Leap Motion, students will be able to use natural hand movements to manipulate the mechanism and observe how the linkages move. This intuitive control coupled with real-time analysis of the model creates an interactive learning environment more effectively than conventional teaching tools. An integral component of this study is Leap Motion: a hardware sensor device that employs three infrared emitters and two infrared detectors to determine the position, orientation and velocity of user’s hands in three-dimensional space [5]. Due to the all-encompassing nature of Leap Motion’s output, the sensor has been used to directly and intuitively control a robotic arm [6]. This study explores two ways to manipulate a virtual mechanism using Leap Motion: through forward kinematics, wherein the input angle to the mechanism is calculated using the user’s fingertip position, and through inverse kinematics, wherein the user’s fingertip position is computed as input for the goal position of the endeffector of the mechanism. The 3D virtual model of the mechanism is designed in Autodesk Inventor, and the input from Leap Motion is processed by a client application and then delivered to the modelling software via Inventor API (Application Programming Interface).
2 Simulation of Mechanisms in Autodesk Inventor Autodesk Inventor is a 3D modelling software that allows a user to create sketches, parts, mechanical assemblies, etc. In the part design context, users can model a part by specifying its shape and physical dimensions. Then, in the assembly design context, users can build mechanical assemblies by constraining the parts with the axes and faces of other parts, to form a full assembly unit. In this study, various mechanisms are modelled in Inventor. To begin, each individual link was designed using Inventor’s part modelling context. Next, these links were imported into the assembly design context and constrained to each other to build a fully constrained mechanical assembly. Examples of a slider–crank assembly and a Whitworth quick return mechanism assembly designed in Inventor are shown in Fig. 1a, b, respectively. They can be described as a single degree-of-freedom (DOF) system. DOF of a system is the number of independent variables that are required
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(a) Slider-crank mechanism
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(b) Whitworth quick-return mechanism
Fig. 1 Virtual mechanism assembly models in Autodesk Inventor software
to fully define the configuration of the system. Hence, one independent variable is present that can be controlled. In both the cases, the orientation of the crank link, i.e., crank angle, can be considered as the controlled variable. Inventor allows user to grasp a link through mouse and move it, and its “Assembly Constraint Manager” allows the corresponding movement of the other links, in an interactive fashion. However, if one has to control a higher DOF mechanism, like a five-bar mechanism, mouse grasp does not deliver the required behaviour. To overcome this inability, one can use “Angle Constraint” as explained next. For a virtual mechanism (say given in Fig. 2), the crank angle can be constrained using “Angle Constraint” in Inventor. Changing the angle of the constraint updates the angle in the assembly, and the “Assembly Constraint Manager” of the software moves the other links as per the kinematic constraints. The “Angle Constraint” can play a major role in the simulation of mechanisms with higher DOF. For example, in a five-bar mechanism, two revolute joints can be assigned with “Angle Constraint” and updating their value, the mechanism gets updated. Figure 2 illustrates two configurations of a five-bar mechanism that has different joint angles in its two Angle Constraints. Thus, the user does not have to solve the kinematic equations of a complicated mechanism, which need not have closed-form solutions, to know the position and orientation of the other links in the given mechanism. However, this
(a) First configuration
(b) Second configuration
Fig. 2 Angle Constraint to update angles of a five-bar mechanism in Autodesk Inventor
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requires updating multiple Angle Constraint’s values manually. These operations can be automated, as explained below.
2.1 Application Programming Interface (API) of Autodesk Inventor Autodesk Inventor exposes its Application Programming Interface (API) that can be accessed from other software applications. Through this, the user can access different functions that allow them to create parts and assemblies, constrain them, change their position and orientation, etc. Almost all operations that can be performed with the Graphical User Interface (GUI) of the software can be executed by other applications, known as plugins or addins. The angles of the “Angle Constraints” in an assembly can also be modified by an addin. One such application was developed by Chittawadigi et al. to determine the Denavit–Hartenberg (DH) parameters from the CAD model of serial robot [7], where the Angle Constraints of a robot were modified, one joint at a time, and the position of the end-effector (last link) was determined using the application, programmatically. It acted as a validation for the research methodology proposed in that reported work. In this work, a similar approach is used, as explained next.
3 Natural Control of Virtual Mechanism Using Leap Motion In this paper, Leap Motion sensor has been integrated with Autodesk Inventor software. A client application, referred to as addin, has been developed using Visual C#. It has a reference to Leap Motion Controller, an interface to read the data received from the Leap Motion device. The addin tracks the index finger of the user, as illustrated in Fig. 3a. The coordinates of the fingertip are measured for every frame of image capture by the device. These measured values are with respect to the coordinate system at centre of the Leap Motion device. However, to obtain meaningful coordinates with regard to controlling the crank angle of a virtual mechanism, an alternate coordinate system is assumed 200 mm vertically above the original coordinate system, as shown in Fig. 3b. Therefore, the tracking of finger is done with respect to the alternate coordinate system and the coordinates of the fingertip are determined. Based on the type of simulation, as explained next, the Angle Constraint(s) value(s) is/are updated in Autodesk Inventor through the API. Thus, the integration of Leap Motion and Autodesk Inventor is completed for an effective visualization of mechanism motion.
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(a) Leap Motion tracking user’s fingertip
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(b) Alternate coordinate system
Fig. 3 Natural control of virtual mechanism using forward kinematics
3.1 Forward Kinematics of One DOF Mechanisms Slider–crank mechanism is one of the most commonly used mechanisms in teaching kinematics. It has one translation joint and three revolute joints, of which one revolute joint between the crank link and the ground/fixed link is usually considered as input link. Angle Constraint is defined for this joint and can be updated programmatically from an addin. The coordinates of the fingertip are determined as the user moves his/her hand over the Leap Motion sensor. The inclination of the position vector to the current point of the fingertip is determined and fed as the joint angle of the crank, through the Angle Constraint. Hence, as the user moves his/her hand in a virtual circle in the XY plane of the alternate coordinate system, the crank angle gets updated. A sample input data for the fingertip movement, as plotted in MATLAB, is shown in Fig. 4a. The corresponding motion of the slider–crank in Autodesk Inventor, through its API, is shown in Fig. 4b. Note that the trace of the finger need not be an exact circle. The proposed methodology has been tested for other one DOF mechanism such as Whitworth quick return mechanism (Fig. 3a). It would work for any virtual
(a) Position of the finger tip
(b) Rotation of crank angle due to finger movement
Fig. 4 Forward kinematics of a slider–crank mechanism
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mechanism which has one Angle Constraint, whose angle can be modified based on the user input over Leap Motion.
3.2 Inverse Kinematics of Five-Bar Mechanism Five-bar mechanism has two DOFs, and hence two Angle Constraints can be defined at the joints that connect the two links connected to the grounded link. Since it has two DOFs in a plane, its end-effector (position of the joint located farthest from the grounded link) can be used to trace any shape on a 2D plane. Inverse kinematics deals with the determination of input joint angles required to reach to the desired end-effector position. The user moves his/her fingertip over the Leap Motion device, and the coordinates in the alternate coordinate system are measured. The X and Y coordinates measured are supplied as input to an inverse kinematics solver, developed based on the formulations discussed in [3]. The solver returns back the corresponding joint angles to be used in the active joints, thus moving the mechanism such that the end-effector point reaches the desired point measured by the device. The end-effector’s position in the Inventor environment is recorded over time, and a sketch is made to connect each consecutive point. An illustration of the positions of the fingertip and the corresponding configurations of the five-bar mechanism in Inventor is shown in Fig. 5a, b, respectively. Leap Motion can also be used to control physical prototype of a mechanism. A similar attempt is made by the second author in [8], where a 3 DOF translational DOF Delta Manipulator is controlled using Leap Motion as input.
(a) Position of the finger tip
(b) End-effector following the path traced by finger
Fig. 5 Inverse kinematics of a five-bar mechanism
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4 Conclusions In this paper, a novel way of teaching kinematics of mechanisms is proposed. It consists of a client software application that can track the human hand as input, and based on the movement of the fingertip, control virtual mechanisms in Autodesk Inventor software. The authors claim that the method allows user to control virtual mechanisms in a natural way. As examples, forward kinematics of slider–crank and inverse kinematics of five-bar mechanism are proposed. Survey on the effectiveness of the proposed methods shall be conducted and reported in future. Leap Motion device can be easily integrated with other similar applications where human hand and fingers act as input to control virtual or physical systems.
References 1. Civelek T, Ucar E, Ustunel H, Aydın MK (2014) Effects of a haptic augmented simulation on K-12 students’ achievement and their attitudes towards physics. EURASIA J Math, Sci Technol Educ 10(6):565–574 2. Cairncross S, Mannion M (2012) Interactive multimedia and learning: realizing the benefits. Innov Educ Teach Int 38(2):156–164 3. Hampali S, Chittawadigi RG, Saha SK (2015) MechAnalyzer: 3D model based mechanism learning software. In: 14th World congress in mechanism and machine science 4. Koul MH, Shahdad I (2017) Towards an open source haptic kit to teach basic STEM concepts. In: Advances in robotics: 3rd international conference of robotics Society of India 5. Leap Motion Website: Accessed in January 2018. https://www.leapmotion.com/ 6. Bassily D, Georgoulas C, Guettler J, Linner T, Bock T (2014) Intuitive and adaptive robotic arm manipulation using the leap motion controller. In: ISR/Robotik 2014: 41st international symposium on robotics 7. Chittawadigi RG, Hayat AA, Saha SK (2013) Geometric model identification of a serial robot. In: The 3rd IFToMM international symposium on robotics and mechatronics 8. Giridharan P, Chittawadigi RG, Udupa G (2019) Intuitive manipulation of delta robot using leap motion. In: 4th International and 19th national conference on machines and mechanisms
Automated Calibration of Cervical Spine Motion Segment Finite Element Model for Physiological Kinematics Dhinesh Natarajan, Jobin D. John, and Gurunathan Saravana Kumar
Abstract Calibration of material model parameters for the validation of spine finite element (FE) model involves the tuning of parameters in multiple spinal components. A method to automate calibration of spine kinematics was developed in this study, which will find application in subject-specific model development and biomimetic mechanisms. Downhill simplex method was used to find the optimum value of twenty-four material parameters in a single motion segment FE model of cervical spine. The model consisted of two vertebrae and the disc and ligaments in between them. Flexion and extension loading cases and combined flexion-extension loading were considered for the automated model calibration. It was seen from the study that a combined consideration of the loading cases is required to obtain a model calibrated in multiple loading directions as spinal components have different contributions in different loading conditions. Keywords Cervical spine kinematics · Finite element simulation · Optimization · Model validation
1 Introduction Spinal column consists of alternating hard and soft tissue structure that facilitates movement and transfer of loads in the human body. The complex-shaped hard tissues (vertebrae) articulate with each other through intervertebral soft tissues (disc and ligaments) and contact surfaces to produce distinct kinematics in different directions like flexion-extension, lateral bending and axial rotation as shown in Fig. 1. Finite element (FE) modelling has become an indispensable tool for understanding the mechanics of spine structure [1]. Not only does FE models allow researchers to study complex loading conditions that are not easily feasible in in vitro experiments, but also is useful for the design of medical devices [2, 3].
D. Natarajan · J. D. John · G. Saravana Kumar (B) Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_124
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Fig. 1 Illustration of a spine motion segment
One of the main steps of FE model development is the validation of the model responses with experimental responses. In the case of spine model, this step is particularly challenging because of the number of components in the spinal structure [4]. Most of the spine model developments utilize manual tuning of the material parameters to obtain validated kinematic responses. This is an arduous and time-consuming task [5, 6]. An automated method for calibration will be required for rapid development of spine models. The objective of this study was to develop an optimization framework to automate material parameter selection, with a focus on tuning the spine motion segment kinematics.
2 Methods 2.1 Finite Element Model of Cervical Spine Motion Segment A validated FE model of single cervical spine motion segment was adapted for geometry and mesh size in this study [7]. The model consisted of two vertebrae and the soft tissues (discs and ligaments) corresponding to C5–C6 spine level (Fig. 2). The trabecular bone in the vertebrae was defined as linear elastic material (E = 400 MPa, ν = 0.29). The intervertebral disc (IVD) is an important component that influences the spine kinematics along with the ligaments and thus a detailed model of IVD as described in was used [8]. The disc was modelled as two separate regions: annulus and nucleus. Annulus fibres were modelled as rebar layers within the annulus elements. Because of the differences in orientation of annulus fibres between the anterior (forward) region and posterior (rearward) region, the annulus was modelled as two separate regions. The annulus fibres in the anterior region were oriented at an angle of 45 degrees and modelled in a criss-cross manner. The fibres in the posterior
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a) Coronal view
b) Sagittal view
c) Capsular ligaments
d) Anterior longitudinal ligament
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Fig. 2 C5–C6 finite element model used in this study
region were oriented in the vertical direction. The annulus ground and nucleus were modelled with linear properties, while the annulus fibres were defined as hypoelastic material [9]. Five cervical spine ligaments were defined in the model: anterior longitudinal ligament (ALL), posterior longitudinal ligament (PLL), capsular ligament (CL), ligamentum flavum (LF) and interspinous ligament (ISL). The ligaments of the spine act only when they are loaded in tension and hence were modelled as truss elements that do not provide resistance in compression. The nonlinear material behaviour of the ligaments was modelled using hypoelastic definitions. The model was simulated in moment-loading conditions: flexion and extension. The model was loaded to moments of 2 Nm in flexion–extension. The lower nodes of C6 vertebrae were fixed in all degrees of freedom. The moment was applied on the upper nodes of C5 vertebra using a multipoint constraint.
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2.2 Optimization Framework for Calibration Material definitions of eleven spinal components were considered in an optimization process. In total, twenty-four parameters associated with the material definitions were investigated. The objective of the optimization process was to minimize the deviation of the model response curves from the experimental curve corridor for flexion and extension from cadaver experiments [10]. The range of material parameter values was taken from literature and initialized at their mean values [11]. For hypoelastic materials, the strain ranges are varied from 0.1 to 0.8. These strain values determine the length of the toe region of the ligaments’ stress–strain responses [12]. The initial values of the parameters and the parameter ranges are given in Table 1. Downhill simplex method was used as the numerical method for the optimization. The spine model was simulated using Abaqus® and optimization framework was developed using Isight® . The framework developed in Isight® is illustrated in Fig. 3. Abaqus® application is initially linked to a loop process to run simulations for four iterations by changing the flexion–extension moment load value from 0.5 to 2 Nm for each iteration. The input parameters to Abaqus® , the load and material properties, are mapped to the loop. In the loop process, a calculator component parameterizes the input variables (hypoelastic materials) for Abaqus® and the output parameters after a simulation in Abaqus® are sent into another calculator component. From these parameters, it calculates the maximum segmental rotation undergone by the model. Each time the loop runs, the script component attached to it writes the rotation value from the calculator to an output file. The list of rotation and corresponding load values stored by the script is accessed by MATLAB® application component that is connected to the optimization loop process (the main loop) and fits a third-order curve for the four simulation points and exports the coefficients to another text file. These coefficients are processed in another script component which sends the response curve discretized as a set of points into the data-matching component. The datamatching component computes the objectives of the optimization by comparing the response curve with the experimental corridor bounds. All the input parameters to Abaqus® and out parameters from the data-matching component are mapped to the optimization process. The developed optimization routine in Isight® was used to consider three load cases for the calibration of material properties of C5–C6 functional spine unit for matching its kinematics under flexion, extension, considered separately and combined flexion–extension with the data available from cadaver experiments.
3 Results and Discussion A method to automate spine FE model calibration was developed in this study. Gradient-free downhill simplex method was used for finding the optimized model
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Table 1 Material parameters and their ranges Spinal component
Material parameter
Range
Nucleus
Results of optimization Flexion loading
Extension loading
Flexion–extension loading
E(MPa)
0.5–1.5
1.069
0.583
0.9
Annulus E(MPa) ground anterior
1.5–4.5
2.012
2.26
2.064
Annulus ground posterior
E(MPa)
1.5–4.5
2.021
2.11
2.141
Annulus fibre anterior
E 1 (MPa)
10–30
24.32
21.51
15.23
E 2 (MPa)
16–48
44.67
34.42
44.58
ε
0.01–0.8
E 1 (MPa)
10–30
13.86
21.12
18.74
E 2 (MPa)
16–48
26.82
31.15
25.77
ε
0.01–0.8
Anterior longitudinal ligament
E 1 (MPa)
7.5–22.5
20.92
16.38
13.63
E 2 (MPa)
15–45
23.81
28.69
29.19
ε
0.1–0.8
Posterior longitudinal ligament
E 1 (MPa)
5–15
14.62
9.02
12.38
E 2 (MPa)
10–30
16.71
17.75
18.93
ε
0.1–0.8
Capsular ligaments
E 1 (MPa)
3.5–10.5
E 2 (MPa)
15–45
ε
0.1–0.8
E 1 (MPa)
2.5–7.5
E 2 (MPa)
5–15
ε
0.1–0.8
0.729
E 1 (MPa)
2–6
E 2 (MPa)
4–12
ε
0.1–0.8
Annulus fibre posterior
Ligamentum flavum
Interspinous ligament
0.334
0.265
0.274
0.306
0.143
0.113
0.25
0.135
0.506
0.432
0.430
0.454
2.86
7.04
6.16
24.29
29.69
31.85
0.274
0.324
0.616
2.13
4.53
3.15
12.95
9.19
9.97
0.251
0.787
1.76
3.75
2.78
2.36
8.14
6.56
0.765
0.332
0.708
parameters. The optimized parameters for the individual load cases (flexion and extension) and the combined flexion–extension load case showed considerable differences. The responses of the optimized model and the experimental corridor for separate flexion and extension load cases are shown in Fig. 4. The response in the case of combined flexion–extension load case is shown in Fig. 5. The optimized parameter values are given in Table 1. The difference in optimized parameters between separate and combined load cases can be attributed to the dissimilar load-bearing functions of the spinal components in different loading directions. For example, most of the
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Fig. 3 Simulation flow in Isight® containing calculators, Abaqus® , MATLAB® , data-matching, optimization, loop and script components
a) Flexion Moment
b) Extension Moment Fig. 4 Response of the optimized FE model in the individual loading cases, flexion (top) and extension (bottom). The shaded region shows the experimental responses
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Fig. 5 Response of the optimized FE model in the combined flexion–extension loading case
ligaments do not have load-bearing function in extension loading. This inconsistency of optimized parameter values between the loading directions shows the need to consider all the loading cases collectively in the objective of the optimization process to obtain a model validated in multiple loading cases. The details of the downhill simplex method for the three loading cases are given in Table 2. Although a single motion segment was considered in this study, it can be easily extended to multiple segments or the whole spine structure. The technique developed in this study can be used along with medical image-driven generation of FE mesh for faster development of subject-specific models of the spine [13, 14]. Rapid development of such models will allow these to be integrated with patient-specific treatment planning tools. The framework developed in this study can also find applications in development of biomimetic mechanisms related to the spine structure. One of the limitations of this study was the implementation of linear material definitions in the disc, which resulted in the linear response of the spine in extension (refer to Fig. 4). In addition, for the model to be validated in all directions of loading, Table 2 Details of downhill simplex optimization procedure in the three loading cases Flexion
Extension
Flexion–extension
Initial simplex size
0.5
0.5
0.5
Initial value of objective function
4.88
0.11
4.99
Final value of objective function
0.0388
0.008
0.14
Time taken (h)
16
4
20
No. of iterations
450
120
240
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lateral bending and axial rotation cases need to be further integrated in the automated calibration process, which will be future study.
4 Conclusions In summary, a method for automated calibration of the spine FE models was developed in this study. The kinematics of a single motion segment model was calibrated with experimental response with a good match in flexion and extension loading. The model can be improved by considering other loading cases namely lateral bending and axial rotation along with additional spine model elements like facet cartilages and muscles which have not been considered in this study.
References 1. Noailly J, Lacroix D (2012) Finite element modelling of the spine. Biomater Spinal Surg 144–234e 2. Oxland TR (2016) Fundamental biomechanics of the spine-what we have learned in the past 25 years and future directions. J Biomech 49(6):817–832 3. Zafarparandeh I, Lazoglu I (2012) Application of the finite element method in spinal implant design and manufacture. The Des Manuf Med Devices 153–183 4. Jones AC, Wilcox RK (2008) Finite element analysis of the spine: towards a framework of verification, validation and sensitivity analysis. Med Eng Phys 30(10):1287–1304 5. Kallemeyn N, Gandhi A, Kode S, Shivanna K, Smucker J, Grosland N (2010) Validation of a C2–C7 cervical spine finite element model using specimen-specific flexibility data. Med Eng Phys 32(5):482–489 6. Fagan MJ, Julian S, Siddall DJ, Mohsen AM (2002) Patient-specific spine models. Part 1: Finite element analysis of the lumbar intervertebral disc—a material sensitivity study. Proc Inst Mech Eng, Part H: J Eng Med 216(July 2016):299–314 7. John JD, Saravana Kumar G, Yoganandan N (2019) Cervical spine morphology and ligament property variations: a finite element study of their influence on sagittal bending characteristics. J Biomech 85:18–26 8. John JD, Arun MWJ, Saravana Kumar G, Yoganandan N (2017) Cervical spine finite element model with anatomically accurate asymmetric intervertebral discs. In: Summer biomechanics, bioengineering, and biotransport conference. Tucson, USA, pp 153–154 9. Erbulut DU, Zafarparandeh I, Lazoglu I, Ozer AF (2014) Application of an asymmetric finite element model of the C2-T1 cervical spine for evaluating the role of soft tissues in stability. Med Eng Phys 36(7):915–921 10. Wheeldon JA, Pintar FA, Knowles S, Yoganandan N (2006) Experimental flexion/extension data corridors for validation of finite element models of the young, normal cervical spine. J Biomech 39(2):375–380 11. Zander T, Dreischarf M, Timm A-K, Baumann WW, Schmidt H (2017) Impact of material and morphological parameters on the mechanical response of the lumbar spine—a finite element sensitivity study. J Biomech 53:185–190 12. Mattucci SFE, Cronin DS (2015) A method to characterize average cervical spine ligament response based on raw data sets for implementation into injury biomechanics models. J Mech Behav Biomed Mater 41:251–260
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13. Zanjani-Pour S, Winlove CP, Smith CW, Meakin JR (2016) Image driven subject-specific finite element models of spinal biomechanics. J Biomech 49(6):919–925 14. Lavecchia CE, Espino DM et al (2018) Lumbar model generator: a tool for the automated generation of a parametric scalable model of the lumbar spine. J R Soc Interface 15(138):20170829
Identification of Inertial Parameters and Friction Coefficients for One-Link Manipulator Anil K. Sharma , S. K. Saha, Virendra Kumar, and Soumen Sen
Abstract Dynamic identification of a manipulator is necessary to know its inertial parameters as well as friction coefficients (Coulomb and viscous) at the kinematic joints. Here, the dynamic parameters and friction coefficients at the revolute joint of a one-link planar manipulator were identified through experimental data. The manipulator was moved in horizontal plane (no joint torque due to gravity) to identify the moment-of-inertia about the joint axis effectively, as the inertia force is dominant. For horizontal plane motion, the Coulomb friction coefficient at the revolute joint was considered as constant, while the viscous friction coefficient varied with respect to the joint rate. The constant viscous friction coefficient leads to the erroneous results for wide range of joint rate. Trapezoidal trajectory (at velocity level) was used to identify the dynamic, Coulomb, and viscous friction parameters through measurement data. The identified dynamic parameters were found in good agreement with the CAD model data of the manipulator. The identified parameters were validated for another trapezoidal trajectory and for a sinusoidal trajectory. From the experimental results, it is shown that the Coulomb friction coefficient is constant for horizontal plane motion, whereas viscous friction coefficient is the function of joint rate. Keywords Dynamic identification · Friction coefficients · Inertial parameters
1 Introduction There is always a disparity between the CAD model and actual installed robot, due to the manufacturing errors, and some parts such as washers, wiring, transmissions, and joint friction which are not modeled with required precision. The identification of inertial parameters and joint friction coefficients is essential mainly for the modelbased control of the multibody systems. Inertial parameters of a rigid link which is the part of a multibody system consist of ten standard inertial parameters (SIP), A. K. Sharma (B) · S. K. Saha Indian Institute of Technology Delhi, New Delhi, Delhi 110016, India V. Kumar · S. Sen CSIR-Central Mechanical Engineering Research Institute, Durgapur, West Bengal 713209, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_125
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known as six components of mass moment-of-inertia, three components of mass moment of center-of-mass (CoM), and mass of the link. There are other unknown parameters such as joint friction coefficients, rotor inertia, gear reduction ratio (GRR), and torque constant. Inertial parameters for the individual body/link can be measured physically, before the assembly of a robot. But after the assembly of the robot, the dynamic parameters of the individual link would change because of the installation of other accessories required for the operation of the robot such as wiring, joint actuators, and other sensors. Further, the estimation of the friction coefficients must be performed in the working condition of the robot. Therefore, the identification of the inertial parameters and friction coefficients should be done through the experimental data. A typical identification methodology consists of modeling of the dynamic system, design of experiments, data acquisition, parameter estimation, and validation of the dynamic model [1]. There are different dynamic modeling techniques such as Newton–Euler and Euler–Lagrange. Regardless of the modeling technique, the equations of motion (EOM) can be represented as linear in unknown parameters. The dynamic parameters can be divided in three categories, namely those which affect the dynamics individually, those which affect the dynamics in groups, and those which do not affect the dynamics. A set of minimum parameters, which affect the dynamics of a multibody system, is known as base parameters [2]. The Gaussian elimination (GE)-based method was used to identify the set of minimum number of dynamic parameters from the vector of SIP [3]. In the literature, the estimated friction coefficients were assumed to be constant for the entire range of the trajectory [3–8], while in our experimental analysis, it was found that the friction coefficients are not constant for entire range which change with respect to the joint rate. Here, the dynamic parameters and joint friction coefficients of a one-link manipulator were identified through experimental data. The key contribution of the paper is the estimation of the Coulomb friction coefficient and viscous friction coefficients with respect to the joint rate of the one-link manipulator.
2 Mathematical Modeling for Dynamic Identification The identification of inertial and frictional parameters is required for precise simulation and control [3]. The philosophy of parameter identification is based on the input–output data of a manipulator under study. The linear-in-parameter (LIP) form of the Newton–Euler equation was used for the identification of inertial and joint friction parameters of the manipulator. The detailed derivation to formulate the LIP form using the DeNOC matrices [9, 10] is presented in [3] for the multibody systems. The LIP form for one-link manipulator is discussed next.
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Fig. 1 One-link manipulator in horizontal plane
2.1 Linear-in-Parameter Form for One-Link Manipulator A one-link manipulator in horizontal plane (without gravity) is shown in Fig. 1. In the horizontal plane motion, inertia is dominant over gravity, and therefore, it was used to identify moment-of-inertia of the link about the joint axis. The LIP model for the one-link manipulator can be expressed as Yχ = τ
(1)
T where Y ≡ θ¨ sgn θ˙ θ˙ is the regressor matrix, χ ≡ I yy f c f v is the vector of unknown parameters for the horizontal motion of the manipulator, I yy is the momentof-inertia about Y-axis of the body-fixed frame, and f c and f v are the coefficients of Coulomb and viscous friction, respectively. Because of the minimal normal force due to gravity in the horizontal plane, the Coulomb friction coefficient was considered as constant, whereas the viscous friction coefficient as the function of joint rate.
2.2 Joint Trajectories for Dynamic Identification and Torque Reconstruction Two different joint trajectories were considered for dynamic identification and torque reconstruction, namely the trapezoidal (at velocity level) and sinusoidal trajectory. The sine wave trajectory as given in the EPOS studio manual was considered, which is written below: θ (t) = a sin 2π nt − π 2 + a
(2a)
θ˙ (t) = 2π na cos 2π nt − π 2
(2b)
θ¨ (t) = −(2π n)2 a sin 2π nt − π 2
(2c)
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where t is the instantaneous time in seconds, a and n are the amplitude (radian) and frequency (Hz) of sine wave, respectively. The trapezoidal trajectory was divided in three phases, i.e., constant acceleration, constant velocity, and constant deceleration. The mathematical formulation for the trapezoidal trajectory is given below: (i)
Constant acceleration phase t ∈ [t0 = 0, t1 ]
Here, t1 is the time duration of constant acceleration phase. To represent constant acceleration, a second-order polynomial is needed, which is given below: θ (t) = a0 + a1 t + a2 t 2
(3a)
θ˙ (t) = a1 + 2a2 t
(3b)
¨ = 2a2 θ(t)
(3c)
The value of unknown coefficients a0 , a1 , and a2 , for the boundary conditions θ (t0 ) = θ0 , θ˙ (t0 ) = θ˙0 , and θ (t1 ) = θ1 is given below: a0 = θ0 ; a1 = θ˙0 ; and a2 = θ1 − θ0 − θ˙0 t1 t12 (ii)
(4)
Constant velocity phase t ∈ [t1 , tf − t1 ]
Here, tf is the final/total time of trajectory. The polynomial form for the constant velocity can be given as θ (t) = b0 + b1 t
(5a)
θ˙ (t) = b1
(5b)
Because of continuity, the joint angle and joint rate at the end of constant acceleration phase must be equal to the joint angle and joint rate in the starting of the constant velocity phase, i.e.,
(iii)
b1 = a1 + 2a2 t1
(6a)
b0 + b1 t = a0 + a1 t1 + a2 t12 ; or b0 = a0 − a2 t12
(6b)
Constant deceleration phase t ∈ [tf − td , tf ]
Here, td is the time duration of deceleration phase. The polynomial form for the constant deceleration phase can be expressed as θ (t) = c0 + c1 t + c2 t 2
(7a)
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θ˙ (t) = c1 + 2c2 t
(7b)
¨ = 2c2 θ(t)
(7c)
The value of unknown coefficients c0 , c1 , and c2 , for the boundary conditions ˙ f ) = θ˙f is given below: θ˙ (tf − td ) = b1 , θ (tf ) = θf , and θ(t c2 = θ˙f − b1 2td ; c1 = θ˙f − 2c2 tf ; and c0 = θf − c1 tf − c2 tf2
(8)
3 Dynamic Identification of One-Link Manipulator The experimental setup of one-link manipulator is shown in Fig. 2. The input parameters for the trapezoidal trajectories are given in Table 1. The joint trajectory ‘Trapezoidal 1’ was used to excite the manipulator for parameter identification. The joint angle, joint rate, and current taken from the EPOS studio software were in quadrature
Fig. 2 Experimental setup of one-link manipulator in horizontal plane
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Table 1 Input parameters for trapezoidal trajectories Joint trajectory
Acceleration duration (s)
Angle at the end of acceleration phase
Deceleration duration (s)
Total time (s)
Time step (s)
Trapezoidal 1
1.2
π/2
0.3
1.5
0.025
Trapezoidal 2
4
π/2
1
6
0.025
count, rpm, and mA, respectively. Further, joint angle, joint rate, and current were converted to rad, rad/s, and Nm, using the conversion factors given in the Maxon motor manual. The joint acceleration was found using the numerical differentiation of the joint rate data. The noise in joint rate, joint acceleration, and joint torque data was removed through zero-phase digital filtering in both forward and reverse direction using the ‘filtfilt(b, a, x)’ command of MATLAB. The mass moment-of-inertia about the joint axis was identified as 0.1321 kg m2 , whereas Coulomb friction coefficient as 0.2. The identified viscous friction coefficient as the function of joint rate is shown in Fig. 3. An eleventh-order polynomial was used to fit the identified data using ‘polyfit(x, y, n)’ command of MATLAB. The polynomial form of viscous friction coefficient is valid for a range of joint rate from 0 to 2.55 rad/s. For the joint rate beyond 2.55 rad/s, the polynomial may lead to a negative viscous friction coefficient, which is incorrect. The validation of the identified parameters was done using joint trajectories ‘Trapezoidal 2’ and a sine wave, Eq. (2a), with amplitude of π 6 and frequency of 0.5 Hz. The results for joint torque reconstruction are shown in Figs. 4, 5, and 6. Fig. 3 Variation of identified viscous friction coefficient with respect to joint rate
Identification of Inertial Parameters and Friction Coefficients … Fig. 4 Joint torque reconstruction for joint trajectory ‘Trapezoidal 1’
Fig. 5 Joint torque reconstruction for joint trajectory ‘Trapezoidal 2’
Fig. 6 Joint torque reconstruction for sinusoidal joint trajectory
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4 Discussion and Conclusions The dynamic and friction coefficients of a one-link planar manipulator were identified through experimental data. A constant Coulomb friction coefficient was proposed for the horizontal motion of the manipulator and found in good agreement with the experimental results. Moreover, the viscous friction coefficient was identified as a function of joint rate. The identified parameters were validated for different joint trajectories and found that the proposed identification strategy for friction parameters is very close to the experimental results. The identification results can be improved using the more precise measuring instruments for joint acceleration and motor current. The numerical differentiation of the joint rate leads to noisy joint acceleration data. The accelerometer mounted on the link can improve the accuracy of joint acceleration, and its numerical integration would be less noisy to get the joint rate and joint angle data. The proposed friction coefficient identification strategy will be implemented to get the variable Coulomb friction coefficient with respect to the joint angle.
References 1. Swevers J, Verdonck W, De Schutter J (2007) Dynamic model identification for industrial robots. IEEE Control Syst 27(5):58–71 2. Khalil W, Dombre E (2004) Modeling, identification and control of robots. ButterworthHeinemann 3. Hayat AA (2017) Identification of kinematic and dynamic parameters of serial manipulator. Ph.D. Thesis, Department of Mechanical Engineering, Indian Institute of Technology Delhi, India 4. Dupont PE (1990) Friction modeling in dynamic robot simulation. In: IEEE international conference on robotics and automation. IEEE, pp. 1370–1376 5. Armstrong-Hélouvry B, Dupont P, De Wit CC (1994) A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30(7):1083–1138 6. Grotjahn M, Daemi M, Heimann B (2001) Friction and rigid body identification of robot dynamics. Int J Solids Struct 38(10):1889–1902 7. Grotjahn M, Heimann B, Abdellatif H (2004) Identification of friction and rigid-body dynamics of parallel kinematic structures for model-based control. Multibody Syst Dyn 11(3):273–294 8. Hayat AA, Saha SK (2017) Identification of robot dynamic parameters based on equimomental systems. In: 3rd International conference on advance in robotics, (AIR-2017), IIT Delhi, June 28–July 2 9. Saha SK (1999) Analytical expression for the inverted inertia matrix of serial robots. Int J Robot Res 18(1):20–36 10. Saha SK (2014) Introduction to robotics, 2nd edn. Tata McGraw-Hill Education, New Delhi
A Task-Based Dimensional Synthesis of an Upper-Limb Exoskeleton: A Hybrid Configuration Sakshi Gupta , Sameer Gupta , Anupam Agrawal, and Ekta Singla
Abstract This paper deals with the human-robot compatibility issue of upper-limb exoskeleton through a dimensional synthesis problem. The work is a contribution to solving misalignment. In this paper, the objective is the task-based dimensional synthesis of a wearable upper-limb exoskeleton for emulating natural human motion. A planar hybrid architecture is used for the purpose, with a four-bar connected to another four-bar in series. The task is selected based upon the standard rehabilitation exercises, only for the planar motion (parallel to sagittal plane). To achieve the proposed objective, the work has consisted of the formulation and solving of a constrained optimization problem, with reachability, design limits and solution continuity as constraints. Genetic algorithm is used for problem-solving. The results are detailed for proposed manipulator for the upper-limb exoskeleton, showcasing variation in design limits and constraints. Keywords Upper-limb exoskeleton · Dimensional synthesis · Hybrid configuration · Elbow misalignment
1 Introduction and Background Stroke is a leading cause of disability in the society in which a person deficits his/her motor-function required to do the daily activities of life (ADL). Neurocognitive rehabilitation is considered the prominent technique to resolve the situation [1, 2] and subacute stage, i.e., within six months of stroke is the best stage for better recovery [3]. Several therapies have been designed to target this critical period. However, due to the repetition of exercises, the manual techniques are monotonous and time consuming for a physiotherapist. Robotic assistance in the direction may help in overcoming the raised issues, and exoskeletons are worked upon for the purpose. Such designs of exoskeleton encounter many challenges, including ergonomically acceptable wearing technology, architectural design, human-motion compatibility, human S. Gupta (B) · S. Gupta · A. Agrawal · E. Singla Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_126
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robot interaction, etc. [4]. Motivation of this work is handling one of the challenges of misalignment. To emulate natural human motion, two aspects had been introduced in the problem formulation. One is task-based dimensional synthesis which involves the data taken from arm motion for standard rehabilitation exercises. The second is the planar architecture with closed loops. A four-bar over a four-bar configuration is used for this purpose which is expected to minimize the misalignment between human and robotic motion. The selection of the four-bar linkage is majorly due to possesses of varying instantaneous centers of the coupler [5, 6]. The work includes exercise-based data collection, kinematic modeling of double four-bar configuration and dimensional synthesis of the mechanism for the selected rehabilitation exercise which is considered as the task.
2 Exercise-Based Data Collection Human motion data extraction had been done for conventional and unconventional techniques, targeting standard rehabilitation exercises as shown in Fig. 1. These exercises have been chosen on the basis of experts recommendations (courtesy: Stroke Rehabilitation Centre Indian Spinal Injury Centre, Delhi) and available anthropometric exercise-based data for rehabilitation purpose in the rehabilitation centers. The selected exercises for presented work, as high-lighted, are assumed to be planar movements. The flexion-extension movement of shoulder and elbow, which is parallel to the sagittal plane, has been retrieved through human motion camera. In the initial phase of the work, this data has been collected for emulating natural human motion. As extracted the selected exercises, the targeted motion tasks are lifting empty hand (liftrighthand, liftlefthand, liftbothhand) and lifting 5 kg load(lift5kgright, lift5kgleft,
Fig. 1 Standard rehabilitation exercises (courtesy: Indian Spinal Injury Centre, Delhi)
A Task-Based Dimensional Synthesis of an Upper-Limb … Table 1 Human motion data Xshoulder Yshoulder 0 0 0 0 0
0 0 0 0 0
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Xelbow
Yelbow
Xwrist
Ywrist
−38.4 −40.8 −60 −48 −31.2
106.6 109.2 109.2 132.6 137.8
−19.2 −76.8 −96 −45.6 −16.8
174.2 166.4 171.6 239.2 257.4
lift5kgboth). The subject data has defined from 25 to 33 years age people and height from 1.6 to 1.9 m. Both male and female gender data have been taken. A sample human motion data is shown in Table 1.
3 Planar Hybrid Configuration 3.1 Double Four-Bar Connected in Series To accomplish the selected tasks, the basic architecture is decided as a hybrid architecture. The work has selected double four-bar connected in series, i.e., a four-bar over another four-bar configuration connected in series which is harness at the wrist as shown in Fig. 2a. The idea to choose four-bar linkage is due to the property of varying instantaneous centers of the coupler[5, 6]. The purpose of the configuration is to provide the assistance flexion-extension movement at shoulder and elbow, i.e., two-DoF, which is parallel to the sagittal plane. The details of the proposed configuration are shown in Fig. 2b.
Fig. 2 Double four-bar mechanism connected with series: a concept design, b configuration
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3.2 Kinematic Modelling and Jacobian Computation The lengths of the configuration are named as l1 to l7 , and all the joint angles are considered in anti-clockwise direction from x-axis marked with θ1 to θ7 as shown in Fig. 2b. The DoF of the system is two, and in this problem, θ1 and θ5 are considered as active joints. To evaluate the performance of the double four-bar configuration, inactive joints θ2 , θ4 , θ6 , θ7 have been expressed in terms of θ1 and θ5 . The location of the end-effector, considered at point P, can be computed using closed-loop equations as. F1 = l1 C1 + l2 C2 − l3 C3 − l4 C4 = 0,
(1)
F2 = l1 S1 + l2 S2 − l3 S3 − l4 S4 = 0, F3 = l5 C5 + l6 C6 − l2 C2 − l7 C7 = 0, F4 = l5 S5 + l6 S6 − l2 S2 − l7 S7 = 0. Px = l1 C1 + l2 C2 + l7 C7 ,
(2)
Py = l1 S1 + l2 S2 + l7 S7 . General formulation of Jacobian of the 2-four-bar system can be computed as J =
δ Px δθ1 δ Py δθ1
δ Px δθ5 δ Py δθ5
(3)
δθ7 2 −l1 S1 − l2 S2 δθ −l S 7 7 δθ1 δθ5 = . δθ7 2 l1 C1 + l2 C2 δθ l7 C7 δθ δθ1 5 2 7 From Eqs. (1, 2), the values of δθ and δθ are computed by first taking their δθ1 δθ5 partial differentials with respect to θ1 and θ5 and then formulating the linear algebraic equations for the system. Finally, the Jacobian of double four-bar loops comes out to be
J=
l1 S(θ1 −θ2 ) S(θ4 ) S(θ2 −θ4 ) S(θ1 −θ2 ) − l1S(θ C(θ4 ) 2 −θ4 )
l5 S(θ5 −θ6 ) S(θ7 ) S(θ6 −θ7 ) S(θ5 −θ6 ) − l5S(θ C(θ7 ) 6 −θ7 )
.
(4)
4 Problem Formulation: An Optimization Approach To emulate the natural human motion and avoid elbow misalignment, the following designing criteria and assumption have been made:
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Assumptions: • Shoulder is considered as a base link. • All link lengths and joint angles are taken as variables and computed through dimensional synthesis. • Task-space locations (TSLs) show the end-effector of the configuration/wrist. • Harness is used to attach the end-effector of the configuration and upper limb. The objective of the problem is to synthesis optimal mechanism for reaching task-space locations (TSLs) while maximizing kinematic performance, which is represented through Jacobian conditioning index. Link length variation and range of motion depend on the anthropometric data for human upper arm. As the problem of finding suitable configuration with good condition number, this kinematic model acts as multi-model and can be solved through genetic algorithm (GA). The objective and the constraints are formulated for emulating natural human movement as given below.
Here, σ = Eigenvalue, n = Number of TSL points, m = Number of joints, L k = kth Link length, θk = kth Joint angle, [xi ,yi ] = ith TSL position, where 1 ≤ i ≤ n, [xc ,yc ] = Current end-effector position, [xs ,ys , θs ] = Shoulder position and orientation, = tolerence limit
5 Results and Discussion: Task-Based Dimensional Synthesis To demonstrate the task-based dimensional synthesis algorithm, MATLAB R2015a is running on an Intel(R)Xeon(R)CPU E5-1607 v2 @ 3.00 GHz 3.00 GHz CPU equiped with 12 GB RAM. The average time to compute the results is 20 h. The formulated problem with double four-bar configuration connected in series represented the mechanism with link lengths l1 , l2 , . . . , l7 . However, lifting left hand has considered as the task and represented the task-space locations (TSLs) as P1 , P2 , . . . , P5 .
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5.1 Case 1: Focus on Ergonomically and Aesthetically Compatible Configuration For the initial analysis, the lower bound and upper bound constraints for the link lengths are set as 2 and 40 cm and joint angles are set as 0.01 and 360◦ , respectively. The illustration of results with the design limits for task-based dimensional synthesis of double four-bar configuration is connected in series, as shown in Fig. 3. The optimal configuration has 0.14, 0.45, 0.28, 0.41 and 0.062 condition index with respect to each TSLs. The results are further improved by modifying the design limits in terms to achieve ergonomically and aesthetically compatible configuration. The modified design lower bound and upper bound constraints for the link lengths are set as lower bound = [2 6 2 2 10 17 2] and Upper bound = [40 12 6 40 20 25 45]. The joint angles are set as 0.01 and 360◦ , respectively, as illustrated in Fig. 4. Moreover, in this iteration introduced joint continuity as another objective of the problem. n,mangle i+1 θ j −i θ j ). Therefore, optimal results for the Joint angle movement = i=1, j=1 ( modified problem are 0.30, 0.45, 0.56, 0.8 and 0.38 with respect to each TSLs. It is observed that the modified optimal results have a better Jacobian conditioning index than previous optimal problem results.
Fig. 3 Optimal results of double four-bar mechanism for the formulated problem
Fig. 4 Optimal results of double four-bar mechanism for the modified design limits
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Fig. 5 Optimal results of double four-bar mechanism avoiding misalignment
5.2 Case 2: Focus on Avoiding Misalignment Issue Furthermore, to avoid misalignment, elbow mapping is introduced as an another constraint to the formulated problem. Here, the condition elbow mapping represents the mapping of instantaneous center of the coupler of the base four-bar mechanism with the instantaneous center of elbow joint. The mathematical equation for the proposed condition is given below: Elbow mapping (xk − xe )2 + (yk − ye )2 ≤ Here, [xk ,yk ] represents kth coupler position, ∀k = 1 to n and [xe ,ye ] represent elbow position. It has been noticed that the obtained Jacobian conditioning index for each TSLs are 0.12, 0.02, 0.13, 0.35 and 0.45, respectively, as shown in Fig. 5 which is not so good as an earlier case but misalignment issue minimized due to target the dimensional synthesis of the base four-bar corresponding to elbow movement.
6 Conclusions A four-bar over another four-bar configuration, connected in series, has been given the flexion-extension movement of shoulder and elbow, i.e., a planar movement parallel to the sagittal plane. This paper presents the kinematic model formulation for task-based synthesis of a two-DoF wearable exoskeleton for rehabilitation exercises. Objective of the problem formulation is to emulate natural human motion within some desirable conditions such as good kinematic condition, least misalignment and joint limits. An optimization problem is formulated and solved for reachability at the collected working points of forearm while minimizing joint movements and misalignment at elbow joint.
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Acknowledgements The authors sincerely acknowledge the grant from Indo-UK industrial project (Department of Science and Technology— Global Innovation and Technology Alliance) for financial support of this work.
References 1. Platz T (2003) Evidenzbasierte armrehabilitation eine systematische literaturubersicht. Der Nervenarzt 74(10):841–849 2. Lum PS, Burgar CG, Shor PC, Majmundar M, Van der Loos M (2002) Robot-assisted movement training compared with conventional therapy techniques for the rehabilitation of upper-limb motor function after stroke. Arch Phys Med Rehabil 83(7):952–959 3. Sørensen L, Månum G (2019) A single-subject study of robotic upper limb training in the subacute phase for four persons with cervical spinal cord injury. Spinal Cord Series and Cases 5(1):29 4. Gupta S, Agrawal A, Singla E (2019) Wearable upper limb exoskeletons: generations, design challenges and task oriented synthesis. In: New trends in medical and service robotics. Springer, pp 134–142 5. Bertomeu JMB, Lois JMB, Guillem RB, Del Pozo ÁP, Lacuesta J, Mollà CG, Luna PV, Pastor JP (2007) Development of a hinge compatible with the kinematics of the knee joint. Prosthet Orthot Intl 31(4):371–383 6. Li Y, Chang S-H, Francisco G, Su H (2018) Interaction force modeling for joint misalignment minimization toward bio-inspired knee exoskeleton design. In 2018 design of medical devices conference American Society of Mechanical Engineers, pp V001T10A011–V001T10A011
Topology Refinement from Design to Manufacturing Using Image Processing-Based Filtration Techniques G. Lakshmi Srinivas
and Arshad Javed
Abstract Topology optimization is an established method for mass reduction of a machine and structural components. Recently, this method is used to reduce industrial manipulator links by considering minimum compliance or stress as an objective function. In the present work, central link of three-DOF articulated manipulator is chosen for topology optimization. The initial material domain is optimized based on solid isotropic material with penalization (SIMP) approach using optimality criteria method. A MATLAB code is developed, which captures performance values such as maximum deflection and von Mises stress of the manipulator link. Further, topology and performance values obtained from MATLAB results are validated using simulation software ANSYS workbench 18.1 packages. The generated optimal topology closely resembles the MATLAB results. The obtained topology contains thin and irregular connections, which results in high-stress values and consumes significant manufacturing time and cost. Here, a methodology is proposed to simplify such elements with reduced stress and simple geometry without adding considerable volume. The proposed methodology is based on image processing techniques that are fused together in a systematic sequence for binary scaled images corresponding to density parameter matrix. The developed filter though this methodology is validated with stress analysis and shows unidirectional stress reduction nature. Keywords Compliance · Image processing · Topology pptimization · Solid isotropic material with penalization · Manipulator link
G. Lakshmi Srinivas (B) · A. Javed Department of Mechanical Engineering, Birla Institute of Technology and Science-Hyderabad Campus, Secunderabad, Telangana 500078, India e-mail: [email protected] A. Javed e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_127
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1 Introduction The topology optimization method helps define connections and holes in the initial material domain subject to boundary conditions. In this way, a novel shape and size of the components are obtained [1]. A significant study on the application of topology optimization was started in 2006; few members of the humanoid were optimized for energy saving [2, 3]. Links were optimized topologically, considering static loading conditions [4, 5]. Few attempts are made on five-bar mechanism, welding robot, and filtration techniques-related optimization [6–8]. Srinivas and Javed provided a comprehensive review of the optimization techniques for industrial manipulators [9]. In the present work, three-DOF articulated manipulator was considered for topology optimization, which is subjected to dynamic loading conditions. In order to apply topology optimization for a rotating mechanical member of an articulated arm, parameters such as initial design space, material properties, and loading conditions need to be identified. For a rotating link, loading conditions are the most complicated and sensitive parameter to define. However, the obtained optimal topology shows a thin structure with tiny holes, which increases the stress values on the overall link and also offers the complexity for the manufacturing process. In the present work, an image processing-based filter is developed and validated through the stress values. The topology is generated using MATLAB as well as ANSYS software for better illustration. The topology optimization process is detailed in Sect. 2. The methodology for filter is provided in Sect. 3, and the validation of the simplified topology is discussed in Sect. 4.
2 Topology Optimization The topology optimization helps to optimize design space by SIMP approach and optimality criteria method [10]. Two different zones are defined in the initial material (design space) domain, specifying the elements that can undergo the topology optimization process, and specifying material zone with a fixed value of density parameters to accommodate assembly and motion constraints. With these considerations, the topology optimization problem is defined, as shown in Eq. (1). ⎫ min : C(xe ) = DT KD ⎪ ⎪ ⎬ Subjected to : V (xe ) − V × f = 0 KD − F= 0 ⎪ ⎪ ⎭ 0 < xe ≤ 1
(1)
where ‘f ’ is the volume fraction, ‘C’ is the compliance, and ‘D’, ‘F,’ and ‘K’ are the global displacement, force vectors, and global stiffness matrix. ‘V (x e )’ and ‘V ’ are the material volume and design domain volume, respectively, and ‘x e ’ is the element design variable.
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2.1 Topology Optimization of Manipulator-Link A three-DOF articulator manipulator with three revolute joints that is an RRR arm configuration with joint variables and free body diagram (FBD) is shown in Fig. 1. Each link has its own self-weight (W) which always acts vertically downwards at center of gravity of the design space. The mass of the motor (M) and normal reaction (N) affect on each link due to contact at the joints. The moment or torque (T) is exerted by the motor at three joints of the manipulator-link. Here, the central link is subjected to the topology optimization, which has length and height, 400 mm and 100 mm, respectively, with thickness 4 mm. The material chosen for manipulator link is mild steel, which has mechanical properties, Young’s modulus of elasticity: 200 GPa, Poisson’s ratio: 0.33, density: 7700 Kg/m3 . The manipulator links one end is subjected to a fixed support, and another end is bearing 30 N payload. The constraint of the volume fraction was selected as 0.5. A MATLAB code is written to adopt the varying boundary condition, element size, dynamic change in center of gravity, and self-weight, with each iteration. The code is made to capture performance values as maximum deflection and von Mises stress. It can update the design variables of topology up to the convergence of the objective function. The final optimum topology along performance values are presented in Table 1 based on stated boundary conditions and constraints after mesh independency test [11].
Fig. 1 Free body diagram of three-DOF articulator manipulator
Table 1 Performance values of the topology obtained from MATLAB Optimal topology of manipulator link
Maximum deflection (mm)
Maximum stress (MPa)
0.01498
5.18
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Topology optimization using ANSYS workbench 18.1. The simulation software package ANSYS is used to validate the results obtained from MATLAB code. The optimal topology and performance values from the simulation software can be obtained in three steps. Step A. Static structural domain can be selected under analysis system in ANSYS workbench. Assignment of material properties in engineering data generates geometry in SpaceClaim-based initial material domain, as stated above. Updating the mesh and apply the boundary conditions as well as constraints on the created geometry. Computation of the maximum deflection and von Mises stress for stated problem. From the results, the region in design space, which is not subjected to the high stress, is removed from the final component. This can be eliminated by introducing a topology optimization toolbox in the schematic, as shown in Fig. 2. Step B. To eliminate the needless, densities from design space topology optimization toolbox added in the ANSYS. Hence, the solution data present in the static structural (A) is transferred to the topology optimization (B). To achieve the similarity in the topology, optimality criteria solver is chosen, and the design and exclusion region of the material domain is specified. The objective function and volume fraction were chosen as minimum compliance and 50% (i.e., 0.5), respectively. The optimization process ends up to the compliance reaches convergence values, and the final optimum topology is generated, as shown in Fig. 3. To achieve this topology, elemental density solver is required 31 iterations. The marginal densities (grayscale) are presented in the topology, which leads to the manufacturing difficulty. The grayscale can be eliminated by the proposed filtration technique with the image processing method.
Fig. 2 Project schematic for topology optimization of an articulated manipulator
Fig. 3 The final optimizaed topology and it’s grayscale element
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Step C. The performance values of the obtained topology can be computed by adding static structural (C) again to the project. Facets are generated in the topology optimization process, which changes to the solids by the use of SpaceClaim. Reapply the boundary conditions and constraints to the geometry. The performance values of the optimum topology are shown in Table 2. The computational time for ANSYS is 14 times more when compared to the MATLAB. The complete simulations are executed in the Dell Precision Tower 5810 XCTO 0.825 KW, 32GB 2.4 GHz RAM, Intel Xeon Processor E5-1650 (6 Core HT, 15MB Cache, 3500 MHz Turbo). The topology and performance values of ANSYS are a good agreement with MATLAB results. Table 2 Performance values of the topology obtained from ANSYS Condition Maximum deflection
Maximum von Mises stress
Optimized topology of manipulator link
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3 Methodology In the obtained topology, various thin members can be observed, as shown in Table 1 and Fig. 3. These members offer high-stress values and are impractical to manufacture. To make it feasible at the manufacturing phase, a filtration method is proposed.
3.1 Filtration Technique Here, a filtration technique is developed using a successive process of reducing the edge contrast, followed by a decrease in the separation between the darkest and lightest area and uniform reduction of each pixel values. Using these levels, the combined effect is defined with parameter ‘P’ expressed, as shown in Eq. 2. P = I1 I2 I3 Ii = 10 20 30 . . . n i = 1, 2, 3
(2)
where I 1 , I 2 , and I 3 are values carrying the various discrete level (n) of individual processing levels for edge contrast, a separation between the darkest and lightest area, reduction of each pixel values, respectively. The effect of process parameters on image processing is shown in Table 3. These three processes are tested manually to obtain this sequence, which is suitable to reduce the thin pixel elements, whether dark or bright, with the surrounding domain of opposite pixel values. Each of these processes is labeled with effect parameters P’s.
4 Results and Discussion The obtained images for each ‘P’ value are further filtered with the volume fraction and von Mises stress values. From these topologies, the stepwise simplification of the topology can be seen with the reduced value of von Mises stress and a marginal increase in the desired volume fraction, which was originally 0.5, as shown in Table 4. The image processing technique based on filtration of process parameters is useful to design to manufacturing phase with tolerant topology optimization [12]. The filtration process eliminates the thin structures and tiny holes present in the initial material domain or space. From results, as volume fraction increases, topology simplified with the elimination of tiny holes and thin members by adding material to the design space. Von Mises stress is reduced to 3.82 MPa for volume fraction 0.52, which is 26.2% less than the initial optimum topology.
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Table 3 Image processing effect on process parameters
I1
I2
I3
Initial topology
80
70
60
50
5 Conclusion A filtration process is proposed here to simplify the topologies for manufacturing process with further increase in the strength with simplified topology. A MATLAB code was developed based on stated boundary conditions and constraints, which generates topology and performance values. From MATLAB numerical simulation of the performance, deflection and von Mises stress were found as 0.01498 mm and 5.18 MPa, respectively. Results obtained from ANSYS were a good agreement with MATLAB regards performance values as deflection 0.01475 mm and von Mises stress 4.87 MPa; however, the computation time for ANSYS was 14 times larger than MATLAB code. The topology obtained from the ANSYS closely resembles the MATLAB results. The image processing technique applied to different combinations of the processing parameters achieved from the proposed filtration process. Von Mises stress is reduced by 26.2% using the image processing method. This method
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Table 4 Image processing technique applied based on processing parameters combination Parameter
Volume fraction
[P1]
Optimal topology of manipulator link
Maximum deflection (mm)
Maximum stress (MPa)
0.505
0.01488
5.02
[P2]
0.510
0.01479
4.56
[P3]
0.515
0.01471
3.96
[P4]
0.520
0.01467
3.82
is useful to design to manufacturing phase where comparatively more uncomplicated geometry is required. Acknowledgements The present work was funded by the “Department of Science and Technology and Science and Engineering research board, India” (Grant Number: ECR/2017/000799).
References 1. M.P. Besndsoe, O.S.: Topology optimization theory, methods and applications. Springer Singapore (2003) 2. Albers, A., Brudniok, S., Ottnad, J., Sauter, C., Sedchaicharn, K.: Upper body of a new humanoid robot - The design of ARMAR III. Proc. 2006 6th IEEE-RAS Int. Conf. Humanoid Robot. HUMANOIDS. 308–313 (2006). https://doi.org/10.1109/ICHR.2006.321289 3. Lohmeier, S., Buschmann, T., Schwienbacher, M., Ulbrich, H., Pfeiffer, F.: Leg Design for a Humanoid Walking Robot. 536–541 (2006) 4. Denkena B, Bergmann B, Lepper T (2017) Design and optimization of a machining robot. Procedia Manuf. 14:89–96. https://doi.org/10.1016/j.promfg.2017.11.010 5. Li, X., Shao, H., Li, G., Liu, W., Liu, C.: Static simulation and structure optimization of key parts of joint welding robots. Proc. 2018 IEEE Int. Conf. Mechatronics Autom. ICMA 2018. 282–287 (2018). https://doi.org/10.1109/ICMA2018.8484469 6. Briot S, Goldsztejn A (2018) Topology optimization of industrial robots: Application to a five-bar mechanism. 120:30–56
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7. Luo H, Fu J, Wang P, Liu J, Zhou W (2019) Design optimization of the ram structure of friction stir welding robot. Mech. Adv. Mater. Struct. 01:1–11. https://doi.org/10.1080/15376494.2018. 1471758 8. Lakshmi Srinivas G, Javed A (2020) Topology optimization of rigid-links for industrial manipulator considering dynamic loading conditions. Mech. Mach. Theory. 153: 9. Lakshmi Srinivas G, Javed A (2020) Optimization approaches of industrial serial manipulators to improve energy efficiency: A review. IOP Conf. Ser. Mater. Sci. Eng. 912: https://doi.org/ 10.1088/1757-899x/912/3/032058 10. Srinivas GL, Javed A (2020) Numerical evaluation of topologically optimized ribs for mechanical components. Mater. Today Proc. https://doi.org/10.1016/j.matpr.2019.12.292 11. G. Lakshmi Srinivas and Arshad Javed: Numerical Simulation and Experimental Study on Lightweight Mechanical Member. In: Advanced Engineering Optimization Through Intelligent Techniques. pp. 631–641. Advances in Intelligent Systems and Computing, Springer Singapore (2020) 12. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech. Sin. Xuebao. 25:227–239. https://doi.org/10.1007/s10409-009-0240-z
Soft Robotic Gripper for Agricultural Harvesting S. M. G. Vidwath, P. Rohith, R. Dikshithaa, N. Nrusimha Suraj, Rajeevlochana G. Chittawadigi , and Manohar Sambandham
Abstract Historically robots have been a compilation of rigid parts or objects which move relative to one another to transfer motion. One of the main drawbacks of this classical method has been the physical restrictions and rigidity of the structure of the components. Conventional robots are made of rigid materials that limit their ability to elastically deform and adapt their shape to external constraints and obstacles. It is also harder to use these robots in any industry which require a certain degree of sensitivity and delicacy when interacting with the environment. Soft Robotics is a relatively new subfield of Robotics with incredible potential to change the industry due to their construction from highly compliant materials. Soft robots have increased flexibility and adaptability as well as improved safety when working around humans. In this work, we have described our attempts at the design and fabrication of few soft robotic grippers. We have experimented with three types of liquid silicone rubber materials with different properties and tried fabricating one-fingered, three-fingered and four-fingered soft pneumatic actuators. We have demonstrated grasping in lifting few objects of different shapes and sizes. The gripper developed has the potential to be used for harvesting in agricultural fields. Keywords Soft robotics · Liquid silicone rubber · Adaptive gripper
1 Introduction Robots can be programmed to do the same task repeatedly with minimal error which is something very difficult for humans. But if these robots are taken out of the factories, where the environments are not perfectly known, then the robots struggle and even fail to do even a simple task which does not require much precision. For many years, S. M. G. Vidwath · P. Rohith · R. Dikshithaa · N. Nrusimha Suraj · R. G. Chittawadigi (B) Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Bengaluru, India e-mail: [email protected] M. Sambandham Green Robot Machinery Private Limited, Bengaluru 560037, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_128
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robots have been designed to emphasize speed and precision, and this translates into a very specific architecture. In this robotic structure, one has to perfectly measure the working environment and one has to perfectly program every movement of the robot joints as even a small error can generate a very large faulty outcome, which can damage the robot or its surroundings or both. Soft robotics is bio-inspired from animals, the way they move and adapt to their surroundings [1]. There is a wide range of medical applications of soft robotics like prosthetic robots and also in food handling equipment. Soft robotics Inc. [2] is a company that deals with food and beverage, advanced manufacturing and ecommerce, and retail. The grippers of these manufacturers are very flexible and do not damage the object. Soft robotic gripper has been developed to pick uneven and soft items such as food packs very effectively [3]. Soft robotics toolkit is a platform that provides people to learn about the design, fabrication, modeling, characterization, and control of soft robotic devices [4]. They hold competitions where researchers and enthusiasts are allowed to use soft robotics in their respective fields, thus making them aware of soft robotics and learn its potential uses. A recent article [5] compares the development and testing of soft gripper developed using soft silicone material and being actuated by different methods, such as pneumatic pressure, SMA (shape memory allow) wire, and electromagnets. One of the verticals or applications where soft robots are useful is in agriculture harvesting. Hiring manual labor for harvesting is one of the significant costs borne by an agriculturist. During peak seasons, there can be a significant demand for labor which results in shooting up of costs. Hence, any automation in harvesting is beneficial. Due to the fragility of the fruits or vegetables, the conventional robots have mostly been out of the scope for harvesting purposes. To achieve the gentle handling which is usually required for harvesting using the conventional rigid robots, force feedback during gripping is a solution, but it is also complicated and expensive. Hence, a soft robot gripper might be a feasible alternative. In this paper, we have made an attempt on developing a few soft robotic actuated grippers using different materials available. Though various methods are available for the development of soft actuator, first we considered fiber-reinforced method, which is explained in Sect. 2. This method had major challenges and hence was not completely successful. Another method based on PneuNets (pneumatic networks) was attempted and has been reported in Sect. 3. This method has a better success rate, considering the materials available with us. Grippers with different number of fingers were developed and they were able to grasp few objects. Section 4 discusses the further scope and conclusions of the work.
2 Soft Actuator with Fiber-Reinforcement Liquid silicone rubber when mixed with suitable agents can be poured into molds. After a certain curing time, the liquid solidifies into an elastic material which has
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many desirable properties for a soft actuator. Solidified elastic material with a hollow space can be injected with compressed air, like a balloon, and the material tries to expand in all possible directions, i.e., in the longitudinal and the lateral direction. This can be explained by considering Fig. 1a, where an elastic material in the form of a finger is subjected to compressed air. The expansion is uniform in all directions. If some form of restraint are placed so that the radial or transverse expansion is restricted, the volume is allowed to expand only longitudinal direction, as shown in Fig. 1b. These restraints are known as fiber-reinforcements. In addition to these reinforcements, if one of the face/side of the finger is stuck with a non-extensible material (bottom side in Fig. 1c), the material on the above part has a provision to expand its volume whereas the bottom side is restricted. Hence, the finger curls down, as shown in Fig. 1c. There are many different ways of obtaining the reinforcement, as explained in [6]. By carefully selecting the dimensions of the mold and the thickness of the wall of the finger, we can obtain certain characteristic relationship between the pressure of the input compressed air and the angle of curl, the maximum contact force the gripper can apply on an object, etc. Soft Robotics Toolkits [4] website and other online resources have a good range of tutorials and DIY (do it yourselves) video tutorials on creating soft robotic actuators. The material used in most of the projects in the above resource is Eco-Flex 00-30. However, when the project was initiated, its availability in Indian market had a longer period for delivery. Hence, before an order was placed, we wanted to explore other materials that were readily available in the market. The first material was “Dow Corning MS-2002”. The silicone rubber and its curing agent were procured and when mixed in the ratio of 1:10, respectively, the mixture was poured in a mold made up of cardboard, as shown in Fig. 2a. By this process, three of the four walls of each finger are formed. After a curing period of 24–30 h, paper is used as the reinforcement or restraining material and another layer of mixture is allowed to cure for another 24–30 h, as shown in Fig. 2b, which forms the fourth and missing wall of each finger. During this process, a provision is made to insert a pipe at the center of the gripper to allow compressed air to pass through and actuate the gripper. After many trials, we were able to get the gripper to curl for lower pressure. At pressure, more than 0.2 bar, the connection between the reinforcement and the main part was getting punctured, as shown in Fig. 2c, hence making this material unusable.
(a) No restraints
(b) Restraint in transverse direction
Fig. 1 Soft actuator with fiber-reinforcement [4]
(c) Non-extensible material at bottom
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(a) Cardboard mold
(b) Paper as restraint
(c) Puncture in
(d) Prototype using
with mixture
for base
Prototype using first material
second material
Fig. 2 Preparation of prototypes with fiber-reinforcements
An alternative material with no formal name was procured from another vendor which had a curing period of 4–6 h. Similar set of operations were carried out and a gripper was molded as shown in Fig. 2d. Here, instead of paper, cloth was used as a restraining material. As this material was less elastic than the first material, it was not able to curl to hold any object. However, we learnt that this material was also not suitable for application in soft robotics. Thereafter, we placed order for Eco-Flex 00-30, which was at least 3 times as expensive as the first two materials. As the volume ordered was lower, we attempted to fabricate the soft gripper using PneuNets method, which consumes lesser material than the reinforcement method. This is explained next.
3 Soft Actuator with PneuNets Methodology PneuNets are a type of soft robotic gripper. A finger has several chambers and passage for compressed air through them, as shown in Fig. 3a. When compressed air is sent through the chambers, the air expands the chambers. If a restraining material is stuck to one of the faces, that face is not allowed to expand, thus resulting in a curl, illustrated in Fig. 3b. This methodology requires accurate mold preparation, as compared to the method explained in the previous section. A thorough overview of
(a) Neutral state
(b) Curled state
Fig. 3 Soft robotic finger with PneuNets methodology [7]
(c) Physical prototype
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(a) 3D CAD model of mold in Autodesk Inventor
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(b) 3D printed mold
(c) Cured griper finger
Fig. 4 Prototype development methodology for single-finger gripper
the mold preparation using a 3D CAD modeling software and subsequent 3D printing of the molds is discussed in [7], which was further implemented as a prototype in Fig. 3c. In this paper, few variants of PneuNets fingers and grippers were fabricated and tested. Single-Finger Gripper: A 3D CAD model as mold was developed using Autodesk Inventor software considering suitable dimensions of the air passage and chambers, as shown in Fig. 4a. The mold was later manufactured using a 3D printer (Fig. 4b). Unlike the first two materials, Eco-Flex 00-30 material is to be mixed in 1:1 ratio for the two solutions, A and B. The mixture was then poured into the mold and is cured to become an elastomer (elastic material). After the first round of curing, restraining material (cloth) was used to cover up the base with additional round of liquid mixture, and allowed to cure. The resulting single finger (Fig. 4c) was flexible and was operational as expected. As compressed air was supplied, the curling of the finger was observed, as illustrated in Fig. 5a. The finger was able to grasp fruits and vegetables such as lemon and tomato, which were of the order of 50 g, illustrated in Fig. 5b, c. Multi-Fingered Gripper: Similar to single-fingered gripper, PneuNets methodology was used to create multi-fingered gripper. First, a four-finger gripper was designed and printed using a 3D printer (Fig. 6a). Note that the fingers are connected at a hub at the center, through which compressed air is sent into individual fingers’ chambers. Hence, all fingers are bound to actuate/curl synchronously. The prototype developed was successful in holding cylindrical-shaped objects, as illustrated in Fig. 6b, c. One problem faced was that the mold and prototype had sharp edges at the joining of neighboring fingers and hence was at times a cause of leakage. To overcome that, fillet and rounding were added at the common edge in a threefingered gripper, as shown in Fig. 6d. Here, the gripper performed better with regard to potential puncture at the common edges.
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(a) Right to Left: With increase in the input pressure, the curl angle increased
(b) Grasping and lifting of a lemon
(c) Grasping and lifting a tomato
Fig. 5 Prototype of single-finger gripper
(a) Mold for four fingered gripper
(b) Holding a jar
(c) Holding cylindrical
(d) Three fingered
object
gripper
Fig. 6 Prototypes of multi-fingered grippers
4 Conclusions Soft robotics deals with usage of elastic material to grasp or manipulate objects. One of the potential applications is in agricultural robotics, particularly for harvesting fruits and vegetables. In this paper, an attempt was made to develop single-finger and multi-finger gripper using available liquid silicone rubber material. Based on the tests and observations, we deduced that Eco-Flex 00-30 gives the best results and should be used in such applications. As the two materials, other than Eco-Flex 00-30, were not able to hold object properly, quantitative comparative tests between the three materials could not be conducted. Tasks such as grasping lemon, tomato and few cylindrical-shaped objects were conducted successfully. In future, the grippers
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would be designed to grasp uneven shaped objects of varying sizes and forms. Also, other alternative materials/household objects that can be used as soft robotic gripper finger shall be explored and reported in the future.
References 1. Laschi C, Cianchetti M, Mazzolai B, Margheri L, Follador M, Dario P (2012) Soft robot arm inspired by the octopus. Adv Robot 26(7):709–727 2. Soft Robotics Inc. www.softroboticsinc.com 3. Wang Z, Torigoe Y, Hirai S (2017) A prestressed soft gripper: design, modeling, fabrication, and tests for food handling. IEEE Robot Autom Lett 2(4):1909–1916 4. Soft Robotics Toolkits. www.softroboticstoolkit.com 5. Karmarkar S, Sarkar A (2019) Design and implementation of bio-inspired soft robotic grippers. In: Advances in robotics: 4th international conference of The Robotics Society 6. Polygerinos P, Wang Z, Overvelde JT, Galloway KC, Wood RJ, Bertoldi K, Walsh CJ (2015) Modeling of soft fiber-reinforced bending actuators. IEEE Trans Rob 31(3):778–789 7. Sun Y, Song YS, Paik J (2013) Characterization of silicone rubber based soft pneumatic actuators. In: IEEE/RSJ international conference on intelligent robots and systems, pp 4446–4453
Surface Profile Accuracy of Deployable Mesh Reflectors Based on Focal Offset Shenoy S. Siddesh, R. Harisankar, and G. K. Ananthasuresh
Abstract A deployable structure based on the tension truss concept is used to develop unfurlable, large-aperture reflectors for space applications. The paraboloid surface of the reflector is approximated by a reflective mesh attached to a cable network consisting of triangular facets. Existing approaches try to minimize the root mean square error between the faceted surface and the local quadratic approximation of the paraboloid for designing the cable net. An alternate approach for computing the surface accuracy of the faceted reflector surface is proposed here based on the proximity of the reflected rays to the receiver of the antenna. The offset distance of reflected rays from the focus or offset error is a more geometrically relevant indicator for determining the accuracy of the facets. An offset paraboloid reflector of 3 m aperture diameter is designed for which the surface errors are computed using both the approaches. These errors are computed for individual facets, and their distribution throughout the reflector surface is compared. Keywords Cable network · Faceting · Surface accuracy · Offset error
1 Introduction In this paper, we consider a deployable mesh reflector antenna whose curved reflective surface is an offset paraboloid. This type of surface helps in focusing electromagnetic rays falling on the reflector from a distant object onto the receiver, which is located at the focus. Such an antenna deployed using a tension truss was first presented by Miura and Miyazaki [1]. It consisted of five components: two cable networks in the front and rear, tension ties connecting corresponding nodes in the front net to those in the rear net, a reflective mesh attached to the front net, and a ring truss. The cable network (front and rear net) is used to approximate the curved surface of the S. S. Siddesh (B) · R. Harisankar · G. K. Ananthasuresh Mechanical Engineering, Indian Institute of Science, Bengaluru, Karnataka 560012, India e-mail: [email protected] G. K. Ananthasuresh e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_129
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reflector [2], whose shape is an offset paraboloid. Morterolle et al. [3] presented a design technique for generating the offset paraboloid. The offset reflector surface is obtained by the intersection of the paraboloid and a cylinder, and then the facets are generated by successive division of this surface into segments at the aperture circle. The lines joining these are connected to the center of the reflector. By doing so, triangular segments are obtained, which are further subdivided. These are then mapped on to the surface using a suitable origin of the coordinate system to obtain the coordinates of vertices of the facets. These flat facets collectively approximate the reflector surface, whose vertices lie on the curved reflector surface and illustrated in Fig. 1. Such an approximation necessitates determination of surface accuracy, which is measured by the deviation between the curved working surface, and the facets are formed by connecting these nodes. This deviation is quantified as the facet error and is a widely adopted approach found in the literature.
Fig. 1 Generation of facets and facet error
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Existing approaches try to minimize the root mean square error between the faceted surface and the local quadratic approximation of the paraboloid for designing the cable net. An alternate approach for computing the surface accuracy of the faceted reflector surface is proposed here based on the proximity of the reflected rays at the receiver (focus) of the antenna. Agarwal [4] discussed a method to choose the initial nominal facet length. Yuan [5] discussed an optimization procedure for finding the optimum position for each node by minimizing the error in the reflector surface, which is different from the offset error presented in this paper. The parameters from Table 1 were used in the design of our reflector with 1176 facets. The faceted reflector is shown in Fig. 2, and the facet error distribution is shown in Fig. 3. The facet error computed for this reflector is shown in Fig. 4 graphically along with a color bar indicating the error (in m) across the reflector. It can be observed that the facet error decreases along the x-axis. This is because the curvature of the reflector decreases gradually, resulting in flatter surface being more accurately approximated by flat facets. Also, the facet error value is symmetric about the y-axis. The RMS value of the facet error is 0.35 mm. Table 1 Input parameters used for designing the mesh reflector Parameter
Value
Parameter
Value
Focal length
2.4 m
Number of bays N b
30
Offset d
0.3 mm
Number of rings (internal) N
14
Aperture diameter (reflector) D
3m
Ring truss minor axis (circumscribed ellipsea )
3379.3 mm
a The
boundary of the offset paraboloid is an ellipse
Fig. 2 Isometric view of faceted reflector
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Fig. 3 Facet error distribution
Fig. 4 Facet error in the top view
2 Focus Offset Error Ideally, the objective of any antenna is to focus all the incoming electromagnetic rays onto the receiver. But due to the aforementioned faceting of the surface, some of the facets (or parts of the facet) do not reflect the rays to the receiver. The offset distance
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by which these rays miss the target point on the receiver (focus) is measured by a factor we call the focal offset error. The offset distance of reflected rays from the focus is a more geometrically relevant indicator for determining the accuracy of the faceted surface than the closeness of the faceted surface to the original surface. A diagrammatic illustration of the offset error is shown in Fig. 5 with an inset illustration of the receiver disk. Incoming parallel rays fall on the reflector surface along the z-direction. Three such rays are denoted by O, O1, and O2 . Consider the ray from O. In the absence of faceted surface, this ray would get reflected from point O* on the surface of the paraboloid, and hence, it would reach the focus as a result of true surface reflection. This is shown by the red colored ray. Similar is the case with point Q lying on the paraboloid except that it happens to be that point at which the tangent plane is parallel to the facet and this will be discussed later to derive offset error. Now, considering the faceted surface, the same ray from O would get reflected from point G, which is the centroid of the facet, and reach a different point on the receiver disk, say J. This is shown by the blue colored ray. Hence, the same incoming ray would reach different points on the focus depending upon the surface of reflection. This distance (FJ) on the receiver disk, by which the ray misses the focus, is called as the focus offset error or simply the offset error. The offset vector is denoted by wp (see Fig. 5). Derivation of the offset error The offset error wp is obtained by the projection of w on the receiver disk shown in Fig. 6, where w is the distance between the reflected rays QF and GJ. Hence, w is calculated first. We obtain w by calculating the distance from the point G to the reflected ray nr passing through point Q as shown in Fig. 7.
Fig. 5 Illustration of the offset error
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Fig. 6 A view of the triangular facet and the receiver disk
Fig. 7 A facet that approximates an orb of the paraboloidal surface
Calculation of w. The coordinates of points G and Q and nr are necessary to evaluate w. G, which is the centroid of the facet can be calculated as the mean of the vertices of the facet denoted by R1 , R2 , and R3 , where ˆ p = 1, 2 and 3 R p = x p iˆ + y p jˆ + z p k;
(1)
is the position vector corresponding to the pth vertex of the facet. Evaluation of coordinates of point Q: By Snell’s rule of reflection, all rays reflected from planes parallel to the facet are parallel. The assumption made earlier of point
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Q is to find the only point on the paraboloid that reflects the ray through the focus in the same direction as the rays reflected from the facet. (i)
The equation of the paraboloid surface is written as H (x, y, z) = 4 f z − x 2 + y 2 = 0
(2)
and the normal n to this surface at any point (x, y) on it is given by n(x, y) = (ii)
−x iˆ − y jˆ + 2 f kˆ ∇H = ∇ H x 2 + y2 + 4 f 2
(3)
Normal to the facet with vertices R1 , R2 and R3 is given by nf =
(R2 − R1 )(R3 − R1 ) (R2 − R1 )(R3 − R1 )
(4)
The coordinates (x Q , yQ , zQ ) of point Q can be obtained by solving n(x Q , y Q ) = n f
(5)
H (x Q , y Q , z Q ) = 0
(6)
The direction nr of the ray QF is given by nr =
−xq iˆ − yq jˆ + ( f − z q )kˆ f −q = f − q xq2 + yq2 + ( f − z q )2
(7)
where f is the position vector of the focus, F. Thus, w can be calculated as the shortest distance between point G from line QF, which is given by w = (g − q) − [(g − q).nr ]nr
(8)
Calculation of wp : By using standard trigonometric identities, the perpendicular component is projected, while the axial component remains unaltered. Therefore, wp =
w − (w · n p )n p + (w · n p )n p nh · nr
(9)
nr × nh nr × nh
(10)
where np =
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and nh is the normal to the focal disk which determines its orientation. The assumed orientation of this disk is such that its normal nh is in the same direction as that of a ray reflected from the center of the paraboloid, to ensure maximum amount of signal received. The magnitude of wp , is the offset error. The rays reflected from O and O2 are parallel to each other as can be seen in Fig. 5. Hence, the rays falling on any point on the facet surface would travel parallel to ray reflected from the centroid of the facet, G. Therefore, the offset error of these rays can be calculated by substituting its position vector for g in Eq. 8. Figure 8 shows the offset error over the entire reflector graphically, wherein the error is indicated by the color. The error is symmetrical about the XZ plane. The rays reflected from the centroids of three facets are shown. They are indicated by their facet numbers 1 (central facet), 600 (intermediate facet), and 1046 (peripheral facet). Figure 9 shows the offset error in the top view. In comparison with the facet error plot in Fig. 4, similar features of symmetry and distribution of error can be observed. Most of the peripheral facets have more offset than others due to their steep change in orientation as they are attached to the ring truss through the anchoring cables. Figure 10 shows the offset error distribution. The average offset error is found to be 0.544 cm. The RMS value of the offset error computed over the entire reflector is 3.59 cm. The ratio of effective area to total area of reflector is 0.91. Effective reflector area Fig. 11 shows how much of the facet surface participates
Fig. 8 Visualization of the offset error
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Fig. 9 Offset error in top view
Fig. 10 Offset error distribution
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Fig. 11 Effective reflector area
in reflecting the incoming rays to the focus. The orange colored portion of the facets represents the points of reflection from where the reflected rays reach the focus and the white portion represents those points where the rays miss the focus. This portrayal of the focal offset error helps to see how the faceted surface and the receiver size can be adjusted to ensure that all of it is capable of reflecting rays to the receiver. Thus, the new measure of error emphasizes the efficacy of the faceted surface rather than its accuracy relative to the paraboloidal surface.
3 Closure In contrast to the computation of the facet error used widely in the literature, the focus offset error offers a practical approach to evaluate the performance of the antenna. How much of the reflected rays are actually able to reach the receiver is given by the focus offset error presented here. The faceting of the reflector surface and the size of the receiver can be optimized using the new error measure. Acknowledgements This work was supported in part by Electronics and Radar Development Establishment, Bengaluru, a laboratory of Defence Research and Development Organization (LRDE-DRDO) India.
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References 1. Miura K, Miyazaki Y (1986) Concept of the tension truss antenna. In: 37th Congress of the International Astronautical Federation 28(6) 2. Tibert AG (2003) Optimal design of tension truss antennas. In: Structures, structural dynamics, and materials conference, Virginia, AIAA 2001-1629 3. Morterolle S, Maurin B, Quirant J, Dupuy C (2012) Numerical form-finding of geotensoid tension truss for mesh reflector. Acta Astronaut 76:154–163 4. Agarwal PK (1981) Preliminary design of large reflector with flat facets. IEEE Trans. Antennas Propag AP-29(4) 5. Yuan S, Yang B Improvement of surface accuracy for large deployable mesh reflectors. In: AIAA/AAS Astrodynamics specialist conference, AIAA 2016-5571. https://doi.org/10.2514/6. 2016-5571
Design and Control of a Low-Cost EMG-Based Soft Robotic Ankle-Foot Orthosis for Foot Drop Rehabilitation Nitish Gudapati , Koushik Kumaran , S. V. Deepak , R. Mukesh Kanna , R. Jinesh , and Himadri Poddar
Abstract Stroke patients often suffer from foot drop, a gait abnormality caused due to the paralysis of the anterior portion muscles of the lower leg, causing an inability or impaired ability to raise the foot at the ankle joint. This condition leads to extremities of the foot being dragged along the ground while walking, causing tripping and other accidents. Braces or splints that fit into shoes are prescribed to help hold the foot in a normal position. For rehabilitation, most patients are trained to walk with canes, and therapists prescribe physiotherapy for a series of short, intensive sessions. These solutions are expensive and slow processes as they require the presence of skilled personnel. In this paper, we present a novel design and control methodology for a 1-DoF Soft Active Ankle-Foot Orthosis (AFO) to address these issues. The AFO is designed to augment the human musculoskeletal system. The AFO is actuated using McKibben muscles (pneumatic artificial muscles), which are driven pneumatically. They are cost-effective and lightweight, offering a significant advantage over motordriven orthoses. The orthosis is controlled using electromyography (EMG) signals from the muscles involved in the motion of the ankle. The use of EMG for control is found to be a better option than existing methods due to its non-invasive nature. Keywords Active foot orthosis · Electromyography (EMG) · Foot drop · Rehabilitation · Soft robotics
1 Introduction According to the World Health Organization, 15 million people suffer stroke worldwide each year. Of these, 5 million people die, and another 5 million are permanently disabled [1]. A common ailment for stroke survivors is foot drop, i.e., the inability to actively perform Dorsiflexion of the foot. This leads to the occurrence of steppage Supported by Institute grants through the Robotics and Machine Intelligence Club, NIT Tiruchirappalli, INDIA. N. Gudapati (B) · K. Kumaran · S. V. Deepak · R. Mukesh Kanna · R. Jinesh · H. Poddar National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_130
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Fig. 1 Hypothetical pattern of recovery after stroke with the timing of intervention strategies. Source [4]
gait, where the foot drags along, requiring the patient to raise their knee higher than usual in the swing phase to counter this [2]. The patient’s foot slaps the ground during the foot strike phase of the gait cycle. Studies have shown that the entire kinematic chain of the lower body is disturbed, resulting in the functioning of the motor system under abnormal load [3]. Patients generally undergo therapy to regain motor control, and the required recovery can take from months to years to complete as shown in Fig. 1 [4]. The available number of medical professionals is far less than the requirement, and they may not be accessible to all those who need it [5]. To this end, robotic devices can significantly decrease the burden on therapists and caregivers and can be used to provide intensive task-oriented practice. Considerable research has gone into developing such robotic devices, especially for upper-body rehabilitation [6–8]. Most of these devices, however, are rigid, heavy and unwieldy. In recent years, the interest in a compliant robotic system has increased exponentially, due to the significant advantages they provide. McKibben muscles or pneumatic artificial muscles (PAM) are a possible actuation method with properties that make them uniquely suited for the application presented. These properties are explored in depth in Sect. 4.
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In this paper, we describe the design and control of a single DOF orthosis, actuated with PAMs and controlled with electromyography (EMG) signals. The EMG sensors are non-invasive, and the design of the orthosis is compliant, lightweight and wearable. Section 2 explains the practices and techniques involved in the rehabilitation of the foot and Sect. 3 explains the existing orthoses and exoskeletons for foot rehabilitation. Section 4 discusses the construction and working of the prototype, and the results are presented in Sect. 5. Some concluding remarks and future scope are mentioned in Sect. 6.
2 Rehabilitation Techniques To understand the requirements and specifications of stroke rehabilitation devices, a brief introduction of common existing techniques is given below [9].
2.1 Physiotherapy The primary technique used is physiotherapy which involves assessment of a physiotherapist using various techniques. The techniques include various exercises. Mobility training is done to stabilize and strengthen to enhance balance and support. Range of Motion Therapy helps in easing muscle tension allowing a good range of motion.
2.2 Technology-Assisted Functional Electrical Stimulation method involves inducing contraction to the weak muscles by application of electricity which helps in training them. Game System Technology is based on the idea of integration of games for rehabilitation. They involve functional activities and keep the patient occupied and motivated with games during the rehabilitation process.
3 Existing Technologies for Rehabilitation Robotic exoskeletons have been employed in assisting individuals suffering from physical disability either for the advancement of therapy or as a permanent assistive tool. They are of various types and can be classified based on their method of actuation. Rigid actuation devices are actuated using DC motors, steppers, etc. Soft
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actuation devices are actuated using compliant methods which generally include pneumatic or using shape memory alloys (SMA) [10–13]. The AFO described in this paper falls under this category.
3.1 Rigid Actuation Anklebot at Massachusetts Institute of Technology (MIT), USA The Anklebot is a 3-DOF wearable robot, which is actuated using two BLDC motors. The linear actuators are mounted in parallel and provide actuation in two of the ankle’s 3-DOF [14]. Position information is provided by encoders. ALEX at University of Delaware, USA This exoskeleton is mounted with a walker to support the device. The controller applies a force-field at the ankle of the subject providing the necessary torque to move it [15]. The disadvantage is that it is not easily portable.
3.2 Soft Actuation Wearable Soft Exoskeleton at Carnegie Mellon University, USA The exoskeleton uses four PAMs connected to the knee and foot braces with the help of artificial tendons [11]. It helps with therapy, but it does not provide much independence for the user. Powered Ankle-Foot Orthosis at University of Michigan, USA This orthosis is actuated by PAMs providing only plantar flexion torque. It uses a real-time computer interface for controlling the air pressure supplied to the PAMs which is determined using a footswitch placed under the foot [12]. Adaptive Control of Actuated Ankle-Foot Orthosis (AAFO) at Paris-Est Créteil University, France AAFO uses a predetermined trajectory of the ankle’s swing phase of the gait cycle, to provide the required assistance [13].
4 Construction and Working 4.1 Design of the Prototype The design of the AFO focuses on achieving the following goals: • User comfort • Minimal weight • Complete range of motion about a single axis
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Fig. 2 AFO’s Layout. Components labelled from (1) to (11): (1) Arduino Uno; (2) motor drivers; (3) solenoid valves; (4) Li-Po; (5) panel for electronic circuitry; (6) knee braces; (7) PAM mount; (8) PAM; (9) foam padding; (10) adjustable strings; (11) orthotic sole
• Simple setup • Minimal cost. Pneumatic Artificial Muscle Design Each PAM consists of a silicone rubber tube, placed within a nylon mesh sleeve. One end is pneumatically sealed, and the other end is used as an inlet for pressurized air. When air flows into the tube, it expands volumetrically. Since the sleeve’s structure only allows it to expand radially while contracting longitudinally, the PAM contracts when pressurized. Multiple studies have been performed on these PAMs, and the force-length characteristics are found to be similar to that of a muscle [16]. The materials used in the PAM are compliant, lightweight and low-cost, making them well suited for the AFO. The analysis of various characteristics of the PAM is presented in Sect. 5. The required length, diameter and pressure of the PAM were decided upon through experiments with various configurations, presented in Sect. 5. Orthosis Design The computer-aided design of the prototype is presented in Fig. 2. The orthosis consists of thigh and shin supports connected by a knee brace or a hinge joint which offers full range of motion of the knee. This framework along with the shin of the leg acts as the ground link for the actuating mechanism. The sole attached to the foot is the end effector which can be modelled as hinged at the shin. The knee brace and the sole are connected using two PAMs in an agonistantagonist configuration, each one providing the necessary dorsiflexion and plantarflexion forces, respectively. The point of the PAM hinged to the sole is dependant
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Fig. 3 Kinematic model of the orthosis
on the required actuation torque and range of motion. Since PAMs are shown to provide a good amount of force [16], but only limited contraction, the choice was made to place them very close to the joint and the length of the force arm was determined experimentally as 3 cm. The corresponding data is presented in Sect. 5. This allows the AFO to achieve the entire Range of Motion which is roughly 20◦ for dorsiflexion and 50◦ for plantarflexion [17]. Some similar devices [18, 19] have been constructed with the ground at the shin, connected to the ankle end effector in a similar fashion. We found that in this configuration, the transverse reaction forces during PAM contraction resulted in skin deformation at the skin-AFO interface causing user discomfort and restriction in the range of motion of the AFO. By placing additional support above the knee, the reaction force is made almost normal to the skin-AFO interface, thereby providing good support with no loss in user comfort. Figure 3 presents the kinematic model of the designed orthosis along with the degrees of freedom of actuation that the orthosis provides. Electronics and Pneumatic Circuitry Each PAM is controlled by a pair of 2position 2-way solenoid valves, where one acts as an inlet and the other acts as an outlet. Each pair of valves is connected to an MCU and an L298N Driver for power
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Fig. 4 Pneumatic circuit
and logic. Two MyoWare EMG sensors are connected to the same MCU. The entire setup is powered by a 12V lithium-polymer battery and placed above the knee to optimize weight distribution throughout the AFO. Both the inlet valves are connected to an air tank with pressurized air at 45 PSI, and the exhaust valves are left open to the atmosphere as shown in Fig. 4. All pneumatic connections were made with 8 mm OD × 6 mm OD polyurethane pneumatic tubes. The control methodologies for the PAMs are described in Sect. 4.2.
4.2 Control of the Orthosis Signal Acquisition In order to detect the intent of the user, the AFO uses MyoWare muscle sensors and EMG electrodes. Each McKibben muscle is placed so as to assist the functioning of a specific muscle. The targeted muscles are: • Tibialis anterior for dorsiflexion • Soleus for plantarflexion. The positions of these muscles are depicted in Fig. 5.
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Fig. 5 Muscles of the leg. Source [20]
Each muscle requires a non-invasive EMG sensor and each sensor requires the placement of three electrodes to capture the differential component of the signal travelling across the muscle. As per EMG electrode placement standards [21], one electrode is placed near the centre of a muscle, one electrode near the end of a muscle and a ground electrode is placed near a bone for reference. The rectified and integrated signal is sampled by the 10-bit ADC of the MCU at a rate 200 Hz. EMG Analysis The rectified and integrated signals (signal envelopes) from the MyoWare muscle sensors produce output with added noise. Studies show that the noise from sEMG signal includes both high frequency components and low frequency components arising from movement artefacts [22]. To overcome this, the signals from both EMG sensors were passed to two independent 1-dimensional Kalman filters. Due to the difficulty involved in creating a dynamic model, a linear predictive model was used, where the filtered output from the previous iteration was taken as the initial estimate for the next iteration. It was found that a value of 0.001 for process noise covariance gave the best results for both sensors while inducing only negligible latency in the filtered output as shown in Fig. 6.
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Fig. 6 EMG signal acquisition, filtering and thresholding
EMG signals for a muscle are highly dependent on a multitude of factors such as electrode placement, fatigue, and the physical and mental state of the user and other EMG artefacts [23, 24]. Hence, real-time calibration is done before every session. This happens automatically on initialization of the AFO. This calibration procedure sets the threshold for each muscle to enable intent estimation, which is described below. This also addresses the issues due to EMG degradation by adjusting the threshold. The rest state characteristics of each muscle are obtained by recording the unfiltered EMG envelope for 10 s. The mean (μrest ) and standard deviation (σrest ) of this data are extracted from this data. Using the collected data, a threshold is calculated for each target muscle using Eq. (1) for dorsiflexion and Eq. (2) for plantarflexion. This threshold is then used to detect activation of each muscle. The muscle is found to be actively contracting when the filtered signal exceeds the threshold. For tibialis anterior muscle, the threshold is determined by Eq. (1), T = μrest + 3 ∗ σrest
(1)
For soleus muscle, the threshold is determined by Eq. (2), T = μrest + 8 ∗ σrest
(2)
Studies have shown that a threshold equal to the mean added to a multiple of the standard deviation gives accurate results when applied to the EMG envelope [25]. The values of the constant multiplier for standard deviation were determined experimentally. The accuracy of intent detection with these values was tested and the results are described in Sect. 5. Intent Detection Logic Using the described method, we determine the state of a muscle in real time. It was determined that the soleus muscle was active during plan-
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Fig. 7 Intent detection Table 1 State of solenoid valves PAM movement Inlet valve Relax (Deflate) Contract (Inflate)
Close Open
Exhaust valve Open Close
tarflexion, and the initial and final stages of dorsiflexion. This is visualized in Fig. 10, which shows plantarflexion, followed by dorsiflexion. Using this information, user intent is detected by the method specified in Fig. 7. Control of PAM with EMG The user closes the feedback loop by attempting to move their ankle by muscle contraction. This eliminates the need of external feedback and allows real-time intent detection. In order to contract a PAM, pressurized air is inflated into it by opening the inlet valve and by closing the outlet valve. In order to relax (expand) a PAM, the air inside it is exhausted by closing the inlet valve and opening the outlet valve. Table 1 summarizes the valve positions for different PAM configurations. In order to ensure smooth movement of the foot, the speed of inflation or deflation of the PAMs is reduced by applying a pulse width modulated (PWM) signal to the valves. Through experiments, it was found that a PWM signal with a time period of half a second and a duty cycle of 10% for 5 s was ideal. The duty cycle can be modified to provide varying angular velocities. Figure 8 depicts the operation of the AFO. The AFO is an open-loop system; however, the user closes the control-loop while using the orthosis. This make the device simple, efficient and cost-effective.
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Fig. 8 Operation flowchart Table 2 State of pneumatic artificial muscles Intent Dorsiflexor PAM Rest state Plantarflexion Dorsiflexion
Relax Relax Contract
Plantarflexor PAM Relax Contract Relax
After detecting user intent, control signals from the MCU are sent to the LM298 drivers that control the pneumatic valves. Table 2 describes the state of each PAM with respect to various user inputs.
5 Results and Statistical Data 5.1 Pneumatic Artificial Muscles Tests The PAMs were tested on an experimental setup consisting of a rigid support to attach the actuator. The other ends of the PAMs were attached to a pan to test contraction under varying loads. The PAMs were connected to an air compressor maintained at a constant pressure while testing. A vernier scale was fixed along the length of the PAM. Initially, PAMs of different lengths and diameters were tested to determine the combination that provides maximum percentage contraction under a minimal load of 1 kg and 45 PSI pressure. These results are presented in Table 3. PAM of 30 cm length with an inner diameter of 1/3 in. was found to be optimal due to its percentage contraction and satisfying the length constraints of the average leg.
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Table 3 Pneumatic artificial muscles tests Inner diameter Outer diameter Length of muscle Length (inches) (inches) (cm) contraction (cm) 1/3 1/2 1/3 1/2
1/2 5/8 1/2 5/8
20 20 30 30
2.5 1.8 5.1 3.0
Percentage contraction (%) 12.5 9.0 17.0 10.0
Fig. 9 Pressure-contraction-load analysis of the PAM designed
Once the PAM characteristics were decided, the length of the force arm was determined by testing the contraction of the PAM under various pressures and loads in the abovementioned experimental setup. This data is presented in Fig. 9. The length of force arm was selected as 3 cm by taking into account the maximum pressure of the tank and the load borne by the actuator, and the required range of motion which is 20◦ for dorsiflexion, 50◦ for plantarflexion [17].
5.2 EMG Control Tests The AFO was tested by three of the authors as the particants for the study. The participants flexed their legs over the entire range of motion for both dorsiflexion and plantarflexion, and the percentage accuracy of intent detection in every case
Design and Control of a Low-Cost EMG-Based Soft … Table 4 Participants were relaxed Participant Dorsiflexion (%) Accuracy Falsepositive 1 2 3
95.00 100.00 96.67
1.67 0.00 1.67
Table 5 Participants were fatigued Participant Dorsiflexion (%) Accuracy Falsepositive 1 2 3
90.00 95.00 91.67
1.67 1.67 3.33
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Falsenegative
Plantarflexion (%) Accuracy Falsepositive
Falsenegative
3.33 0.00 1.67
90.00 98.33 91.67
8.33 1.67 6.67
Falsenegative
Plantarflexion (%) Accuracy Falsepositive
Falsenegative
8.33 3.33 5.00
90.00 91.67 90.00
6.67 5.00 5.00
1.67 0.00 1.67
3.33 3.33 5.00
was calculated and presented. Table 4 presents the data when they were relaxed and Table 5 after a while when fatigued. The control scheme achieves a mean dorsiflexor accuracy of 97.22% and inaccuracies being 1.11% false-positive and 1.67% false-negative. The mean plantarflexion accuracy is 93.33%, and the inaccuracies are 1.11% false-positive and 5.56% falsenegative. The control scheme achieves a mean dorsiflexor accuracy of 92.22% and inaccuracies being 2.22% false-positive and 5.56% false-negative. The mean plantarflexion accuracy is 90.56% and the inaccuracies are 3.88% false-positive and 5.56% falsenegative. From the results mentioned in Tables 4 and 5, it is found that the simple control scheme presented is able to achieve a mean dorsiflexion accuracy of 94.72%, and inaccuracies being 1.66% false-positive and 3.62% false-negative. The mean plantarflexion accuracy is 91.94% and the inaccuracies are 2.5% false-positive and 5.56% false-negative. It is worth noting that the accuracy decreases minimally when the participants are fatigued. Performance The orthosis exhibited a good response time to the changes in the EMGbased muscle activation signal with minimal lag due to the simple control logic. The absence of a feedback control signal also eliminates the time delay introduced by sensors and error detectors. The activation function’s output for both the muscles is shown in Fig. 10. An ergonomic and compliant prototype of the orthosis was made as a proof of concept. The completed prototype is shown in Fig. 11.
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Fig. 10 Voltage-time graph depicting filtered sensor data and the corresponding thresholds Fig. 11 Prototype of the ankle-foot orthosis
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6 Conclusions The design and control of a 1-DOF Soft Ankle-Foot Orthosis presented in this paper shows promise to be a therapeutic alternative for foot drop rehabilitation. The AFO uses EMG signals which are reliable and easy to calibrate. The main purpose of the work is to present the scope of soft-robotic devices as an alternative for rigid and heavy hardware previously used for robotic rehabilitation. The AFO uses PAMs, which are pneumatically actuated, low-cost and lightweight. The torque provided by the AFO can be increased by enhancing air pressure driving techniques, and this could allow the use of the device as an exoskeleton. Also, machine learning algorithms could be used to estimate the intent of the user with greater accuracy. The device could also be further developed by incorporating more degrees of freedom. This would allow the AFO to perform inversion and eversion of the leg.
References 1. World Health Report—2002 from the World Health Organization Retrieved on: 2019 from http://www.who.int/whr/2002/en/ 2. Walking abnormalities MedlinePlus Retrieved on: 2019 from https://medlineplus.gov/ency/ article/003199.htm 3. Wiszomirska I, Bła˙zkiewicz M, Kaczmarczyk K, Brzuszkiewicz-Ku´zmicka G, Wit A (217) Effect of drop foot on spatiotemporal, kinematic, and kinetic parameters during gait. Appl Bionics Biomech. 2017:3595461. https://doi.org/10.1155/2017/3595461 4. Langhorne P, Bernhardt J, Kwakkel G (2011 ) Stroke rehabilitation, stroke care 2, The lancet, vol 377 5. Khor KX, Rahman HA, Fu SK et al (2014) A novel hybrid rehabilitation robot for upper and lower limbs rehabilitation training. In: Internatioinal conference on robot pride (ConfPRIDE) 2013–2014, procedia computer science, vol 42, pp 293–300 6. Asokan A, Vigneshwar M, Design and control of an EMG-based low-cost exoskeleton for stroke rehabilitation. In: 2019 fifth Indian control conference (ICC), New Delhi, India, pp 478–483. https://doi.org/10.1109/INDIANCC.2019.8715555 7. Kim B, Deshpande AD (2017) An upper-body rehabilitation exoskeleton harmony with an anatomical shoulder mechanism: design, modeling, control, and performance evaluation. Int J Rob Res (IJRR) 36(4):414–435 8. Perry JC, Rosen J, (26) Design of a 7 degree-of-freedom upper-limb powered exoskeleton. The First IEEE, RAS-EMBS Interenational conference on biomedical robotics and biomechatronics, (2006) BioRob 2006. Pisa 2006:805–810 9. Stroke rehabilitation: What to expect as you recover (2017) Retrieved on 2 June 2019, from https://www.mayoclinic.org/diseasesconditions/stroke/in-depth/strokerehabilitation/art-20045172 10. Ikuta K (1990) Micro/miniature shape memory alloy actuator. In: Proceedings IEEE international conference on robotics and automation, vol 3. Cincinnati, OH, USA, pp 2156–2161. https://doi.org/10.1109/ROBOT.1990.126323 11. Park Y-L, Chen B, Pérez-Arancibia NO, Young D, Stirling L, Wood RJ, Goldfield EC, Nagpal R (2014) Design and control of a bio-inspired soft wearable robotic device for ankle-foot rehabilitation. Bioinspiration Biomimetics 9(1). https://doi.org/10.1088/1748-3182/9/1/016007 12. Gordon KE, Sawicki GS, Ferris DP (2006) Mechanical performance of artificial pneumatic muscles to power an ankle-foot orthosis. J Biomech 39(10):1832–1841. https://doi.org/10. 1016/j.jbiomech.2005.05.018
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13. Arnez-Paniagua V, Rifai H, Amirat Y, Mohammed S (2017) Adaptive control of an actuatedankle-foot-orthosis. In: 2017 International conference on rehabilitation robotics (ICORR). https://doi.org/10.1109/icorr.2017.8009474 14. Roy A, Krebs HI, Williams DJ, Bever CT, Forrester LW, Macko RM, Hogan N (2009) Robotaided neurorehabilitation: a novel robot for ankle rehabilitation. IEEE Trans Rob 25(3):569– 582. https://doi.org/10.1109/tro.2009.2019783 15. Banala SK, Kim SH, Agrawal SK, Scholz JP (2009) Robot assisted gait training with active leg exoskeleton (ALEX). IEEE Trans Neural Syst Rehabil Eng 17(1):2–8. https://doi.org/10. 1109/tnsre.2008.2008280 16. Jakub T, Laski P, Slawomir B, Gabriel B, Dawid P (2016) Determining the static characteristics of pneumatic muscles. Measur Control 49:62–71. https://doi.org/10.1177/0020294016629176 17. Generally accepted values for normal range of motion (ROM) in joints (2019) Retrieved on 8 June 2019 from https://www.verywellhealth.com/what-is-normal-range-of-motion-ina-joint-3120361 18. Ferris DP, Czerniecki JM, Hannaford B (2005) An ankle-foot orthosis powered by artificial pneumatic muscles. J Appl Biomech 21(2):189–197 19. Zhu Y (2017) Design and web-based control of a soft ankle foot orthosis 17:82–88 20. Appendicular Muscles of the Pelvic Girdle and Lower Limbs. Retrieved on 8 July 2019 from https://cnx.org/contents/[email protected]:y9_gDy74@5/Appendicular-Musclesof-the-Pelvic-Girdle-and-Lower-Limbs 21. Datasheet Retrieved on 8 June 2019 from https://cdn.sparkfun.com/datasheets/Sensors/ Biometric/MyowareUserManualAT-04-001.pdf 22. De Luca CJ, Gilmore LD, Kuznetsov M, Roy SH (210) Filtering the surface EMG signal: movement artifact and baseline noise contamination. J Biomech 43(8):1573–1579. ISSN 00219290. https://doi.org/10.1016/j.jbiomech.2010.01.027 23. Muhammad J (2012) Signal acquisition using surface EMG and circuit design considerations for robotic prosthesis. In: Computational intelligence in electromyography analysis–a perspective on current applications and future challenges. https://doi.org/10.5772/52556 24. Reaz MBI, Hussain MS, Mohd-Yasin F (2006) Techniques of EMG signal analysis: detection, processing, classification and applications. Biol Proced Online 8(1):11–35 25. Özgunen KT, Çelik U, Kurdak SS (2010) Determination of an optimal threshold value for muscle activity detection in EMG analysis. J Sports Sci Med 9:620–628
Comparison of PPC and LQR Controller for Stabilization of Cart Pendulum System: Simulation and Real-Time Study Gurminder Singh and Ashish Singla
Abstract In the present work, mathematical modeling, simulation, and real-time study has been carried out for stabilization of cart pendulum system. Euler-Lagrange method is used to derive the mathematical dynamic model of the system. The actuator dynamics is included with developed model of the system to achieve more realistic model. Two different feedback controllers, Pole Placement and Liner Quadratic Regulator are used to control the inverted pendulum at unstable equilibrium system. Simulink environment is used to carry out the analytical simulation and the experimental work was conducted on Googoltech Linear Inverted Pendulum (GLIP) setup. The experiment results found in close agreement with analytical solutions. Both controllers have been compared to check the efficacy of the developed controllers. It has been found that LQR controller has 55.5% less amplitude of oscillations of pendulum at unstable position and 47.9% less control input as compared to Pole Placement controller. Keywords Cart pendulum · Motor dynamic · Stabilization · Pole placement controller · LQR controller
1 Introduction The mechanical systems having less control inputs as compared to degree of freedom (DOF) of the system comes under the underactuated system family. These systems have diverse applications in the field of aerospace, marine engineering, flexible manipulators, mobile robots, etc. The use of such kind of systems is not limited up to the cost reduction of the system but they play significant role in fault-tolerant design, lower power consumption, lower environment impact, etc. Although, these systems have several advantages, but these systems are difficult to control due to G. Singh (B) School of Mechanical and Materials Engineering, University College Dublin, Dublin, Ireland A. Singla Mechanical Engineering Department, Thapar Institute of Engineering and Technology, Patiala, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_131
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lesser number of control actuators. Cart pendulum system is a native member of underactuated system family. The cart pendulum system is nonlinear in nature and treated as benchmark example to study the different types of controllers. In this system, three major control tasks are available to address, such as to swing up the inverted pendulum from downward position to unstable upwards position [1, 2], to stabile the inverted pendulum at upward position [3, 4] and to control the track of stabilize pendulum on cart from one position to other position [5]. Stabilization of inverted pendulum at upward unstable equilibrium has several real-time applications like biped walking, tank missile launcher, rocket propellers, etc. In the present literature, several types of controllers are found to address the stabilization problem of cart pendulum system. These controllers are categorized into three set of controllers such as classical, modern and intelligent control theories. More than 12 different types of experiments can be executed on the cart pendulum system (77.8% with classical control theory and 100% with modern control theory) [6]. The classical control theory controllers are difficult to apply on the nonlinear systems. However, these types of controllers are cheap and productive in computation calculation. Correspondingly, the modern control theory controllers are idealized to control nonlinear system and consume less time as compared to intelligent controllers. Song et al. [7] studied the comparison between PID (Proportional-Integrate-Derivative), pole placement and fuzzy logic controller to control the stabilization problem of cart pendulum system. They concluded that the pole placement controllers give better dynamic and steadystate performance as compared to other two controllers. Also, the pole separation factor was studied to control the stabilization problem of cart pendulum system [8]. However, dynamic modeling of the actuator motor was neglected during mathematical modeling of the system. The integration of actuator dynamics plays significant role in the simulation study of the underactuated system, which is further important to match real-time behavior of the system [9, 10]. In the present study, Pole placement (PPC) and Linear Quadratic Regulation (LQRC) controllers are used to address the stabilization problem of cart pendulum system. As both controllers have shown different advantages and disadvantages in literature, the comparison study has done on two important responses such as control input and pendulum behavior at unstable position.
2 Experimental Setup The experimental setup Googoltech Linear Inverted Pendulum (GLIP) (purchased from Googoltech, Hong Kong) was used for the real-time stabilization control study. The actual setup along with schematic diagram is shown in Fig. 1a. The setup consists of two DOF system: cart moving in linear horizontal direction and a pendulum free to move in a vertical and horizontal direction in a plane. An AC servo motor with feedback encoder was used to give input to the system to drive the cart. The timing pulley belt drive arrangement was used to move the cart in linear direction. Further, the linear movement of the cart rotates the pendulum freely to both equilibriums.
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Fig. 1 a Actual setup and b schematic diagram of Googoltech Linear Inverted Pendlum (GLIP)
The rotary encoder was placed on the pendulum for the rotating feedback considering pendulum downward position as zero reference and left/right motion as positive/negative magnitude. As it is a closed loop system, first the initial input to the motor was provided with the help of Simulink interface with the help of motion controller card. Further, signal data transfers to the AC motor drive which amplifies the data and gives command to the motor. In the same manner, the feedback from both encoders reverts to the Simulink interface with help of driver and controller card. The conversion factors were used to convert the signal into angular and linear data. The parameter used for GLIP was shown in Table 1. Table 1 Parameters of Googoltech cart-pole system [11, 12] Parameter
Units
Symbol
Value
Cart mass
kg
M
1.096
Pendulum mass
kg
m
0.109
b
0.1
Coefficient of friction Inertia of pendulum
kgm2
I
0.0034
Pendulum length
m
l
0.25
Acceleration due to gravity
m/s2
g
9.8
Motor rotor inertia
kgm2
Im
1.4 × 10−5
Motor viscous-friction coefficient
Nms
Bm
0.03
Motor torque constant
kgm2
Km
2
Back EMF constant for
Ns/rad
Kb
0.1
Resistance of motor
Rm
2.5
Radius of timing pulley
m
r
0.0195
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3 Modeling of Cart Pendulum System The Euler-Lagrange is used to drive the analytical model of the system. Figure 2b depicts the schematic line diagram of the two DOF system. The cart movement is restricted to a particular plane to move in horizontal direction and pendulum is free to move in vertical and horizontal direction in a plane. Consider that M and m are the mass of cart and pendulum, l is the distance between the center of pendulum and the pendulum cart joint, b is the friction coefficient for the cart. Moreover, x(t) and φ(t) are horizontal and angular position of the cart and inverted pendulum from stable equilibrium position and f (t) is the horizontal force acting on the cart. The state vector and its derivative of the system are represented in Eqs. (1) and (2). Further, the total kinetic energy is calculated given as Eq. (3). x − l sin φ xp = , z(t) = yp l cos φ x˙ − l φ˙ cos φ x˙ p = . z˙ (t) = −l φ˙ sin φ y˙ p
(1)
(2)
K = K Cart + K Pend = 21 (M + m)x˙ 2 + 21 I φ˙ 2 + 21 ml 2 φ˙ 2 − ml cos φ˙ x˙
(3)
The potential energy of the system only depicts from the pendulum movement and contains only the strain energy stored in the pendulum and given as Eq. (4).
Fig. 2 a Pendulum, b cart displacement and c control input using PPC
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P = Pcart + PPend = 0 + (−mg(l − l cos φ)
(4)
Further, Lagrange L can be written as the subtraction of both calculated kinetic and potential energies, given as L=
1 1 1 (M + m)x˙ 2 + I φ˙ 2 + ml 2 φ˙ 2 − ml cos φ˙ x˙ − mg 2 2 2
(5)
The damping factor due to friction between cart and shaft is calculated by Rayleigh’s dissipation function and given as Eq. (6) R = 21 b x˙ 2 .
(6)
The equation of motion of the present system can be written as (M + m)x¨ − ml cos φ φ¨ + ml sin φ φ˙ 2 + b x˙ = f,
(7)
I + ml 2 φ¨ − ml cos φ x¨ − mgl sin φ = 0.
(8)
The dynamic model of the AC servo motor is coupled with the derived mathematical model of the system to develop a realistic dynamic model of the system. M +m+
Im K B K K ¨ + ml sin φ φ˙ 2 + b + m + m b x˙ = m Vc . x ¨ − ml cos φ φ r2 r2 Rm r 2 Rm r (9)
where, Im is the inertia of the rotor, Bm is the viscous-friction coefficient, K m and K b are the torque and back EMF constants,Rm and Vc (t) are the resistance of motor and voltage applied to the AC servo motor and r is the timing pulley radius. The developed mathematical model is described by Eqs. (7) and (9). The developed equations can be linearized to address linear stabilization problem. The square term can be neglected and sin φ and cos φ can be written as φ and 1. The derived equations are converted into state-space representation with respect to controller design. z˙ (t) = Az(t) + Bu(t), y(t) = C z(t) + Du(t)
(10)
T where z(t) = x xφ ˙ φ˙ , is called as the state vector, y(t) = [xφ]T as the output vector, u(t) is known as the input vector and A, B, C and D are called as stateweighting coefficient matrices, which are given as
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⎡
⎤ 0 1 0 0 ⎢ 0 − m 2 l 2 Rm r 2 − 2 m 2 gl 2 Rm r 0 ⎥ ⎢ ⎥ 1 1 3 k m 21 3 km A=⎢ ⎥, ⎣0 0 0 1⎦ mgl −ml2 0 0 1 3 3 ⎡ ⎤ 0 ⎢ ml+1 ⎥ 1000 0 ⎢ ⎥ B = ⎢ 1 ⎥, C = , D= . 0010 0 ⎣ 0 ⎦ ml 1 3
Im Rm r Bm K m K b Rm r , 2 = b + 2 + , 1 = M + m + 2 r Km r Rm r 2 K m m 2 l 2 Rm r 3 = I + ml 2 − 1 km
4 Controller Design PPC method is the classical and cheap approach to stabilize the unstable closed loop system. In this method, a feedback controller is designed to place the eigenvalues of the closed loop system at the required place. The controller is designed to satisfy the transient and steady-state configuration of the system. Pole Placement control design consists of multiple steps. First, the coefficient vector a = [a1 , a2 , . . . , an ] for the polynomial of state matrix A is calculated as given in Eq. (11). The state-space equation transforms into the controller companion form by using T given in Eq. (12). In addition, the desired polynomial is calculated by using desired eigenvalues from Eq. (13) and determine the coefficient vector α = [α1 , α2 , . . . , αn ]. Further, the state feedback gain matrix K can be calculated from Eq. (14). |s I − A| = s n + a1 s n−1 an−1 an.
(11)
T = PW
(12)
⎡
an−1 an−2 ⎢ an−2 an−3 ⎢ ⎢ .. W = ⎢ ... . ⎢ ⎣ a1 1 1 0
⎤ 1 0⎥ ⎥ .. ⎥ .⎥ ⎥ ... 0 0⎦ ... 0 0
. . . a1 1 0 .. .. . .
(13)
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(s − μ1 )(s − μ2 ) . . . (s − μn ) = s n + α1 s n−1 + · · · + αn−1 s + αn. = 0
(14)
K PPC = (α − a)T.
(15)
To design optimal controller for a closed loop system, Linear Quadratic Regulator (LQR) controller is used. However, the given controller is difficult to customized and required heavy calculations. The LQR controller based on the minimization of the integration of the sum of state weighting (Q) and control cost matrix (R) of the system. The object function used in LQR control design is elucidated in Eq. (16). Further, the optimal feedback control matrix (K LQRC ) can be calculated as Eq. (17). The control matrix depends upon the algebraic Riccati Equation given by Eq. (18). ∞
(z T (τ )Qz(τ ) + u T (τ )Ru(τ )dτ,
(16)
K LQRC = R −1 B T M0 ,
(17)
M0 A + A T M0 − M0 B R −1 B T M0 + Q = 0.
(18)
J∞ (t) =
0
It is observed that the feedback control matrix directly depends upon Q and R. These matrices have to be determined by manual tuning, which increase the time cost of the controller.
5 Results and Discussions Simulation and experimental studies are demonstrated in the present section to check the efficacy of both controllers. The calculated eigenvalues for the state matrix A are [0, −250.86, −5.64, 5.64]. The present system is unstable having one of the positive eigenvalue. Thus, a closed loop controller is required to design to shift the eigenvalue to the negative direction. The step response of the system is calculated to find out the gains of the PPC controller. The settling time of the step response is constraint √to be within√in 2 s. Therefore, the desired poles are selected as [−10, −10, −2 +j 3, −2 − j 3]. PPC controller gains are found to be KPPC = [−54.4218, −24.4898, 93.2739, 16.1633]. PPC with designed configurations is used to stabilize the inverted pendulum and both experimental and simulation are compared. To decide the initial position for φ for stabilization, the pendulum lifted in anti-clock direction manually to area of unstable equilibrium (±20◦ from the upper reference point). It is found that the controller starts work at −19.5 and stabilize the inverted cart pendulum at upper equilibrium point in 2 s. The overshoot value for the experimental results found to be 5° (Ref. Fig. 2a). Figure 2b depicts the displacement of cart to stabilize
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the pendulum with PPC. It has found that the 0.15 m overshoot distance is taken by cart from reference point to stabilize the system. The whole system stabilizes within 2 s. In analytical results pendulum and cart overshoots to 11° and 0.3 m for the stabilization of system, which is higher as compared to experimental results. The gap between analytical and experimental results is due to the real-time activation of linear controller. Figure 2c depicts the control input comparison between analytical and experimental results. The control input magnitude found to be 50 units for analytical study and 48 units for experiment. The maximum input magnitude of both analytical and experimental results is found to be near, which validate the mathematical modeling of the system. To develop LQR controller for the present system, the Q and R matrices are decided manually by hit and trail in order to settle the system with 2 s. The gain matrix for LQRC is calculated as K LQRC = [−31.628, −20.1507, 72.718, 13.152] using Eqs. (16)–(18). To stabilize the pendulum at upright position, the next test run is executed with LQR controller, in the same manner, as explained earlier with Pole Placement controller. The pendulum stabilizes at its inverted position with a settling time of 2 s and a marginal overshoot difference of 6° between experimental and analytical results, as shown in Fig. 3a. It can be observed from Fig. 3b that the maximum overshoot value for cart displacement is 0.17 m for analytical results and 0.12 m for experimental results. LQR controller is able to stabilize the cart to its desired position within 2 s. The comparison of control input is shown in Fig. 3b, where the maximum control input magnitude is found as 25 for the experimental run and 20 for the simulation run. The comparison between two controllers is discussed. Both the controllers can stabilize the cart-pole system. From the experimental and analytical results, it is observed that the settling time for both controllers is near 2 s for stabilization. For
Fig. 3 a Pendulum, b cart displacement and c control input using LQR
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Fig. 4 Comparison between PPC and LQRC for a control inputs and b pendulum position at stabilize equilibrium
the case studied, the control input magnitude by both the controllers from experimental results is compared in Fig. 4a. From the results, it is found that the maximum control input magnitude for stabilization of inverted pendulum using Pole Placement controller is 48 units, whereas, it is found as 25 units with the LQR controller. Hence, LQR controller performs better than the Pole Placement controller, as it has resulted in 47.9% reduction in maximum control input. Further, experimental results show that the pole is oscillating marginally around its upward position. It can be seen from Fig. 4b that the pendulum oscillates in the range of −0.6° to 0.6° using LQR controller and between −1.35° and 0.6° using Pole Placement controller. Therefore, low oscillations motion was observed in case of LQR controller as comparison with Pole Placement controller. In other words, the LQR controller results in 55.5% reduction in the oscillation of the inverted pendulum.
6 Conclusions This paper has investigated the real-time control of cart-pole system, which is underactuated in nature. Two linear controllers (PPC and LQRC) has been used to solve this problem. Both the controllers have been designed to stabilize the system within 2 s. The actuator dynamics has been incorporated in the mathematical model to ensure the accuracy of the analytical model, which is validated through the close agreement of the simulation results with the experimental results. Control input magnitude by both controllers to the system is compared, which results in the LQRC outperforming the PPC as it results in 47.9% reduction in maximum control input magnitude and 55.5% reduction in the oscillations of the inverted pendulum at unstable equilibrium position.
References 1. Mason P, Broucke M, Piccoli B (2008) Time optimal swing-up of the planar pendulum. IEEE Trans Autom Control 53(8):1876–1886
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2. Yang J, Shim S, Seo J, Lee Y (2009) Swing-up control for an inverted pendulum with restricted cart rail length. Int J Control Autom Syst 7(4):674–680 3. Mus N, Tovornik B (2006) Swinging up and stabilization of a real inverted pendulum. IEEE Trans Industr Electron 53(2):631–639 4. Solihin MI, Akmeliawati R (2010) Particle swam optimization for stabilizing controller of a self-erecting linear inverted pendulum. Int J Electr Electron Syst Res 3(June):13–23 5. Chaturvedi NA, Mcclamroch NH, Bernstein DS (2008) Automatica stabilization of a 3D axially symmetric pendulum. Automatica 44:2258–2265 6. Googoltech (2016) Inverted pendulum experimental manual suitable for GLIP Series 7. Song Z, Song X, Liu C, Zhao Y (2013) Research on real-time simulation and control of linear 1-stage inverted pendulum. J Comput 8(4):896–903 8. Roshdy AA, Wang T (2012) Stabilization of real inverted pendulum using pole separation factor. In: Proceedings of 2012 international conference on mechanical engineering and material science (MEMS). (Mems), pp 711–715 9. Singla A, Singh G, Virk GS (2016) Matlab/simMechanics based control of four-bar passive lower-body mechanism for rehabilitation. Perspect Sci 8:351–354 10. Singh G, Singla A, Virk GS (2016) Modeling and simulation of a passive lower-body mechanism for rehabilitation. In: Conference on mechanical engineering and technology (COMET-2016), IIT (BHU), Varanasi, India 11. Singla A, Singh G (2017) Real-time swing-up and stabilization control of a cart-pendulum system with constrained cart movement. Int J Nonlinear Sci Numer Simul 18(6):525–539 12. Singh G, Singla A (2017) Modeling, analysis and control of a single stage linear inverted pendulum. In: 2017 IEEE international conference on power, control, signals and instrumentation engineering (ICPCSI), pp 2728–2733 (2017)
Effective Education Using a 2-DOF Five-Bar Mechanism Shourie S. Grama, Prithvi Bharadwaj Mellacheruvu, S. Prasanth, V. Prathosh Kumar, S. Vignesh, and Rajeevlochana G. Chittawadigi
Abstract Innovative methods to teach kindergarten students are on the rise which use virtual environment like a mobile phone or a tablet. However, the students who get great joy in using these gadgets are moving away from actual writing on a paper or slate. In this paper, the authors have proposed a simple mechanism which can be controlled to draw a shape, a numeral, or a letter. It consists of a five-bar mechanism which has 2 degrees-of-freedom (DOF) and can control the position of the end-effector (EE). The novelty of the proposed mechanism is that it has an iris mechanism at the EE. The student has to hold a pen/pencil and start following the motion of the EE, which draws as per the command from a computer. The pen/pencil has a contact patch and if it touches the inner circle of the iris mechanism, the computer notes it down and the student loses a point. The iris mechanism can be opened and closed to have different difficulty levels for the students, thus making its usage more challenging. A working prototype was tested with few children, and based on their feedback, a newer version is being developed and has also been reported in this paper. Keywords Five-bar mechanism · Writing mechanism · Kindergarten education
1 Introduction In earlier days, teachers used to hold the wrist of the students in the entry level (now kindergarten students) and guide them on writing shapes, numerals, or alphabets on slates by using chalks. Though this tradition still exists in rural India, most of the schools in urban areas have resorted to books which have dotted lines on which students have to write and practice. Though it is convenient for teachers, the students tend to take more time to master the art of tracing on dotted line. Also with the exposure of smartphones and tablets, a plethora of mobile applications are available S. S. Grama · P. B. Mellacheruvu · S. Prasanth · V. Prathosh Kumar · S. Vignesh · R. G. Chittawadigi (B) Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_132
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which let students trace letters on the screen of the gadgets using their finger. Some of these applications also have a game element to it to motivate the students to perform better. The authors have tried to bring together the aspects of writing on a paper and a game environment in the proposed mechanism. Mechanisms to write on a paper/surface have been developed by a number of researchers. Since the end-effector of a mechanism has to trace a curve in 2D space, 2 degree-of-freedom (DOF) mechanisms can achieve this. They can be broadly classified as open-loop and closed-loop mechanisms. Open-loop mechanisms can be in the form of 2R (R: revolute) or 2P (P: prismatic) planar mechanisms. The forward kinematics in such mechanisms is straight forward but the inverse kinematics usually has multiple solutions. Though the construction of these mechanisms is simple, they are prone to inaccuracies due to lesser rigidity and bending of links with time. An example of a serial mechanism used for writing is reported in [1] and a commercially available 2P plotter is AxiDraw [2]. Closed-loop mechanisms can also be used to write on a paper. These have unique inverse kinematics but might have multiple solutions for forward kinematics. Though they have better rigidity, they have lesser range or workspace as compared to a serial chain of similar dimensions. A widely used example of a closed-loop mechanism for writing is a 2-DOF pantograph or five-bar mechanism. Collaborative Haptics and Robotics in Medicine (CHARM) Laboratory at Stanford University has a dedicated Web resource on the kinematic analysis of the five-bar mechanism [3]. The five-bar mechanism was used in [4] to write letters by taking input from the user, by creating a human–machine interface (HMI). Another category of mechanisms uses cables for driving the motion of the end-effector link. H-Man cable-driven mechanism is one such variant [5]. In this paper, the authors have used a 2-DOF five-bar mechanism with some modifications at the end-effector. These modifications include using an iris mechanism and letting the student trace the shape/letter/numeral by keeping a pen/pencil within the inner circle of the iris mechanism. The details of the mechanism, the integration of the system, and results are discussed in the following sections.
2 Kinematics of Five-Bar and Iris Mechanism A five-bar mechanism consists of five links of which one is grounded. The other four moving links are connected as a closed loop and have five revolute joints, as shown in Fig. 1a. The degree-of-freedom (DOF) of the mechanism can be determined as 2 using the Grubler–Kutzbach equation as DOF = 3(N − 1) − 2P1 − P2
(1)
where N = number of links, i.e., 5; P1 = number of joints with 1 relative DOF, i.e., 5; and P2 = number of joints with 2 relative DOF, i.e., 0. Though any two joints can be chosen to be driven or active joints, the joints on either side of the fixed link, i.e.,
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O
S X Y R
P
Q T (a) Five-bar mechanism
(b) Iris mechanism
Fig. 1 Mechanisms used in the proposed methodology
J1 and J5 , are used as it enables the placement of the motors on the fixed link, thus reducing the moving mass of the mechanism. The link length of links is a1 through a5 . The Link 3 is extended by a3 beyond Q to allow fixing of an iris mechanism at point T. A mathematical model of the mechanism, known as the kinematic model, exists to correlate the joint angles at joint J1 and J5 , i.e., θ1 and θ5 , and the position of the point T, as explained next. Iris mechanism on the other hand has one link that can be fixed and several curved links connected to it through revolute joints. The curved links have an extrusion which go inside slots of another floating part. The geometry is such that floating part forms a virtual revolute joint with the fixed link. This is due to the synchronous motion of the curved links with respect to the central axis. Hence, by controlling the movement of the floating component, the radius of the circle (polygon) formed by the curved links can be changed. In Fig. 1b, the circular disk is the fixed link and the curved links are colored. The floating part is not shown for better visualization. In this paper, the iris mechanism is used in a novel way, as explained later. Forward Kinematics: For the given joint angles θ1 and θ5 , the determination of the coordinates of point Q and thereafter point T is known as forward kinematics. Two solutions of the point Q exist for given values of θ1 and θ5 . This can be visualized as the intersection of two circles with radii a2 and a3 , centered at P and R, respectively. One of the intersection points is the Q as shown in Fig. 1a. The other solution is the other intersection points of the two circles. The formulations are explained in [6]. Inverse Kinematics: Determination of the angles θ1 and θ5 for joints J1 and J5 for a given location of point Q (or T ) is known as inverse kinematics. If the coordinates of T are known, one can solve the inverse kinematics by drawing a circle centered at T with (a3 + a3 ) as radius and another circle with a4 as radius and S as its center. These two circles intersect at two points (hence two solutions of θ5 ). Of these two solutions, one of them would be point R in Fig. 1a. Q point can be determined using the values of a3 and a3 . To determine point P, draw a circle with radius a2 centered
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at Q and another circle with radius a1 centered at O. Similar to determination of point R, two solutions exist for point P. Hence, four solutions (combination of two for θ5 and corresponding two solutions for θ1 ) exist, of which one solution has to be selected such that the mechanism does not pass through a singularity condition. Singularity is a serious problem in closed-loop mechanism which has to be dealt with care [7]. Due to space constraint, the kinematic formulations and the singularity determination are not included in this paper. Motion Planning: For drawing a shape, letter or a numeral, a start-point is provided for which inverse kinematics is performed to move the mechanism to that configuration. Between the start-point and an end-point, several via-points can be defined. Between any two points, linear interpolation is done and for each of the intermediate points, inverse kinematics is performed. From the multiple solutions, the one that has least deviation from the current active joint angles has to be selected and moved.
3 Prototype I The links of the five-bar mechanisms were created using aluminum sheet metal, as shown in Fig. 2a. The joints J1 and J5 were powered using two servomotors. These motors were rigidly fixed to the ground and the distance between the axes of the motors acted as dimension a5 . A readymade iris mechanism was procured and attached at the end of the Link 3. The iris mechanism had a lever, rotation of which caused the opening and closing of the inner curved metallic links, thus changing the radius of the circle inside the iris mechanism. The servomotors were connected to an Arduino Uno controller board, as shown in Fig. 2b. The inner links of the iris mechanism were connected to one of the digital
(a) Physical prototype with iris mechanism
Fig. 2 Prototype I developed using servomotors
(b) Circuit diagram
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Fig. 3 Numerals drawn using a pen clamped in iris mechanism to check the motion trace
pin of Arduino board and a pen with a metallic strip on its body was connected to a digital input. Whenever a contact is made between the metallic strip and the iris links, the circuit is completed and an LED is connected to show the output as a contact. The Arduino is connected to a computer through a USB cable. On the computer, a client application was developed using Visual C#, which would send out the angles (θ1 and θ5 ) to the Arduino board through serial port communication. Using the inverse kinematics formulations discussed earlier, motion planning was performed in the client application for each of the numerals (0–9). To test the working, a pen was grasped tightly in the iris mechanism and the mechanism was controlled from the computer. The drawing of the pen for few of the numerals is shown in Fig. 3.
3.1 Effective Education Using Proposed Mechanism The kids enjoy to play games and through the proposed mechanisms, and the authors are trying to integrate learning with a gaming element. For any given numeral (decided by the computer), the end-effector/iris mechanism moves to trace the selected numeral. The user (a student) has to hold the pen (with a metallic strip on its body) and has to follow the iris mechanism. If there is a contact between the pen with the inner circle of the iris, it is noted as a penalty or error by the user. In the initial trials, the user has a tendency to make more mistakes, and for the repeated trials, the number of errors should reduce. In addition to the above feature, the iris mechanism also allows the user to select different levels of complexity. If the inner radius of the iris is more, the user has more tolerance to make an error and with smaller radius, and the error tolerance is strict. Hence, there are multiple ways in which the performance of a user can be modified. The prototype developed was tested over 5 students in the age group of 7–12 years. The number of errors made by students for fully open (big ring) and semi-open (small ring) conditions for five trials was noted down. It was observed that for the big ring, the number of errors with usage was reducing. However, with the small ring, the number of errors did not have a direct correlation with the usage. This could be attributed to the inherent issue of the servomotors lacking good resolution. Sample results of two students, Person 1 and Person 2 (both 8-year-old female writing numeral 5), are shown in Fig. 4a, b, respectively.
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(b) Person 2
(a) Person 1
Fig. 4 Game results: left column is for big ring and right column is for small ring
4 Prototype II The prototype explained in the previous section had servomotors with limited resolution. Also, the workspace available for writing was restricted to a small region. A new software application was developed in Visual C# to simulate the writing of different numerals and letters, and this is illustrated in Fig. 5a. Also, in the newer version of the prototype, high resolution DC stepper motors are being used. To have an increased area of the workspace [8], the length of the fixed link was made zero such that the axis of the joints J1 and J5 are collinear, as shown in Fig. 5b. The links and connectors available in MAKIT toolkit were used to develop the prototype. The integration of the software application (developed using Visual C#) and the hardware (mechanisms) are underway. The pedagogy results with more users shall be reported in the due course.
(a) Simulation software writing numeral 9
(b) Physical prototype II
Fig. 5 Software and hardware for prototype II version
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5 Conclusions In this paper, five-bar and iris mechanisms have been used to develop a novel methodology to provide effective education. The letters and numerals to be learnt are fed into the mechanism to trace and the user has to follow the iris mechanism attached on one of the links of the mechanism. Whenever the user’s pen touches the inner circle of the iris mechanism, error is noted, thus bringing in the elements of gaming and education together. A prototype was developed and was used by a few students. Based on the feedback and observations, a new prototype is being developed which shall be evaluated and reported in the due course.
References 1. Tresset P, Leymarie FF (2013) Portrait drawing by Paul the robot. Comput Graph 37(5):348–363 2. AxiDraw: Website accessed in Jan 2018. https://www.axidraw.com/ 3. CHARM Lab Project Page: Website accessed in Jan 2018. https://charm.stanford.edu/ME327/ JaredAndSam 4. Campolo D, Tommasino P, Gamage K, Klein J, Hughes CM, Masia L (2014) H-Man: a planar, H-shape cabled differential robotic manipulandum for experiments on human motor control. J Neurosci Methods 235(1):285–297 5. Can FC, Sen H (2017) Real time controlled two dof five bar robot manipulator. In: Proceedings of the AzC-IFToMM international symposium of mechanism and machine science 6. Hampali S, Chittawadigi RG, Saha SK (2015) MechAnalyzer: 3D model based mechanism learning software. In: 14th World Congress in Mechanism and Machine Science 7. Zhao L, Yen AKW, Coulombe J, Bigras P, Bonev IA (2014) Kinematic analyses of a new medical robot for 3D vascular ultrasound examination. Trans Canad Soc Mech Eng 38(2):227–239 8. Koul MH, Rabinowitz D, Saha SK, Manivannan M (2013) Synthesis and design of a 2-DOF haptic device for simulating epidural injection. In: 13th World Congress in Mechanism and Machine Science
Renewable Energy System Using Thermoelectric Generator (RESTEC) Ritwik Dhar, Param Shah, Parth Kansara, and Niti Doshi
Abstract Continuous variable transmission (CVT) systems are widely used in allterrain vehicles (ATV). Over long runs, the sheaves of CVT generate sufficient amount of heat energy and get damaged due to friction between the sheaves and belt mainly due to clutch slippage during acceleration and deceleration. This further deteriorates the condition and performance of the vehicle, and breakdown is also discovered as a worst-case scenario. A novel method has been developed and implemented by creating a renewable energy system to maintain the temperature of CVT and improve longevity. The renewable energy system makes use of thermoelectric generators (TEG) working on the principle of Seebeck effect to utilize the heat dissipated from the exhaust chamber of engine which comprises of 80% of the waste energy of the ATV and is used for cooling the CVT. Keywords CVT · TEG · RPM · Renewable energy system · ATV
1 Introduction A continuously variable transmission (CVT) is a transmission that can change through an infinite number of effective gear ratios between maximum and minimum values [1]. One of the unique features of a CVT allows the driving shaft to maintain a constant angular velocity over a range of output velocities. Better fuel efficiency can be obtained due to such features as compared to other transmission systems by allowing the engine to run at its most efficient revolutions per minute (RPM) for a range of speeds. This infinite set of gear values accounts for large selection of ratios. R. Dhar (B) · N. Doshi Electronics and Telecommunication Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai 400056, India P. Shah Mechanical Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai 400056, India P. Kansara Information Technology Engineering, Dwarkadas J. Sanghvi College of Engineering, Mumbai 400056, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_133
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Common in internal combustion engines is the conservation of fuel at high speeds, and ease of shifting to torque peak or power peak when needed according to the terrain. Heat generation is not desirable for the components inside CVT housing [2]. Material properties of the components depend on temperature. It changes thermophysical properties on material. Temperature has an impact on strength/elasticity of drive belt. Belts tend to lose its elastic property at higher temperature. More slippage will occur due to less tension in belt, giving rise to loss in transmission efficiency. Due to increase in temperature, thermal stresses develop in the components which reduce its service life. To overcome this, especially in continuous runs spanning over a long period of time where breakdown might become an inevitable option, a design model was proposed and its efficiency was experimented. A Gaged GX9 CVT is used for the experimental analysis which is used on an all-terrain vehicle that competes for the annual BAJA SAE competition. The model follows a basic two-step energy conversion from heat energy to electrical energy which is converted to cool air with the help of a Peltier thermoelectric generator.
2 Previous works 2.1 CVT Heating Dhongde et al. [3] performed an experiment to map the temperature from CVT by placing thermocouple sensors at various locations inside the CVT housing and measuring CVT components temperatures with the help of IR sensor as shown in Fig. 1. This would help in better understanding the airflow [4] and also help in Fig. 1 Sensor placement
Renewable Energy System Using Thermoelectric Generator (RESTEC) Table 1 Placement information
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IR Temperature sensor 1. Primary sheave 2. Secondary sheave Thermocouple sensor 3. Driver region 4. Central region 5. Driven region
optimum positioning of the thermoelectric generators further discussed in Sect. 3 (Table 1). A similar test was conducted in dynamic condition while the all-vehicle was accelerated on a straight patch until the CVT goes to full shift range. The CVT components temperatures were measured using the contactless IR sensor MLX906014, and the airflow temperature around pre-determined regions in the CVT housing was measured using air temperature RTD probe. The test results of the CVT component temperature at various instances of time during continuous runs are shown in Fig. 2. It was observed that the component temperature increased at a much higher rate. The driver region, driven region and the central region also showed considerable increase in temperature with respect to time. Therefore, it was targeted to reduce and prevent the sheaves from increase in temperature in order to increase the durability of belt and transmission subsystem.
2.2 Thermoelectric Generator A thermoelectric power generator is a solid-state device based on Seebeck/Peltier effect for its working [5, 6]. A thermoelectric power generator also known as Peltier module provides direct energy conversion from thermal energy into electrical energy based on Seebeck effect by creating a temperature difference between the junctions of dissimilar metals. The module working with charge carriers follows the basic laws of thermodynamics and closely resembles the working of a conventional heat engine. The Peltier effect occurs whenever electrical current flows through two dissimilar conductors; depending on the direction of current flow, the junction of the two conductors will either absorb or release heat. In the world of thermoelectric technology, semiconductors (usually Bismuth Telluride) are the material of choice for producing the Peltier effect because they can be more easily optimized for pumping heat [7]. Using semiconductor such as Bismuth Telluride, a Peltier module can be constructed in its simplest form which is soldered to electrically conductive material on each end. In this configuration, the second dissimilar material required for the Peltier effect is actually the copper connection paths to the power supply. Ismail and Wael [8] saw the major drawback of thermoelectric power generator is their relatively low conversion efficiency (typically ~5%). This has been a major
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cause in restricting their use in electrical power generation to specialized fields with extensive applications where reliability is a major consideration and cost is not. An easy of improving efficiency and longevity of thermoelectric generators is to keep the temperature difference between the two sides as low as possible which can be done with the help of heatsinks [9]. Figure 3 shows the heatsink used on the thermoelectric generators during competition to improve cooling.
3 RESTEC 3.1 Design The design basically follows a three-stage methodology where Stage 1 consists of heat-to-electric conversion, Stage 2 consists of cooling of CVT and recharging of battery, Stage 3 consists of the feedback mechanism updating the sensor subsystem. STAGE 1: The aim of the first stage is to successfully utilize the excess heat energy that is generated from the engine with the help of thermoelectric generators. The engine taken into consideration is a Briggs and Stratton 305 cc model. A set of two pairs of thermoelectric generators are placed in contact with the exhaust chamber of the engine with the help of thermal adhesive to improve the heat transfer between the two bodies. Two pairs of thermoelectric generators were chosen as a trial setup which gave good results and were also placed easily with the surface area available on the top face of exhaust chamber. With the help of heatsink calculations from Sect. 2.2, the hot side of the thermoelectric generator is placed with a heatsink using thermal adhesive. The thermoelectric generators here work on Seebeck or reverse Peltier effect to use the heat energy from engine in creating a temperature difference between the sides of the generator. The electric energy produced here is passed on to a power electronics circuit in order to amplify the current, maintain a stable input for the second stage of the design and remove any surge currents from passing onto further components. The power electronics circuit basically consists of a current amplifier and voltage regulator and battery isolator. STAGE 2: The experiment discussed in Sect. 2.1 comes into use in the placement of the thermoelectric generators used for cooling of the continuously variable transmission (CVT). The sets of thermoelectric generators used here work on Peltier effect as the electricity produced from the modules in Stage 1 is used as input after amplification to the modules on the CVT housing for cooling of the system. The modules can handle up to 80 °C temperature difference without the semiconductors inside getting damaged. Therefore, a maximum temperature difference of around 50–60 °C is to be maintained so that the modules are not damaged as well as also prolonging the lifetime of the modules. The lesser the temperature difference, the more will be the performance of the modules, i.e., more efficient cooling. A constant output of 5 V
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at 1.5 A is to be maintained which provided sufficient results after trial runs with variations of V max and I max of modules over a span of 1 h. The electricity generated is used as power source for cooling of CVT but it also has another application, i.e., recharging of lithium polymer (LiPo) battery which is used in the all-terrain vehicle as power source of brake light as well as other sensors. A battery isolator is used to reduce battery exhaustion with time. STAGE 3: The main goals of the design are completed within the first two stages as explained above. The third stage deals with improving the system to adapt to the condition of sensors as well as system to increase or decrease efficiency depending upon the dynamic state. The feedback loop is defined as shown in Fig. 4, which makes use of a microcontroller unit that reads the incoming sensor readings to analyze the current state, i.e., the temperature of CVT components and LiPo battery level. A microcontroller unit is used which regulates the power supply input given to Stage 2 thermoelectric generator modules. The contactless IR temperature sensor is used for measuring the CVT component temperature at the primary sheave and secondary sheave. A voltage detector is used as feedback from LiPo battery to the microcontroller unit. This voltage detector sensor feeds when the battery is charged above 70% to microcontroller unit, which on receiving this data stops the charging and all the electricity generated is focused on CVT cooling. Similarly, the contactless IR sensor at sheaves of CVT continuously monitors the temperature of sheaves in real time, and if the CVT is running in the proper temperature range, the electricity generated is shifted toward battery recharging only. Although during the test runs, this case would never occur as the temperature of CVT needed to be constantly maintained within the limited temperature range as it would heat up instantly after long runs.
3.2 Implementation The RESTEC design makes use of TEC1-12706 Peltier modules for thermoelectric generation which has a maximum operating temperature of 138 °C. The temperature of various parts of the engine was checked using contactless IR temperature sensor, and the top face of exhaust chamber was chosen as suitable option for placement as it did not exceed beyond 120 °C at maximum. Along with help from data acquired from Fig. 2, the placement of TEC on the CVT was on spread over the top of primary sheave and central region as it gave comparatively better results by hit and trial method. An Arduino Uno was used as the microcontroller unit for making the system adaptive. Figure 5 shows the basic setup on the all-terrain vehicle.
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Fig. 2 Temperature versus time
Fig. 3 1. Heatsink implementation and 2. heatsink used
4 Observation RESTEC was successfully tested and gave decent results in test runs. The changes in hardware with respect to the feedback loop were initially slow for which changes in the algorithm were made to provide faster refreshing and response rate. A comparative analysis of the all-terrain vehicle in two conditions, i.e., with the model and without the model mounted was considered to determine the efficiency of the system at three positions. The three positions taken for readings were the temperature at primary sheave, secondary sheave using MLX906014 sensor, and region between the sheaves using thermocouple sensor as noted by region 4 (central region) in Fig. 1. Table 2 shows the temperature difference and output efficiency of
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Fig. 4 TEC placement (exhaust), TEC placement (CVT casing)
Fig. 5 System implementation on ATV
the system taken over a continuous period of 1 h of testing and noted at four intervals of 15 min.
3
72.11
85.83
92.68
98.91
15 minutes
30 minutes
45 minutes
60 minutes
94.24
87.52
79.33
69.26
62.66
55.47
46.21
39.40
Central region (°C)
94.81
87.9
81.02
66.37
Primary sheave (°C)
1
Secondary sheave (°C)
2
1
Primary sheave (°C)
Interval
With RESTEC
Without RESTEC
Table 2 System results during test run
90.0
83.34
74.72
63.93
Secondary sheave (°C)
2
59.86
51.76
42.32
35.45
Central region (°C)
3
4.1
4.78
4.81
5.74
1
4.24
4.18
4.61
5.89
2
2.8
3.71
3.89
3.95
3
Temperature difference (°C)
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5 Conclusion The system was considered successful after assessing the difference in temperature without its implementation. The all-terrain vehicle performed at annual BAJA SAEINDIA competition with the system incorporated and no CVT-related breakdowns occurred during the competition. The system was also presented at the Innovation Design Competition at BAJA SAEINDIA and was awarded the first prize out of 120 teams performing from all over the country. Table 2 gives the test readings of the system. Although the system is successful in accomplishing its main objective, it can be seen that the system can be seen deteriorating linearly with time. This may be due to the continuous working of semiconductors under harsh conditions. Further works to improve the overall system would involve working on custom thermoelectric generators with different semiconductor materials to improve maximum operating temperature without affecting longevity and also a more packed transmission housing to maintain a distinct temperature enclosure.
References 1. Aaen O (2011) Olav Aaen’s clutch tuning handbook. Racine, WI, AAEN performance 2. Wurm J, Fitl M, Gumpesberger M, Väisänen E, Hochenauer C (2017) Advanced heat transfer analysis of continuously variable transmissions (CVT). Appl Thermal Eng 114:545–553. ISSN 1359-4311 3. Dhongde SV, Chandran V (2014) Experimental study of cooling of continuously variable transmission (CVT) in scooter. https://doi.org/10.4271/2014-01-2003 4. Vaishya A, Phadnis S (2013) Experimental investigations of forced air cooling for continuously variable transmission (CVT). SAE Technical Paper 2013-32-9073 5. ElCosnier W, Gilles M, Lingai (2008) An experimental and numerical study of a thermoelectric air-cooling and air-heating system. Int J Refrig 31:1051–1062 6. Jugsujinda S, Vora-ud A, Seetawan T (2000) Analyzing of thermoelectric refrigerator performance. In: Proceedings of the 2nd international science, social-science, engineering and energy conference, vol 25, pp 154–159 7. Yadav N, Meha N (2013) Review on thermoelectric materials and applications. Int J Sci Res Dev 1:413–417 8. Ismail BI, Ahmed WI (2010) Thermoelectric power generation using waste-heat energy as an alternative green technology. Recent Patents Electr Eng 2. https://doi.org/10.2174/187447611 0902010027 9. Chein R, Chen Y (2005) Performances of thermoelectric cooler integrated with microchannel heat sinks. Int J Refrig 28:828–839
Bending and Free Vibration Analysis of Exponential Graded FG Plate Using Closed-Form Solution Dheer Singh, Yogesh Kumar, and Ankit Gupta
Abstract This article is intended to study the bending and vibration response of the exponentially graded FGM plate using the closed-form solution. The governing equations of motions are formulated on the basis of Hamilton’s principle, whereas the Navier method has been used to solve the governing equation with simply supported boundary conditions. The results have been validated for non-dimensional deflection values of a square plate subjected to transverse loading. Apart from these, the exponential gradation model has been considered for calculating the effective material properties of a functionally graded plate and its mechanical properties are assumed to vary in the thickness direction. Bending analysis and non-dimensional frequency has been evaluated for various modes of vibration of functionally graded thin and thick plates. Keywords Composite Materials · Functionally graded (FG) plate · Bending analysis · Vibration analysis · Close-form solution · Hamilton’s principle · Navier solution
1 Introduction Functionally graded materials (FGMs) are the materials in which the properties of the materials vary exponentially about the thickness coordinates. Extensive literature has been published in the field of functionally graded material structures. Koizumi [1] introduced the concept of FGMs. Reddy [2] presented the closed-form solution to investigate the dynamic response of FGM plates based on the third-order shear deformation theory (TSDT). Abrate [3] proposed that the functionally graded (FG) plates can be treated like homogeneous plates after considering the suitable reference surfaces. Mantari and Guedes [4] investigated the flexural response of exponential graded plates using trigonometric higher-order shear deformation theory (HSDT). Thai and Kim [5] examined the bending and vibration characteristics of FG plates D. Singh · Y. Kumar · A. Gupta (B) School of Engineering, Shiv Nadar University, Gautam Buddha Nagar, Greater Noida, Uttar Pradesh 201314, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_134
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using HSDT. Sburlati [6] presented the structural response of FG multi-layered structures. Chakraverty and Pradhan [7] examined the vibration characteristics of exponential FG plates with various boundary conditions. Zenkour [8] investigated the effect of thermo-mechanical loading on structural response of exponentially graded thick plates. Ebrahimi and Mokhtari [9] analyzed exponentially graded Timoshenko beams. Gupta and Talha [10] provided an extensive literature review on the recent development on the structural responses of FG structures. Akavci [11] studied the mechanical behavior of FG sandwich plates resting on elastic foundations. Li et al. [12] investigated the thermal analysis of exponentially graded plates. Pandey and Pradyumna [13] analyzed FG sandwich plates using higher-order layer-wise theory. Zenkour and Alghanmi [14] employed refined quasi-3D shear normal and deformation theory to investigate the structural characteristics of functionally graded plates. Zenkour and Alghanmi [15] analyzed exponentially graded plates with piezoelectric composite actuators on elastic foundations. Mantari and Guedes [16] have investigated the exponentially graded plate using trigonometric higher-order shear deformation theory.
Fig. 1 Geometric coordinates of exponentially graded FGM Plates
Table 1 Material properties of functionally graded plates [5] S. No.
Properties
Metal
Ceramic
(SUS304)
(Al2 O3 )
(ZrO2 )
1
E (GPa)
208
380
200
2
ρ (kg/m3 )
8166
3800
5700
3
ν
0.3
0.3
0.3
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2 Mathematical Formulation The functionally graded plate of thickness h, length a and width b corresponding coordinates x, y and z are shown in Fig. 1 and Table 1 shows the different material properties of functionally graded plates.
2.1 Displacement Field The displacement field employed in the present study is expressed as [5]: ⎫ ⎧ ⎫ ⎪ ⎧ ∂ w˜ b ∂ w˜ s ⎪ u(x, ˜ y, t) − z − f (z) ⎪ ⎪ (x, y, z, t) u ˜ ∂x ⎬ ⎨ ⎨ 1 ∂x ⎬ ∂ w ˜ ∂ w ˜ b s u˜ (x, y, z, t) = v(x, ˜ y, t) − z ∂ y − f (z) ∂ y ⎪ ⎭ ⎪ ⎩ 2 ⎪ ⎪ ⎭ ⎩ u˜ 3 (x, y, z, t) w˜ (x, y, t) + w˜ (x, y, t) b
s
f (z) =
(1)
3
4z 3h 2
where h is the thickness of the plate, u˜ and v˜ represents the displacements in the x and y direction in midplane of functionally graded plate. w˜ b and w˜ s represent the bending and shear component of transverse displacements [5] ε˜ x = ε˜ y = γ˜x y = γ˜x z = γ˜yz =
∂ u˜ 1 ∂x ∂ u˜ 2 ∂y ∂ u˜ 1 ∂y ∂ u˜ 1 ∂z ∂ u˜ 2 ∂z
∂ u˜ ∂ 2 w˜ b 4z 3 ∂ 2 w˜ s = −z − 2 ∂x ∂x2 3h ∂x2 2 3 2 ∂ v˜ ∂ w˜ b 4z ∂ w˜ s = −z − 2 ∂y ∂ y2 3h ∂ y2
∂ u˜ 2 ∂ u˜ ∂ v˜ ∂ 2 w˜ b 8z 3 ∂ 2 w˜ s + = + − 2z − 2 ∂x ∂y ∂x ∂ x∂ y 3h ∂ x∂ y 2 ∂ u˜ 3 4z ∂ w˜ s + = 1− 2 ∂x h ∂x 2 ∂ u˜ 3 4z ∂ w˜ s + = 1− 2 ∂y h ∂y
(2)
Equation (2) shows the transverse shear strains (γ˜x z , γ˜yz ) are zero at the top (z = +h/2) and bottom (z = −h/2) surfaces of the plate.
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2.2 Energy Equations
A.
Strain Energy: Strain energy of FG plate can be given as: + 2 h
δU =
(σ˜ x δ ε˜ x + σ˜ y δ ε˜ y + σ˜ x y δ γ˜x y + σ˜ x z δ γ˜x z + σ˜ yz δ γ˜yz )dAdz A
(3)
− h2
Values of n, m and P are stress resultants are shown in Eq. (4) + 2 h
(n˜ x , n˜ y , n˜ x y ) =
σ˜ x , σ˜ y , σ˜ x y dz
− h2
+ 2 h
(m˜ bx, m˜ by, m˜ bx y, ) =
σ˜ x , σ˜ y , σ˜ x y zdz
σ˜ x , σ˜ y , σ˜ x y f (z)dz
− h2
+ 2 h
(m˜ sx, m˜ sy, m˜ sx y, ) = − h2 + h2
P˜x y =
4z 2 1 − 2 σ˜ x y dz h − h2
+ 2 4z 2 1 − 2 σ˜ yx dz P˜yz = h h
(4)
− h2
B.
Work done δV : Work done can be given as: pδ(w˜ b + w˜ s )dA
δV = −
(5)
A
C.
Kinetic Energy δ K : Kinetic energy variations are shown in Eq. (6) δK = V
(u˙˜ 1 δ u˙˜ 1 + u˙˜ 2 δ u˙˜ 2 + u˙˜ 3 δ u˙˜ 3 )ρ(z)dAdz
(6)
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ρ(z) Represents mass density, Ii represent mass inertias in Eq. (7) + 2 h
(I0, I1, I2, I3, I4, I6 ) =
1, z, z 2 , z 3 , z 4 , z 6 ρ(z)dz
(7)
− h2
D.
Hamilton’s Principle: Governing equation for the structural response of FG plates can be given as: T (δU +δV − δ K )dt
0=
(8)
0
In this equation, δU represent strain energy, δV represent work done by external forces and δ K represent of kinetic energy. When substituting all the value of δU , δV and δ K in Eq. (8). Integrating Eq. (9) by parts, after that collected δ u, ˜ δ v, ˜ δ w˜ b and δ w˜ s coefficients then willget the governing equations of motion are shown in Eq. (10). Where c = 3h4 2 ;
⎧ ∂ 2 δ w˜ b ∂ 2 δ w˜ s ∂δ u˜ b s ⎪ n ˜ × − m ˜ × − m ˜ × x ⎪ 2 2 x x ∂ x ∂ x ∂ x ⎪
⎪ ⎪ 2 2 ⎪ ⎪ + n˜ y × ∂δv˜ − m˜ by × ∂ δw2˜ b − m˜ sy × ∂ δw2˜ s ⎪ ∂y ⎪ ∂y
∂y
⎪ ⎪ ⎪ ∂ 2 δ w˜ b ∂ 2 δ w˜ s ∂δ u˜ ∂δ v˜ b s ⎪ + n ˜ − 2 m ˜ + × − 2 m ˜ xy × x y ⎪ x y ∂ y ∂ x ∂ x∂ y ∂ x∂ y ⎪
⎪ ⎪ ⎪ ∂δ w˜ s ∂δ w˜ s ⎪ + P × × + P ⎪ x z yz ∂x ∂y ⎪ ⎪ ⎪ + w ˜ ) − pδ( w ˜ ⎪ ⎪ b s ⎨ ˜˙ u˜˙ + vδ ˜˙ v˜˙ + (w˜˙ b + w˜˙ s )δ(w˜˙ b + w˜˙s )] −I0 [uδ ⎪ ˙ ˙ ˙ ∂δ w˙˜ b ⎪ ˙ +I1 u˜ ∂ x + ∂∂w˜xb δ u˙˜ + v˙˜ ∂δ∂wy˜ b + ∂∂w˜yb δ v˙˜ A ⎪ ⎪ ⎪ ˙ ⎪ ⎪ ˙ ˙ ˙ ⎪ ⎪ −I2 ∂∂w˜xb × ∂δ∂wx˜ b + ∂∂w˜yb × ∂δ∂wy˜ b ⎪ ⎪ ⎪ ⎪ ⎪ ˙˜ ∂δw˙˜ s + ∂ w˙˜ s δ u˙˜ + v˙˜ ∂δw˙˜ s + ∂ w˙˜ s δ v˙˜ ⎪ u +cI 3 ⎪ ∂x ∂y ∂y ⎪ ∂x ⎪ ⎪ ⎪ −cI ∂ w˙˜ b × ∂δw˙˜ s + ∂ w˙˜ s × ∂δw˙˜ b + ∂ w˙˜ b × ∂δw˙˜ s + ∂ w˙˜ s × ⎪ 4 ∂x ⎪ ∂x ∂x ∂x ∂y ∂y ∂y ⎪ ⎪ ⎪ ⎩ −c2 I ∂ w˙˜ s × ∂δw˙˜ s + ∂ w˙˜ s × ∂δw˙˜ s 6 ∂x ∂x ∂y ∂y
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ dA = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂δ w˙˜ b ⎪ ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎭ (9)
∂ w¨˜ s ∂ w¨˜ b I0 u¨˜ + I1 u¨˜ − cI3 ∂x ∂x
¨˜ b ¨˜ s ∂ n˜ y ∂ n˜ x y ∂ w ∂ w + = I0 v¨˜ − I1 − cI3 δ v˜ = ∂x ∂y ∂y ∂y
δ u˜ =
∂ n˜ x ∂ n˜ x y + ∂x ∂y
=
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∂ 2 m˜ by ∂ 2 m˜ bx y ∂ 2 m˜ bx +p δ w˜ b = + +2 ∂x2 ∂ y2 ∂ x∂ y ¨˜ ¨˜ ∂ u ∂ v 2 2 = I0 (w¨˜ b + w¨˜ s ) + I1 + − I2 ∇ w¨˜ b − cI4 ∇ w¨˜ s ∂x ∂y ∂ 2 m˜ sy ∂ 2 m˜ sx y ∂ 2 P˜yz ∂ 2 m˜ sx ∂ 2 P˜x z + + +p + +2 δ w˜ s = ∂x2 ∂ y2 ∂ x∂ y ∂x ∂y ¨˜ ¨˜ ∂ u ∂ v 2 2 2 = I0 (w¨˜ b + w¨˜ s ) + cI3 + − cI4 ∇ w¨˜ b − c I6 ∇ w¨˜ s ∂x ∂y
(10)
2.3 Constitutive Equations Exponential law distribution is used to calculate Young’s modulus E (z) shown in Eq. (11) 2z E(z) = E c e−δ1 k (1− h )
2z ρ(z) = ρc e−δ2 k (1− h )
(11)
where δ1 =
Ec ρc 1 1 , δ2 = ln . ln 2 Em 2 ρm
In the FG plate E is the Young’s modulus and parameter k which is indicates the material variation in the thickness direction. Em and Ec represents the Young’s modulus of the metal and ceramic, respectively [7] ⎫ ⎧ ⎡ ⎪ σ˜ x ⎪ 1v 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢v 1 0 ⎪ ⎬ ⎨ σ˜ y ⎪ ⎢ E(z) ⎢ σ˜ x y = ⎢ 0 0 1−v 2 ⎪ ⎪ 1 − v2 ⎢ ⎪ ⎪ ⎪ ⎣0 0 0 σ˜ x z ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ σ˜ ⎪ 00 0 yz
0 0 0 1−v 2
0
Substitute Eq. (2) into Eq. (12)
⎧ ⎫
∂ 2 w˜ b ∂ u˜ 4z 3 ∂ 2 w˜ s ⎪ − z − ⎪ ⎪ ⎤⎪ 2 2 2 ⎪ ⎪ ∂x ∂x 3h ∂ x
⎪ ⎪ ⎪ ⎪ 0 ⎪ ∂ v˜ 2 2 3 ⎪ ∂ w ˜ ∂ w ˜ 4z b s ⎪ ⎪ − z ∂ y 2 − 3h 2 ∂ y 2 ⎪ ⎪ ⎥ ⎪ ⎪ ∂ y 0 ⎥⎨ ⎬
2 2 3 ⎥ ∂ u˜ ∂ w˜ b ∂ w˜ s ∂ v˜ 8z (12) 0 ⎥ ∂ y + ∂ x − 2z ∂ x∂ y − 3h 2 ∂ x∂ y ⎪ ⎥⎪
⎪ 2 ⎪ ⎪ 0 ⎦⎪ ∂ w ˜ 4z ⎪ ⎪ ⎪ 1 − h 2 ∂ xs ⎪ ⎪ 1−v ⎪ ⎪ ⎪
⎪ ⎪ 2 2 ⎪ ⎪ ∂ w˜ s 4z ⎩ ⎭ 1 − h2 ∂ y
Bending and Free Vibration Analysis of Exponential Graded …
⎧ ⎫ ⎡ ⎤⎛ ⎧ ∂ u˜ 1v 0 ⎨ n˜ x ⎬ ⎨ ∂x ⎜ = ⎣ v 1 0 ⎦⎝d0 ∂∂ vx˜ n˜ y ⎩ ⎭ ⎩ ∂ u˜ n˜ x y 0 0 1−v + 2 ∂x ⎧ ⎫ ⎡ ⎤⎛ ⎧ ∂ u˜ b ⎪ 1v 0 ⎨ m˜ x ⎪ ⎬ ⎨ ∂x ⎜ m˜ by = ⎣ v 1 0 ⎦⎝d1 ∂∂ vx˜ ⎪ ⎩ ∂ u˜ ⎩ m˜ b ⎪ ⎭ 0 0 1−v + xy 2 ∂x ⎧ ⎫ ⎡ ⎤⎛ ⎧ ∂ u˜ s ⎪ 1v 0 ⎨ m˜ x ⎪ ⎬ ⎨ ∂x ⎜ m˜ sy = ⎣ v 1 0 ⎦⎝cd3 ∂∂ vx˜ ⎪ ⎩ ∂ u˜ ⎩ m˜ s ⎪ ⎭ 0 0 1−v + xy 2 ∂x
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⎫ ⎬
⎫ ⎪ ⎬
∂ v˜ ∂x
⎧ ∂ 2 w˜ b ⎪ ⎨ − ∂2x 2 + d1 − ∂∂ yw˜2b ⎪ ⎭ 2 ⎩ −2 ∂∂ x∂w˜ yb
⎧ ∂ 2 w˜ s ⎪ ⎨ − ∂2x 2 + cd3 − ∂∂ yw˜2s ⎪ ⎪ 2 ⎭ ⎩ −2 ∂∂ x∂w˜ ys ⎫ ⎪ ⎬
∂ v˜ ∂x
⎧ ∂ 2 w˜ ⎫ b ⎪ ⎨ − ∂2x 2 ⎬ ∂ w˜ b + d2 − ∂ y 2 ⎪ ⎭ 2 ⎩ −2 ∂∂ x∂w˜ yb
⎫ ⎪ ⎬
∂ v˜ ∂x
⎧ ∂ 2 w˜ ⎫ b ⎪ ⎨ − ∂2x 2 ⎬ ∂ w˜ b + cd4 − ∂ y 2 ⎪ ⎭ 2 ⎩ −2 ∂∂ x∂w˜ yb
⎧ ∂ 2 w˜ s ⎪ ⎨ − ∂2x 2 ∂ w˜ s + cd4 − ∂ y 2 ⎪ ⎪ 2 ⎭ ⎩ −2 ∂∂ x∂w˜ ys
⎫⎞ ⎪ ⎬ ⎟ ⎠ ⎪ ⎭ (13) ⎫⎞ ⎪ ⎬ ⎟ ⎠ ⎪ ⎭
⎧ ∂ 2 w˜ s ⎪ ⎨ − ∂2x 2 ∂ w˜ s 2 + c d5 − ∂ y 2 ⎪ ⎪ 2 ⎭ ⎩ −2 ∂∂ x∂w˜ ys
(14) ⎫⎞ ⎪ ⎬ ⎟ ⎠ ⎪ ⎭ (15)
˜ Px z P˜yz
( = As
10 01
) ∂ w˜ s ∂x ∂ w˜ s ∂y
(16)
where d0 , d1 , d2 , d3 , d4 , d5 and As are the stiffness coefficients defined in Eqs. (17a)– (17b) [5] + 2 h
(d0 , d1 , d2 , d3 , d4 , d5 ) = − h2
E(z) 1, z, z 2 , z 3 , z 4 , z 6 dz 1 − v2
+ 2 4z 2 E(z) s 1− 2 dz A = h 2(1 − v)
(17a)
h
− h2
Equations of Motion in Terms of Displacements are Shown in Eq. (18)
∂ 2 u˜ ∂ w˜ b ∂ w˜ s 1 − v ∂ 2 u˜ 1 + v ∂ 2 v˜ d0 − d1 ∇ 2 − cd3 ∇ 2 + + ∂x2 2 ∂ y2 2 ∂ x∂ y ∂x ∂x ∂ w¨˜ b ∂ w¨˜ s = I0 u¨˜ − I1 − cI3 ∂x ∂x
2 ∂ v˜ ∂ w˜ b ∂ w˜ s 1 − v ∂ 2 v˜ 1 + v ∂ 2 u˜ − d1 ∇ 2 − cd3 ∇ 2 d0 + + ∂ y2 2 ∂x2 2 ∂ x∂ y ∂y ∂y ∂ w¨˜ b ∂ w¨˜ s = I0 v¨ − I1 − cI3 ∂y ∂y
(17b)
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∂ u˜ ∂ v˜ + ∂x ∂y
− d2 ∇ 4 w˜ b − cd4 ∇ 4 w˜ s + p ∂ u¨˜ ∂ v¨˜ ¨ ¨ + = I0 (w˜ b + w˜ s ) + I1 − I2 ∇ 2 w¨˜ b − cI4 ∇ 2 w¨˜ s ∂x ∂y
˜ ∂ v˜ 2 ∂u + − cd4 ∇ 4 w˜ b − c2 d5 ∇ 4 w˜ s + As ∇ 2 w˜ s + p cd3 ∇ ∂x ∂y ¨˜ ¨˜ ∂ u ∂ v + = I0 (w¨˜ b + w¨˜ s ) + cI3 − cI4 ∇ 2 w¨˜ b − c2 I6 ∇ 2 w¨˜ s ∂x ∂y d1 ∇ 2
(18)
2.4 Analytical Solution Navier solution has been used to find the bending response of functionally graded plate, and initial solutions can be assumed as shown in Eq. (19). u(x, ˜ y, t) =
∞ * ∞ *
U˜ mn eiωt cos αx sin βy
m=1 n=1
v(x, ˜ y, t) =
∞ * ∞ *
V˜mn eiωt sin αx cos βy
m=1 n=1 ∞ * ∞ *
w˜ b (x, y, t) =
W˜ bmn eiωt sin αx sin βy
m=1 n=1
w˜ s (x, y, t) =
∞ * ∞ *
W˜ smn eiωt sin αx sin βy
(19)
m=1 n=1
√ where i = −1, α = mπ and β = nπ , angular frequency (ω). (U˜ mn , V˜mn , W˜ bmn and a b W˜ smn ) all are coefficients. Transverse load P has been expanded into double Fourier sine series as shown in Eq. (20) and in Eq. (21), P˜mn represents the load coefficient. p(x, y) =
∞ * ∞ *
P˜mn sin αx sin βy
(20)
m=1 n=1
The coefficients P˜mn are given below for the some typical loads P˜mn
4 = ab
a b p(x, y) sin αx sin βy 0
0
(21)
Bending and Free Vibration Analysis of Exponential Graded …
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Substitute Eqs. (20), (21) and (22) in motion Eq. (18), the analytical solution can be obtained from: K˜ Q˜ mn − ω2 M˜ = P˜mn (22) 4×4
4×4
4×1
4×1
Q˜ mn is the displacement coefficient matrix where (U˜ mn ,V˜mn , W˜ bmn and W˜ smn ) are coefficients of this matrix. Stiffness matrix coefficients and mass matrix coefficients of FG plates are calculated from the above Eq. (22). 1−v 1+v d0 β 2 , k12 = d0 αβ, k13 = −d1 α(α 2 + β 2 ), 2 2 1−v d0 α 2 + d0 β 2 , k23 = −d1 β(α 2 + β 2 ) = −cd3 α(α 2 + β 2 ), k22 = 2 = −cd3 β(α 2 + β 2 ), k33 = d2 (α 2 + β 2 )2 , k34 = cd3 (α 2 + β 2 )2 ,
k11 = d0 α 2 + k14 k24
k44 = c2 d5 (α 2 + β 2 )2 + As (α 2 + β 2 ). m 11 = m 22 = I0 , m 12 = 0, m 13 = −α I1 , m 14 = −αcI3 , m 23 = −β I1 , m 24 = −βcI3 , m 33 = I0 + I1 α 2 + β 2 , m 34 = I0 + cI4 α 2 + β 2 , m 44 = I0 + c2 I6 α 2 + β 2 .
3 Results and Discussion Firstly, the validation study has been carried out to confirm the accuracy of the present methodology. Table 2 shows the non-dimensional (ND) frequency of exponentially graded Al/Al2 O3 plates. The obtained results have been compared with the results given by Chakraverty and Pradhan [7]. Results obtained for verification purpose have been found to be in good agreement with previously published works. Table 2 Comparison of ND frequency of exponentially grade Al/Al2 O3 plate Exponential index (k)
0.1
0.5
1
a/h
b/a
HSDT [7]
Present
HSDT [7]
Present
HSDT [7]
Present
2
1
0.6362
0.6299
0.5194
0.5192
0.4011
0.4004
2
1.2775
1.2825
1.0441
1.0448
0.8086
0.8018
3
1.534
1.5291
1.254
1.257
0.9719
0.971
1
0.3602
0.3644
0.2949
0.2938
0.2295
0.2274
2
0.8325
0.8345
0.6819
0.6822
0.5319
0.5311
3
1.0345
1.0285
0.8459
0.8433
0.6601
0.6578
4
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Table 3 ND frequencies ω SUS304/Si3 N 4 material E c /E m
1
1.5
2
2.5
3
ND frequencies ω SUS304/Si3 N4 material
a/h
Mode
5
1(1, 1) 0.1501
0.1353 0.1256 0.1185 0.1131
2(1, 2) 0.3280
0.2954 0.2741 0.2587 0.2467
3(2, 2) 0.4743
0.4270 0.3963 0.3739 0.3566
1(1, 1) 0.0410
0.0370 0.0344 0.0325 0.0310
2(1, 2) 0.0978
0.0882 0.0819 0.0773 0.0738
3(2, 2) 0.1501
0.1353 0.1256 0.1185 0.1131
1(1, 1) 0.0105
0.0095 0.0088 0.0083 0.0080
2(1, 2) 0.0260
0.0234 0.0218 0.0206 0.0196
3(2, 2) 0.0410
0.0370 0.0344 0.0325 0.0310
1(1, 1) 0.0017
0.0015 0.0014 0.0013 0.0013
2(1, 2) 0.0042
0.0038 0.0036 0.0034 0.0032
3(2, 2) 0.0068
0.0061 0.0057 0.0054 0.0051
1(1, 1) 0.0004
0.0004 0.0004 0.0003 0.0003
2(1, 2) 0.0011
0.0010 0.0009 0.0008 0.0008
3(2, 2) 0.0017
0.0015 0.0014 0.0013 0.0013
10
20
50
100
In this section, the effect of Young’s modulus on ND frequencies for different modes of functionally graded square plate of SUS304/Si3 N 4 and SUS304/ZrO2 have been studied. The thickness of the FG square plate has been varied from moderately thin to thick plates for different Young’s modulus ratios. Reported results in Table 3 and Table 4 show that as the ratio of Ec /Em increases the ND frequency for different modes at different thickness ratios decreases. Reduction 15–20% in the ND frequency values has been reported when the Ec /Em ratio increased from 1 to 3. The effect of density ratios on ND frequencies for different modes of functionally graded square plate of SUS304/Si3 N4 and SUS304/ZrO2 has been analyzed. The thickness of the FG square plate has been varied from moderately thin to thick plates for different density ratios. Reported results in Tables 5 and 6 show that as the ratio of ρ c /ρ m increases the ND frequency for different modes at different thickness ratios decreases. Increment of 20–30% in the values of ND frequency has been reported when the Ec /Em ratio is increased from 1 to 3. The effect of Young’s modulus ratio on non-dimensional displacement in thickness direction has been reported for different thickness ratios. It has been found that the effect of Ec /Em ratio on the displacement is significant as shown in Table 7.
Bending and Free Vibration Analysis of Exponential Graded …
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Table 4 ND frequencies ω SUS304/ZrO2 material E c /E m
1
1.5
2
2.5
3
0.1738
0.1615
0.1526
0.1456
0.3802
0.3534
0.3339
0.3187
0.6095
0.5501
0.5114
0.4833
0.4614
1(1, 1)
0.0526
0.0475
0.0441
0.0417
0.0398
2(1, 2)
0.1255
0.1133
0.1053
0.0994
0.0949
3(2, 2)
0.1926
0.1738
0.1615
0.1526
0.1456
1(1, 1)
0.0135
0.0122
0.0113
0.0107
0.0102
2(1, 2)
0.0333
0.0301
0.0279
0.0264
0.0252
3(2, 2)
0.0526
0.0475
0.0441
0.0417
0.0398
1(1, 1)
0.0022
0.0020
0.0018
0.0017
0.0016
2(1, 2)
0.0054
0.0049
0.0046
0.0043
0.0041
3(2, 2)
0.0087
0.0078
0.0073
0.0069
0.0066
1(1, 1)
0.0005
0.0005
0.0005
0.0004
0.0004
2(1, 2)
0.0014
0.0012
0.0011
0.0011
0.0010
3(2, 2)
0.0022
0.0020
0.0018
0.0017
0.0016
1.5
2
2.5
3
a/h
Mode
ND frequencies ω SUS304/ZrO2 material
5
1(1, 1)
0.1926
2(1, 2)
0.4213
3(2, 2) 10
20
50
100
Table 5 ND frequencies ω SUS304/Si3 N 4 material ρ c /ρ m
1
a/h
Mode
ND frequencies ω SUS304/Si3 N4 material
5
1(1, 1)
0.1893
0.2089
0.2230
0.2341
0.2432
2(1, 2)
0.4143
0.4573
0.4883
0.5125
0.5324
3(2, 2)
0.5995
0.6619
0.7068
0.7418
0.7705
1(1, 1)
0.0517
0.0570
0.0609
0.0639
0.0664
2(1, 2)
0.1233
0.1361
0.1453
0.1525
0.1584
3(2, 2)
0.1893
0.2089
0.2230
0.2341
0.2432
1(1, 1)
0.0133
0.0146
0.0156
0.0164
0.0170
2(1, 2)
0.0327
0.0361
0.0385
0.0404
0.0420
3(2, 2)
0.0517
0.0570
0.0609
0.0639
0.0664
1(1, 1)
0.0021
0.0024
0.0025
0.0026
0.0027
2(1, 2)
0.0053
0.0059
0.0063
0.0066
0.0068
3(2, 2)
0.0085
0.0094
0.0100
0.0105
0.0109
1(1, 1)
0.0005
0.0006
0.0006
0.0007
0.0007
2(1, 2)
0.0013
0.0015
0.0016
0.0017
0.0017
3(2, 2)
0.0021
0.0024
0.0025
0.0026
0.0027
10
20
50
100
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Table 6 ND frequencies ω SUS304/ZrO2 material ρ c /ρ m
1
1.5
2
2.5
3
ND frequencies ω SUS304/ZrO2 material
a/h
Mode
5
1(1, 1) 0.2133
0.2352 0.2511 0.2634 0.2735
2(1, 2) 0.4668
0.5147 0.5491 0.5760 0.5980
3(2, 2) 0.6754
0.7446 0.7943 0.8331 0.8648
1(1, 1) 0.0583
0.0643 0.0686 0.0720 0.0748
2(1, 2) 0.1390
0.1533 0.1636 0.1717 0.1783
3(2, 2) 0.2133
0.2352 0.2511 0.2634 0.2735
1(1, 1) 0.0149
0.0165 0.0176 0.0185 0.0192
2(1, 2) 0.0369
0.0407 0.0434 0.0456 0.0473
3(2, 2) 0.0583
0.0643 0.0686 0.0720 0.0748
1(1, 1) 0.0024
0.0027 0.0028 0.0030 0.0031
2(1, 2) 0.0060
0.0066 0.0071 0.0074 0.0077
3(2, 2) 0.0096
0.0106 0.0113 0.0119 0.0123
1(1, 1) 0.0006
0.0007 0.0007 0.0007 0.0008
2(1, 2) 0.0015
0.0017 0.0018 0.0019 0.0019
3(2, 2) 0.0024
0.0027 0.0028 0.0030 0.0031
10
20
50
100
Table 7 Exponential gradation FGM plate deflection values for metal/ceramic E c /E m
1
1.5
2
2.5
3
a/h
Mode
Deflection values
5
SUS304/Si3 N4
0.0332
0.0406
0.0467
0.0521
0.0569
SUS304/ZrO2
0.0332
0.0406
0.0467
0.0521
0.0569
10
SUS304/Si3 N4
0.0215
0.0263
0.0304
0.0340
0.0373
SUS304/ZrO2
0.0215
0.0263
0.0304
0.0340
0.0373
SUS304/Si3 N4
0.0185
0.0227
0.0262
0.0294
0.0323
SUS304/ZrO2
0.0185
0.0227
0.0262
0.0294
0.0323
50
SUS304/Si3 N4
0.0177
0.0217
0.0251
0.0281
0.0309
SUS304/ZrO2
0.0177
0.0217
0.0251
0.0281
0.0309
100
SUS304/Si3 N4
0.0176
0.0215
0.0249
0.0279
0.0307
SUS304/ZrO2
0.0176
0.0215
0.0249
0.0279
0.0307
20
4 Conclusions In this study, the bending and free vibration analysis of functionally graded square plate have been analyzed using HSDT. The study provided the valuable results in terms of vibration characteristics and effects of Young’s modulus on displacement in the thickness direction. The results showed that the change in Young’s modulus ratio
Bending and Free Vibration Analysis of Exponential Graded …
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at constant densities of metal and the ceramics has pronounced effects on frequencies of the used materials. Significant reductions in frequencies have been reported in results. Similarly, the change in density ratio has also led to the significant increment in frequency of the materials.
References 1. Koizumi M (1993) The concept of FGM. Ceram Trans FGM 34:3–10 2. Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Methods Eng 47:663–684 3. Abrate S (2008) Functionally graded plates behave like homogeneous plates. Compos Part B 39(1):151–158. https://doi.org/10.1016/j.compositesb.2007.02.026 4. Mantari JL, Guedes Soares C (2012) Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos Struct 94. https:// doi.org/10.1016/j.compstruct.2012.01.005 5. Thai H-T, Kim S-E (2013) A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos Struct 96:165–173. https://doi. org/10.1016/j.compstruct.2012.08.025 6. Sburlati R (2014) Three-dimensional analyses of functionally graded multi-layered systems. Proc Eng 88:235–241. https://doi.org/10.1016/j.proeng.2014.11.150 7. Chakraverty S, Pradhan KK (2014) Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions. Aerosp Sci Technol 36:132–156. https://doi.org/10.1016/j.ast.2014.04.005 8. Zenkour M (2015) Thermo-mechanical bending response of exponentially graded thick plates resting on elastic foundations. Int J Appl Mech 7(4):1550062. https://doi.org/10.1142/S17588 25115500623 9. Ebrahimi F, Mokhtari M (2015) Vibration analysis of spinning exponentially functionally graded Timoshenko beams based on differential transform method. J Aerosp Eng 0(0):1. https:// doi.org/10.1177/0954410015580801 10. Gupta A, Talha M (2015) Recent development in modeling and analysis of functionally graded materials and structures. Prog Aerosp Sci 79:1–14. https://doi.org/10.1016/j.paerosci.2015. 07.001 11. Akavci SS (2016) Mechanical behavior of functionally graded sandwich plates on elastic foundation. Compos Part B 96:136–152. https://doi.org/10.1016/j.compositesb.2016.04.035 12. Li D, Deng Z, Chen G, Ma T (2018) Mechanical and thermal buckling of exponentially graded sandwich plates. J Therm Stress 41(7):883–902. ISSN: 0149-5739. https://doi.org/10.1080/ 01495739.2018.1443407 13. Pandey S, Pradyumna S (2018) Analysis of functionally graded sandwich plates using a higherorder layer wise theory. Compos B. https://doi.org/10.1016/j.compositesb.2018.08.121 14. Zenkour AM, Alghanmi RA (2018) Bending of functionally graded plates via a refined quasi3D shear and normal deformation theory. Curved Layer Struct 5:190–200. https://doi.org/10. 1515/cls-2018-0014 15. Zenkour AM, Alghanmi RA (2019) Bending of exponentially graded plates integrated with piezoelectric fiber-reinforced composite actuators resting on elastic foundations. Eur J Mech A Solids. https://doi.org/10.1016/j.euromechsol.2019.03.003 16. Mantari JL, Guedes Soares C (2012) Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos Struct 94(6):1991– 2000. https://doi.org/10.1016/j.compstruct.2012.01.005
Structural Responses of Geometrically Imperfect Functionally Graded Plates with Microstructural Defects Under Hygrothermal Environment Ankit Gupta and Mohammad Talha
Abstract This article deals with the structural analysis of geometrically imperfect functionally graded plates (FGM) with microstructural defects (porosity) under a hygrothermal environment. The mathematical formulation is based on nonpolynomial-based quasi-3D higher-order structural kinematics developed by the authors. The comparative studies have been carried out by comparing the obtained results with the published results. Finally, the influence of various parameters such as moisture contents, thermal environment, and various defects on the vibration and bending of FGM plates has been studied. Keywords FGM plates · Thermal environment · Moisture contents · Geometric imperfection · Porosity inclusion
1 Introduction Functionally gradient materials (FGMs) are a heterogeneous material which has been progressively applied for ultra-modern engineering structures subjected to extreme temperature gradient [1]. Since the last few decades, remarkable studies have been performed on the structural response of FGM structures. Nguyen-Xuan et al. [2] used an isogeometric formulation with the use of NURBS to investigate the structural response of the FGM plate. Zenkour [3] investigated the vibration response of FGM plates using a non-polynomial-based deformation plate theory. Gupta and Talha [4– 6] analyzed the linear/nonlinear structural response of gradient plates with geometric and microstructural imperfection. Wu et al. [7] studied the structural response of the FGM plate subjected to dynamic load using the modified Pagano method. In the present study, the objective is to illustrate the vibration and the bending response of imperfect porous FGM plates under a hygrothermal environment. The A. Gupta (B) Mechanical Engineering Department, Shiv Nadar University, Chennai, India e-mail: [email protected] M. Talha School of Engineering, Indian Institute of Technology, Mandi, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_135
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A. Gupta and M. Talha
geometrical imperfection has been induced in the plate using imperfection function, which is capable to model the plate with various types of geometrical imperfections. The material property of FGM plates with microstructural defects (porosity) has been estimated using the modified rule of mixture. The mathematical formulation has been done using previously developed displacement field by the authors. The influence of the various parameters like geometric imperfection, porosity inclusion, moisture contented, and temperature gradient on the vibration and the bending response of FGM plates have been investigated in detail.
2 Mathematical Formulation 2.1 Displacement Field Structural kinematics developed by the authors’ [8] has been employed in the present study as given below, ⎧ ⎪ ⎪ U i (x1 , x2 , x3 ; t) = u i (x1 , x2 ; t) − x3 αxi ⎪ ⎨ + f (x )β (x , x ; t) + g(x )θ (x , x ; t) 3 xi 1 2 3 xi 1 2 ⎪ Where, for i = 1, 2; {θi } = 0 ⎪ ⎪ ⎩ for i = 3; {αxi , βxi } = 0 ⎧
hcosh2 (/2) ⎪ −1 x 3 ⎪ ⎪ f (x sinh − x3 3) = − ⎪ −1 ⎨ h h 1 + 2 /4 −1
⎪ ⎪ x x ⎪ ⎪ ⎩ g(x3 ) = cosh2 h
The field variables are represented as {D} = u i αxi βxi θxi .
2.2 Strain–Displacement Relation The nonzero strains are given as follows: {ε l } =
∂u 1 ∂u 2 ∂u 3 ∂u 2 ∂ x1 ∂ x2 ∂ x3 ∂ x3
+
∂u 3 ∂u 1 ∂ x2 ∂ x3
+
∂u 3 ∂u 2 ∂ x1 ∂ x1
+
∂u 1 ∂ x2
2 0 1 + zεxi − C sinh−1 xh 3 − h x3 εxi , for i = 1, 2 εxi = εxi 0 for i = 3 εxi = εxi
(1)
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⎧
x3 ⎪ 0 1 2 ⎪ − x3 γx1x2 + x3 γx1x2 − C sinh−1 γx1x2 = γx1x2 ⎪ ⎪ ⎪ h h ⎪ ⎪ ⎛ ⎞ ⎪ ⎨
1 0 ⎝ ⎠γ 1 + cosh2 x3 βz (x, y, t)γ 2 , γ = γ + C h − 1 ⎪ xi x3 xi x3 xi x3 xi x3 ⎪ h 2 ⎪ ⎪ 1 − x3 h ⎪ ⎪ ⎪ ⎪ ⎩ for i = 1, 2
⎧ nl/0 nl/1 nl/2 nl 2 x 3 2 4 x 3 ⎪ ⎪ ε ε εxi = ε + cosh + cosh for i = 1, 2 ⎨ xi xi xi h h
⎪ x3 x3 nl/0 nl/1 nl/2 ⎪ nl ⎩ γx1x2 γx1x2 + 2 cosh4 γx1x2 = γx1x2 + cosh2 h h
where ⎧ ∂u i 0 ⎪ ⎪ ⎪ εxi = ∂ xi ⎪ ⎪ ⎪
⎨ ∂αxi C ∂βxi 1 =− εxi − ⎪ ∂ xi h ∂ xi ⎪ ⎪ ⎪ ⎪ ∂β ⎪ ⎩ ε2 = C xi xi ∂ xi
for i = 1, 2
2.3 Stress–Strain Relations The properties of the porous FGM plate are presumed to vary gradually through the thickness as given in Eq. 2.
2z + h n 2|z| + Em − (λ/2)[E c + E m ] 1 − E(z, T ) = [E c − E m ] 2z h
2|z| 2z + h n + ρm ρ(z) = [ρc − ρm ] − (λ/2)[ρc + ρm ] 1 − 2z h
2z + h n 2|z| + αm α(z, T ) = [αc − αm ] − [αc + αm ] 1 − 2z h
2z + h n kt (z) = [ktc − ktm ] + ktm 2z {σ}6x1 = Qi j {ε} 6x1 − {α}6x1 T − {β}6x1 C
(2) (3)
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Q 11 Q 12 Q 44
⎧ ⎫ ⎡ ⎤ ⎪ Q 11 Q 12 Q 13 0 0 0 ⎪ σx x ⎪ ⎪ ⎪ ⎪ ⎢Q Q Q ⎪ ⎥ ⎪ ⎪ σ yy ⎪ ⎢ 21 22 23 0 0 0 ⎥ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥ σzz 0 0 0 ⎥ ⎢Q Q Q = ⎢ 31 32 33 ⎥ ⎪ τx y ⎪ ⎪ ⎢ 0 0 0 Q 44 0 0 ⎥ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 0 Q 55 0 ⎦ τ yz ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ τx z 0 0 0 0 0 Q 66 ⎧ ⎫ ⎫ ⎧ ⎫ ⎞ ⎛⎧ ⎪ ⎪ ⎪ 1⎪ εx x ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎪ ⎪ε ⎪ ⎪ ⎜⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1⎪ 1⎪ yy ⎪ ⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎬ ⎨ ⎪ ⎜⎨ ⎬ ⎨ ⎬ 0 0 ⎟ ⎜ εzz β(z, C)⎟ α(z, T ) − − ×⎜ ⎟ ⎪ ⎪ ⎪ ⎪ ⎜⎪ ⎪ 1 1 γ ⎪ ⎪ ⎟ ⎪ ⎪ ⎪ xy ⎪ ⎪ ⎪ ⎪ ⎜⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎠ ⎝⎪ 0 0 γ yz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ⎩ ⎭ ⎩ ⎪ ⎭ ⎪ 0 0 γx z E(z, T ) 1 − υ 2 = Q 22 = Q 33 = 1 − 2υ 2 − 2υ 3 E(z, T )υ(1 + υ) = Q 21 = Q 13 = Q 31 = Q 23 = Q 32 = 1 − 2υ 2 − 2υ 3 E(z, T ) = Q 55 = Q 66 = 2(1 + υ)
3 Finite Element Formulation In the present analysis, FE formulation has been employed using C0 isoparametric element with nine DOFs per node. The shape functions of the selected element are as follows:
⎧ 1 ⎪ ξ 2 + ξi ξ η2 + ηi η , for i = 1, 2, 3, 4 ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ 1 − ξ 2 η2 + ηi η , for i = 5, 7 2 Ni = 1 ⎪ ⎪ ⎪ ξ 2 + ξi ξ 1 − η2 , for i = 6, 8 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ 1 1 − ξ 2 1 − η2 , for i = 9 2
(4)
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3.1 Expression for Characteristics Equation The characteristics equation for vibration, and bending response of the FGM plate is obtained using Lagrange equation as follows: d dt
$
∂
˙i ∂ D
%
+
∂ ∂{Di }
+
∂V ∂{Di }
= 0 for i = 1, 2, . . .
(i) For free vibration analysis is as follows: [K ]{D} = λ[M]{D},
(5)
With, λ = ω2 where ω is a natural frequency. (ii) For static response is: [K ]{D} = {F}
(6)
4 Results and Discussion To validate the present solution, some illustrations have been given in the first part. The initial geometric imperfection has been included using the following function [9].
X1 X1 − ψ1 cos μ1 π − ψ1 ζ = hς sech δ1 a a
X2 X2 − ψ2 cos μ2 π − ψ2 × sech δ2 b b where ‘ς ’ is the imperfection amplitude (Fig. 1; Table 1).
4.1 Comparison Study In the first section, the comparison study has been performed. The ND frequency has been calculated and compared with reported results given by Zhu and Liew [10] as shown in Fig. 2.
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Fig. 1 Perfect and sine type geometrically imperfect plate
Table 1 Imperfection parameters Type
Imperfection parameters
Perfect type
P
δ1 = δ2 = 0, μ1 = μ2 = 0, ψ1 = ψ2 = 0
Sine type
S
δ1 = δ2 = 0, μ1 = μ2 = 0.5, ψ1 = ψ2 = 1
4.2 Parametric Study In this segment, the influence of different parameters like moisture contents, porosity inclusions, and thermal environment on the deflection and vibration response have been explored. Ti–6AL–4V/Si3 N4 FGM plate has been considered for the analysis. The following non-dimensional (ND) parameters are used in the present study.
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Fig. 2 Convergence and validation of the present finite element solution
ND frequency ω=ω
12(1 − ν 2 )ρc a 2 b2 /π 4 E c h 2
ND deflection w = 10wE c h 3 Qa 4 In Fig. 3, the ND frequency of the FGM plate under hygrothermal environment has been computed. T has been considered as 0 K, 200 K, and 700, whereas the moisture content (C) is considered as 0.1. An S-type imperfect plate has been used for the analysis. ND frequency of the plate has been plotted against imperfection amplitude. It is noteworthy that if the imperfection amplitude ‘ς ’ is zero, then the plate is perfect, whereas as its value increases from 0 to 0.4, the intensity of the geometric imperfection increases. It is evident from Fig. 3 that as T increases from 0 to 700 K, the ND frequency decreases. It is also found that the ND frequency increases with the ‘ς ’. In Fig. 4, it is observed that the ND frequency decreases as the value of ‘n’ and ‘λ’ increases. This is anticipated because of the large value of ‘n’ means the plate has a more metallic component. It is also perceived that the existence of porosity will result in the decrement of the stiffness of the plate. Figure 5 shows the variation of ND frequency with the moisture contents (C). The T is taken as 200 K. It is clear from the figure that as the moisture content increases, the ND frequency decrease for all values of ‘n’. Hence, the moisture content
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Fig. 3 Influence of T and ‘ς’ on ND frequency of FGM plate
Fig. 4 Influence of ‘λ’ and ‘n’ on the ND frequency of FGM plate
is also an important parameter while dealing with the FGM structures subjected to the hygrothermal environment. In Fig. 6, the ND deflection of FGM plate under the hygrothermal environment has been computed. The T has been considered as 200 K, whereas the moisture content (C) is considered as 1. The central deflection of the FGM plate has been plotted against ‘ς ’ for different various values of ‘n’. It is evident from Fig. 6 that
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Fig. 5 Influence of moisture contents (C) and ‘n’ on ND frequency of FGM plate
Fig. 6 Influence of ‘ς’ and ‘n’ on ND deflection of FGM plate
as ‘ς ’ increases from 0 to 0.4, the central deflection decreases. It is also found that the ND deflection decreases as ‘n’ increases. Therefore, it is again verified that the geometric imperfection will lead to make the plate stiffer. In Fig. 7, the ND central deflection has been plotted for the different values of ‘n’ and ‘λ’. Again the T and C are considered as 200 K and 1, respectively. It is found that the ND deflection increases as ‘n’ and ‘λ’ increases. Therefore, it is
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Fig. 7 Effect of ‘λ’ and ‘n’ on ND deflection of FGM plate
concluded that the porosity inclusion in the structures will result in the increment of the flexibility of the plate. Figure 8 depicts the change of ND deflection with the moisture contents (C). T is taken as 200 K. It is obvious from the figure that as the moisture content increases, the ND deflection increase for all values of ‘n’.
Fig. 8 Effect of ‘C’ and ‘n’ on ND deflection of FGM plate
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5 Conclusion In this study, a non-polynomial HSNDT is implemented for the bending and vibration response of geometrically imperfect porous FGM plate under a hygrothermal environment. It is concluded that the consideration of the moisture contents as well as temperature gradient during the structural analysis of the FGM plate is important because it influences the mechanical behavior of the plates significantly.
References 1. Koizumi M (1997) FGM activities in Japan. Compos Part B Eng 28(1–2):1–4 2. Nguyen-xuan H, Tran LV, Thai CH, Kulasegaram S, Bordas SPA (2014) Isogeometric analysis of functionally graded plates using a refined plate theory. Compos Part B 64:222–234 3. Zenkour AM (2005) A comprehensive analysis of functionally graded sandwich plates: Part 2-buckling and free vibration. Int J Solids Struct 42(18–19):5243–5258 4. Gupta A, Talha M (2018) Imperfection sensitivity of the post-buckling characteristics of functionally gradient plates using higher-order shear and normal deformation theory 5. Gupta A (2018) Static and stability characteristics of geometrically imperfect FGM plates resting on pasternak elastic foundation with microstructural. Arab J Sci Eng 6. Gupta A, Talha M (2017) Influence of porosity on the flexural and vibration response of gradient plate using nonpolynomial higher-order shear and normal deformation theory. Int J Mech Mater Des 14(2):1–20 7. Wu CP, Chen SJ, Chiu KH (2010) Three-dimensional static behavior of functionally graded magneto-electro-elastic plates using the modified pagano method. Mech Res Commun 37(1):54–60 8. Gupta A, Talha M (2016) An assessment of a non-polynomial based higher order shear and normal deformation theory for vibration response of gradient plates with initial geometric imperfections. Compos Part B 107:141–161 9. Yang J, Huang X-L (2007) Nonlinear transient response of functionally graded plates with general imperfections in thermal environments. Comput Methods Appl Mech Eng 196(25– 28):2619–2630 10. Zhu P, Liew KM (2011) Free vibration analysis of moderately thick functionally graded plates by local Kriging Meshless method. Compos Struct 93(11):2925–2944
Simplified Aerodynamic Modeling of a Bird Robot Using the DeNOC Matrices Anil K. Sharma , Sasanka S. Sinha, Rajesh Kumar, and S. K. Saha
Abstract To design an efficient flapping wing aerial vehicle, a simplified aerodynamic model of a bird robot is presented in this paper. A bird robot was divided into three modules, namely, main body, right wing, and left wing. The main body of the bird was considered as spheroidal prolate, and the wings as rigid plate with airfoil cross-section. The robotic bird was considered as tree-type multibody system, where the main body is parent and wings are children. Each wing was connected to the main body using a two-degree-of-freedom (DoF) joint, which provides twisting and flapping motions to the wings. The kinematic configuration of the bird model was represented using the modified Denavit–Hartenberg (DH) parameters. The flapping and twisting of the wings generate both lift and forward thrust for the bird flight. The equations of motion of the robotic bird were derived using the DeNOC matrices. The constant drag and lift coefficients were considered for the main body of robotic bird; however, variable drag and lift coefficients were used for the aerodynamic modeling of the wings. The sinusoidal trajectory was considered as the desired joint motion for twisting and flapping of the wings. A proportional-derivative (PD) controller was used for the force forward simulation of the robotic bird. A relation between the flapping frequency and lift force on the main body was established. Keywords Aerodynamics · Bird robot · DeNOC matrices · Dynamic modeling
1 Introduction The perfection of bird flight is always a source of attraction to the researcher. The physics of bird flight is a complex phenomenon and still a hot area of research. A bird has large lungs to flap the wings at higher frequency for longer time and hollow bones to get the high buoyant mass. The wing’s structure of a bird is like a human hand and mainly contributes to the dynamic control of the bird flight. The wings can be divided into primary and secondary feathers. During the upstroke, primary feathers are rotated and separated, and therefore allowing air to pass through them. This phenomenon A. K. Sharma (B) · S. S. Sinha · R. Kumar · S. K. Saha Indian Institute of Technology Delhi, New Delhi, Delhi 110016, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_136
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reduces the air resistance on the wing and provides some thrust during the upstroke of the flapping flight. However, during the downstroke, the primary feathers are closed and hit the surrounding air to produce the lift and thrust forces. The secondary feathers provide lift force by creating the airfoil structure of the wings. The aerodynamic characteristics of bird wings were studied at Reynolds number of 1 − 5 × 104 , and a correlation with morphological parameters such as nose radius, camber, and aspect ratio was established in [1]. The researchers from zoology and biology have examined many complicated aerodynamics phenomena that happens during the flapping wing flight of birds and insects [2–4]. The maneuverability and ability of bird flight and inertial power required for wing beat are related to the mass moment-of-inertia of the wings [5]. In [5], strip theory was used to estimate the moment-of-inertia of the wings of 29 birds and three bats. A study on upward stroke of wingbeat was presented in [6]. A computational simulation of a robotic bird and its prototype was developed in [7, 8] for different flight modes such as take-off, landing, and cruising. In [9], modified strip theory was used to formulate the aerodynamic model of a semi-elliptical wing, and parametric study was done to describe the effect of different parameters on lift, thrust, and drag force. A nonparametric mathematical model of a flying bird was developed in [10] from flight test data using the time domain identification technique. The singular value decomposition (SVD) and modal analysis were performed to get the minimal order system identification model. The flapping wing aerodynamics for the three-class of species, i.e., birds, bats, and insects were studied in [11]. The motion of the flapping wing is quite complex and is characterized by unsteady, threedimensional flow, where the state of the wings change every instant of flapping flight [12]. A detailed study on aerodynamics of a bird flight, such as induced drag, lift, and thrust forces due to wing-generated vortices, airfoil shape of the secondary wing, spread out of the primaries at the tip, etc., was discussed in [12]. A comprehensive sizing method for flapping wing micro-air vehicle (FWMAV) based on statistical and theoretical analyses was proposed in [13], which was verified with the help of a FWMAV, namely, Thunder I. The design and operation of a microrobot capable of swimming, flying, and transitioning between water and air was presented in [14]. The multibody dynamic approach for the mathematical modeling of space vehicle was proposed in [15, 16] using the DeNOC matrices [17, 18]. In this paper, the kinematic configuration of the bird model is represented using the modified DH parameters and dynamic modeling using the DeNOC approach. A simplified aerodynamic modeling of a robotic bird is presented, where variable drag and lift coefficients were used to perform the dynamic simulations. The joint torques for the desired sinusoidal trajectory of the wingbeat were generated through PD controller.
2 Mathematical Modeling for Simplified Bird Model The multibody dynamic approach is used to model a bird flight. The dynamic model of a flying bird is required for precise simulation and control of bird flight [8]. Here,
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a robotic bird was divided into three modules, namely main body, right wing, and left wing. The kinematics and dynamics of the bird model are presented next.
2.1 Kinematic Modeling of Robotic Bird A simplified kinematic structure of a robotic bird is shown in Fig. 1. The six-DoF of the main body were represented with six one-DoF joints, i.e., revolute or prismatic. Three prismatic joints and three revolute joints were used to represent the translational and rotational motion of the main body as shown in Fig. 1, where dashed line represents the virtual links. The geometry of the main body was considered as prolate spheroid and that of wing as rectangle plate with aerofoil cross-section. The semi-major and minor axes of the main body are represented as ba , and bb = bc , respectively. The modified DH parameters with respect to the attached DH frames are listed in Table 1. The twist vector [17] for the above kinematic configuration can be expressed as ti ≡ pi q˙i for i = 1
(1a)
ti ≡ Ai,i−1 ti−1 + pi q˙i for i = 2, 3, . . . 8, 10.
(1b)
ti ≡ Ai,6 t6 + pi q˙i for i = 9
(1c)
Left wing
X9 , X10
Z6
X2
Z0 , Z1 X1 X0
b1
b3 Z2
b2
Z
X4
X3 Z 3
Z5
Z4 , X5 , X6
X7 , X8 Z8
Main body of robotic bird
Y0 Z7
Z9
Right wing
Fig. 1 Simplified kinematic and modified DH frame representation of a bird model
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Table 1 Modified DH parameters for bird model S. No.
b
θ
a
α
1
b1 (JV)
0
0
0
2
b2 (JV)
3π 2
0
3
b3 (JV)
0
0
4
0 0
θ4 (JV) 3π 2 + θ5 (JV)
0
5 6
0 bc
θ6 (JV) π 2 + θ7 (JV)
0
7 8
0 bc
θ8 (JV) 3π 2 + θ9 (JV)
0
9 10
0
θ10 (JV)
0
0 0 0
3π 2 π 2 3π 2 3π 2 3π 2 π 2 π 2 3π 2 3π 2
JV: Joint variable
0 1 O , and Ai,i−1 ≡ , in which 0 is three-dimensional where pi ≡ ei ai,i−1 × 1 1 vector of zeros, ei is the unit vector along ith joint axis, 1 and O are 3 × 3 identity and null matrices, respectively. The three-dimensional vector ai,i−1 represents the position vector between two consecutive origins. The set of generalized coordinates T ˙ q ≡ b1 b2 b3 θ4 θ5 θ6 θ7 θ8 θ9 θ10 and its time derivative are represented as q. The twist vector for the complete system can be expressed as
t = Nl Nd q˙ or t = Nq˙
(2)
The matrices Nl and Nd are known as the decoupled natural orthogonal complement matrices for the bird model under study, whose combined form was originally proposed for the serial-chain system in [19] as natural orthogonal complement matrix N.
2.2 Dynamic Modeling of Robotic Bird The dynamic equations of motion for an ith rigid body using the Newton–Euler equations can written as Iio ω˙ i + m i di × o¨ i + ωi × Iio ωi = nio m i o¨ i − m i di × ωi − ωi × (m i di × ωi ) = fio
(3)
where Iio is the inertia tensor about the origin, m i is mass, ωi is angular velocity, oi is position vector of origin with respect to the inertial frame, di is the position
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vector from origin to the center-of-mass (CoM). The three-dimensional vector nio and fio are the moment about and force at the origin of the ith body, which are the combination of gravity and buoyancy, drag and lift, and external moments and forces. The matrix–vector form of the above equations can be represented as Mi t˙i + Wi Mi Ei ti = wi ; where wi = wib + wid + wie
(4)
where Mi ≡
mi d × 1 Iio ωi × 1 O 1 O ; Wi ≡ and Ei ≡ . −m i d × 1 m i 1 O ωi × 1 OO
The wrench vector wi is the combination of gravity and buoyancy wrench wib , drag and lift wrench wid , and external wrench wie . The Newton–Euler equations for the complete system can be expressed as M t˙ + W MEt = w
(5)
where the 6n × 6n matrices M and W are respectively the generalized aerodynamic mass matrix, and the generalized aerodynamic matrix of Coriolis and convective inertia terms and n is DoF of the bird model. They have the following representations: M ≡ diag.[M1 , M2 ..., Mn ], and W ≡ diag.[W1 , W2 ..., Wn ]
(6)
Also, the 6n-dimensional generalized wrench vectors due to gravity and buoyancy, wb , drag, wd , and external moments and forces, we , are given by T T T T ; wd ≡ w1d T · · · wnd T ; and w1b · · · wnb T we ≡ w1e T · · · wne T
wb ≡
(7)
2.3 Reduced-Order Equations of Motion The reduced-order equations of motion are obtained by pre-multiplying Eq. (5) with the transpose of the natural orthogonal matrix N ≡ Nl Nd given by Eq. (2). The resulting equations are as follows: NT M t˙ + W ME t = NT w
(8)
˙ q. ˙ Substituting t and t˙ in terms of q˙ and R where t = N q˙ and t˙ = Nq¨ + N q into Eq. (8) yields
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˙ = τ; and τ = τb + τd + τe I(q)q¨ + h(q, q)
(9)
where I(q) ≡ NT MN is the n × n generalized aerodynamic inertia matrix ˙ + WMEN q˙ is the n-dimensional vector of aerody˙ ≡ NT MN (GAIM), h(q, q) namic convective inertia (VACI) terms, τb ≡ NT wb is the n-dimensional vector of gravity and buoyancy forces (VGBF), τd ≡ NT wd is the n-dimensional vector of aerodynamic drag and lift forces (VADF), and τe ≡ NT we is the n-dimensional vector of generalized actuator forces (VGAF), which has three driving forces to translate the vehicle, three moments to orient it, and n joint torques to drive the manipulator on the vehicle. Further, (9) can also be written as Iq¨ = ϕ; and ϕ = τ − h
(10)
Equation (10) will be used to perform forward dynamics where q¨ need to be solved.
3 Dynamic Simulation of Robotic Bird The dynamic simulation of the ten-DoF robotic bird was performed by integrating Eq. (10) numerically in MATLAB. The physical parameters of the bird model are summarized in Table 2. The initial state vector for the dynamic simulation is given below: T y = y0T y˙ 0T ; y0 = 0 0 0 0 3π 2 0 π 2 0 3π 2 0 y˙ 0 = 2 2 0 0 0 0 0 0 0 0
(11)
The desired trajectory for the flapping was considered as sinusoidal, which is written below: θ8d = π 6 sin(2π f t); θ˙8d = π 6 (2π f ) cos(2π f t)
(12)
The joint torque for the flapping wing can be calculated through proportionalderivative controller as Table 2 Physical parameters of a robotic bird
S. No
Parameters
Main body
1
Mass (kg)
0.355
2
Wingspan (m)
0.75
3
Wing surface (m2 )
4
Amplitude of flapping frequency (rad)
0.12 π 6
5
Flapping frequency (Hz)
6
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τ8 = k p θ8d − y8 + kd θ˙8d − y18
(13)
The same joint torque input was given to the flapping joint of the left wing, i.e.,θ10 . The value of k p and kd were taken as 5 and 2, respectively. The results for the lift and forward motion are shown in Figs. 2 and 3. The lift force generated through flapping of the wing is shown in Fig. 4. Fig. 2 Variation of the lift motion of the robotic bird
0.5
Lift of the bird (m)
0.4
0.3
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.3
0.4
Time (s)
0.8
Forward move of the bird (m)
Fig. 3 Variation of the forward motion of the robotic bird
0.6
0.4
0.2
0 0
0.1
0.2
Time (s)
Fig. 4 Lift force generated through flapping of the wing
A. K. Sharma et al. Lift force generated through flapping (N)
1444 15
10
5
0
-5
0
0.1
0.2
0.3
0.4
Time (s)
4 Discussion and Conclusions The kinematic configuration and dynamic modeling of a robotic bird were presented using the modified DH parameters and DeNOC approach, respectively. The sinusoidal joint trajectories were used to get the joint torques using PD controller. The proposed dynamic and aerodynamic model was able to provide enough lift and forward thrust to move the robotic bird. There is a little drop of the bird during the upward stroke of the wings as shown in Fig. 2. Figure 3 shows that there is minimal loss of kinetic energy in the forward direction due to streamline characteristics of main body of the bird. The lift is also increasing in each cycle of wingbeat, and the mean of the lift force is very close to the weight of the robotic bird. The proposed aerodynamic model is used to generate preliminary simulation results and later will be used for the simulation and control of a more complex robotic bird.
References 1. Withers PC (1981) An aerodynamic analysis of bird wings as fixed aerofoils. J Exp Biol 90:143–162 2. Norberg UM (1990) Vertebrate flight: mechanics, physiology, morphology, ecology and evolution. Springer, New York. ISBN 13:978-3-642-83850-7 3. Spedding GR (1992) The aerodynamics of flight. Mech Animal Locomot 11:52–111 4. Tobalske BW, Dial KP (1996) Flight kinematics of black- billed magpies and pigeons over a wide range of speeds. J Exp Biol 199:263–280 5. Berg C, Rayner J (1995) The moment of inertia of bird wings and the inertial power requirement for flapping flight. J Exp Biol 198(8):1655–1664 6. Poore SO, Sanchez-Haiman A, Goslow GE Jr (1997) Wing upstroke and the evolution of flapping flight. Nature 387(6635):799 7. Wu JC, Popovi´c Z (2003) Realistic modeling of bird flight animations. In: ACM Trans Graph (TOG) 3:888–895
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8. Couceiro MS, Figueiredo CM, Ferreira NF, Machado JT (2009) The dynamic modeling of a bird robot. In: Proceedings of the 9th conference on autonomous robot systems and competitions (Robotica’09) 9. Malik MA, Ahmad F (2010) Effect of different design parameters on lift, thrust, and drag of an ornithopter. In: Proceedings of the World Congress on Engineering, vol 2, pp 1460–1465 10. Shepherd S, Valasek J (2012) Modeling and analysis of eagle flight mechanics from experimental flight data. In 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 2012, p 27 11. Chin DD, Lentink D (2016) Flapping wing aerodynamics: from insects to vertebrates. J Exp Biol 219(7):920–932 12. Dvoˇrák R (2016) Aerodynamics of bird flight. In: EPJ Web of Conferences, vol 114, p 01001, EDP Sciences 13. Hassanalian M, Abdelkefi A, Wei M, Ziaei-Rad S (2017) A novel methodology for wing sizing of bio-inspired flapping wing micro air vehicles: theory and prototype. Acta Mech 228(3):1097–1113 14. Chen Y, Wang H, Helbling EF, Jafferis NT, Zufferey R, Ong A, Ma K, Gravish N, Chirarattananon P, Kovac M, Wood RJ (2017) A biologically inspired, flapping-wing, hybrid aerial-aquatic microrobot. Sci Robot 2(11) eaao5619 15. Saha SK (1993) Modeling and simulation of space robots. In: Proceedings of 1993 IEEE/RSJ International conference on intelligent robots and systems (IROS’93), vol 3, pp 2033–2040 16. Saha SK (1996) A unified approach to space robot kinematics. IEEE Trans Robot Autom 12(3):401–405 17. Saha SK (1999) Analytical expression for the inverted inertia matrix of serial robots. Int J Robot Res 18(1):20–36 18. Saha SK (2014) Introduction to robotics, 2nd edn. Tata McGraw-Hill Education, New Delhi 19. Angeles J, Lee S (1988) The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement. Trans ASME J Appl Mech 55:243–244
Parallel Mechanism-Based Master–Slave Manipulation Ravinder Kumar, S. K. Sinha, T A. Dwarakanath, and Gaurav Bhutani
Abstract The paper presents the development of a parallel mechanism-based master–slave (M-S) manipulation. The paper discusses the generalized 6 degree of freedom (DoF)-based M-S arrangement. The feasibility of architectures possessing specific advantages is discussed. The cases starting from two to six DoF parallel manipulators are dealt. The aspect of one-to-one joint space correspondence, workspace mapping and issues related to direct kinematic problem is discussed. The mechanical master–slave manipulator design, motion transmission and control of stationary-active (St-Ac) axis parallel manipulators are demonstrated. The importance of parallel mechanism-based M-S arrangement over serial mechanism-based M-S in complementing the application space is shown. Keywords Parallel manipulator · Master–slave manipulation · Multi-DoF telemanipulation
1 Introduction There exist two types of the manipulators by architecture design. (1) Serial manipulator (SM), (2) parallel manipulator (PM). The links are connected in articulated chain with end-effector at the tip in SMs, whereas PM is a closed-loop kinematic mechanism whose end-effector (platform) is connected to the base by independent serial kinematic chains. These chains are similar and possess one controllable DoF. R. Kumar (B) · S. K. Sinha · T. A. Dwarakanath · G. Bhutani Division of Remote Handling and Robotics, Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra 400085, India e-mail: [email protected] S. K. Sinha e-mail: [email protected] T. A. Dwarakanath e-mail: [email protected] G. Bhutani e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_137
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SMs have large workspace but they are very sensitive to kinematic positioning errors because error of each joint adds up in sequence. The load handling capability of SMs is less in comparison with PMs because the load is distributed among several parallel links [1]. PMs can provide higher accuracy and can be faster in manipulation; therefore, they are preferred for the tasks which require precision and agility. This paper deals with kinematic analysis and construction of a 6 DoF parallel mechanism-based master–slave (M-S) framework. Both the master and slave are based on parallel kinematic configuration. The operator through master serves to provide joints trajectory and control. The direct kinematics problem in serial chain will yield a unique solution at the task space of the slave manipulator. The simpler direct kinematic problem apart from their higher workspace is also the reason for the popularity of the serial mechanism-based servo M-S manipulators which is one of the earliest architecture. However, low payload to weight ratio, low stiffness, accumulation and amplification of positioning errors leading to low accuracies are certain weak factors that led to an alternative framework for M-S arrangements. The parallel architecture-based mechanism which exhibits reciprocal properties to serial chain is the obvious choice. The limited working volume, complexity in singularity identification and highly complex direct kinematic problem are the factors for its absence in M-S arrangement. The paper deals with the ways of addressing the above factors and proposes methods to overcome the issues.
2 Parallel Mechanism-Based Master–Slave Manipulation 2.1 Mechanical Master–Slave Parallel Manipulators Figures 1 and 2 show a stationary-active arrangement of the parallel mechanismbased M-S arrangement [2, 3]. It provides an uncoupled and direct transmission Fig. 1 Parallel mechanism-based 3 DoF M-S arrangement
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Fig. 2 Parallel mechanism-based 6 DoF M-S arrangement
compared to the master–slave arrangement of the serial architecture, which involves, multiple axis (arm) motion transmission. However, the limitation of this type is its limited workspace. A prototype development of mechanical M-S arrangement of limited degree of freedom is presented in Fig. 3. The master receives the trajectory input through its end-effector and the slave reflects the trajectory as output at the endeffector. The transmission of force and motion from the master to slave is through kinematic chain in a mechanical arrangement and through servo loop of each kinematic chain in electro-mechanical arrangement. The method of achieving one to one
Fig. 3 Prototype of parallel mechanism-based 3 DoF M-S arrangement
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correspondence of master and the slave connectors along the active axis for identical motion response at the slave for the given motion input at the master is shown.
2.1.1
6 DoF Master–Slave Manipulation
The 6 DoF manipulator is a multi-axis independent electro-mechanical servo system. The multiple joints of manipulator can be synthesized to have translational and rotational degree of freedom at the tool (or end-effector). The mechanical arrangement should be such that the mechanism should possess easier solution to both direct kinematic problem (DKP) and inverse kinematic problem (IKP). In other words, both DKP and IKP solutions should be computable within a fraction of a real time. The co-ordinate frame attached to the tool is often referred as task space and the co-ordinate frames attached to joints are referred to as join space. There are few options to choose from, (i) serial mechanism-based open architecture, (ii) parallel mechanism based on closed-loop architecture or (iii) hybrid of (i) and (ii). Statically most stable geometry for a manipulator is arrived by optimizing the determinant of a Jacobian, J, which is the mapping between joint space velocity and a velocity of the tool defined in task space. The dynamics of the manipulator are obtained by considering payload, inertia as well as motion parameters of the tool frame and its relationship with the joint velocities and forces in joint space. The elements of the Jacobian matrix (see Eq. 1) of a serial manipulator are nonlinear functions of all the joint co-ordinates comprising of the mechanism, and the joint space is coupled with more than one joint axis parameters. Therefore, it has a complex transformation matrix to resolve the task space wrench in to joint torques and forces because of the coupled nature of the Jacobian elements [4]. ⎡ J (q) = ⎣
0 0 ∂ r0,t ∂ r0,t ∂q1 ∂q2 0 0 ∂ ϕ0,t ∂ ϕ0,t ∂q1 ∂q2
.. ..
0 ∂ r0,t ∂q6 0 ∂ ϕ0,t ∂q6
⎤ ⎦
sˆ1 . . sˆ5 sˆ J (q) = 0 b0 × sˆ0 b1 × sˆ1 . . b5 × sˆ5
(1) (2)
0 The position vector of the reference point on the tool, r0,t , and orientation of 0 the tool, ϕ0,t , depend on positions and orientations of all the joint axis of the serial manipulator. On the other hand, parallel mechanism-based architecture has a much simpler Jacobian transformation matrix (see Eq. 2). The joint space is nicely decoupled making it straight forward to resolve the task space wrench into joint forces. The parallel architecture presents high positional accuracy because of the closed-loop nature of the construction. They offer relatively easier solution to separate inertial forces from the external forces; the inertial forces acting on various links are not coupled as in the case of serial architecture. The parallel architecture also gives high dynamic stability (high eigenfrequency). Hence, a parallel architecture presents
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Fig. 4 6 DoF master–slave manipulation
competitive merits suitable for applications where active and real-time control is required. Serial mechanism-based master–slave manipulators are very popular and widely used for various applications. But parallel mechanism-based master–slave manipulation is yet to be established because of the unavailability of closed-form solution of FKP of 6 DoF parallel mechanism and limited workspace. But there are many applications where the workspace is not the constraint like neurosurgery, etc. We have developed an approach for master–slave manipulation with dissimilar structures of master and slave arm as shown in Fig. 4. We have developed the 6 DoF passive master manipulator. The magnetic linear encoders are attached with each leg for feedback of 6 leg lengths. We compute the leg lengths of 6 links using FPGA-based encoder interface. After the computation of leg lengths, the forward kinematics algorithm computes the task space which is transmitted to the slave manipulator after proper scaling the task space required for dissimilar structures. We have developed and tested many algorithms of the forward kinematics for fast and accurate convergence, out of which one algorithm is used in this master–slave manipulation.
2.2 Forward Kinematics of Parallel Mechanisms The forward and the inverse kinematic equations of serial and parallel manipulators show duality. Whereas the forward kinematics of SM is comparatively easy to
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Table 1 Computation time and avg. no. of iterations for FKP algorithm 1
Average computation time for FKP algorithm
50 ms per pose
2
Average no. of iterations (real-time implementation)
20 per pose
compute, the same for PM is challenging issue to solve. But the scenerio is viceversa in case of inverse kinematics. The forward kinematic equations of parallel manipulators are highly nonlinear, coupled and exhibit multiple solutions for the same input. A closed-form solution is not available generally. Many schemes require further simplification and reformulation of equations. Numerical methods with fast computations are used to get time optimal accurate solution [5]. The kinematic equations for the 6DoF UPS-based parallel manipulator are formulated and converted to reduce canonical form to implement a Newton–Raphson (N-R) solver. The output from the N-R solver is not necessarily the desired solution. Due to the highly coupled and nonlinear nature of the equations, the output may be a pose that is significantly far away (in terms of Euclidean distance) from the actual pose of the manipulator. This happens if the input trajectory is complex and/or when the seed-point given to the solver falls outside the sphere of influence of the desired solution. To check for correct convergence, the previous converged pose is checked against the pose that the N-R solver converges to. In the case of incorrect convergence, i.e., a considerably large Euclidean distance between the previous converged pose and the converged pose, the resolution of the trajectory is doubled by re-formulating the input arguments for the N-R solver. This was done by taking the mid-point of the leg lengths for previous pose (correctly converged) and current leg lengths. The resolution is doubled till a correct convergence is achieved. These intermediate values are stored separately. Once any particular convergence occurs correctly (as per the set target leg lengths), the target leg lengths are then set to be the actual leg lengths. This process is continued till the output from the N-R solver correctly converges to the actual pose. The intermediate values inserted in between the original pose values in the trajectory forms the modified trajectory. This modified trajectory ensures a smoother motion of the manipulator and correct convergence of the forward kinematics during operation by creating the required resolution in the trajectory. The following observances made have been shown in Table 1. The computer used for FKP computation has a 3.30 GHz i3-3220 processor by Intel and 4 GB RAM.
3 Communication Channel Currently, Ethernet has been used as the channel to establish communication between the master and the slave node [6].
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Basic algorithm of the master communication with the slave: A user-defined header file has been created for the master and slave communication which has the following member functions. A.
B.
C.
Connect to Slave (): Request for a connection with the slave IP. Check for socket error in it or any other type of error. If no error, only then connection is established. This is set up only once. SEND (posture): Sends posture (translation and rotation) to the slave if connection is established. The data is sent as a character buffer, which means that data needs to be converted to a character buffer as well. Close Connection (): This is done in case of any error in connecting to the slave or after the task has been completed, this needs to be performed only once.
4 Slave Manipulator There is a computer simulated model of 6 DoF parallel manipulator which accepts the posture (translation and rotation) inputs and produces the movements of all six legs to attain that posture [7]. This model acts as a slave manipulator for the M-S manipulation. The generated data of leg lengths of passive master manipulator is passed to FKP algorithm for computation of the posture which is then transmitted to slave model using Ethernet communication after proper scaling of the posture. In this way, we are establishing the master–slave manipulation based on parallel mechanisms. The slave model is replaced by actual 6DoF parallel manipulator and the scheme of working is shown in Fig. 5.
5 Conclusion This paper details the development and description of a parallel mechanism-based 6 DoF master device, development of a 6 DoF parallel slave manipulator and description of the framework for the master–slave arrangements. The advantage of the parallel mechanism-based M-S arrangement is in its reciprocal performance properties to that of the serial mechanism and hence can complement the gap in the application space.
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Fig. 5 Prototype of 6 DoF master and 6 DoF slave manipulator in M-S arrangement
References 1. Patel YD, George PM (2012) Parallel manipulators applications—a survey. Mod Mech Eng 2:57–64 2. Lagoo KD, Dwarakanath TA, Badodkar DN (2015) Single actuator shaker design to generate infinite spatial signatures. In: 2nd International and 17th National Conference on Machines and Mechanisms iNaCoMM, p 55 3. Dwarakanath TA, Lagoo KD, Badodkar DN (2017) 6-PSS based parallel manipulators. In: Corves B, Lovasz EC, Hüsing M, Maniu I, Gruescu C (eds) New advances in mechanisms, mechanical transmissions and robotics. mechanisms and machine science, vol 46. Springer, Berlin 4. Dwarakanath TA, Bhutani G (2011) Beam type hexapod structure based six component forcetorque sensor. Mechatronics 21(8):1279–1287 5. Merlet J-P (2004) Solving the forward kinematics of Gough-type parallel manipulator with interval analysis, vol 23, pp 221–236 6. Advanced Master Slave Manipulators. https://barc.gov.in/technologies/msm/msm.html 7. Bhutani G (2015) Modeling design and development of frameless stereotaxy in robot assisted neurosurgery. HBNI, Mumbai
Comparative Stiffness and Damping Analysis for Various Flow Controlling Devices of Hole Entry Worn Hybrid Conical Journal Bearing Under the Variation of Speed Vikas M. Phalle and Sanjay R. Pawar Abstract The choice of proper flow controlling device plays a significant part in determining the performance parameters of hybrid conical journal bearings. During the operation, they are subjected to variation from low to high speed to meet the need of today’s industrial demand. The objective of this paper is to study the comparative dynamic performance analysis of capillary, CFV and orifice compensated, hole entry, worn hybrid conical journal bearing under varying speed by analytical method. The finite element analysis technique is used to solve the modified Reynolds equation. The numerically simulated results show the appreciable change in the performance characteristics due to change in the compensating device with variation in speed with worn/unworn condition of bearing. Keywords Hybrid · Conical · Compensating device · Wear · Speed parameter
1 Introduction Hybrid bearing finds wide spread application in modern machineries. The performance of these hybrid bearing system is greatly depends upon the type of flow controlling device used in the system [1, 2]. In the beginning, Raimondi and Boyd [3] as well as Shaw and Mack’s [4] have studied the analytical and experimental performance characteristics of hydrostatic bearing compensated with capillary and orifice restrictor. The characteristics of membrane-type variable restrictor compensated externally pressurized bearings were studied by Cusano [5] He reported that the maximum pressure generated and stiffness coefficients are strongly dependent on the membrane compliance, the pressure ratio, and the eccentricity ratio. He with Conry [6] also analyzed pocketed externally pressurized bearings from minimum total power loss, point of view. Donoghue [7] solved the problems of earlier system by using parallel feed system. Pande and Somasundaram [8] reported that the variable restrictors with position sensing arrangement are superior than conventional restrictors. V. M. Phalle · S. R. Pawar (B) VJTI, Mumbai 400019, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_138
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Yoshimoto et al. [9] suggested a hydrostatic bearing compensated with a selfcontrolled restrictor using a floating disk. Further, theoretical and experimental work of Palzewicz [10] reported hydrostatic journal bearings without a pressure chamber between the bush and the journal. Jain et al. [11] performed a theoretical study to compare the performance of a multi-recess flexible hydrostatic/hybrid journal bearing with different flow controlling devices. Study reports that the proper choice of restrictor, with the value of deformation coefficient, may give a better bearing performance. Very comprehensive studies were made by Ripple and his associates which have been systematically reported in a ‘Design Manual’ [12] edited by Rippel. The manual contains useful information and data on the thrust pad and journal bearing. Also, the role of the restrictors (capillary, orifice and constant-flow valve) has been highlighted in the manual. During the transient period of start/stop operation, a change in the operating parameter affects the rotating speed and load, so most of the bearing fails during this time due to wear. This wear has dominant effect in deciding life of machine elements. So, it is a vital need to study clearly the variation in the performance of the bearing in worn condition using various flow control devices at various speed parameters. So, the focus of this work is to understand the comparative dynamic performance of worn conical hybrid journal bearing at various speed with different restrictors.
2 Mathematical Formulations The developed Reynolds equation for lubricant flow is given as [9]. 1 ∂ r ∂r
r 3 ∂p h 12µ ∂r
3 ω j ∂h 1 h 1 ∂p ∂h ∂ + = + 2 ∂ϕ ∂t sin2 γ ∂ϕ 12μ r 2 ∂ϕ
(1)
2.1 Finite Element Formation The pressure is represented approximately as p=
4 j=1
Nj P j
(2)
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2.2 Fluid Film Thickness The fluid film thickness h in non-dimensional form for a conical journal bearing is given as h = 1 − X j cos α − Z j sin α cos γ
(3)
2.3 Worn Zone Model Abrasive wear model suggested by Dufrene is used to considerwear in this bearing. This defect value (∂h) is added to the calculate film thickness h given by Eq. (3)
2.4 Restrictor Flow Equation The fluid flow rate through (a)
(b)
(c)
Capillary restrictor is given as Q R = C s2 1 − P c
(4a)
1 Q R = Cs2 1 − p c 2
(4b)
Q R = Q sp
(4c)
Orifice restrictor is given as
CFV restrictor is given as
3 Solution Procedure A mathematical model is established to find the result for the characteristics of hole entry worn hybrid conical journal bearing and FORTRON 77 coding language is used. Based on analysis, due to unavailability of literature, the obtained results are compared with the outcomes of Stout of Rowe [13]. The comparative performance parameters have been shown for the double rows of 12 holes symmetrically arranged on worn hybrid conical journal bearing in terms of stiffness and damping coefficients.
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4 Results and Discussion The computed results are obtained using the following operating and geometrical parameters of non-recessed hole entry worn hybrid conical journal bearing. λ = 1, θ = 30°, IR = 0, 1 and 2ab = 0.25, β* = 0.5, U = 10o (Fig. 1).
4.1 Variation of Stiffness S11 and S22 Coefficient with Speed Figure 1 represents the performance of various restrictors in worn and unworn condition under the variation of speed in the range of 0–2. Very small or negligible amount of change is observed in the stiffness coefficient S 11 for CFV and capillary restrictor, when they are subjected to variation in the speed parameter. However, it is reducing when the bearing is subjected wear. For orifice restrictor, the stiffness coefficient remains the same in unworn condition and reduces progressively when speed parameter varies in the range of 0–2. CFV restrictor is found to be better for acquiring the good stiffness coefficients followed by orifice and capillary restrictor. The graphical 2 plot describes the stiffness behavior of various flow controlling devices in worn/unworn condition under the variation of speed. Stiffness coefficient S 22 is found to be more for CFV restrictor than orifice and capillary in the worn out and new of non-recessed hybrid conical journal bearing. The reduction in the stiffness coefficient S 22 due to wear is also more in case of CFV restrictor followed by orifice and capillary restrictor. Fig. 1 Stiffness coefficient S 11 versus speed parameter
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Fig. 2 Stiffness coefficient S 22 versus speed parameter
4.2 Variation of Damping Coefficient C 11 and C 22 with Speed Parameter Ω Figure 3 depicts the variation of damping coefficients C 11 with respect to the change in the speed parameter ( = 0–2) for various flow controlling devices in worn and unworn conditions of the non-recessed hybrid conical journal bearing. Very small amount of change in the damping coefficient with change in the speed is observed for all the three types of flow controlling devices. CFV restrictor gives better performance as compared to the orifice restrictor followed by capillary restrictor. So, the comparative performance C 11 for non-recessed hole entry hybrid conical journal bearing may be noted as C 11 CFV > C 11 orifice > C 11 capillary. Figure 4 indicates the plot of the damping coefficient C 22 verses the speed parameter for 100 semi-cone angle hybrid conical bearing using different flow controlling devices in worn and unworn condition. Fig. 3 Damping coefficient C 22 versus speed parameter
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Fig. 4 Damping coefficient versus speed parameter
It is observed that small reduction in the damping coefficient up to speed parameter
= 0.5 and then it stabilizes for further rise in the speed parameter in case of capillary, orifice and CFV restrictor. Reduction in the damping coefficient due to wear is more in CFV restrictor than capillary and orifice restrictor. However, it is the same for capillary and orifice restrictor. So again, the performance of CFV restrictor is better compared to orifice and followed by capillary restrictor.
5 Conclusion 1.
2.
The stiffness coefficient S 11 and S 22 for CFV restrictor is highest followed by orifice and capillary. It remains constant for unworn condition and reduces as speed increases for worn condition. The damping coefficient C 11 and C 22 are highest for CFV restrictor followed by orifice and capillary. They are remaining almost constant with variation of speed.
So, it has been observed that flow control devices, when properly selected and tuned, can deliver very good bearing performance.
References 1. Malanoski SB, Loeb AM (1961) The effect of method of compensation on hydrostatic bearing stiffness. Trans. ASME, J. Basic Eng. 83:179–187 2. Ling TS (1962) On the optimization of the stiffness of externally pressurized bearings. Trans. ASME J Basic Eng 84:119–22 3. Raimondi AA, Boyd J (1954) An analysis of orifice and capillary compensated hydrostatic bearings. In: Proceedings of ASME/ASLE Joint Conference on Lubrication, 18–19 Oct 1954, Baltimore, Md
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4. Mayer JE, Shaw MC (1963) Characteristics of externally pressurized bearing having variable external flow restrictors. ASEM J Basic Eng 85:291 5. Cusano C (1974) Characteristics of externally pressurized journal bearings with membrane type variable flow restrictor as compensating elements. Proc IMechE 188:527–536 6. Cusano C, Conry TF (1974) Design of multirecess hydrostatic journal bearing for minimum total power loss. Trans ASME J Lubr Technol 94(1):226–232 7. O’Donoghue JP (1972) Parallel orifice and capillary control for hydrostatic journal bearings. Tribology 5:81–82 8. Pande SS, Somasundaram S (1979) Analysis of a four-pocket hydrostatic journal bearing with a position-sensing variable restrictor. Wear 54:331–341 9. Yoshimoto S, Anno Y, Amari K (1990) Static characteristics of hydrostatic journal bearings with a self-controlled restrictor employing floating disk. Trans JSLE 56:3360–3367 10. Palzewicz A (1992) Hydrostatic journal bearing without pressure chambers between bearing surfaces. Wear 159:31–38 11. Jain SC, Sinhasan R, Sharma SC (1992) Analytical study of a flexible hybrid journal bearing system using different flow control devices. Trib Int 25:387–395 12. Rippel HC (1965) Cast bronze hydrostatic bearing design manual, 2nd edn. Cast Bronze Bearing Institute, Cleveland, U.S.A., Apr 1965 13. Stout KJ, Rowe WB (1974) Externally pressurised bearings—design for manufacture Part-1(a). J Bearing Sel Tribol Int 7/3 98–106. Part-3(b) Design for liquid externally pressurized bearings for manufacture including tolerancing procedure. Tribol Int 7(3):195–212
A Comparative Study of Three Methods for the Computation of Determinants of Univariate Polynomial Matrices V. Safar , Anirban Nag , Bibekananda Patra , and Sandipan Bandyopadhyay
Abstract This paper compares three different methods for computing the determinant of a univariate polynomial matrix. From a systematic empirical study involving 100 trials of each numerical experiment, it appears that the method of FFT-IFT-based interpolation and evaluation performs the best in terms of both speed and accuracy of computation. The utility of this method is further demonstrated by a successful application to the classical problem of forward kinematics (FKP) of the general Stewart-Gough platform manipulator (SPM). Keywords Polynomial matrix · Determinant computation · Stewart platform
1 Introduction Problems in manipulator kinematics often lead to systems of multivariate polynomial equations. Some of the common methods for solving such systems involve the computation of resultants, e.g., using Sylvester’s dialytic elimination (see [7], p. 74). A key step in the process of computing such resultants is the computation of the determinant of certain matrices, (whose entries are univariate polynomials referred to henceforth as “polynomial matrices”). Therefore, the evaluation of the determinant of a polynomial matrix constitutes a central problem in studying the kinematics of robot manipulators. On the surface, the problem may seem fairly trivial, as the dimensions of the matrices involved are not very large, i.e., between 6 and 64 in many cases, and rarely above 100. Determinants of much larger matrices with purely numerical entries can be computed accurately and very quickly in standard computational systems. However, the polynomial nature of the matrices introduces the requirement that the entries of the matrix be treated symbolically, which necessitates the use of a computer algebra system (CAS). The symbolic nature of the computation not only reduces the speed greatly but also requires large amounts of RAM. Apart from the requirements V. Safar · A. Nag · B. Patra · S. Bandyopadhyay (B) Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India e-mail: [email protected] URL: https://ed.iitm.ac.in/~sandipan © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_139
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of computational speed and need for special hardware/software resources, accuracy is also an issue, as the roots of a polynomial of large degree can be highly sensitive to very small perturbations in the coefficients (see, e.g., [8]). Unfortunately, not all algorithms suited for symbolic computations are appropriate for a symbolic-numeric hybrid situation, which is the case here. Given such diverse and rather conflicting requirements, it is difficult to follow the computational path for finding the solutions of a polynomial system via computing resultants. However, since such a path offers several advantages, e.g., the ease of formulation and simplicity of the algorithm, the present paper makes an attempt to identify appropriate alternative computational strategies to obtain the said determinant in a fast and accurate manner, specifically for matrices of sizes which are common in kinematics. In this work, the CAS Mathematica is used to demonstrate the difficulties faced by a general-purpose “black-box” routine, namely, Det, in the scenario explained above. As can be seen in Sect. 3, this built-in routine can be used successfully only for very small polynomial matrices. Thereafter, two different alternatives are studied in detail. The first uses a recursive formula arising out of Newton’s identities to compute the invariants of a matrix, the last among which is the determinant. The recursive algorithm used here has an order of complexity O(n 4 ) when the input matrix of size n × n has completely sparse polynomial entries [4]. The other method is based on the evaluation-interpolation concept [5], which seems to perform accurately at high speed. Numerical experiments are conducted on matrices of size n = 3, . . . , 28 with cubic polynomials generated with randomly chosen real coefficients as their entries, to ascertain the computational speed and accuracy of the methods. Each experiment is repeated 100 times, and the results are presented visually via errorbars (see Fig. 1). Finally, the best-performing method is applied to the problem of the forward kinematics problem of a general Stewart platform manipulator, confirming its utility in such cases. The rest of the paper is organised as follows: Sect. 2 presents a brief description of the methods studied in this paper. The numerical experiments conducted are described in Sect. 3, followed by the application to the forward kinematics problem of a general Stewart platform manipulator in Sect. 4. The conclusions of the paper are presented in Sect. 5.
2 Mathematical Preliminaries The three methods used in this paper for the computation of the determinant of a matrix with (univariate) polynomial entries are described in brief below.
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2.1 Method 1: Direct Expansion of the Determinant Inside a CAS Most CASs offer in-built routines for the symbolic expansion of determinants. In this work, the routine Det available in the CAS Mathematica has been used primarily to establish the level of complexity of the calculations involved, which, in turn, indicates the necessity of customised algorithms, such as the following ones.
2.2 Method 2: Using Newton’s Identities Newton’s identities can be used to recursively compute the invariants (denoted by Ii ) of a matrix A ∈ Rn×n , starting with I1 = tr( A) and leading to det( A) = In , using the formula (see, e.g., Appendix B of [1]): k−1 k−1 (−1)k+1 i Ik = , I1 = tr( A), k ≤ n. (−1) Ii tr A tr( A) + k+1 i=1
(1)
2.3 Method 3: Evaluation-Interpolation Method with FFT and IFT This method relies upon the fact that the required determinant of a matrix A, whose elements are polynomials in a single variable (say, x) is also a polynomial in x. Moreover, the degree of the determinant polynomial (say, m) can be computed with the knowledge of the degrees of its entries alone. To compute the m + 1 coefficients of the determinant polynomial, it is evaluated at m + 1 distinct values of x. This results in a system of linear equations in the unknown coefficients, which is solved subsequently. The key appeal of this method lies in its simplicity. It is indeed trivial to choose m +1 distinct, random values of x, and evaluate the determinant at these values. Additionally, Horner’s rule can be used to speed up such computations [3]. However, a practical difficulty encountered frequently in the implementation of this method is that the Vandermonde matrix appearing as the coefficients of the unknown vector in the linear system described above is often numerically ill-conditioned. A solution to this problem has been presented in [5], in which the concepts of fast Fourier transformation (FFT) and inverse Fourier transformation (IFT) have been used to compute the unknown coefficients (details may be found in [5], which are not reproduced here due to lack of space).
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3 Numerical Experiments The methods described above are employed systematically to a set of problems in this section, which allows a quantitative comparison of the efficacy of the methods in terms of both speed and accuracy.
3.1 Setup of the Numerical Experiments The experiments involve a set of matrices, An , of sizes n × n, where n = 3, . . . , 28, each element of which is a cubic polynomial in x, with real coefficients. The steps of the experiments conducted with each An follow. 1. 2. 3.
4.
Coefficients of all the elements of An are generated randomly, in the range [10−3 , 103 ] using the function RandomReal of Mathematica. The determinant of An is computed using the three methods (whenever feasible) described in Sect. 3 and the corresponding CPU times are noted. The roots of the resulting determinants polynomials are then obtained using the gsl_poly_complex_solve routine of the GSL library for C/C++ (see [2], p. 29). Finally, the accuracy of each method is quantified in terms of the parameter n defined as the maximum absolute value of the determinant (computed via each method) over all its roots: n = maxi {| det( An )(xi )|}, i = 1, . . . , 3n.
(2)
Each method 100 trials for every value of n. However, for Method 1, underwent 7 being of O 106 , experiments for larger values of n are conducted only for Methods 2 and 3. The results are discussed in greater details below.
3.2 Comparative Analysis of the Results, for Accuracy Figure 1 depicts the erosion of accuracy with increasing n, for small values of n. As can be seen clearly, Method 1 degenerates unacceptably, and hence it is not studied further for larger matrices. The remaining two methods show comparable trends (see Fig. 2) in accuracy till n = 15, after which errors in Method 2 grow at a much faster rate, while those in Method 3 remain relatively unaffected by the increase in n. These observations may be explained by the fact that the number of arithmetic operations increases as O(n 4 ) in Method 2, leading to greater accrual of truncation errors.
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Fig. 1 Degeneration of accuracy for n = 3, . . . , 7 in all the methods
Fig. 2 Degeneration of accuracy for n = 3, . . . , 28 for Methods 2 and 3
3.3 Comparative Analysis of the Results, for Computational Speed Methods 2 and 3 are implemented in C++ and compiled using the -O2 optimiser of GCC compiler(Version 7.4.0) on a PC with an Intel Core i7-4930 CPU running at 3.40 GHz. The execution times shown in Fig. 3 are the CPU times consumed for the computation of det( An ) only, and not the previous or subsequent computations. As expected from the above discussions, up till n ≈ 15, the time taken by the two methods are comparable, but afterwards, Method 2 becomes much slower in
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Fig. 3 Variations of the execution times of Methods 2 and 3 for n = 3, . . . , 28
comparison with Method 3. In particular, for n = 28, the time taken by Method 2 is almost six times as that of Method 3.
4 An Application to Kinematics: FKP of the General SPM It appears from the above empirical study that Method 3 is likely to be the best method for applications in kinematics, e.g., in solving the FKP of the general SPM following [6] where n = 28. The details are skipped for the want of space, but it may be noted that when the problem is solved for the data given in [6], the determinant expansion takes on an average of 1.45 ms, with a standard deviation of 40.0 µs. The mean and standard deviation of n are 2.35 ×10−4 and 1.26 ×10−6 , respectively. For the same data, method 2 takes an average of 15.4 ms (over 100-trials), with a standard deviation of 39.6 µs. The mean and standard deviation of n are 6.89 ×10−1 and 5.21 ×10−3 , respectively.
5 Conclusions Studies of the accuracy and speed of computation of determinants of univariate polynomial matrices are presented in this paper, using three different methods. From the empirical results, the method of interpolation-evaluation using FFT and IFT emerges to be the fastest as well as the most accurate. An application to a benchmark problem in kinematics, i.e., the FKP of the general SPM, confirms the practical utility of this method.
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References 1. Bandyopadhyay S, Ghosal A (2004) Analytical determination of principal twists in serial, parallel and hybrid manipulators using dual vectors and matrices. Mech Mach Theory 39(12):1289–1305 2. Gough B (2009) GNU scientific library reference manual, 3rd edn. Network Theory Ltd. 3. Horner WG (1819) A new method of solving numerical equations of all orders, by continuous approximation. Philos Trans R Soc Lond 109:308–335 4. Horowitz E, Sahni S (1975) On computing the exact determinant of matrices with polynomial entries. J ACM 22(1):38–50 5. Hromˇcík M, Šebekt M (1999) New algorithm for polynomial matrix determinant based on FFT. In: 1999 European control conference (ECC). Karlsruhe, Germany, pp 4173–4177 6. Lee TY, Shim JK (2003) Improved dialytic elimination algorithm for the forward kinematics of the general Stewart-Gough platform. Mech Mach Theory 38(6):563–577 7. Salmon G (1876) Lessons introductory to the modern higher algebra. Hodges, Foster, And Co., Dublin, 3rd edn. 8. Wilkinson JH (1959) The evaluation of the zeros of ill-conditioned polynomials. Part I. Numerische Math 1(1):150–166
A Comparative Study of Different Numerical Scanning Strategies for Finding the Safe Working Zone of a 3-DoF Parallel Manipulator Bibekananda Patra , V. Safar , and Sandipan Bandyopadhyay
Abstract Identification of the safe working zone (SWZ) of a parallel manipulator can be an important part of its path-planning process or design. Computationally efficient and accurate methods are required to make these demanding calculations practically feasible. This paper looks at the Cartesian and polar variants of a 2-D numerical scanner which are applicable for 3-degrees-of-freedom (DoF) planar or spatial parallel manipulators and presents a detailed analysis of their relative speeds, advantages and disadvantages, based on a case study involving the spatial 3-RRS manipulator. Results of such a study provide the analysts with a deeper understanding of the functioning of the scanner, which in turn, helps the computation of the SWZ of other manipulators in a fast and reliable manner. Keywords Parallel manipulator · Safe working zone (SWZ) · Numerical scanning
1 Introduction Safe working zone of a parallel manipulator implies a subset of its workspace in which the manipulator is free to move without encountering either the gain- or the loss-type singularities, interferences among the links, or physical limits on the motions of the passive joints [1, 6]. Identifying such regions inside the workspace of a manipulator renders the task of path-planning (inside the SWZ) rather trivial.1 Also, the concept provides a mathematical basis for the design of parallel manipulator for certain a user-specified SWZ [2, 7]. Computation of the SWZ needs to be performed only once for an existing manipulator with a fixed geometry. However, it is a non-trivial task, as the various boundary functions (denoted by Si ), which demarcate the regions free of each of the above1 Obstacles
inside the SWZ are not considered in this case.
B. Patra · V. Safar · S. Bandyopadhyay (B) Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India e-mail: [email protected] URL: https://ed.iitm.ac.in/∼sandipan © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_140
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mentioned issues, are typically not available in analytical forms—these can only be evaluated numerically at the points of interest. Therefore, as explained in [1, 2, 6], one needs to scan (a subset of) the ambient space of the workspace of the manipulator to identify the regions free of each of the issues, and eventually leading to the identification of the SWZ. Such a scanning involves the overlaying of a grid on the region of interest and identifying the blocks of the grid which sees the boundary surfaces (defined by Si = 0), pass through them (an odd number of times for a given side of the block, to be precise). Several variations are possible, based on the chosen geometry of the grid, and the logic behind the expansion and contraction of their regions of coverage. These details may appear to be trivial on the prima facie; however, indeed, these determine the feasibility of the scanning scheme as a component of a design process, such as the ones described in [2, 7]. The computational requirements of such a design can be very high, as the SWZ needs to be computed typically for tens of thousands of candidate designs, so as to identify the optimal ones. In this context, this paper presents a comparative study between two scanning schemes for the plane: the Cartesian, used in [1], and the polar, used in [5]. The example of the spatial 3-RRS manipulator in [5] is re-worked in both the methods, and the results are compared for computational efficiency. Detailed analysis of the relative advantages and disadvantages is presented in a quantitative manner. Such a study is hoped to be highly useful for researchers in the field while choosing the appropriate scanner for their work, as per the architecture of the manipulator is concerned. The rest of the paper is organised as follows: Sect. 2 presents a brief description of the numerical scanning techniques studied in this paper. The case study on a spatial 3-RRS manipulator is described in Sect. 3. Finally, the paper is concluded in Sect. 4.
2 Numerical Scanning Techniques The scanners mentioned above operate on a simple strategy: they identify the zerolevel sets2 of the boundary functions, Si , one at a time (see [1] for the details; in particular, see Fig. 4, where the progressive refinement of the scanning grid is shown). The discrete nature of the grid allows the identification of only a finite number of points of these boundary curves (in the case of 2-D scanners, which is the case in point here). The point closest to the origin3 among the said points is identified next, which lies on the largest circle enclosing the points inside the boundary curve. Such circles are stacked up at a desired resolution along the axis orthogonal to the 2-D slices considered, and together, they define the 3-D region is free of the issue modelled by the associated boundary function. The process is then repeated for the next such issue, inside this region, and so on. 2 The
zero-level set of a function in a plane refers to a region which is a subset of the workspace of the manipulator. 3 Following the reason presented in [1], the SWZ is considered to be centred at the origin in this paper.
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In the following, two different scanners are described and compared, namely, the Cartesian and the polar, which follow the same general ideas as described above, but differ in the geometry of the grids used.
2.1 Cartesian Scanner In this case, the chosen search region is divided into a number of rectangular (typically square) blocks, as per a chosen initial resolution. If a boundary curve crosses (for an odd number of times) any side of a rectangle (or a grid line, equivalently), then the signs of the corresponding boundary function, when evaluated at the vertices delimiting the said grid line would differ. This indicates that the block is of interest, and hence it is subdivided, and the process is repeated in a recursive manner. This is done for all the blocks in the grid, till sizes of the smallest blocks in the grid reduce to a desired limit, at which point the geometric centre of the each of the smallest blocks is taken to be a point on the boundary curve.
2.2 Polar Scanner The grid, in this case, consists of concentric circles and radial lines, enclosing blocks which are of the form of pieces of an annulus. The peripheral dimension of the blocks would increase linearly as the distance of the blocks from the origin increase; hence, the number of divisions in each of the annuli is increased progressively with the distance from the origin, so as to keep the peripheral dimensions nearly constant in all the blocks, depicted in Fig. 1. The computational strategy is identical with the Cartesian scanner in all other aspects.
3 Case Study: Application to a Spatial 3-RRS Manipulator The two scanners described above are employed in this section to compute the SWZ of a spatial 3-RRS manipulator shown in Fig. 2, having the same geometry as described in [3]. The three legs have RRS architecture with identical dimensions. The fixed and moving platforms are in the form of equilateral triangles having a circumradius of b and a, respectively.
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Fig. 1 Change in peripheral resolution to keep the sizes of the blocks (B1 and B2 ) close together, that is, r1 δθ1 = r2 δθ2 , where r1 and r2 are the radii of the circles C1 and C2 , respectively
Fig. 2 Kinematic diagram of the spatial 3-RRS manipulator (taken from [3])
3.1 Setup of the Numerical Experiments The manipulator has 3-DoF, which can be considered to manifest as the roll (denoted by α), pitch (denoted by β), and heave (denoted by z) of the moving platform. The region of interest in the heave direction is z ∈ [0.1 m, 1.475 m]. A total of 2751 instances of the α-β planes are considered, orthogonal to the Z -axis, and spaced equally. The ranges of scanning in the 2-D slices mentioned above differ as per the geometry of the scanner. In the Cartesian scanner, only half of the desired rectangles in the α-β plane are scanned, since there is a symmetry in all the boundary functions about β = 0◦ ; hence, scanning for the region {α, β} ∈ [−180◦ , 0◦ ] × [−180◦ , 180◦ ] is sufficient. The initial resolution in α, β for the scanner is set at 0.5◦ , and after two stages of refinement, the final resolution becomes 0.125◦ . On the other hand, there
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is a three-way symmetry4 in the architecture of the manipulator, which is exploited naturally in the polar scanner, by scanning only for {r, θ } ∈ [0◦ , 180◦ ] × [0◦ , 120◦ ], where r, θ are the polar coordinates for the α-β plane, i.e. α = r cos θ and β = r sin θ . The initial resolution δθ for the outermost radius is 14.32◦ and final resolution after two stages of refinement is 0.06◦ . The resolution in r is 0.5◦ , which remains constant during the refinements of the grid. This shows that apart from the distinctions in their grid geometries, the two scanners also differ in the ways they can exploit various symmetries in the problem to achieve computational economies to different extents. Both the scanners have been implemented in the C programming language, and compiled using the gcc compiler, version 7.4.0, including the optimiser flag -O2. A single core of an Intel Core i7-4790 CPU with a speed of 3.60 GHz has been used for all the computations.
3.2 Details of the Boundary Functions The boundary functions are the same as those described in [5]. In brief, S1 = 0 demarcates the boundaries of the workspace; S2 = 0 defines the set of points where the manipulator suffers from gain-type singularities; S3 ≥ 0 identifies the regions in which at least two of the links interfere. The limitations on the motions of the passive joints are ignored for the sake of consistency in comparison, as these were not considered in [5], either. However, the definition of the function S3 is different in this case, as both the proximal and distal links in each limb are considered to be enclosed in bounding boxes of cylindrical shape (for the ease of detection of interference between these) in this paper, as opposed to the cuboidal bounding boxes used for the distal links in [5]. This approximation leads to much faster computations (following the approach presented in [4]) of interferences between the links, albeit being a bit more conservative.
3.3 Numerical Results The radii of the safe circles, ri , i = 1, 2, 3, corresponding to the boundary functions, Si , i = 1, 2, 3, are plotted against the extent of the heave motion of the top platform, as shown in Fig. 3. For any given value of the z, the difference between rSWZ computed from the Cartesian scanner differs from that the computed using polar scanner by 0.375◦ at the most. It has also been confirmed that the plots in Fig. 3 agree with the corresponding ones in [5].
4 The
manipulator has a three-way symmetry because of the fixed and moving platforms are equilateral triangles and the legs are identical in their architecture.
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Fig. 3 Variations of the safe radii along the heave direction
3.4 Comparison of Computation Times The analysis of the computational efficiency of the two scanners needs to be done carefully, for two reasons: (a) the extent of the search domains are different in the two scanners for the differences in their inherent symmetries (see Sect. 3.1 for the details) and (b) the total time taken in each scanner is a sum of the time taken for the generation of the initial grid, the refinements thereof and the consequent calls to the different boundary functions. In terms of the total time, the Cartesian scanner takes 214.22 s, while the polar variant consumes 309.86 s, both the numbers being averaged over 50 runs each (see Fig. 4d). This is in sharp contrast with Fig. 4a, where it can be seen that the polar scanner makes a much lesser number of function calls than the Cartesian for the first two boundary functions, and comparable number for the third. Further, Fig. 4b shows that the times per the function call are comparable between the two scanners in all three functions. Therefore, as seen in Figs. 4c, d, the total time taken in the calls to the respective functions Si consumed comparable times in the two scanners. This discrepancy is resolved in Fig. 4d, where it can be seen that the polar scanner takes a lot more time in the generation and refinement
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Fig. 4 Comparison of the Cartesian and polar scanner
of the grid; hence, the Cartesian scanner emerges as more economical in the overall sense. This observation is significant, since from the relative sizes of the scan regions alone, one may be lead to believe otherwise.
4 Conclusions This paper has presented a comparison of the Cartesian and polar scanners for the purpose of computing the SWZ of spatial parallel manipulators. The spatial 3-RRS manipulator is used as an example for a detailed case study. Various aspects of the scanners, e.g., certain symmetries in their grid structures and the manipulator’s architecture, geometric features, such as uniform versus non-uniform grid size, are brought out clearly. The comparisons show that for the resolutions and scan dimensions used in the paper, the Cartesian scanner requires more number of function calls in the initial stages of the scan, i.e. for the function S1 . However, since the run-time consumed by this function is very small, the overall time taken for the computation of
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all the boundary functions is comparable between the two scanners. The polar scanner, on the other hand, takes a longer time for the generation and the refinement of its grid. On the whole, the Cartesian scanner seems to be a better choice in this case, which is somewhat non-intuitive, as the Cartesian scanner utilises only a two-way symmetry as opposed to the three-way symmetry of the polar scanner.
References 1. Karnam MK, Baskar A, Srivatsan RA, Bandyopadhyay S (2020) Computation of the safe working zones of planar and spatial parallel manipulators. Robotica 38(5):861–885. https://doi.org/ 10.1017/S0263574719001139 2. Kilaru J, Karnam MK, Agarwal S, Bandyopadhyay S (2015) Optimal design of parallel manipulators based on their dynamic performance. In: Proceedings of the 14th IFToMM world congress. Taiwan, pp. 406–412. https://doi.org/10.6567/IFToMM.14TH.WC.OS13.129 3. Muralidharan V, Bose A, Chatra K, Bandyopadhyay S (2020) Methods for dimensional design of parallel manipulators for optimal dynamic performance over a given safe working zone. Mech Mach Theory 147. https://doi.org/10.1016/j.mechmachtheory.2019.103721 4. Nag A, Bandyopadhyay S (2019) Analytical determination of a sphere inside which the stewart platform translates without suffering any leg interference. In: Lenarcic J, Parenti-Castelli V (eds) Advances in robot kinematics 2018. Springer International Publishing, Cham, pp 74–82.https:// doi.org/10.1007/978-3-319-93188-3_9 5. Patel D, Kalla R, Tetik H, Kiper G, Bandyopadhyay S (2017) Computing the safe working zone of a 3-RRS parallel manipulator. In: Wenger P, Flores P (eds) New trends in mechanism and machine science, vol 43. Springer International Publishing, Cham, pp 113–120. https://doi.org/ 10.1007/978-3-319-44156-6_12 6. Srivatsan RA, Bandyopadhyay S (2014) Determination of the safe working zone of a parallel manipulator. In: Thomas F, Perez Gracia A (eds) Computational kinematics, vol 15. Springer Netherlands, Dordrecht, pp 201–208. https://doi.org/10.1007/978-94-007-7214-4_23 7. Tamilmani E, Bandyopadhyay S (2019) Computation of singularity free region and design optimisation of a four-degrees-of-freedom hybrid arm. In: Proceedings of the advances in robotics 2019. AIR 2019, Association for computing machinery, New York. https://doi.org/10.1145/ 3352593.3352673
Motion Control of a Phalange Using Tendon-Based Actuation System: A Bond Graph Approach Sandeep Kumar Uppal and Anand Vaz
Abstract Daily interactions with the environment by the hands such as grasping, manipulating, writing, pull/push, carry, and playing music require specific forces and some precise movements provided by the various elements of the human hand. Skeletal muscles play a significant role in locomotion of upper as well as lower limbs. The muscle tendon unit involves nonlinear characteristics. A biomechanical model has been proposed to control the motion of a phalange, actuated through the muscle tendon unit by a virtual phalange, a part of the control by the central nervous system. The authors propose a bond graph based mathematical model to control the motion of a phalange through a virtual system using tendon actuation. Simulation results validate the proposed bond graph model. Keywords Bond graph · Skeletal muscle · Nonlinear behavior · Muscle tendon unit · Virtual control
1 Introduction We perform most of our daily interactions with the environment by the hands such as grasping, manipulating, writing, pull/push, carry, playing music, and expressing. All these interactions require specific forces and some precise movements provided by the various elements of the human hand. Biomechanics of the human hand comprises the rigid bones termed as phalanges and muscle tendon structure. It is important to understand the musculoskeletal actuation system of the human hand to analyze its motion. It is difficult to understand the behavior of the muscle tendon unit (MTU) in-vivo as well as in-vitro. Most of the studies have been carried out on specific static posture aspects without considering the dynamics of the system. This work presents a bond graph [1, 2] -based mathematical model to control the motion of a phalange through muscle tendon actuation by taking inputs from a virtual system considered as a part of the control by the central nervous system (CNS). The CNS itself determines S. K. Uppal (B) · A. Vaz Dr. B. R. Ambedkar NIT, Jalandhar, Punjab 144011, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_141
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the required effort to be applied by the MTU at the respective insertion point of the tendon on the real phalange.
2 Proposed Model A system analogous to the real biomechanical system [3] has been proposed here for the development of the model. In real-life biomechanical system, every locomotion action which is to be performed by the body whether we talk of upper limbs or lower limbs movements requires some initiation at the level of brain. It is the brain which decides upon what action is needed for a specific motion. The brain then stimulates the CNS, which further send the specific controlled signals through the network of nerves to the corresponding skeletal muscles for the required actuation at their end. Then after receiving the controlled signal from the CNS, the skeletal muscles generate specific efforts to bring the body into motion. In order to describe the concept developed, we have considered one of the phalanges of the hand which is to bring in motion through muscle tendon actuation for the desired motion. The phalange has been considered as a cylindrical body for simplicity. In order to control the motion of this phalange considered as real phalange, another phalange with same specifications has been considered as a virtual phalange. There are no MTU in the virtual system but the insertion locations of the tendons on the virtual phalange have been considered and taken at the same locations as that in the real phalange. Figure 1 shows the system considered. The frame {1} is fixed at the bottom center of phalange, and the frame {0} also initially coincides with the frame {1}. In order to actuate the real phalange, four muscle tendons have been considered on the real phalange at symmetrical locations as shown in figure. These MTUs need to be actuated to provide the desired motion to the real phalange after getting the input stimuli from the virtual system. Corresponding to each MTU, there is a hook point in the system which is fixed with respect to the inertial frame, through which the muscle tendon freely glides like a pulley system. Tendons are attached to the insertion locations in the real phalange and are made to pass through these hook locations to provide specific direction of effort application and moment generation.
2.1 Muscle Tendon Units and Their Modeling Major part of the body is composed of skeletal muscles which is the prime mover for locomotion [4]. Skeletal muscles are viscoelastic in nature having a nonlinear behavior. These muscles possess very high strength to hold high tensile loads. A well-developed and extensively used concept based on Hill’s model [5] has been
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utilized to model the biomechanics of the skeletal muscles, referred here as MTU, i.e., muscle tendon unit, is shown in Fig. 2. Bond graph model of Hill’s model is shown in Fig. 2. Here T CE is the contractile element which develops the effort during active movement. There is no contribution from this element while in passive motion or in resting position. K PE is the parallel elastic member which resist passive stretching in the muscle and has a strain-dependent nonlinear behavior as represented by Eq. (1) below. f PE = 0.00163(exp7.66ε −1)
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Fig. 2 Physical and bond graph of Hill’s muscle model (MTU)
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Fig. 3 Force versus strain plot for nonlinear element (K PE ) of Hill’s model
There is a dynamic resistive element Bs considered as viscous damping element. And there is forth element in Hill’s model which is series elastic element K SE which represent the tendon interface with the bone. This K SE is much stiffer than K PE element. And this is the element through which the effort is applied to the bone. A passive model has been considered for the system actuation. The flow activation is provided at one end of the MTU, and the other end of the MTU applies the effort to the phalange at the tendon insertion locations. In our developed biomechanical model, a desired motion is given as an input to the tip of the virtual phalange. The input to the virtual phalange is considered as the initiation from the brain for the desired motion. This input brings the virtual phalange into motion as desired. As the task is to make the real phalange to follow the virtual phalange, there is a controller unit in the system which compares the flow available at the corresponding tendon insertion locations on the virtual and real phalange, respectively. The input to the controller is the flow difference from the virtual and the real phalange, based upon which the controller generates corresponding controlled effort signal to minimize the flow difference. The controlled effort signal thus generated is given as an input at the free end of the MTU where a small mass δm is attached to stimulate the MTU. The small mass of negligible inertia has been attached to the free end of the MTU is just to measure the flow corresponding to the effort signal being supplied by the controller unit. The flow from this small mass is given to the MTU which compares this flow with the flow at tendon insertion points on the real phalange. After comparing the two flows, MTU generates the required effort based upon the nonlinear behavior of the muscles, to be applied at the corresponding tendon insertion points to control the motion of real phalange. By providing this effort generated by the MTU at the insertion locations, the real phalange moves to the desired position and follows the motion of the virtual phalange
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as desired. Multibond graph approach has been used to model the biomechanical system.
2.2 Multibond Graph Model A detailed multibond graph model [6] based on rigid body dynamics has been demonstrated for real and virtual phalange system as shown in Fig. 4. The figure below shows the real phalange connected with single MTU, and similarly, the real phalange is connected to other MTUs. The same muscle tendon structure has been used for the other MTUs. Figure 4 clearly demonstrates the virtual and real phalange rigid body mechanics and the controller part to control the motion of real phalange through muscle tendon actuation. MTF modulus C0 1 r O11v × is the skew symmetric T 0 matrix obtained from position vector C0 1 r O11v = C0 1 x O11v C0 1 y O11v C1 z O11v . Similarly, other MTF moduli have been evaluated. MTF modulus 0p rˆ¯H is the unit vector describing the direction along which the effort is applied. The desired flow is provided to the virtual phalange through bond 91v. Flow information from the virtual and real phalange is taken from bond 107v and 107, respectively. The controller compares these flows and generates an effort at bond 12 through the C and R elements (PD Control) connected to it at bond 10 and 11, respectively. This controlled effort signal is then passed on to the MTU through the small mass δm connected at free end of the muscle tendon. MTU then develops an effort based upon the nonlinear behavior of the muscle tendon, to be applied at the insertion location on the real phalange through bond 107. By this effort, the real phalange moves and tracks the virtual phalange as desired.
3 Simulation and Results The biomechanical behavior [4] of the phalange motion using tendon-based actuation, with the help of proposed model, has been simulated. Simulation code has been written from the bond graph model itself using MATLAB. The mass of the virtual and real phalange is taken as 0.0055 kg. The phalanges have been considered as cylindrical bodies having radius of 0.005 m. The behavior of the model is studied under two conditions using MATLAB coding. Case I: The virtual phalange is dragged in −ve X-direction in inertial frame in a trajectory given by polynomial equation 2. The motion in rotational domain is permitted only about Z-axis, that has been considered as flexural axis of phalange and the rotation in other two axes is constrained. Corresponding to movement of virtual phalange, the real phalange tracks the trajectory of virtual phalange through muscle tendon actuation. Figure 5 shows the position trajectory of the center of mass of the virtual and real phalange over the time period w.r.t inertial frame.
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Fig. 4 Multibond graph model of virtual and real phalange with controller and MTU
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x(t) ˙ = 0.0031 t 7 − 0.0219 t 6 + 0.0525 t 5 − 0.0437t 4 , 0 ≤ t ≤ 2s
(2)
Case II: Initially, the virtual phalange is dragged in −ve X-direction in inertial frame in flexure motion by using polynomial trajectory mentioned in equation 2 above and then is rotated in inertial X–Z plane about the initial axis of symmetry of phalange with a radius of 10 mm. The real phalange follows the trajectory of the virtual phalange. Figure 6 shows the position trajectory of the center of mass of the virtual and real phalange over the time period w.r.t inertial frame, and Fig. 7 shows the forces applied to the real phalange by the muscle tendon unit. 10
-7
2
0.02 Real x-cord.
Virtual x-cord. Virtual y-cord. Virtual z-cord.
0.01
1
Real y-cord. Real z-cord.
0 SE1 SE2 SE3 SE4
-1 0 -2 -3
-0.01 0
1
2
3
4
5
6
Fig. 6 Position of center of mass of phalange
7
0
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3
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-3
10
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1 0.8
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0.2
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1
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0 0
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Fig. 7 Force in muscle tendons unit
4 Conclusion It is difficult to understand the behavior of the MTU in-vivo as well as in-vitro. The proposed model provides a novel alternative to the in-vivo as well as in-vitro studies. The real phalange tracks the motion of the virtual phalange according to the desired trajectory. The model developed can be used to describe the force pattern and deformation pattern of the nonlinear MTU for providing a desired kind of motion through muscle tendon actuation. The model has been simulated to study the muscle tendon behavior for different motions given to the phalange.
References 1. Karnopp DC, Margolis D, Rosenberg RC (2000) System dynamics: modeling and simulation of mechatronic systems. Wiley-Interscience, New York 2. Mukherjee A, Karmakar R (2000) Modeling and simulation of engineering systems through Bondgraphs. Narosa Publishing House, New Delhi 3. Valero-Cuevas FJ, Elise Johanson M, Towles JD (2003) Towards a realistic biomechanical model of the thumb: the choice of kinematic description may be more critical than the solution method or the variability/uncertainty of musculoskeletal parameters. J Biomech 36:1019–30 4. Fung YC (1993) Biomechanics, mechanical properties of living tissues. Springer 5. Vaz A, Singh K, Dauphin-Tanguy G (2015) Bond graph model of extensor mechanism of finger based on Hook—string mechanism. MAMT 91:187–208 6. Mishra N, Vaz A (2017) Bond graph modeling of a 3-joint string-tube actuated finger prosthesis. Mech Mach Theory 117:1–20
Taguchi Optimization for Wear Behaviour of Drum Brake Shoe Interface Vaibhav A. Kalhapure and H. P. Khairnar
Abstract The effect of parameters on wear behaviour of drum brake shoe is optimizing using Taguchi method. Pin samples made from brake shoe material is tested on pin-on-disc test rig against disc of same material as of drum. The effects of variation in load (40, 60 and 80 N), sliding speed (300, 600, 900, 1200 rpm) and track diameter (40, 60, 80 mm) on wear rate of shoe surface were examined. From signalto-noise ratio (S/N) and analysis of variance (ANOVA) of experimental results, the major parameter responsible for wear of drum brake shoe is found and different conditions are suggested for improvement in wear life and reliability of brake. Keywords Wear volume · SN ratio · Taguchi optimization · ANOVA
1 Introduction The most substantial safety aspect of a vehicle is its braking system, which retard the motion of vehicle quickly and reliably under varying operating conditions. Brake shoe of an automotive drum brake system indicates one of the most complex composite materials, since they have numerous constituents that are diverse in different properties. The effect of various operating parameters causes wear of shoe material. Different researchers studied the Taguchi optimization for the influence of operating parameters on wear behaviour of materials. Deuis et al. studied the impact of load, sliding conditions like distance, speed, time and volume percentage on wear of different composites under dry sliding condition [1]. Xu et al. examined that the sliding motion had major influence by contact of parts and behaviour of material under different tribological condition. The formulation of suitable composite for improving wear-resistant properties, evaluation of relation between various parameter and wear rate have been necessary [2]. Montgomery suggested that the Taguchi method has been used for optimizing designs and to achieve different manufacturing parameters among the various techniques of experimental design [3]. This technique was specially designed to minimize the number of experiments and to investigate the V. A. Kalhapure (B) · H. P. Khairnar Veermata Jijabai Technological Institute, Mumbai 400019, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_142
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influence of interaction between different factors on performance of system [4, 5]. Manivel et al. investigated the selection and optimization of cutting parameters for machining of austempered ductile iron using Taguchi L18 orthogonal array [6]. The Taguchi optimization was used by many researchers for different types of industrial and other applications like hard turning tool wear, and wear process parameters on 16-Cr Ferritic ODS steel. Similarly, variance analysis was used to analyze the effect of these parameters on wear performance of different applications [7–9]. The objective of current work is to optimize wear behaviour of drum brake shoe due to the effect of various operating parameters. Different sliding speed, applied load and track diameter are selected as control factors, and the response parameter is wear volume. The experimental test design is basically done by using L25 orthogonal array. By the use Taguchi method testing of optimal parameters is determined for wear volume. Experimentation is conducted on pin-on-disc test rig for various operating conditions. Finally, analysis of variance (ANOVA) is performed to know the impact of individual and multiple parameters on response variable.
2 Experimental Design 2.1 Materials The pin samples used for wear test, which was made up of commercial automotive brake liner material. The friction materials of brake liner were attached to mild steel pin with the help of strong epoxy adhesive. The pin samples were machined hemisphere shape at end as per ASTM G-99 standards, such that each pin was of 30 mm in length and 12 mm circular in cross section. The cast iron disc was casted with expendable mould in dry sand casting of dimension of 162 mm in diameter and 8 mm in width as that of same material of drum of automotive vehicles. In wear test, the pin samples rotating against cast iron disc to examine the wear behaviour of brake samples at different operating conditions.
2.2 Selection of Process Variables and Design Levels The drum brake performance was strongly influenced by applied load, sliding speed and interface radius, and the performance was affected by these design parameters. The optimum value of each parameter is necessary for increasing the wear behaviour and performance of braking system, and the below parameters are considered for the experiments, as listed in Table 1. The main parameters used are load, the sliding speed and the track radius. Parameters and their levels are shown in Table 1.
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Table 1 Levels of different parameters Parameters
Levels 1
Applied load (N)
2
3
4
5
40
60
80
Sliding speed (rpm)
300
600
900
1200
1500
Track radius (mm)
20
30
40
50
60
2.3 Experimental Setup The wear experiments were carried out on the pin-on-disc tribometer for studying the dry sliding behaviour of brake friction material as shown in Fig. 1. The wear tests were conducted by varying the load from 40 to 80 N, speed from 300 to 1500 rpm and track radius from 20 to 60 mm, while keeping the sliding distance constant at 1000 m. Wear volume of pin samples was calculated by using the formula, Wear Volume mm3 =
πh 6
3d 2 + h2 4
2 1/2 where h = r − r 2 − d4 d is the wear scare diameter (mm) and r is the pin end radius (mm).
Fig. 1 Pin-on-disc test rig
(1)
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3 Results and Discussion There are different orthogonal arrays available for the design of experiments; each of the arrays is indicating the definite number of independent variables which has to be design and their levels. In this study, experimental test is designed for the three and for five levels, then total number of experiment is 243. The total number of results provide by the system which is about 170 exact results. In association with the above method, Taguchi method provides the list 25 experimental combinations based on the orthogonal array in a specific order which cover all factors. Those twenty-five experimental test results will give 99.96% accurate result. By using Taguchi method of optimization, number of experiments reduced to 25 instead of 243 with almost similar accuracy. Statistical analysis of data collected from experimental data was done using Minitab V.17.1.0. Analysis of test results is done in five stages. The relation between response variable in the form of mathematical formulation is developed in terms of wear volume as a control variables using regression analysis in the first stage. In second stage, ANOVA is done for investigating the significant parameters affecting the wear volume of brake sample. In the last stage, optimal values for control variables for best response factor are recognized using Taguchi technique. It is applied to the 25 experimental test results conducted as per L25 orthogonal array. Table 2 endows orthogonal array along with experimental test results and S/N values for the response variable. Equation (1) predicts this relationship with an adjusted R2 of 98.21%. The R2 value is a measure of goodness of fit between observed data and modelled data. According to the model developed, 98.21% variability in the dependent variables is predictable from the independent variable. Hence, it ensures that predicted model gives fairly good clarification of correlation between input variables and the response. Wear volume = −4.39123 ∗ 10−8 + 1.66169 ∗ 10−9 . load + 3.52618 ∗ 10−11 . Sliding Speed + 1.19016 ∗ 10−9 . Interface Radius
(2)
Table 3 gives the results of ANOVA test for wear volume of brake sample at 95% confidence level. The results illustrate that load and sliding speed are significant factors as their p-value is less than 0.05, whereas interface radius is not significantly influencing response variable. Further, it is observed that the applied load has maximum influence (35.40%) on wear volume. The other factors like sliding speed (33.88%) and interface radius (29.67%) have less influence on response variable (Figs. 2 and 3).
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Table 2 Experimental design using L25 orthogonal array Run No
Applied load (N)
Sliding speed (rpm)
Track radius (mm)
Wear volume (mm3 )
SNR
1
40
300
20
2.206190E−08
153.127
2
40
600
30
3.077870E−08
150.235
3
40
900
40
5.179890E−08
145.714
4
40
1200
50
8.507420E−08
141.404
5
40
1500
60
1.265410E−07
137.955
6
60
300
30
3.957570E−08
148.051
7
60
600
40
7.286360E−08
142.750
8
60
900
50
9.539690E−08
140.409
9
60
1200
60
1.420365E−07
136.952
10
60
1500
20
7.736500E−08
142.229
11
80
300
40
5.969990E−08
144.481
12
80
600
50
1.078155E−07
139.346
13
80
900
60
1.585744E−07
135.995
14
80
1200
20
7.502410E−08
142.496
15
80
1500
30
1.416027E−07
136.979
16
40
300
50
4.024230E−08
147.906
17
40
600
60
6.190361E−08
144.166
18
40
900
20
3.366200E−08
149.457
19
40
1200
30
5.519380E−08
145.162
20
60
1500
40
1.200239E−07
138.415
21
60
300
60
7.024150E−08
143.068
22
60
600
20
4.321440E−08
147.287
23
80
900
30
7.189900E−08
142.866
24
80
1200
40
1.414072E−07
136.991
25
80
1500
50
2.068510E−07
133.687
Table 3 Results of ANOVA for wear volume Source
DF
Seq SS
Adj SS
Adj MS
F-value
P-value
Contribution (%)
Load
2
209.17
134.58
67.29
151.97
0.000
35.40
Sliding Speed
4
200.27
201.24
50.31
113.62
0.000
33.88
Interface Radius
4
175.80
175.80
43.95
99.26
0.000
29.67
Error
14
6.19
6.19
0.44
Total
24
592.44
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V. A. Kalhapure and H. P. Khairnar Main Effects Plot for SN ratios Data Means
Load
15.5
Interface Radius
Sliding Speed
Mean of SN ratios
15.0 14.5 14.0 13.5 13.0 12.5 40
60
80
300
600
900
1200
1500
20
30
40
50
60
Signal-to-noise: Smaller is better
Fig. 2 Effect plots for SN ratios on wear behaviour of brake friction materials
Main Effects Plot for Means Data Means
Load
0.25
Interface Radius
Sliding Speed
Mean of Means
0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 40
60
80
300
600
900
1200
1500
20
30
40
50
60
Fig. 3 Effect plots for means on wear behaviour of brake friction materials
4 Conclusions Wear behaviour of drum brake friction lining sample is inspected by conducting sliding wear test for 25 experimental test combinations as per L25 orthogonal array. The effect of three control features like applied load, sliding speed and interface radius, wear volume is studied during these tests by varying their levels after applying the load as a response variable. Based on the test results of brake specimens, the following conclusions are summarized. • The wear volume of brake friction material sample is optimized depends on dry sliding wear test by implementing Taguchi method effectively. Empirical multiple
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linear regression equation is formulated with adjusted R2 of 98.21% for predicting the response variable, wear rate. • 80 N applied load, 1500 rpm sliding speed, and 50 mm interface radius are the optimum situations, which attains minimum wear volume. • As per ANOVA test results, the load is the most significant factor and having maximum influence (35.40%) on wear volume followed by sliding speed (33.88%) and interface radius (29.67%).
References 1. Deuis L, Subramanian C, Yellup M (1996) Abrasive wear of aluminium composites—a review. Wear 201:132–144 2. Xu W, Ma X, Tang N, Zhu L, Li W, Ding Y (2015) Effect of post-welding heat treatment on wear resistance of cast-steel die with surfacing layer. Manuf Rev 2:25 3. Montgomery DC (1997) Design and analysis of experiments. Wiley, New York 4. Peace GS (1993) Taguchi methods: a hands-on approach. Addison- Wesley, Reading 5. Kim SJ, Jang H (2000) Friction and wear of friction materials containing two different phenolic resins reinforced with aramid pulp. Tribol Int 33:477–484 6. Manivel D, Gandhinathan R (2016) Optimization of surface roughness and tool wear in hard turning of austempered ductile iron (grade 3) using Taguchi method. Measurement 93:108–116 7. Mia M, Dey PR, Hossain MS, Arafat MT, Asaduzzaman M, Ullah MS, Zobaer ST (2018) Taguchi S/N based optimization of machining parameters for surface roughness, tool wear and material removal rate in hard turning under MQL cutting condition. Measurement 122:380–391 8. Kuntoglu M, Saglam H (2019) Investigation of progressive tool wear for determining of optimized machining parameters in turning. Measurement 140:427–436 9. Dharmalingam G, Mariappan R, Prasad MA (2019) Optimization of wear process parameters on 16-Cr Ferritic ODS steel through Taguchi approach. Materials Today: Proceedings
Analysis of a Hydrodynamic Journal Bearing of Circular Cross Section Lubricated by a Magnetomicropolar Fluid Debasish Tripathy and Kingshook Bhattacharyya
Abstract An infinite journal bearing of circular cross section lubricated with a fluid containing magnetic micropolar inclusions has been considered for this study. Unlike most analyses dealing with ferrofluid bearings where the magnetic forces are mainly dealt with as separate entities with their own physics, in this study the idea of micropolar fluids has been extended to consider magnetic inclusions. A rotating coordinate system has been used for deriving the dynamic parameters of the bearing. In addition to the above, the electric field and flow rate relations have been perturbed by use of electromagnetic boundary conditions to eliminate the same from the Reynolds equation. The basic Reynolds equation and its perturbed forms have been solved analytically to derive the stiffness and damping coefficients in the rotating coordinate frame. Keywords Magnetomicropolar · Bearing · Dynamic stiffness · Damping
1 Introduction Magnetohydrodynamic bearings first evoked interest at the beginning of the second half of the last century. Initial studies by Kuzma [1, 2] and Sasada et al. [3] considered a fully flooded bearing. Kamiyama [4] introduced Reynolds boundary condition, thus considering a partially flooded bearing. Malik and Singh [5] extended such analyses to the case of a finite bearing. Stability behavior of finite magnetohydrodynamic journal bearings was reported by Kulkarni and Rao [6]. One of the earliest attempts at analyzing micropolar lubrication of journal bearings was by Prakash and Sinha [7]. Dynamic characteristics of such a bearing were studied by Huang and Weng [8]. Detailed stability analysis of such a bearing in a rotating coordinate frame was done by Das et al. [9]. The momentum conservation equations for a micropolar magnetic suspension are discussed in the works of Tanahashi and Nakai [10] and Ido [11]. The present study is an effort to incorporate these ideas and present a unified D. Tripathy · K. Bhattacharyya (B) Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721302, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_143
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Fig. 1 Schematic diagram of MMHD journal bearing
model of magnetomicropolar lubrication of a journal bearing. Perturbed forms of the electromagnetic boundary conditions and the consequent flow rate and electric field relationships have been derived as also the stiffness and damping coefficients.
2 Formulations 2.1 Assumptions The bearing is assumed to be infinitely long and of circular cross section as shown in Fig. 1. The radius of the shaft is R, the center to center distance between the bearing and the shaft is e, with the maximum being the clearance C, while the attitude angle is φ. W is the weight of the shaft. The x, y coordinate system is as described in the figure. The film thickness h is a function of x. The shaft rotates with a speed of ω and precesses with a speed of ωp . The external magnetic field is radial and is given by the relation B = −B0 Rr −1 , where r is the radial distance of a point within the bearing from the origin. The viscosity coefficients of the lubricant are μ and χ electrical conductivity is σ, pressure of the fluid film is p, and ez is the electric field in z direction.
2.2 Non-dimensionalization The non-dimensional variables are as follows y h u pC 2 x ,η = , H = ,U = ,P = , R h C ωR μω R 2 1 γ σ σ C μ M= B0 C, E = ez , N = ,L = μ μ ωR 2μ + χ C 4μ
θ=
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N: coupling number (ratio of rotational and Newtonian viscous forces), L: nondimensional microstructural length parameter.
2.3 System Equations ∂ At the onset, we define the operators Dη = ∂η ,Dθ = ∂θ∂ and Dτ = ∂τ∂ . The non-dimensionalized coupled linear and angular momentum conservation equations for Couette flow are
H −2 Dη2 U + 2N 2 H −1 Dη − 1 − N 2 M 2 U = 1 − N 2 Dθ P − 1 − N 2 M E (1) −1 H −2 Dη2 − 0.5L −2 N −2 − 1 2 + H −1 Dη U = 0 (2) The fluid velocity in a rotating frame at a section with thickness H is obtained as U = M −2 (Dθ P − M E){F8 (η) − 1} + F7 (η) − F8 (η)Dτ φ
(3)
Table 1 sequentially describes the parameters and functions required to evaluate U. The non-dimensional flow rate is obtained as Table 1 Parameters and functions required to evaluate U A = 1 − N 2 M 2 + N 2 L −2 , B = (1 − N )2 M 2 − N 2 L −2 (1 + N )2 M 2 − N 2 L −2 √
√
α = 0.5 A + B , β = 0.5 A − B , K α = αL 2 N −2 − 1 − α −1 , K β = β L 2 N −2 − 1 − β −1
1 (H ) = 2K α K β {1 − cosh(α H )cosh(β H )} + K α2 + K β2 sinh(α H )sinh(β H ) F1 (α, β, H ) = K α sinh(α H )sinh(β H ) + K β {1 − cosh(α H )cosh(β H )} F2 (α, β, H ) = K α sinh(α H )sinh(β H ) + K β (1 + cosh(α H ))(1 − cosh(β H )) F3 (α, β, H ) = F1 (β, α, H ), F4 (α, β, H ) = F2 (β, α, H ) F5 (α, β, H ) = K β sinh(α H ) cosh(β H ) − K α sinh(β H ) cosh(α H ) F6 (α, β, H ) = K α sinh(β H ) − K β sinh(α H ) + F5 (α, β, H ) F7 (η, H )1 = K α {F1 cosh(α H η) + F5 sinh(α H η)} + K β {F3 cosh(β H η) − F5 sinh(β H η)} F8 (η, H )1 = K α {F2 cosh(α H η) + F6 sinh(α H η)} + K β {F4 cosh(β H η) − F6 sinh(β H η)} FB (H ) = F1 M 1 K α α sinh(α H ) +
F5 α (cosh(α H ) − 1)
+ Kβ
F5 F3 β sinh(β H ) + β (cosh(β H ) − 1)
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1 Q=
U H dη = M −2 (Dθ P − M E){2FB (H ) − M H } + FB (H )(1 − 2Dτ φ)
0
(4) The non-dimensional Reynolds equation in rotating frame is Dθ Q + Dτ H = Dθ M −2 (Dθ P − M E){2FB (H ) − M H } + FB (H )(1 − 2Dτ φ) + Dτ H = 0 (5) Following references [4, 8], for an insulated bearing in open-circuit condition we have θc
θc H (θ )dθ = M Q
E 0
dθ
θc : cavitation boundary
(6)
0
2.4 Perturbed Equations and Solutions We introduce the following perturbations which are standard in stability analysis of hydrodynamic bearings. Additionally, the electric field and flow are also perturbed. H0 = 1 + ε0 cos θ, H = H0 + cos θ ε1 eiλ R τ + sin θ ε0 φ1 eiλ R τ , ε = ε0 + ε1 eiλ R τ , φ = φ0 + φ1 eiλ R τ , P = P0 + P1 ε1 eiλ R τ + P2 ε0 φ1 eiλ R τ , E = E 0 + E 1 ε1 eiλ R τ + E 2 ε0 φ1 eiλ R τ ; Q = Q 0 + (Q 1Re + i Q 1Im0 + i Q 1Im sin θ)ε1 eiλ R τ + (Q 2Re + i Q 2Im0 + i Q 2I m cos θ)ε0 φ1 eiλ R τ We get the perturbed flow rates and the corresponding equations from (3) and (4) Q 0 = M −3 (Dθ P0 − M E 0 )g0 + M −1 f 0 , Dθ Q 0 = 0
(7)
Q 1Re = M −3 (Dθ P1Re − M E 1Re )g0 + 0.5M −1 f 0 + M −3 (Dθ P0 − M E 0 )g0 cos θ, Dθ Q 1Re = 0
(8)
Q 1Im0 + Q 1Im sin θ = M −3 (Dθ P1Im − M E 1Im )g0 , Dθ [Q 1Im0 + Q 1Im sin θ ] = −λ R cos θ
(9)
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Q 2Re = M −3 (Dθ P2Re − M E 2Re )g0 + 0.5M −1 f 0 + M −3 (Dθ P0 − M E 0 )g0 sin θ, Dθ Q 2Re = 0 Q 2Im0 + Q 2Im sin θ = −
(10)
1 (Dθ P2Im − M E 2Im )g0 M3
2 λR f 0 , Dθ [Q 2Im0 + Q 2Im sin θ ] + λ R sin θ = 0 M ε0
(11)
The functions and variables used above are described in Table 2. The following expressions for electric fields are then obtained using (5) E 0 = Mθc Q 0 , E 1Re = Mθc Q 1Re − Mθc sin θc 2 Q 0 , E 1Im = Mθc Q 1Im0 + M(1 − cos θc )Q 1Im E 2Re = Mθc Q 2Re − Mθc (1 − cos θc )2 Q 0 , E 2Im = Mθc Q 2Im0 + M sin θc Q 2Im (12) Applying Reynolds boundary conditions P0 (0) = P0 (θc ) = Dθ P0 (θc ) = 0, we now get M f 0 (θc ) − E 0 g0 (θc ) , M2 P0 (θ ) = M 3 Q 0 F01 (θ ) − M 2 F02 (θ ) + M E 0 θ
Q0 =
(13)
Applying boundary conditions Pi (0) = Pi (θc ) = 0 for the perturbed pressures, we get Table 2 Variables and functions related to flow equations and pressure profiles = (θc + ε0 sin θc )−1 , f i = FB (Hi ), gi = 2FB (Hi ) − M Hi , F01 (θ) = θ 0
θ 0
g0−1 dθ, F02 (θ) =
f 0 g0−1 dθ
θ F1Re1 (θ) = − (Dθ P0 − M E 0 )g0−1 g0 cos θdθ
F1Re2 (θ) =
0
F1Im2 (θ) =
θ 0
F2Re1 (θ) = −
M 2 Mg0−1 sin θ + (1 − cos θc ) dθ
θ 0
(Dθ P0 − M E 0 )g0−1 g0 sin θdθ
F (θ) =
2Im2 θ 3 g −1 cos θ + sin θ − 2 f ε −1 g −1 dθ M c 0 0 0 0 0
θ 0
F1Im1 (θ) =
0.5M 2 g0−1 f 0 cos θdθ
θ 0
θ
M 2 Mg0−1 + θc dθ
0.5M 2 go−1 f 0 sin θdθ
θ F2Im1 (θ) = 0 M 2 Mg0−1 + θc dθ
F2Re2 (θ) =
0
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M 2 θc2 2 sin θc Q 0 + F1Re2 (θc ) − F1Re1 (θc ) , F1Re3 (θc ) + M 2 θc2 F1Im2 (θc ) , Q 1Im2 = −λ R , = λR F1Im1 (θc )
Q 1Re = Q 1Im1
M 2 θc 2 (1 − cos θc )Q 0 + F2Re2 (θc ) − F2Re1 (θc ) , F2Re3 (θc ) + M 2 θc2 F2Im2 (θc ) , Q 2Im2 = −λ R = λR F2Im1 (θc )
(14)
Q 2Re = Q 2Im1
P1Re (θ ) = F1Re1 (θ ) − F1Re2 (θ ) + Q 1Re M 3 F01 (θ ) + M 2 θc θ − M 2 θc sin θc Q 0 2 θ F1Im2 (θc ) F1Im1 (θ ) − F1Im2 (θ ) P1Im (θ ) = λ R F1Im1 (θc ) P2Re (θ ) = F2Re1 (θ ) − F2Re2 (θ ) + M 3 F01 (θ ) + M 2 θc θ Q 2Re − M 2 θc (1 − cos θc )2 Q 0 F2Im2 (θc ) f 2Im1 (θ ) − λ R f 2Im2 (θ ) P2Im (θ ) = λ R F2Im1 (θc )
(15)
(16)
(17)
The stiffness and damping coefficients may now be evaluated according to expressions provided in [9].
3 Results and Discussion The variation of non-dimensional pressure profile, stiffness and damping coefficients (for M = 8, ε = 0.8, λ R = 1, L = 0.1) with N is presented in Fig. 2. It can be seen that a value of N > 0.7 significantly changes the nature of both stiffness and damping
Fig. 2 Effect of variation of coupling number on pressure P0 , stiffness (S ij ) and damping (Dij )
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Fig. 3 Effect of variation of length parameter on pressure P0 , stiffness (S ij ) and damping (Dij )
and causes a significant drop in peak pressure. Additionally, it may be noted that with increase in N, S Rφ and both DφR and Dφφ which are already negative become drastically more so after N = 0.7. This indicates that while increasing N beyond 0.7 may apparently lead to a significant increase of load bearing capacity, the bearing may become unstable. Same variations with L (for M = 8, ε = 0.8, λ R = 1, N = 0.8) are presented in Fig. 2. Here, the critical value of L is 0.1. S φR , DφR and Dφφ which are already negative decrease sharply till this value. Pressure profile also remains largely unaltered beyond this value indicating that increasing L beyond this value may not be needed to improve load bearing capacity and stability will also not be affected if the bearing is stable up to this point. Hence, choosing L beyond this value does not help much (Fig. 3). The presence and development of negative stiffnesses and dampings indicate that simply opting for higher values of N and L while apparently indicating a benefit in terms of increased load bearing capacity may lead to instability. Hence for optimal bearing design, a detailed stability analysis considering the effects of M, L, N is required to obtain the optimal values of these parameters for the lubricant to get optimal performance with assured stable operation.
References 1. Kuzma DC (1963) The magnetohydrodynamic journal bearing. J Basic Eng TRANS ASME Ser D 85(3):424–427 2. Kuzma DC (1964) The finite magnetohydrodynamic journal bearing. J Basic Eng TRANS ASME Ser D 86(4):445–448 3. Sasada T, Kurosaki Y, Honda K, Kamijo K (1974) MHD journal bearing in a magnetic field perpendicular to its axis 1st report, analysis of an infinitely long bearing. Bull JSME 17(114):1645–1651 4. Kamiyama S (1969) Magnetohydrodynamic journal bearing (Report 1). Trans ASME J Lub Tech 91:380–386 5. Malik M, Singh DV (1980) Analysis of finite magnetohydrodynamic journal bearings. Wear 64:273–280
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6. Kulkarni PA, Rao BVA (1977) Stability behaviour of finite MHD journal bearings. Mech Mach Theory 12:293–302 7. Prakash J, Sinha P (1975) Lubrication theory for micropolar fluids and its application to a journal bearing, ht. J Eng Sci 13:217–232 8. Huang TW, Weng CI (1990) Dynamic characteristics of finite-width journal bearings withmicropolar fluids. Wear 141:23–33 9. Das S, Guha SK, Chattopadhyay AK (2005) Linear stability analysis of hydrodynamic journal bearings under micropolar lubrication. Tribol Int 38:500–507 10. Tanahashi T, Nakai T (1995) Fundamental nonlinear theory for micropolar electrically conducting fluids. JSME Int J Ser B 38(2):273–279 11. Ido Y (2005) Basic equations and constitutive equations of micropolar magnetic fluids with E-B analogy and the Abraham expression of electromagnetic momentum. JSME Int J Ser B 48(3):273–279
Twin-Plate Turbine Using Parallel Four-Bar Mechanisms Bhanu Vardhan Chennoju, Sai Vikas Coca, and Rajeevlochana G. Chittawadigi
Abstract Turbines are used for the generation of electric power using hydel, air flow or wind. In a conventional wind turbine, the angle of attack of the wind changes as the cups or plates rotate. However, if the angle of attack remains perpendicular to the plate, the effort of the wind on the turbine rotation would be higher. In this paper, a novel concept of a turbine is proposed that ensures that the plates always remain perpendicular to the direction of wind flow. This is achieved by using parallel four-bar mechanisms. A simplified mathematical model is presented along with that of Persian turbine, of similar form. The analyses were compared, and the proposed mechanism was found to generate more power than Persian turbine. Physical prototypes of both the turbines were fabricated and tested, and the proposed turbine was found to support the theoretical calculations. Keywords Turbine · Four-bar mechanism · Static force analysis
1 Introduction Renewable energy generation is the need of the hour. Wind or air is one such source of energy. Over the years, wind energy is being used for domestic and small-scale applications. The Persian turbine, one of the earliest of its kind, was used for grinding grains and pumping water depending on the necessity. Wind turbines are broadly classified as horizontal axis wind turbine (HAWT) and vertical axis wind turbine (VAWT). Persian turbine is a VAWT that consists of plates or blades that are in the radial direction of a rotating column [1, 2], as shown in Fig. 1a. Half of the width is exposed to the wind, and therefore, wind applies force on each plate for half of the revolution. Its plates receive the wind at distinct angles of attacks on the plate areas, which results in two different components of forces. The component that is tangential to the plate of the turbine is responsible to create turning moment, and the radial component results B. V. Chennoju · S. V. Coca · R. G. Chittawadigi (B) Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_144
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(b) Aerogenerator Exawind [3]
(c) Wind turbine with
(d) Feathering Sidewheel [5]
Fig. 1 Vertical axis wind turbines and paddling using four-bar mechanisms
in the reaction force on the vertical column. Many researchers and organizations have worked on means of improving the efficiency in the form of different shapes of plates or blades and plates that orient themselves during each revolution to have larger turning moment. The former adopts airfoils instead of flat plates to produce pressure difference which drives the rotor, e.g. Savonius, Darrieus, etc. The latter deals with the concepts of mechanisms and has been explored in this paper. To get higher turning moment, the plates should be aligned such that the angle of attack should be such that the direction of wind is perpendicular to the plate surface. Few commercial products such as aerogenerator exawind [3] exist which have means of making the plates orient using some mechanisms, as shown in Fig. 1b. Thang [4] has proposed wind turbine with flipping airfoils, where the airfoils/blades orient themselves based on the rotation angle. However, there is a grove in which the airfoil links move, thus making the system susceptible for frictional losses. It is shown in Fig. 1c. The authors conceptualized a mechanism that used parallel four-bar mechanism, and in their literature survey, they found a similar mechanism used in paddling of steamers or boats [5], shown in Fig. 1d. When the plate is at the bottom region, within water, it should remain vertical for maximum thrust. The plates while entering water or exiting should be aligned such that there is minimal splash and hence saving of energy. The paddle wheel though uses four-bar mechanisms to orient the plates, they do not ensure that the plate remain parallel to one of the directions, which is what the authors have proposed in their novel design named ‘Twin-plate Turbine’, which is discussed next. The novel design uses several parallel four-bar mechanisms whose coupler links act as plates or blades of the turbine. The parallel four-bar ensures that the coupler’s orientation always remain parallel to the fixed link, as shown in Fig. 2.
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Fig. 2 Coupler (BC) link in parallel four-bar mechanism remains parallel to fixed link (AD)
2 Twin-Plate Turbine The proposed turbine consists of two circular discs, vertical and parallel to each other. The discs are constrained with a support link such that the axes of the circular discs have an offset in the vertical direction and have revolute joint with it. Side view of the CAD model is shown in Fig. 3a. Plates with Z shape are used to connect between the two discs, each end of the plate forming a revolute joint with the corresponding circular disc. The distance between the revolute joint axes of the plate is kept same as the vertical offset between the centres of the circular discs. If the support link is considered as fixed link, the circular discs form the crank and the follower of a fourbar mechanism and the plate link forms the coupler. Note that the coupler link would always remain vertical due to the fact that all these links form a parallel four-bar mechanism. Similarly, holes on the circular discs can be created and identical plates can be inserted between them, forming a series of parallel four-bar mechanisms, as shown in Fig. 3b. The lower half of the disc (i.e. relatively higher, referred to as ‘Higher disc’) is covered on the sides and ensures that if wind is flowing, it interacts only with the
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plates that are instantaneously above the cover. The wind pushes the plates that are in its line of action (above the cover) and the plates then rotate the circular discs. The axis of the ‘Higher disc’ is considered as output, and a generator can be connected to it to extract electrical energy. The support link and the cover can in turn be mounted on another base link through a vertical revolute joint, which would let the turbine orient itself about the vertical direction such that wind’s direction is perpendicular to the surface of the plates. These are shown in Fig. 3a. Working of the turbine mainly depends on various physical parameters such as radius of rotation, number of plates and plate area. Optimizing these physical parameters also plays a major role in performance. Radius of the turbine is the crank rotation radius since all the points have the same radius of rotation. Number of plates can be increased to generate more power and to reduce the fluctuation in the system. Area of plates or dimensions of the plates should be designed geometrically to get the maximum output. The proposed setup has horizontal axis of power generation. However, if the circular discs are considered horizontal, the output shaft would be vertical and hence would fall under VAWT category. For easier fabrication, horizontal version is selected.
3 Mathematical Model The simplified mathematical model of the Persian turbine and the proposed turbine are covered in this section. The wind direction is assumed to be parallel to the horizontal plane. As the area of plate perpendicular to the wind direction may be different, pressure is considered and the effective force on the plate is considered as pressure upon projected area of the plate. The striking force acting on the plate has been assumed to be acting at the centre of the plate, which would make the analyses simpler. Consider the Persian turbine in Fig. 4a. The turning moment on the output axis is due to the tangential component of the force. At an instant when the plate
Fig. 4 Turning moment calculation using static force analysis
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makes θ with respect to the horizontal, the turning moment on the output shaft can be determined as b sin2 θ (1) T = Pbl r + 2 where P is the pressure, b and l are the breadth and length of the plate, respectively, r is the radius of the crank or link. The force applied is considered as pressure × area, i.e. P × (b × l). The turning moment is a function of θ and has maximum value when θ is 90°. For the proposed turbine, static force analysis method is used to determine the turning moment on the output shaft, connected to the ‘Higher disc’, as shown in Fig. 4b. The free-body diagrams of each link of a parallel four-bar mechanism are shown in Fig. 4c, for the crank making θ with horizontal. The internal forces acting on the links are considered such that follower link (CD) is a two-force member, coupler (BC) is a three-force member and the crank (AB) is a two-force and one moment member. For the coupler link, since geometric symmetry exists, the force polygon is an isosceles triangle, and hence, the solution is obtained analytically. Hence, the direction and the magnitude of the forces and hence the moment due to these at the output shaft can be determined by knowing the perpendicular distance between the forces acting on the crank link. Due to space constraint, the brief derivation is shown in Fig. 4c. The turning moment is determined as T = Pblr sinθ
(2)
The static force analysis is a simpler method where inertia effects of the moving links are not considered. The formulations derived for the Persian turbine and the proposed turbine were implemented as a MATLAB program. The physical parameters considered are P = 1 MPa, l = 0.15 m, b = 0.1 m, r = 0.15 m. The turning moment due to one plate in each of the turbine for the range of motion of crank from 0° to 180° is compared in Fig. 5a. It can be observed that the turning moment in the proposed turbine is larger than that in Persian turbine of same size. When six plates are considered equally distributed in an angular fashion, the effect of wind acting on
Fig. 5 Comparison of turning moment plots for Persian and twin-plate turbines
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the plates is only considered for the range of θ between 0° to 90°. This is because for the plates beyond 90°, some other plate maybe obstructing the direct impact of the wind on the plate under consideration. The comparative plots for six plates are shown in Fig. 5b. Similarly, Fig. 5c compares the analysis for eighteen plates. The fluctuations observed can be reduced by using higher number of plates or with by using a flywheel on the output shaft. It can be observed that the proposed method has larger area under its curve, compared to Persian turbine and hence can be concluded to be more efficient.
4 Physical Prototype Physical prototype of the Persian and the proposed twin-plate turbines were developed using circular discs made up of plywood, plates in the form of plastic boxes and PVC plastic pipes for connecting the plates with circular discs. The dimensions used in the previous section were used for the fabrication, and six plates were used in the prototype. The twin-plate turbine in action is shown in Fig. 6. Air blower was used to generate the action of wind on the plates and for similar conditions. A dynamo (used in a cycle) was connected to the output shaft, so that the electrical output can be measured. The physical prototypes are shown in Fig. 7a, b. A voltmeter was connected across the dynamo, and an ammeter (multimeter) was connected in series. The air blower was held close to the plates and the turbines rotated due to the force acting on the plates. The current and voltage measured for both the prototypes are reported in Fig. 7c. The output power in the proposed ‘Twin-plate’ turbine was found to be around 2.5 times more that of Persian turbine. The results compliment the analyses obtained by the mathematical model proposed earlier. The physical prototypes were developed using common household items, and the joint axes were not aligned exactly as per the design. Hence, the rotation of the turbines caused vibration of the support. An exact model of the proposed turbine is being developed, and the results shall be reported in the due course.
Fig. 6 Physical prototype of twin-plate turbine with rotating plates (numbered 1–6)
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(a) Persian turbine (b) Twin-plate turbine
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Fig. 7 Physical prototypes of turbines and the electrical output generated as results
5 Conclusion In this paper, a novel turbine design is proposed which uses a set of parallel fourbar mechanisms to keep the plates or blades of the turbine always perpendicular to the direction of the wind. Static force analysis was performed to determine the turning moment produced by the mechanism, and it was found to be higher than that obtained by Persian turbine of similar size. Physical prototypes of both variants were developed, and the generated power was measured in the two cases. The power generated in the proposed method was found to be higher than that obtained in Persian turbine. In future, more accurate model of the prototype is being planned and realistic formulation considering the effect of drag and power coefficient of wind would be considered.
Reference 1. Gasch R, Twele J (eds) (2011) Wind power plants: fundamentals, design, construction and operation. Springer 2. Bennert W, Werner UJ (1989) Windenergie (Wind Energy), VEB Verlag Technik 3. Exawind: https://www.bastantecnologies.com. Last accessed 2018/12/01 4. Wind turbine with Flipping airfoils: https://www.youtube.com/watch?v=zDW7jGOZGo0. Last accessed 2018/12/10 5. The Feathering Sidewheel: https://nautarch.tamu.edu/PROJECTS/denbigh/WHEEL.HTM. Last accessed 2019/01/16
Intuitive Manipulation of Delta Robot Using Leap Motion P. Giridharan , Rajeevlochana G. Chittawadigi , and Ganesha Udupa
Abstract Delta robots are parallel manipulators used extensively in industries to perform pick-and-place and sorting operations. While the latter is usually integrated with a vision system, the former requires users to teach robot positions and corresponding actions that it has to repeatedly follow, using teach pendants. The teach pendants are generally bulky and require thorough training to be used upon. In this paper, the authors propose usage of Leap Motion, a hand tracking device, to manipulate a delta robot. The kinematics of the robot and the methodology used to integrate Leap Motion and the manipulator developed by them are presented in this paper. The robot was successfully controlled, and it was found to be intuitive for the end-user. Keywords Delta robot · Inverse kinematics · Leap Motion
1 Introduction Robots in industries can be broadly classified as serial and parallel robots or manipulators. The former have links connected from the base link to the last link (end-effector) in series, generally using single degree-of-freedom (DOF) joints. The latter have the base and the end-effector connected through a set of parallel links or arms. The serial robots have larger workspace and are simpler to design and develop. However, they are prone to inaccuracies due to the hanging of one link with respect to its previous link. On the other hand, the parallel robots have better rigidity and accuracy but have poor workspace coverage and are bound to get into singularity. Serial robots are used to perform operations that require more movement of the robot arm link pick and place, palletizing, painting, welding, etc. These robots are generally taught positions and the corresponding motion between the positions using P. Giridharan · G. Udupa Department of Mechanical Engineering, Amrita Vishwa Vidyapeetham, Amritapuri, Clappana, India R. G. Chittawadigi (B) Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_145
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a teach pendant, which is a handheld device with few buttons and display unit. Some simple robot programs can also be developed using a teach pendant, and for complicated robot programs, proprietary environments are provided by the robot manufacturers to simulate, test and feed the program to the robot controller. However, it requires thorough training of robot technicians before they can attempt to teach and control robot using a teach pendant. Delta robots or manipulators are one of the commonly used parallel robots in industries. Primarily, they are used to sort objects moving on a conveyor belt, where an image processing unit identifies the objects moving and accordingly controls the robot. Delta robots are also used to pick objects from one place and place onto another. In such applications, again teach pendants are used for teaching or demonstration purpose by a technician. The robot has three DOFs, and its architecture (discussed later in the paper) allows only translation of the end-effector along the three principal axes. Hence, to control using a teach pendant, which just has buttons (X+ , X−, Y + ,Y −, Z+ and Z−), it may require clear understanding of the Cartesian frame with respect to which the motion is defined. For intuitive control of robots, many alternative methods have been proposed. Master–slave manipulation is one such scheme where the user wears an exoskeleton to control a physical robot, acting as a slave. One such work is reported in [1]. Another intuitive method is to have identical kinematic chains for master and slave manipulators. Joint encoders on master send out data to the slave manipulator to copy, as reported in [2]. Vision-based sensors such as Microsoft Kinect, as reported in [3], are also used to intuitively control a robot arm. Leap Motion is another vision-based sensor that has very good accuracy of tracking human hands. Few researchers have also successfully used it to control serial robots, as reported in [4]. The authors have proposed to use Leap Motion to intuitively control a delta robot which is reported in this paper. Recently, it was found out that a related work was also published at [5].
2 Kinematics of Delta Robot Delta robot is one of the commonly used parallel manipulators, whose architecture has been used in commercial robots like ABB FlexPicker. The robot consists of a base platform in the shape of an equilateral triangle, which is fixed to the ground and has three arms. A revolute joint connects the base with bicep link of each arm. The bicep link is connected to a pair of identical links through universal joints. The identical links are individually connected to an equilateral triangle (of lower dimension than the base) and referred to as the moving platform, through another set of universal joints. The connection of base link till the moving platform exists through three parallel chains, and hence the robot is classified as a parallel manipulator. The architecture of the robot ensures that the identical links between a bicep arm and moving platform always remain parallel. Overall, the moving platform is constrained to only have translation motion with respect to the base link and hence the moving platform always remains parallel to the base link. Thus, this robot architecture is ideal for
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Fig. 1 Delta manipulator with 3 DOFs for translation along X, Y and Z directions
pick-and-place operation where the end-effector or gripper has to be moved along the three coordinate axes, without any change in its orientation. In this work, base link is considered at the bottom and the moving link at the top. The robot can also be inverted to perform any real-life pick-and-place application. The 3D model of the delta robot developed in Autodesk Inventor software is shown in Fig. 1a. M1 , M2 and M3 correspond to the motors (rotational actuators), providing the 3 DOFs required to translate the end-effector along X, Y and Z directions. The universal joints have been replaced by spherical joints for easier fabrication, though these joints introduce a redundant degree of freedom in each of the identical links (to have rotation about its own axis), which does not affect the motion of the overall systems. The important dimensions of the mechanism are mentioned in Fig. 1a, whose values considered in the design are Wb = 173 mm, Wp = 29 mm, L = 150 mm and l = 300 mm. The kinematics of the mechanism are explained below. Inverse Kinematics: If the position (X, Y and Z) of the midpoint of the platform (or end-effector) is given, determining the angles θi for each motor or rotational joint is known as inverse kinematics. Delta robot has a closed-form solution for inverse kinematics, which provide at most two possible solutions. In this paper, the formulations used in [6] are followed. If a robot is already in a given configuration, if the platform has to be moved to some other configuration, the relative displacement vector between the initial and final points has to be determined. A linear interpolation of the displacement vector would give various via points. Inverse kinematics has to be performed on individual via points in the sequence to determine the joint angles for the motors to be actuated. If multiple solutions exist for a via point, the solution that is closest in the individual joint space has to be selected; i.e., the solution which has least incremental motion at the joints has to be selected for that via point. Inverse kinematics is therefore important to control parallel robots, and hence only few architectures of parallel robots exist which have closed-form solutions.
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Forward Kinematics: For given values of θi , i.e., individual joint angles, determination of the position (X, Y and Z) of the end-effector or platform is known as forward kinematics. Parallel robots generally have multiple solutions for the forward kinematics as well, and they are complicated to solve. It is useful for simulation purposes to check collision between links, to check for singularity, to determine the position of the end-effector so that effective control strategies can be implemented.
3 Physical Prototype The base link was made in a triangular shape using plywood. The bicep links were fabricated using 3D printing, where the extensions on the bicep arm had spherical shaped ends, for the connection with forearm links. The spherical joints were created using tie-end rods, which were connected to the either ends of forearm links. The platform was fabricated using 3D printing and again had extensions with spherical shape for the tie-end rods to be connected. The assembled robot is shown in Fig. 1b. The inverse kinematics of the robot was implemented as a Python program on a system running Windows 8.1 operating system. The program made available online [7] takes the X, Y and Z coordinates of the end-effector and determines the joint angles. These joint angles are then sent to Arduino Mega 2560 Controller, connected through USB in the form of serial port communication. The Arduino has a program to read the incoming data and convert to corresponding step value to be fed to TB6560 motor driver, which further sends data to DC stepper motors (of NEMA 23 make) to rotate the bicep arm with respect to the base link. A 24V:3A external power supply is connected to the stepper motor driver to actuate the motors. The circuit diagram depicting the connections between the components is given in Fig. 2. Several test motions like incremental jogging along X, Y and Z and circular motion of the end point in a plane parallel to XY plane were developed as Python programs and executed. The robot was found to move as expected. The integration of robot with Leap Motion is covered next.
4 Integration with Leap Motion Leap Motion is a vision-based sensor available for a low cost (around 80 USD) and has revolutionized the gaming industry. It has 3 LEDs and two cameras, and the technology has very good accuracy to track the human hands in its hemispherical view. The device comes with a proprietary software to be installed on a computer. The device is connected to the computer through USB. The software gets the data of the position of human hands (10 fingers) and interpolates position of all important joints and bones in human hand. Though it has found good acceptance in the gaming and virtual reality-based applications, many researchers have used it in their research.
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Fig. 2 Circuit diagram of the delta robot prototype
The second author has proposed its application to interact with virtual mechanisms, as reported in [8]. In this work, Python program has been developed to track the human hand and the position information of the index finger of the right hand is recorded for every time frame. A coordinate system is attached at the center of Leap Motion device, and coordinates measured are in this frame. The incremental motion of the fingertip is provided as input to the inverse kinematics module/method of the program, which determines the joint angles to be supplied to the delta robot. The new set of joint angles are sent through Arduino to the drivers, and motors move correspondingly. The integration of the Leap Motion with the computer and then the delta robot was successfully completed. Few tests conducted by a user, as illustrated in Fig. 3, are reported next.
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Fig. 3 Delta manipulator with 3 DOFs for translation along X, Y and Z directions
The intent of this work was to intuitively control a delta robot. To achieve this, several trials of human interaction were performed. The user moved his hand approximately along the X, Y and Z directions, with respect to the frame attached on the sensor. The position of the fingertip for few of these tests and the corresponding joint trajectories, after inverse kinematics for each of the position, are shown in Fig. 4. The resultant motion of the delta robot was observed to follow the same trend as the input motion. However, due to lack of encoders (at joints or at end-effector), the accuracy of the prototype was unable to be determined. In the future, suitable sensors would
(a) Finger-tip along Z
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Fig. 4 Results of intuitive control of delta robot
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be used for accurate motion through feedback control. Finally, it was observed that the control of the robot was intuitive as compared to conventional teach pendants.
5 Conclusions In this paper, a physical prototype of delta parallel robot developed by the authors has been integrated with Leap Motion sensor. The sensor tracks a fingertip of a user, and the incremental motion of the fingertip is mapped onto the corresponding motion of the moving platform of the robot. The proposed methodology was found to be intuitive for the user to effectively control the robot. In the future, encoders at joint and for the end-effector would be used for achieving better accuracy.
References 1. Arun K, Nair AP, Chittawadigi RG, Venkatesh K, Patnaik S (2018) An efficient methodology to determine the usability of exoskeleton to control a serial manipulator. In: Proceedings of 5th IFToMM Asian conference on mechanism and machine science 2. Ajith A, Nambiar NM, Akshay VP, Ajit A, Ramachandran R (2018) SAKSHA-self automated kinematic smart haptic arm. Procedia Comput Sci 133:711–717 3. Du G, Zhang P, Mai J, Li Z (2012) Markerless kinect-based hand tracking for robot teleoperation. Int J Adv Robot Syst 9(36) 4. Bassily D, Georgoulas C, Guettler J, Linner T, Bock T (2014) Intuitive and adaptive robotic arm manipulation using the Leap Motion controller. In: Proceedings of 41st international symposium on robotics 5. Zhang X, Zhang R, Chen L, Zhang X (2019) Natural gesture control of a delta robot using Leap Motion. In IOP J Phys Conf Ser 1187(3) 6. Williams II RL (2016) The delta parallel robot: kinematics solutions, internet publication, www. ohio.edu/people/williar4/html/pdf/DeltaKin.pdf 7. Code: https://github.com/giridharanponnuvel/Delta-Robot-Inverse-Kinematics (2019) 8. Pullil S, Chittawadigi RG (2019) Natural control of virtual models of mechanisms using Leap Motion for interactive learning. In: 4th international and 19th national conference on machines and mechanisms
A Computation Model of Contact Interaction Between the Scaphoid and Its Neighboring Bones Using Bond Graph Approach Arvind Kumar Pathak and Anand Vaz
Abstract Wrist modeling, as an idealized mathematical representation, is increasingly important in the analysis of complex wrist biomechanics. Contact mechanics at a wrist joint is intricate due to non-uniform geometry of mating bones and intermediate cartilage layer. A bond graph-based model of contact interaction of the scaphoid bone, one of the carpal bones, with its neighboring bones is presented in this work. The geometries of mating bones are considered in the form of a point cloud. The behavior of the intervening cartilage layer is modeled using a nonlinear stiffness and damping field. The scaphoid bone is connected through ligaments with its neighboring bones. Each ligament is modeled as a linear, tension only spring element in combination with a linear damper. The scaphoid bone is under the action of gravitational force. Investigations of the contact area between mating bones and generalized forces on scaphoid bone and its ligaments are discussed. The proposed model may also be an alternating iterative process between FE-models and kinematic rigid body models. Keywords Wrist joint · Cartilage layer · Contact mechanics · Bond graph
1 Introduction The human wrist represents the most complex joint system in the human body. The wrist joint consists of eight small, intricately, and unique shaped carpal bones, interposed between the distal end of radius and ulna and base of the five metacarpals. The extrinsic and intrinsic ligaments provide stability to wrist joint. The morphologic variations of the carpal bones and their contacts with soft cartilage layer have its own significance. In regard to biomechanical modeling of natural joint systems, there are two typical ways to create a computational model of a biological joint system: finite element analysis (FEM) [1] and multi-body simulation (MSB) [2]. However, the existing models of the wrist had not considered the physiological path of the A. K. Pathak · A. Vaz (B) Department of Mechanical Engineering, Dr. B R Ambedkar National Institute of Technology, Jalandhar, G.T. Road, Bye Pass, Jalandhar, Punjab 144011, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_146
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implemented ligaments. The cartilage layers of the different wrist bones and wrist joint systems, respectively, have also not been considered. Also, the order of the system modeled based on FEM depends on the number of nodes and turns out to be extremely large in cases of mating bones with intervening soft cartilage. However, the large order lays a severe burden on the computation involved [3]. Alternatively, modeling of the biological joints requires a simpler model which requires less computational time. The objective of this work is to present one such alternative model of cartilage layer at a synovial joint. It is of interest to understand the effects of change in the contact area between the bone and the cartilage layer, the deformation of the soft cartilage between the bones, and the force distribution as the contact area changes, during dynamic interactions. To demonstrate the simplified conceptual model and its implementation, a bond graph model of contact interaction of the scaphoid, one of the carpal bones, with its neighboring bones is presented. The surface geometry of each bone is represented by point cloud. The intervening cartilage layer between bones has nonlinear characteristics, and its constitutive relations have been logically and systematically derived. The ligaments connected between the scaphoid and its neighboring bones are modeled as linear, tension only spring element in combination with a linear damper. The equations governing the dynamics of the system have been derived from the bond graph model [4]. This paper is organized as follows. Section 2 elaborates the bones geometry, the proposed nonlinear model of the cartilage layer, and the ligament modeling. The multibond graph model for the proposed system is discussed in Sect. 3. The simulation results are analyzed in Sect. 4. Concluding remarks and potential for future work are discussed in Sect. 5.
2 Proposed Model This work proposes a simplified alternative computational model to emulate the contact interaction between the scaphoid bones and its neighboring bone is presented using the bond graph approach. The scaphoid bone (S) is one of the carpal bones on the thumb side of the wrist, just above the radius bone. It is the largest bone of the proximal row of wrist bones. The mass of scaphoid bone is 3.35 g. The neighboring bones of the scaphoid are radius (R), lunate (L), trapezium (TM), trapezoid (TD), and capitate (C) which are fixed and are connected through ligaments with it. The complete arrangement of bones, the contact area comprises the cartilage layer between the scaphoid and its neighboring bones, and respective ligaments are shown in Fig. 1. The detailed explanation about bone geometries, intervening cartilage layer between mating bones and ligaments is presented in the following subsections.
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Fig. 1 Scaphoid and its neighboring bone
2.1 Bone Geometry In the present study, the bones are considered to be rigid in nature. The surface geometries of the mating bones are represented by 3D point clouds. The point cloud, representing the irregular surface geometry of all bones, is taken from a digital database of carpal anatomy developed and published by Moore et al. [5]. Threedimensional Voronoi and Delaunay triangulation-based surface reconstruction algorithm have been used to generate the planar triangulated patches over 3D point cloud [6]. Surface normals are computed at respective centroids of these patches. With reference to the frame fixed in the respective bone, the directions of these surface normals are always fixed. Multibond graph sub-models are used to represent their rigid body mechanics of all bones [7].
2.2 Cartilage Layer Instead of modeling the soft material using the conventional finite element method (FEM), which can prove to be an elaborate task, a nonlinear stiffness field is used to characterize the behavior of the intervening soft cartilage layer in the normal direction at the patch. The detailed bond graph model for the same had been presented by Pathak and Vaz [8]. In this study, the contact force model does not account the contribution of the contact area between the bone and the cartilage lyaer. In the present study, the
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areas of patches lying in the contact region between the bone and cartilage layer are considered. A suitable candidate function, representing the cartilage layer stiffness, is proposed as, F¯ B1i K ⎧ ⎪ ⎪ ⎨ T = nˆ B1i K ⎪ ⎪ ⎩
B2 j
⎛
⎫ ⎪ ⎪ ⎬
⎞
2 ⎜ ⎟ A B2 j B1i q B2 j ⎜ ⎟ A B1 B2 r¯ nˆ B1i . i j B1i q 2 ⎝ ⎠ ⎪ B1i B2 j ⎪ A + A 2 B2 j B1i ⎭ B1 q B2max 1 − B1 q j=1
B2max
i=1
(1) where K is a positive constant which depends on the cartilage material, B1i and B2 j are the centroid of the ith patch of bone 1 and jth patch of bone 2, respectively, which are in contact region, A B1i and A B2 j are the areas of the ith patch of bone 1 and jth patch of bone 2, respectively, B1i q B2 j = TC − B1i d B2 j , is the cartilage deformation, TC is the initial cartilage thickness, B1i d B2 j is the distance between the ith patch of bone 1 and jth patch of bone 2, B2 j r B1i is the relative position vector between ith and jth patch, B1 q B2max is the maximum permissible deformation of cartilage layer, nˆ B1i is the unit surface normal at the ith patch on the bone 1. The constitutive relation for the stiffness of the cartilage material is considered such that the force developed by the cartilage layer at the area of contact interaction between the two bones increases asymptotically as the surface of bone 1 approaches closer to the predefined clearance limit with respect to the surface of the bone 2. The damping characteristics of the cartilage layer is, ⎧ ⎪ ⎪ ⎨ R B2 j r˙ B1i nˆ B2 j F B1i D = B1i ⎪ ⎪ ⎩ where R is positive constant and
⎫ ⎪ ⎪ ⎬
2 B1i q B2 j
2 B1 q B2max
˙
q 2 AC A B1i ⎪. B1 B2 ⎪ ⎭ 1 − B1 qi j
B2 j r B1i nˆ B1i
(2)
B2max
is a component of relative velocity
between B1i and B2 j , along the direction of nˆ B1i .
2.3 Ligament Ligaments are passive structures connecting bones, stabilizing joint systems, guiding and constraining the joint motion. In the present study, the behavior of each ligament is modeled using linear, tension only spring element in combination with a linear damper [9], as shown in Fig. 2. The information of anatomical directions and locations of insertion points on the bone surfaces, for each ligament, is identified and
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Fig. 2 Ligament model
KLig Insertion Point Bone 2
Insertion Point Bone 1 RLig
approximated based on anatomical atlases and literature data. The ligament offers constraints to bones only when it is in tension. The force offered by the ligament is, F Lig = K Lig qLig B1L rˆB1L + R Lig
˙
B1 L r B1 L
B1 L rˆ B1 L
(3)
where q Lig is the extension in the ligament, B1 L rˆ B1 L is the unit vector along the direction joining the insertion points of the ligament onthe respective bone, ˙ B1 L r B1 L B1 rˆB1 is the component of relative velocity between the insertion points L L of the ligament on the respective bone, along the B1L rˆB1L . The information about different ligaments connected between the scaphoid and its neighboring bones has been taken from [10].
3 Bond Graph Model A bond graph model is developed for the interaction between a pair of two mating bones separated by intervening cartilage layer and a ligament connected between them and is shown in Fig. 3. Similarly, all the bones interacting with the scaphoid and respective ligaments are modeled using bond graph. The model is initiated using flow mapping which represents the kinematics of the system under consideration. The are considered to be rigid and modeled based on rigid body dynamics. 0 two bones C B2 r B2 L × is the skew symmetric matrix, represented as MTF, obtained from posiT tion vector C0 B2 r B2L = C0 B2 x B2L C0 B2 y B2L C0 B2 z B2L . Similarly, all the MTF has been formulated. The stiffness and damping characteristics of intermittent cartilage layer are same as discussed in the Sect. (2.2). The junctions 0 F B1 and 0 F B2 are reprei j senting the force offered by the cartilage layer at the ith and jth patch, respectively. The junctions 0 F B1 and 0 F B2 are representing the ligament insertion points on the L L bone 1 and bone 2, respectively. The force offered by the ligament is the effort on the junction 0 B2 F B1 . This effort is shared by all bonds connected to this junction. DirecL L tions of power bonds clearly tell that the two bones experience the same magnitudes of forces but in opposite directions.
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Fig. 3 Bond graph model for contact interaction between two mating bones and a ligament connected between them
4 Result and Discussion The dynamics of scaphoid bone while interacting with its neighboring bones has been simulated in this work, based on the model shown in Fig. 3. The model is general and is valid for motion in three dimensions. The simulation is for a duration of 1 s. A force due to acceleration due to gravity is acting on the scaphoid bone. The articulation between the scaphoid and its neighboring bones and the minimum distance between them are mentioned in Table 1. An initial movement of 2 mm is given to a point PS on the surface of scaphoid bone along the positive y-direction of inertial frame {0}. The desired smooth trajectory of that point is given in (4). r PS (t) = 0, t45
Low arched
Flat
Intermediate arched
Normal
[0, 20)
[20–30]
High arched
Index values
Foot type
12
4
2
3
14
5
1
1
1
3
0
0
1
0
R
1
BMI [15–20]
R
L
Total group
Table 1 Chippaux-Simark Index (CSI) (R-right foot) (L-left foot)
3
0
0
0
1
L
4
2
2
1
0
R
BMI [20–25]
6
3
0
0
0
L
3
1
0
0
1
R
BMI [25–30]
2
2
1
1
0
L
2
1
0
1
0
R
BMI [30–35]
3
0
0
0
0
L
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3.2 Foot Types Based on Hallux-Valgus Angle (HVA) The Hallux-Valgus angle (HVA) measures the existence over foot of both HalluxValgus, the misalignment of first toe as well as Hallux-Varus which may be defined as misalignment of first toe on exterior side. The obtained data of angle β the outcome exhibits 5% participants have Hallux-Valgus condition, 5% in left foot and 9% in right foot have the potential risk to occur it (Table 2).
4 Conclusion In demand towards emerging a balanced organization method of foot assertion grounded on constraints resulting as of the plantar footmark, these researches directed. In the direction towards calculation, and to be able to record it, in a similar mode intended for all disciplines, the writers have recognized regulation of calculating the central part of a plantar footmark. Hallux-Valgus Angle and ChippauxSimark Index were used in course to categorize the foot assortment. Disciplines would be separated into five groups. Variances amongst oldness have been identified by means of arithmetical investigation, T-test and Levene. Representing the requirement of demonstrating and scheming customized shoe lasts, footwear and prophylactic mechanisms conferring towards the age of the discipline (Fig. 3). In demonstrating and scheming prophylactic footwear, in dropping the amount of examinations of corporal models, in the successive examination of foot anthropometrical constraints, the requirement of the generous of studies is originated. Consequently, it is also raising the productivity of the footwear technical method. No concerns were raised by the author(s) regarding the conflict of interest in this study. It is declared that no funding or financial support was available for this study.
Index value
>15
[10, 15)
[0, 10)
0, then
If εb < 0, then also e14 = K c qc + Rc f 14 ; K c = 0, Rc = 0
(4)
e14 = 0; K c = 0, Rc = 0
(5)
Else,
In (3) and (4) it can be noticed that the effort e14 will act only in case the link 2 tries to go beyond the minimum or maximum limits. When it moves between the limits effort e14 will be zero (5) as the values of K c and Rc are modulated to zero. In this case, only effort e17 will act which is due to the internal damping of the joint. All the WBGOs of cRC in Fig. 2 work on the same principle, and for each joint, corresponding joint limits of motion have been taken for simulation. The internal damping of the joint has been taken into account in both virtual and the actual sub-models.
4 Controller Between Virtual and Actual Sub-models Based on the information of the joint angle from the virtual sub-model, joint torque is provided on the joint of the actual sub-model through PD controller as shown in Fig. 4. One of the WBGOs of the cRC of virtual and that of the actual sub-model are
Fig. 4 PD controller between virtual and actual sub-models. a WBGO of a cRC of the virtual sub-model. b WBGO of a cRC of the actual sub-model
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shown in Fig. 4a, b, respectively. In Fig. 4a, 2v 1v w2vz represents the z-component of the relative angular velocity of link 2 with respect to link 1 expressed in the body frame of link 2 for virtual sub-model. Similarly, 2a 1a w2az represents for the actual sub-model as shown in Fig. 4b. The difference in these angular velocities gives the derivative of 2a error which is defined as a difference in angular positions 2v 1v θ2vz and 1a θ2az . The PD controller gives an output torque which will be equal to the summation of a term proportional to the error and a term proportional to the derivative of the error. This output torque has been applied at the joint of the actual sub-model. As a result, the actual sub-model will move along the same desired trajectory which has been given to the virtual sub-model. All the joint angles of the actual sub-model are controlled by applying the same principle.
5 Results This section presents the MATLAB simulation results of the developed bond graph model. As shown in Fig. 1, initially, physical system is considered in a sitting posture on the chair. For SiTSt and StTSi motions, points 4e and 4f of the virtual sub-model are moved along a desired circular arc. As discussed earlier, the actual sub-model will track the motion of the virtual through PD controllers and therefore, moves along the same desired trajectory. In the simulation, SiTSt motion starts at 0 s and ends at 2.5 s. From 2.5 s to 3.5 s, the system remains in the standing position. Further, StTSi motion starts at 3.5 s and ends at 6.5 s. Figure 5 shows the ankle, knee and hip joint angles of the virtual and that of the
Fig. 5 Joint angles of the virtual and actual sub-models
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Fig. 6 Joint torques applied by the PD controller at the joints of the actual sub-model
actual sub-model. It can be observed that the ankle and knee joints of the actual submodel almost track the corresponding joints of the virtual sub-model. The hip joint of the actual sub-model does not track that of the virtual sub-model completely and a small error can be observed. The error can be further removed by fine-tuning of PD gains of the controller between the virtual and the actual sub-model. Figure 6 shows the joint torques applied through the PD controllers at the joints of the actual submodel. It can be noticed that the torque profile during the SiTSt motion is smooth, and there are no transients. At the start of the StTSi motion, transients can be observed which are shown as encircled portions. These transients are caused due to discontinuities in the desired trajectory profile and can be omitted by tuning the PD gains of the controller between the virtual and actual system.
6 Conclusion In this work, a seven-rigid-link bond graph model has been developed to model the SiTSt and StTSi motions of the human body. For a desired trajectory, information of joint angles has been taken from the virtual sub-model which is considered to be a part of the CNS. The internal damping of the natural joints has been modeled using an R element. Conditional rotational couplings (cRC) have been used to account for the natural constraints in the motion of the ankle, knee and hip joints. The actual sub-model, through PD controllers, tracks the desired movement trajectory given to the virtual sub-model. Transients have been observed in the torque profiles and are caused due to discontinuity in the desired trajectory profile. Tuning of PD gains of the controller can be used to control these transients. Estimated torque requirements
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at the joints would be helpful for the development of assistive devices for human beings. Future scope of the present work would include, simulating the developed bond graph model for the exact trajectory of natural SiTSt and StTSi motions and fine-tuning of the PD.
References 1. Lutz W, Sanderson W, Scherbov S (2008) The coming acceleration of global population ageing. Nature 451:716–719 2. Rupal BS, Rafique S, Singla A, Singla E (2017) Lower-limb exoskeletons : Research trends and regulatory guidelines in medical and non-medical applications. Int J Adv Robot Syst 14(6):1–27 3. Taghvaei S, Tavasoli A, Feizi N, Rajestari Z, Abdi M (2017) A control-oriented dynamic model for sit-to-stand motion with fixed support. Proc Inst Mech Eng Part K J Multi-body Dyn 232(2):265–273 4. Pop C, Khajepour A, Huissoon JP, Patla AE (2003) Experimental/analytical analysis of human locomotion using bondgraphs. J Biomech Eng 125(4):490–498 5. Selk Ghafari A, Meghdari A, Vossoughi GR (2007) Modeling of human lower extremity Musculo-Skeletal structure using bond graph approach. in Volume 9: Mechanical Systems and Control, Parts A, B, and C:1489–1498 6. Mishra N, Vaz A (2017) Bond graph modeling of a 3-joint string-tube actuated finger prosthesis. Mech Mach Theory 117:1–20 7. Plagenhoef S, Evans FG, Abdelnour T (1983) Anatomical data for analyzing human motion University of Massachusetts -Amherst. Res Q Exercise Sport 54(2):169–178
Detecting Cancerous Cells Using Data Augmentation In Deep Cascaded Networks Akshay Jain , Pallavi Chaturvedi , and Lalita Gupta
Abstract In this article, an approach has been introduced for detecting cancerous cells. Image processing techniques have been used, based on cancer cell area using CNNs. A very intriguing aspect of this experiment was that from a very small image dataset, a large number of images were generated using information augmentation which was then taken as the training set data. The suggested scheme detects cancer behaviors through a convolutional neural network in images of celled samples. Previously, the same attempts failed to stay away from the database dependencies, which were somewhat proportional to the number of images in datasets, so we used a method called data augmentation on smaller sets of images. The scheme preprocesses the input image by grayscale, binarization, inversion, median filtering, and flood-fill procedures. Depending on the sort of feature to be identified, the preprocessed image is then cancerous cell detected. This methodology was used for several sets of pictures, and the system was optimized with the feedback from those tests. For independent cancer cell detection with narrower datasets, the suggested technique can be efficiently used, which will greatly accelerate the study of cancer and open greater dimensions. Keywords Cancer cell · Convolutional neural network · Data augmentation · Deep learning · Image processing
1 Introduction Cancer is a collection of associated diseases in which the cells of the body begin dividing without stopping and spread into surrounding tissue and form growths called tumors [1]. It is one of the leading causes of deaths worldwide, and the number of new cases is expected to rise to 23.6 million by 2030 [2]. Computed tomography (CT) and magnetic resonance imaging (MRI) are two popular methods used in the diagnosis of cancer.Through detection, the tumors are classified into two categories: A. Jain · P. Chaturvedi · L. Gupta (B) Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh 462003, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_155
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(i)non-cancerous tumors (benign) and (ii)cancerous tumors (malignant). From all the researches available in the literature, it can be safely said that neural network plays an important role in the recognition of cancer cells in between the normal tissues, which gives us a very successful tool and also beats traditional AI in image detection and classification. The more data an ML algorithm has access to, the better the results it can produce [3]. Even when the information is of lower quality or has only a few aberrations, algorithms can perform way better, as long as some information can be mined out by the algorithm. Although enormous and unstructured datasets for cancer cells are accessible from different sources, sources with no correct information about the exactness of cancerous cell image, the mission becomes one of the discovering structures within a pool of unstructured data [4]. Instead of beginning with an exceptionally big collection of unstructured and unlabeled data, we can take a tiny, curated collection of structured information instead and augment it in a manner that improves the efficiency of models using it for training purposes [5]. Hence, in this paper, we present a technique primarily based on convolutional neural network (CNN) to classify tumors as malignant or benign using a comparatively narrower dataset and to check the efficacy of dissimilar data augmentation techniques in an image classification task.
2 Proposed Methodology CNN is used in this paper to identify cancer based on CT images. This study’s prime objective is to assess whether the tumor is malignant or benign. We also need to increase pictures along with the goal in order to achieve better outcomes when training. We will first use basic training for the potential comparison and then use the bigger enhanced information set to detect cancer cells.
2.1 Basic Image Classification To execute a CNN image classification, we will train our own tiny dataset. We will conduct the tumor classification with a narrower set of information. We will then use typical methods for information augmentation and retrain our models.
2.2 Data Augmentation and Performing Image Classification Data augmentation usually has two distinct methods. The first strategy is to achieve added data before the classifier is trained. The second strategy tries to learn to augment through a neural net that is prepended. This neural network takes two random pictures from the set of training data at the moment of training and delivers a single picture
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so that this picture fits either in fashion or in context with a defined picture from the training set. In addition to the initial training data, this output, which represents an improved picture created by the network, is fed into the second classification network. The training loss is then back-propagated to train the network’s expanding layers as well as the network’s classification layers [5]. Traditional Transformations Traditional transformations consist of manipulating the training data by using a mixture of affine transformations. We produce a “duplicate” picture for each input picture by shifting, zooming in/out, rotating, flipping, distorting, or shading it with a hue. The neural net is supplied with both picture and duplicate. We create a 2N size dataset for a size N dataset [6]. Generative Adversarial Networks We can choose a style picture from a subset of six different styles for each input picture: Cezanne, Monet, Ukiyo-e and Van Gogh. A styled transformation is generated of the original image. In order to train the net, both the original and the styled image are fed [7].
2.3 Dataset A small amount of tiny pathological pictures were given with the dataset used to classify from Kaggle Platform. A binary label showing the existence of cancer cells is annotated for each picture. A favorable label shows that a patch’s core region of 32 × 32px includes at least one pixel of tumor tissue and thus makes the cell malignant, whereas if the label was negative, there were no cancer cells present and therefore a benign one. The dataset used has been taken from https://www.kaggle. com/c/histopathologic-cancer-detection/data [8].
2.4 Convolutional Neural Networks (CNNs) Deep learning enables multi-layered computational models to learn information representations with various abstraction levels. These techniques in generally used for voice recognition, visual object recognition, object detection, and many other fields such as drug discovery and genomics have dramatically enhanced the technologies. Deep learning discovers complex structures of data in big information sets by using the back-propagation algorithm to show how a machine should alter its inner parameters in order to achieve a better result in the next iteration, using back-propagation representation in the past layer to calculate the representation in each layer [9]. A convolutionary neural network is a neural network class that is specialized in processing data with a grid-like topology, like the image. A digital picture represents the visual data on a binary basis. It contains a number of grid-like pixels, with pixel values to indicate how luminous and color each pixel is. The three layers of CNN are: a convolutional layer, pooling layer, and fully connected layer [10].
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Fig. 1 Architecture of CNN
CNN shares weights in the convolutional layer reducing the memory footprint and increases the performance of the network. The important features of CNN lie with the 3D volumes of neurons, local connectivity, and shared weights. By combining different sub-areas of the input image with a kernel, a feature map is produced by the convolution layer. The ReLu layer then uses the anonymous activation feature to increase the convergence characteristics when the error is small. A pixel with the maximum value or average values is chosen as a representative pixel for the picture/feature map area, and a grid of 2 × 2 or 3 × 3 is reduced to a single scalar value in the pooling layer. The sample size is thus greatly reduced. In combination with the convolutionary layers toward, the output phase traditional fully connected (FC) layer is sometimes used. The convolution layer and the pool layer are generally used in some conjunction in CNN architecture. Two kinds of activities generally take place in the pooling layer—max pooling and means pooling. In mean pooling, the average area is calculated within the characteristics and calculated within the maximum of feature points in the maximum pooling. Mean pooling decreases the mistake created by the size of the area and maintains background data. Max pooling decreases the estimated error of the convolution layer parameter induced by medium deviation [11] (Fig. 1).
2.5 Different Layers Of CNNs Convolution Layer The key construction block of the CNN is the convolution layer. It holds the primary computational load of the network. This layer produces a dot product between two matrices in which one matrix is a set of learning parameters known as a kernel and the other matrix is the limited part of the receptive area. The kernel is smaller, but in-depth than an image. This implies the kernel width and height, but the depth, when the picture is made up of three channels (RGB), is spatially low. The kernel slides across the height and width of the picture and represents the receptive region on the front. This results in a two- dimensional representation of the
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picture known as an activation map, which shows the kernel response in each image space position. The kernel’s sliding size is known as a stride [10]. Pooling Layer The pooling layer replaces the network output at certain places by obtaining an overview of the outputs in the vicinity. This helps to reduce the spatial dimension of the image, which reduces the necessary calculations and weights. The pooling procedure is carried out separately on each section of the representation. Some pooling functions are such as the rectangular district average, the L2 rectangular district norm, and the weighted average from the core pixel distance. The most common method, however, is max pooling, which reports the highest output [10] (Figs. 2 and 3). Fully Connected Layer In this layer, fully connected layer neurons have complete connectivity to all neurons in the layer above and below. For this reason, a matrix multiplication and a bias impact can be calculated as normal. The FC layers help to map the image from input to output [10].
Fig. 2 Convolution operation
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Fig. 3 Pooling operation
Non-linearity Layer Because convolution is linear and images are far from linear, nonlinearity layers can often be positioned right after the convolution layer so that nonlinearity appears on the activation map. Nonlinear activities have various kinds, most of which are sigmoid, Tanh, ReLU, etc. [10].
3 Experiment We conducted information experiments to test the efficiency of numerous augmentations. In the following table, the findings of the tests are presented. All tests are conducted with Adam Optimization for 40 epochs at a learning pace of 0.00001. The best score is recorded for the greatest test accuracy of all times. The first experiment consisted of the narrower dataset images and therefore had a very small precision on all epochs. In the second experiment, the pictures were first augmented by traditional transformations. After that, operations on newly discovered picture set with a large amount of pictures was carried out by the same CNN model. About the convoluted neural network• 3 layers of ConvNet with 32 channels and 3 × 3 filters. Relu activations and one layer of Max pooling with 2 × 2 stride and 2 × 2 filters. • 3 layers of ConvNet with 64 channels and 3 × 3 filters. Relu activations and one layer of Max pooling with 2 × 2 stride and 2 × 2 filters. • 3 layers of ConvNet with 128 channels and 3 × 3 filters. Relu activations and one layer of Max pooling with 2 × 2 stride and 2 × 2 filters. • FC(fully connected) with output dimension. Dropout. • FC(fully connected) layer with output dimension 2.
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4 Results The neural network based on convolutional and watershed segmentation has been implemented in Jupyter Notebook using Python and the system is trained with sample datasets for the model to understand and familiarize the cancer cell (Fig. 4). A sample image has been fed as an input to the trained model, and the model at this stage is able to tell the presence of cancer. In case malignancy is present, a message indicating 1 will be displayed which indicates the presence of cancer cells. While the major focus of the experiment was to show how even with smaller sets of images, we can achieve great accuracy. The following table shows the highest accuracy or the best score in all the 40 epochs (Table 1).
Fig. 4 Output-1 (malignant) Table 1 Results Experiment No. 1 2
Best score
Final score
0.674 0.9071
0.629 0.8776
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5 Future Work A convolutional neural network-based system was implemented to detect the malignancy cells present in the input image. The proposed system is able to detect the presence and absence of cancerous cells with accuracy of about 90% with very small sets of images. Data augmentation has been shown to produce promising outcomes to enhance the precision of machine learning duties and classification of images . Although traditional increase without a particular neural net has been very efficient, information augmentation method empowered by CycleGAN and similar neural networks can be very promising and greater task precision with little information can be obtained [12]. We would expect that such data augmentation techniques might be used to benefit only classification tasks lacking sufficient data, but even in the case of plethora of data, it can help improve the current state-of-the-art algorithms for classification. query Please check whether the sentence ‘Sometimes, such data augmentation could help us when the classes...’ conveys the intended meaning. Sometimes, such data augmentation could help us when the classes are unbalanced, yet this will be highly case dependent as in majority of cases where all cases have been equally covered there seems no requirement of creating more datapoints [13]. In the near future, the system will be trained with small datasets converted to larger datasets to diagnose the type of cancerous cell with its size and shape. The overall accuracy of the system can be improved using 3D convolutional neural network and also by improving the hidden neurons with deep network. To set a conclusion, we can clearly say that data augmentation even in its traditional form helped severely in improving the test results, and if used with CycleGAN, it can majorly improve accuracy.
References 1. NCI Homepage. https://www.cancer.gov/about-cancer/understanding/what-is-cancer 2. Thun MJ, DeLancey JO, Center MM, Jemal A, Ward EM (2010) The global burden of cancer: priorities for prevention. Carcinogenesis 31(1):100–110 3. Song Q, Zhao L, Luo X, Dou X (2017) Using deep learning for classification of lung nodules on computed tomography images. J Healthcare Eng 4. Halevy A, Norvig P, Pereira F (2009) The unreasonable effectiveness of data. IEEE Intell Syst 24(2):8–12 5. Perez L, Wang J (2017) The effectiveness of data augmentation in image classification using deep learning. arXiv:1712.04621 6. Bjerrum EJ (2017) Smiles enumeration as data augmentation for neural network modeling of molecules. arXiv:1703.07076 7. Zhu J-Y, Park T, Isola P, Efros AA (2017) Unpaired image-to-image translation using cycleconsistent adversarial networks. In: Proceedings of the IEEE international conference on computer vision, pp 2223–2232 8. Kaggle. https://www.kaggle.com/c/histopathologic-cancer-detection/data 9. LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 10. Datascience.https://www.datascience.com/blog/convolutional-neural-network
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11. Sasikala S, Bharathi M, Sowmiya BR (2018) Lung cancer detection and classification using deep CNN. Int J Innov Technol Explor Eng (IJITEE) 8(2S) 12. Chaturvedi P, Gupta L (2018) Study and detection of eye blink artifacts in EEG signals,. In: 2018 IEEE international students’ conference on electrical, electronics and computer science (SCEECS), Bhopal, India.https://doi.org/10.1109/SCEECS.2018.8546907 13. Makwana G, Gupta L (2018) The unreasonable effectiveness of data. IEEE Intell Syst 7(2.16):29–32
Free Vibration Analysis of the Sandwich Curved Panels with the Gradient Metallic Cellular Core Mohammad Amir and Mohammad Talha
Abstract In this study, free vibration analysis of the sandwich curved panels with the gradient metallic cellular core is presented. The present finite element model is established on higher-order shear deformation theory using curvilinear coordinate system. The sandwich curved panel consists of two isotropic face sheets and an FG metallic cellular core layer. The internal pores in the core layer follow different types of distributions. The material properties of the gradient metallic cellular core layer of the sandwich curved panel vary in the thickness direction as a function of porosity coefficient and mass density. The present model is validated with the limited results available in the open literature, and few new results are also discussed that can be utilized as a benchmark solution. The influence of porosity coefficient (e0 ) and pore distribution types on the free vibration characteristics of the sandwich curved panel with the gradient metallic cellular core are also analyzed. Keywords Metallic cellular core · Sandwich curved panels · Pore’s distributions
1 Introduction The curved panels are very commonly used in several engineering applications like civil engineering, aerospace, ship engineering, high-pressure vessels, and chemical industry and so on [1]. The sandwich curved panels with gradient metallic cellular core have exceptional mechanical properties of the lightweight structures which makes it more suitable to use it in aircraft, spacecraft and space structures. Therefore, for designing such types of structures, the vibration analysis of these structures is required. Zhao et al. [2] scrutinized the free vibration behavior of the gradient metallic cellular shallow curved panels with general boundary conditions. Amir and Talha [3] investigated the nonlinear vibration behavior of the functionally graded (FG) porous curved panels. Pandey and Pradyumna [4] presented a layer-wise finite element M. Amir (B) · M. Talha School of Engineering, Indian Institute of Technology Mandi, Suran, Himachal Pradesh 175005, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_156
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formulation for studying the vibration responses of the functionally graded sandwich curved panels. MagnuckaBlandzi [5] proposed mathematical modeling of the rectangular sandwich porous plate based on the poroelasticity. He considered the core as gradient metallic cellular with properties varying across its thickness. Liu et al. [6] scrutinized the blast resistance and dynamic response of hollow sandwich cylinders with gradient metallic cellular cores using the finite element method and found that the sandwich structures with thinner inner face sheets are superior to the sandwich structure with thicker face sheets of the cylinders. Jing and Zhao [7] studied the blast resistance, dynamic response and energy absorption capabilities of the sandwich curved panel with a gradient metallic cellular core, under the air-blast loading. Xiang et al. [8] studied the impulsive response of the clamped– clamped sandwich plate with FG core under blast impulsive loadings. MagnuckaBlandzi and Magnucki [9] calculated the effective design of the optimal dimensionless parameters of a porous sandwich beam using a theory of minimum total potential energy. Amir and Talha [10] studied the vibration characteristics of the porous curved beams/arches. Free vibration analysis of the sandwich curved panels with gradient metallic cellular core has not been reported so far. So, the present study focuses on the influence of porosity coefficient (e0 ) and pore distribution types on free vibration characteristics of sandwich curved panels with the gradient metallic cellular core. The internal pores in the core layer follow different types of distributions. The material properties of the gradient metallic cellular core layer of the sandwich curved panel vary in the thickness direction as a function of porosity coefficient and mass density.
2 Theoretical Formulations 2.1 Displacement Fields Consider a sandwich curved panel having gradient metallic cellular core, and the upper and the lower faces are made of isotropic metals. The total thickness h and the dimensions along the orthogonal curvilinear axes (ξ, ζ ) are a and b, respectively, and the principal radii of the curvature are denoted by Rξ and Rζ , respectively (Fig. 1). The displacement field based on higher-order shear deformation theory is taken as: z u + zφ1 + z 2 ϕ1 + z 3 ψ1 u = 1+ Rξ z (1) v + zφ2 + z 2 ϕ2 + z 3 ψ2 v = 1+ Rζ w=w
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Fig. 1 Geometry and configuration of the sandwich curved panel
where (u, v, w) are displacements of a random point along with the orthogonal curvilinear coordinates (ξ, ζ, z) and (u, v, w) are the corresponding displacements at the mid-plane and (φi , ϕi , ψi ) are higher-order rotation parameters of a point. The linear strain–displacement relationship for the sandwich curved panel is given as in [3] ⎧ ⎫ 1 ∂u w ⎪ + ⎪ ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ (1+z/R ) ∂ξ R ξ ξ ⎪ ⎪ ⎪ ⎪ ⎪ εξ ξ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂v w ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ ⎪ ⎪ ⎪ (1+z/R ) ∂ζ R ζ ζ ⎨ ⎬ ⎨ εζ ζ ⎪ ⎬ ⎪ ⎪ 1 ∂u 1 ∂v (2) = (1+z/Rζ ) ∂ζ + (1+z/Rξ ) ∂ξ γξ ζ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γξ z ⎪ ⎪ 1 ∂w ⎪ ⎪ − Ruξ + ∂u ⎪ ⎪ ⎪ ⎩ γ =⎪ ⎭ ⎪ (1+z/Rξ ) ∂ξ ⎪ ⎪ ∂z ⎪ ⎪ ⎪ ζz ⎪ ⎪ ∂w v ∂v 1 ⎩ ⎭ + − (1+z/Rζ )
∂ζ
Rζ
∂z
2.2 Modeling of Cellular Core The sandwich curved panel with the upper and the lower faces having thickness hf is made of isotropic metal. The metallic cellular core of the curved panel has a thickness hc with varying mechanical properties (Young’s modulus E c , density ρc , and Poisson’s ratio νc ) due to the distribution of the pores. Two types (type 1 and type 2) of pore distributions are taken in the modeling of the core layer. Figure 2a represents the type 1 pore distribution, in which mid-plane is highly porous compared to the top and bottom surface. The mechanical properties (i.e., density and Young’s modulus) have a minimum value (P0 ) at the mid-plane and maximum value P1 at the upper and lower surfaces. Figure 2b illustrates type 2 pore distribution in which the top is less porous and the bottom surface has the highest porosity. The mechanical
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Fig. 2 Cross sections and patterns of pore distributions: a Type 1 and b Type 2
properties P(z) vary with the thickness and have minimum value P0 at the bottom surface and maximum value P1 at the top. The material properties of the functionally gradient metallic cellular core with different pore distributions are expressed as in [2] For distribution type 1 E c (z) = E c1 [1 − e0 cos(π ζ )] ρc (z) = ρc1 [1 − em cos(π ζ )]
(3)
For distribution type 2 E c (z) = E c1 [1 − e0 cos(π ζ /2 + π/4)] ρc (z) = ρc1 [1 − em cos(π ζ /2 + π/4)]
(4)
where ζ = z/ h c e0 = 1 − E c0 /E c1 (0 < e0 < 1) and em = 1 − ρc0 /ρc1 (0 < em < 1)
(5)
where e0 is the porosity coefficient, em is the porosity coefficient for mass density, E c1 and ρc1 are the maximum values of Young’s modulus and density, E c0 and ρc0 are the minimum values of Young’s modulus and density, respectively.
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3 Results and Discussions To ensure the accuracy and efficiency of the present finite element model, two numerical results are compared. First of all,√the convergence and the comparison of the first three frequency parameters (λ = ωh ρ1 /E 1 ) of the simply supported (SSSS) spherical panel with those results given by Zhao et al. [2] are mentioned in Table 1. The geometric parameter of the spherical panel, a = b = 1 m, h = 0.02 m, Rζ = 2 m, Rξ = 2 m, and material properties, E 1 = 70 Gpa, ρ 1 = 2702 kg/m3 , ν = 0.3, and pore distribution type 2 with e0 = 0.5 is considered. From the table, it is found that there are a good convergence and excellent agreement between the present and the reference results. Due to a lack of results available for the present type of formulation, we compared the results of the frequency parameters of the sandwich cylindrical panels with FG (Al/Al2 O3 ) core at the a/h = 10, Rζ /a = 0.5, n = 0.2 and thickness proportions 1-81. The geometry and material properties are taken from Pandey and Pradyumna √ [3]. The non-dimensional frequency parameter is expressed as: λ = ωL 2 / h ρ0 /E 0 , where E 0 = 1 GPa and ρ0 = 1 Kg/m3 . The convergence and comparison of the present results are given in Table 2, and the present results are compared to the results calculated by Pandey and Pradyumna [3] using different shell theories. From the table, it is observed that there is a good agreement with reference results. Now, consider a sandwich spherical panel with gradient metallic cellular core and isotropic face layers having a proportion of the thickness 1:10:1. The material properties are E 1 = 200 GPa, ρ1 = 7800 Kg/m3 , ν = 1/3 and geometric properties of the spherical sandwich panel are a/b = 1, a/h = 100. The two distinct types of pore distributions and clamped–clamped (CCCC) are considered. The dimensionless √ frequency parameter is defined as:λ = ωh ρ1 /E 1 . Table 3 presents the frequency parameters of the clamped sandwich panels with curvature ratio (R/a = 20, 50, 100) and porosity coefficient (e0 = 0 and 0.7), for both types of pore distributions (type 1 and type 2). The effect of curvature ratio is Table 1 Comparison of the frequency parameters of a porous FG cylindrical panel at e0 = 0.5 Mesh size
Frequency parameters λ1
λ2
λ3
λ4
2×2
0.0091
0.0113
0.0114
0.0129
3×3
0.0093
0.0110
0.0111
0.0129
4×4
0.0094
0.0108
0.0108
0.0126
5×5
0.0094
0.0107
0.0107
0.0125
6×6
0.0094
0.0107
0.0107
0.0124
7×7
0.0094
0.0106
0.0106
0.0124
Ref. [2]
0.0093
0.0105
0.0105
0.0123
% Difference
1.1
0.9
0.9
0.8
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Table 2 Comparison of the frequency parameter of the sandwich panel (1-8-1) at the a/h = 10, Rζ /a = 0.5, and n = 0.2 Mesh size
Frequency
2×2
2.7686
3×3
2.7918
4×4
2.8036
5×5
2.8100
6×6
2.8139
7×7
2.8163
Sendor’s theory [2]
2.7746
Love’s theory [2]
2.7797
Donnell’s theory [2]
2.8709
Table 3 Influence of curvature ratio on frequency parameters sandwich panels R/a
Distribution type 1
Distribution type 2
e0 = 0
e0 = 0.7
e0 = 0
e0 = 0.7
20
0.001263
0.001281
0.001263
0.001228
50
0.001087
0.001139
0.001087
0.00107
100
0.001059
0.001117
0.001059
0.001044
observed that on increasing the values of R/a, the frequency parameters decrease for both types of pore distributions. Table 4 shows the effect of the porosity coefficient on frequency parameters of the clamped sandwich panels at the curvature ratio (R/a = 20, 50). From the table, it is observed that the effect of the porosity coefficient depends on the type of pore distribution taken. For distribution type 1, on increasing the value of e0 , the frequency parameters increase, while for distribution type 2, the frequency parameters decrease with the increase in porosity coefficient. For a better understanding, the effect of porosity coefficients for both distribution types on frequency parameters Table 4 Effect of porosity coefficients on frequency parameter sandwich panels e0
Distribution type 1
Distribution type 2
R/a = 20
R/a = 50
R/a = 20
R/a = 50
0
0.001263
0.001087
0.001263
0.001087
0.2
0.001259
0.001092
0.00125
0.001079
0.4
0.00126
0.001103
0.001238
0.001073
0.6
0.00127
0.001123
0.001229
0.001069
0.8
0.001302
0.001164
0.001232
0.001075
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Fig. 3 Effect of porosity coefficients for both distribution types on frequency parameters
of the clamped sandwich panels (at R/a = 100) is displayed in Fig. 3. The same conclusions can be drawn from the figure as it is observed from Table 4.
4 Conclusions A C0 finite element method in conjunction with higher-order shear deformation theory is utilized for the present formulation. The influence of porosity coefficient (e0 ), curvature ratio and pore distribution types on free vibration responses of the sandwich curved panels with the gradient metallic cellular core are studied. It is found that the effect of the porosity coefficient depends on type of distribution taken. On increasing the values of R/a, the frequency parameters decrease. Acknowledgements The authors would like to acknowledge financial support by the DST-Science and Engineering Research Board, Govt. of India, under Project No. YSS/2015/001290, dated-09th Nov 2015.
References 1. Pang F, Li H, Cui J, Du Y, Gao C (2019) Application of flügge thin shell theory to the solution of free vibration behaviors for spherical-cylindrical-spherical shell: a unified formulation. Eur
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J Mech A Solids 74:381–393 2. Zhao J, Xie F, Wang A, Shuai C, Tang J, Wang Q (2018) A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions. Compos Part: B Eng 156:406–424 3. Amir M, Talha M (2019) Nonlinear vibration characteristics of shear deformable functionally graded curved panels with porosity including temperature effects. Int J Press Vessels Pip 172:28–41 4. Pandey S, Pradyumna S (2015) A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells. Compos Struct 133:438–450 5. Magnucka-Blandzi E (2011) Mathematical modelling of a rectangular sandwich plate with a metal foam core. J Theor Appl Mech 49:439–455 6. Liu X, Tian X, Lu TJ, Zhou D, Liang B (2012) Blast resistance of sandwich-walled hollow cylinders with graded metallic foam cores. Compos Struct 94(8):2485–2493 7. Jing L, Zhao L (2016) Blast resistance and energy absorption of sandwich panels with layered gradient metallic foam cores. J Sandwich Struct Mater 0(0):1–19. https://doi.org/10.1177/109 9636217695651 8. Xiang C, Qin Q, Wang F, Yu X, Wang M, Zhang J, Wang TJ (2018) Impulsive response of rectangular metal sandwich plate with a graded foam core. Int J Appl Mech 10(6):1850064 9. Magnucka-Blandzi E, Magnucki K (2007) Effective design of a sandwich beam with a metal foam core. Thin-Walled Struct 45(4):432–438 10. Amir M, Talha M (2018) Thermoelastic vibration of shear deformable functionally graded curved beams with microstructural defects. Int J Struct Stab Dyn 18(11):1850135, 24 p
A Study on Clean Coal Technology in the Indian Context Swayam Sampurna Panigrahi and Purna Chandra Panigrahi
Abstract Coal was, is, and will continue to be the backbone of global electricity generation. This is because coal as a fuel is one of the most cost-effective ways to provide affordable, safe, and reliable electricity at the scale that is needed to achieve genuine access to modern electricity services world-wide. However, the conventional processes involved in mining are associated with quite few accidents rendering injuries to the workers. In addition, a huge amount of greenhouse gases (GHG) is also emitted during the process. Thus, the process of mining coal must undergo a paradigm shift in terms of the machinery and equipment employed which will not only bring about a safe working environment for the workers, but also an efficient, economic, and eco-friendly method. In this regard, the present article analyses the initiatives adopted by an Indian coal mining organization located in Odisha to promote workplace safety, environment conservation, and economic viability which are also the three major aspects of sustainable development. A comparative study between the conventional coal mining process and the cleaner way of coal mining expressed in the form of the energy consumption has also been presented by deploying the surface miner technology. The subsequent savings in the form of energy, cost, and reduction of carbon emissions have been highlighted. Keywords Clean coal technology · Surface miner · Ergonomics · GHG emissions · Sustainable development
1 Introduction At present, population explosion and per capita energy consumption are the two important aspects in the globe. Coal is the major fossil fuel in the planet earth. Coal as a fossil fuel is a major player of energy security of world. It is considered as a S. S. Panigrahi (B) Operations Management, International Management Institute, Bhubaneswar, India e-mail: [email protected] P. C. Panigrahi Mahanadi Coalfields Limited, A Subsidiary of Coal India Limited, Sambalpur, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_157
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significant source and material for electricity generation, especially for the industries which are energy-intensive like steel or cement industry [2]. It supplies energy for the development of the society. Hence, coal mining cannot be avoided. A number of greenhouse gases (GHGs) are emitted in the processes of coal mining including carbon dioxide. Coal mining also has a significant contribution towards greenhouse effect which leads to climate change and global warming. In 2013, more than 83% of the total carbon dioxide emission originated from the combustion of coal [3]. An amount of almost 79% of sulphur dioxide (SO2 ), 57% of nitrogen oxides (NOx ), and 44% of particulate matter (PM) originated out of the direct coal combustion of coal. Also, 93% of SO2 , 70% of NOx , and 67% of PM emissions originated from all kinds of usage of coal constituting direct combustion emission and emission from coke stoves and other industrial furnaces [2]. Owing to which, an effective means for the reduction of carbon emissions is of extreme importance in the current scenario. And in order to facilitate coal mining in ecologically and economically viable way, clean coal technologies (CCTs) are essential. Hence, application of CCTs in mining, be it coal or non-coal mining, is the need of the hour. The biggest coal reserves are found in the USA, Russia, China, and India. As per the BP Statistical Review of World Energy [7], the top coal producers in 2016 are indicated in Table 1. Clean coal technologies involve a wide range of coal production and utilization related technologies, including green mining, surface miners, coal purification, highefficiency power generation, advanced coal conversion, pollution control, and carbon capture, utilization, and storage. As the present article is concerned with the coal mining organization located in Odisha, the following are some figures about the availability of the same in the state. The total mineable coal reserve available in Odisha is 75.8 BTe, with two coalfields, i.e. one is Talcher coalfield and the other is Ib coalfield. Mineable coal reserves of Talcher coalfield and Ib coalfield are 50.968 BTe and 24.830 BTe, respectively. The coal producing company under study is a pioneer in the introduction of clean and new technologies. It was also the first company to introduce surface miners for coal production in India. The organization was one of the early birds to augment its clean coal production programme. It has been the trendsetter in introducing blast-free technology for winning coal in open cast mine by surface miner. There also exists a SILO operating at one of the mines owned by this organization. The organization has transformed itself from one with the smallest capacity (23 MTY) in 1992 to the one with the largest capacity (100MTY) in 2010–11. With the growing demand of clean and high-quality coal as the measures of energy security, this organization has always been looked as viable, productive, and growing coal company since its inception. Table 1 Top coal producers in 2016 Country
China
USA
Australia
India
Indonesia
Russia
Production (in BTe)
3.411
0.660
0.493
0.692
0.434
0.385
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2 Clean Coal Technology There is a tremendous pressure on the organizations to reduce the carbon footprints from their manufacturing processes. The mining sector is a major contributor to air pollution; hence, it requires a paradigm shift from traditional technologies of mining to the latest ones. It has been reported that the mining sector still engages in violation of human rights as well as environmental protection (Human rights watch 2012). It is high time to focus on clean and sustainable mining practices in the Indian context. India is the fourth-largest coal producer in the world. Coal will have a 48% share in the energy consumption in India by the year 2040. The coal sector alone has its unique set of environmental challenges to handle. Some of them are mine accidents, land acquisition, mining waste disposal, and last but not the least air pollution [14]. This study proposed a conceptual framework that addresses the above issues through adoption of clean technologies and processes and land restoration measures. There are some techniques like the underground coal gasification (UCG) which are considered to be clean technologies that minimize air pollution in coal mining [13]. In emerging economies like China and India, which have large coal deposits, coal is the major source of energy generation. Owing to the fact that, the dependency on coal cannot be withdrawn overnight, it is imperative to adopt measures that support clean and sustainable mining [8]. Focused research needs to be conducted with regards to such nations where coal serves as the primary energy source. In view of this, authors from China have probed into certain breakthroughs in coal mining under the umbrella of China Coal Industry 4.0 and 5,0 which includes concepts like the intelligent mining, ultra-low emission mining, etc. [10]. A systematic review on the sustainable mining concept covering 2562 articles from the literature over the time horizon of 1998–2017 has been presented. The authors highlighted the fact that for sustainable mining, it is crucial to be flexible towards innovations and latest technologies [1]. A study conducted on mining sector in Finland highlighted the need for top management support for implementing clean and green mining technologies [9]. They also reported that there exists a disparity between management commitments and actual implementation. Appropriate business strategies that bridge this gap must be developed by the mining enterprises. The authors have developed a novel approach for the underground waste rock disposal to address the waste generated during the mining process [11]. It is important to understand the current status and future of coal production. A study has been conducted in this regard which reported that although the overall coal production will be reduced in the upcoming decades, coal will still be required to meet the global energy demands. This necessitates the need to develop clean coal technologies that minimize air pollution, ensuring workforce safety along with economic viability. The following sub-sections discuss various technologies adopted by the focus company.
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2.1 Blast-Free Mining Technology Surface miners were first employed in the Indian context on a limestone deposit in the year. The surface miner was introduced for the first time in coal mining in the year 1999 in Lakhanpur OCP which is one of the mines of this company. Currently, at least more than 100 surface miners have been employed in the Indian mining sector owing to the benefits of higher productivity as well as eco-friendliness associated with it [5]. The surface mining technology was introduced in 1970s which was inspired from the road milling machines which is based on cutting the old road surface for road construction [4]. Surface miner finds its natural applications in opencast projects where operations such as drilling, blasting, and crushing are prohibited. The surface miner produces coal with desired size as demanded by the consumers. It also segregates the parting/dirt band which is responsible for reduction in the quality of coal. This help in providing high-quality coal generating huge revenues. Apart from the above, one most important aspect is the safety feature which comes with the surface miner as it eliminates drilling, blasting, and crushing operations. Figure 1a represents a surface miner and (b) represents the cutting of coal by the surface miner. One of the reasons for deploying surface miners in the Indian context was strategic advantage. In conventional mining, owing to the depletion of surface deposits, the surface mines have now become deeper. Moreover, the operating surface area required is becoming larger. India is a highly populated country and the land available for mining is less than the nation’s mineral demand. Now, the operating surface mines have gotten closer to the human settlements where conventional drilling and blasting technique is prohibited in anticipation of ground vibration [6]. Hence, in the Indian context, employing the surface miners has become popular. The following sub-sections below discuss the advantages obtained by using the surface miner for coal mining: 1.
Reduction in dependency on human factor and reduction in stress/ strain / health / hygiene / safety of manpower:
Fig. 1 a Surface miner and b cutting of coal by surface miner at the coal face
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• Surface miner reduces dependency on manpower as a surface miner alone produces coal more than a few workers will produce. Surface miner delivers more than 90% target size coal (2–40 mm). Also, surface miner is designed to produce high volume of coal with integrated and enhanced safety features. • Surface miner avoids blasting, drilling, and crushing which helps in reduction of stress and strain caused to workers due to dust and noise produced by these operations. • Blasting and drilling operations produce noise and dust which also has impact on health of workers, and the latest machineries have helped in minimizing the effect of this on workers’ health. • Safety is enhanced as no explosive is involved in this type of mining also the latest equipment are inbuilt with latest safety features which raises the level of safety while working. • Use of surface miner results in improved slope stability, reduced rockfall, and no production delays like interruptions for blasting. • Use of surface miner results in no fire hazards to coal seam as it never leaves behind any material amenable to spontaneous heating. 2.
Ergonomics/helping in environment protection: • Different types of environment pollution like noise and air caused by drilling, blasting, and crushing are reduced with the help of surface miner. • Energy consumed by conventional method (drilling, blasting, and crushing) is 11.444 MJ/Te of coal whereas energy consumed by deployment of surface miner is 7.128 MJ/Te, thus taking total energy conserved per Te of coal as 4.316 MJ/Te which will also reduce total carbon emission. • Conveyor system has reduced the exposure of workforce in loading and transportation of coal; this prevents accidents, controls pollution, and saves energy. • SILO with RLS expedites the transporting and loading operations which saves huge amount of energy eliminating a large fleet of vehicles/trucks and loading machines. This quantity of energy saved is manifested in reduction of carbon emission which is need of the day. • The surface miner’s anti-vibration cabin offers better ergonomics and optimized working conditions for the machine operators. • The use of surface miner also has a positive effect on water management at opencast mine with improved drainage resulting in reduced seepage in ground. • The coal processing through washery will reduce the ash contents to the desired level, and at the same time, it will reduce the weight of the yield by 7–10%. This will result in decrease in transportation cost and in turn reduces the fuel consumption. Thus, the coal processing will reduce carbon emission.
3.
Quality and quantity of production: • The latest equipment has helped in increasing the production as machines are more efficient helping in raising production levels
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• Surface miners have combined different operations earlier which were part of mining activities resulting in high productivity and reduced idle time. • Surface miner produces coal with less coal loss and dilution, raising the quality and quantity of coal produced. • Surface miner also segregates the parting/dirt band which decreases the quality of coal, thereby providing quality coal generating huge revenues and increasing productivity and profitability of the company. • Improved coal recovery especially in areas sensitive to blasting results in increased quantity of coal produced.
2.2 Case Study for Comparison of Energy Consumption A study has been done on energy conservation considering three broad aspects/operations, i.e. drilling, blasting, and crushing, which are eliminated due to deployment of surface miner in opencast coal mines of the organization being dealt with. 1.
Surface miner energy consumption considering the following: (a) (b) (c) (d) (e)
Size of the Patch = 100 × 100 × 25 m3 Total no. of tonnes of coal production = 100 × 100 × 25 × 1.69 = 422,500 Te Surface miner production on average basis (Practical) = 500 Te/h Average diesel consumption of Surface miner (Practical) = 90 l/h No. of hours surface miner has to operate for production of above said patch = 422,500/500 = 845 h. So, the total diesel consumption = 90 × 845 = 76,050 l. Energy conversion factor = 1 ltrs of diesel (Industrial) ~ 39.6 MJ. Energy surface miner = 76,050 × 39.6 = 3,011,580.00 MJ Energy consumption of surface miner per Te of coal = 3011580.00 MJ/422500.00 Te of coal = 7.128 MJ/Te
2.
(1)
Drilling operation energy consumption considering the following: (a) (b) (c)
Spacing = 5 m Burden = 5 m Depth = 1 m. Coal production through drilling operation per 1 m depth = 5 × 5 × 1 × 1.69 = 42.25 Te/m depth. Drilling operation norms (Practically through diesel drill) = 1.2 l/m. Total diesel consumption = 10,000 m × 1.2 Ltrs/m = 12,000 Ltrs.
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Energy conversion factor = 1 l of diesel (Industrial) ~ 39.6 MJ. Energy consumption through drilling operation = 12,000 × 39.6 MJ = 475,200 MJ Energy consumption through drilling operation per Te of coal = 475200 MJ/422500 = 1.124 MJ/Te 3.
(2)
Blasting operation energy consumption considering the following: (a) (b) (c)
Powder Factor = 4.45 Te/Kg of explosive Total explosives required for 100 × 100 × 25 m3 patch = 422,500/4.45 = 94,943.82 kg Total explosives required in Tonnage (approximately) = 95 Te Energy conversion factor = 1 Te of Explosives ~ 4184 MJ. Energy consumption through blasting operation = 95 × 4184 MJ = 397,480 MJ Energy consumption through blasting operation per Te of coal = 397,480/422,500 = 0.940 MJ/Te
4.
(3)
Coal handling plant (CHP) energy consumption considering the following (a) (b)
CHP CKT crushing capacity = 200 Te/h No. of hours CHP CKT has to operate for crushing 422,500.00 Te of coal = 422,500/200 = 2112 h. (i) Crusher motor = 160 kW, (ii) Belt conveyor motor = 110 kW, (iii) Chain conveyor motor = 90 kW, (iv) Cooling fan motor = 5 kW, and (v) Total kW = 160 + 110 + 90 + 5 = 365 kW. Energy consumption for 1 hr CHP CKT operation taking 70% efficiency into consideration = 365 kW/0.7 = 255.5 kW
Energy conversion factor = 1 kWH = 3.6 MJ. Energy consumption for 1 h CHP CKT operation = 521.4 × 3.6 MJ = 1877.04 MJ. Energy consumption for 2112.5 h = 1877.04 × 2112.5 = 3,965,247 MJ Energy consumption through CHP crushing operation per Te of coal = 9.380 MJ/Te
(4)
It can be observed that the total energy conserved per Te by deployment of surface miner instead of conventional methods (drilling, blasting, and crushing) for coal production is given by:
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Table 2 Quantification of carbon emission reduction by using surface miner Year
Total coal production (MTe)
Coal production by surface miner (MTe)
% share of surface miner
Total energy saved (MJ)
Total carbon emission reduced (Kg)
Total carbon emission reduced (Ton)
2010–11
98.485
54.916
56
2011–12
103.118
59.125
57
237,017,456
17,988,284.960
17,988.285
255,183,500
19,366,985.000
19,366.985
2012–13
107.895
73.832
2013–14
110.439
86.463
68
318,658,912
24,184,409.920
24,184.410
78
373,174,308
28,321,820.280
28,321.820
2014–15
121.379
2015–16
137.901
106.821
88
461,039,436
34,990,286.760
34,990.287
125.683
91
542,447,828
41,168,723.480
2016–17
139.208
41,168.723
127.809
92
551,623,644
41,865,116.040
41,865.116
Energy consumed by conventional method − Energy consumed by deployment of surface miner = [(2) + (3) + (4)]−(1) = (1.124 + 0.940 + 9.380) − (7.128) = 11.444 − 7.128 = 4.316MJ/Te of coal = 0.004316 GJ/Te Table 2 represents the reduction in carbon emissions in the present mining organization obtained by using the surface miner instead of the conventional methods.
2.3 Case Study for Cost Savings A study has been made by analysing the cost incurred in different operations involved in the conventional method of mining in opencast coal mines and that of in the surface miner. Table 3 below represents the cost savings per tonne of coal made by the organization by using surface miner compared to the conventional method of mining.
3 Clean Processing Technology The coal deposit available in the company in focus is of thermal grade. The raw coal ash as per the test results of CMPDI is 36.3%, and the likely average monthly raw coal ash considering dilution has been arrived as 40.1%. Hence, the main washing process
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Table 3 Calculation of cost saving due to use of surface miner S. No
Cost comparison
Conventional (Rs/Te) Surface miner (Rs/Te)
1
Drilling cost
7.5
0
2
Blasting cost
9.37
0
3
Crushing cost
52
4
Transportation in CHP 5
0
5
Cutting/loading cost
10
30
6
Transportation cost up 35 to siding
40
7
Net cost at siding
70
118.87
Cost saving per Te
48.87
Cost saving for 127.8 MTe (in Cr.)
624.559
Table 4 Time line for commencement of different washeries
0
Name of washery
Time line for commencement
Basundhara Washery
January, 2020
Ib-Valley Washery
July, 2019
Jagannath Washery
December, 2020
Hingula Washery
December, 2020
for the washery has been selected on the basis of likely average monthly raw coal ash, i.e. 40.1% and qualitative requirement of washed coal of the thermal power stations. The ash percentage of washed coal has been kept around 33.5 ± 0.5%. Further, the study of test results revealed that washing of 50–13 mm coal and mixing of untreated −13 mm size fraction are suitable for achieving the targeted washed coal ash. The process of the proposed washery for coal processing is based on closed water circuit system. All the water fed into the system will be collected after use and re-circulated after treatment in various units, and no effluent will be allowed to escape into the natural drainage system. The coal processing through washery will reduce the ash contents to the desired level, and at the same time, it will reduce the weight of the yield by 7–10%. This will result in decrease in transportation cost and in turn reduces the fuel consumption. Thus, the coal processing will reduce carbon emission. The company is planning to set up four washeries of 10 MTY capacity each, shortly in different phases; the details of time line for commencement are mentioned in Table 4.
4 Silo In order to save energy, control pollution, and prevent accidents by eliminating exposure of workforce in loading and transporting operation, it is desirable to transport
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Fig. 2 SILO at Bharatpur OCP, Talcher Coalfield
coal and over burdens through conveyor system. This will also lead to substantial amount of investment which will result in long-term savings on transportation cost both in departmental and outsourced areas. The SILO with rapid loading system (RLS) was commissioned in the year 1989, and coal is being dispatched to NALCO through the RLS from Bharatpur OCP as shown in Fig. 2. SILO with RLS expedites the transporting and loading operations which saves huge amount of energy eliminating a large fleet of vehicles/trucks and loading machines. This quantity of energy saved is manifested in reduction of carbon emission which is the need of the day.
5 Conclusion Our planet’s atmosphere is overloaded with heat-trapping carbon dioxide, methane, nitrous oxide, water vapour, etc., which threatens large-scale disruptions in climate with disastrous consequences. It is crucial to spur the adoption of cleaner technology and energy sources in all fields. Greenhouse effect is one of the major threats leading to climate change and global warming. Worldwide efforts are now being made to control the emission of greenhouse gas like carbon dioxide and thereby reduce the pressures for accelerated climate changes. The governments in various nations are introducing stricter policies in order to curb this large-scale pollution. There is a need of adopting the technologies as an initiative with clean coal management approach. The present article elaborates the initiatives made by a coal mining organization located in Odisha to promote workplace safety, environment conservation, and economic viability in coal mines. The article describes the use of surface miner instead of the conventional methods for coal mining. Various advantages associated with surface mining as compared to the conventional methods have been brought forward with the help of few case-studies. The subsequent savings in the form of energy, cost, and reduction of carbon emissions have been highlighted. In addition, coal processing technology and use of SILOs have also been elaborated.
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References 1. Aznar-Sánchez JA, Velasco-Muñoz JF, Belmonte-Ureña LJ, Manzano-Agugliaro F (2019) Innovation and technology for sustainable mining activity: a worldwide research assessment. J Clean Prod 221:38–54 2. Chang S, Zhuo J, Meng S, Qin S, Yao Q (2016) Clean coal technologies in China: current status and future perspectives. Engineering 2(4):447–459 3. International Energy Agency (IEA). CO2 emissions from fuel combustion 2015 edition [Internet]. OECD/IEA, Paris; c2016 [cited 2016 Jul 21]. Available from https://wds.iea.org/ WDS/Common/Login/login.aspx 4. Dey K, Ghosh AK (2008) Predicting “ cuttability” with surface miners-A rockmass classification approach 5. Dey K, Bhattacharya J (2012) Operation of surface miner: retrospect of a decade journey in India. Procedia Eng 46:97–104 6. Dey K (1999) Performance analysis of continuous surface miner in indian surface coal mines— A case study. Unpublished M. Tech thesis, Indian School of Mines, Dhanbad, p 40 7. Dudley B (2017) BP statistical review of world energy 2016. British Petroleum Co., London, UK 8. Jiachen W, Lei W, Yang Y, Shengli Y (2015) Science mining and clean coal technology in China. J Clean Energy Technol 3(6) 9. Ruokonen E, Temmes A (2019) The approaches of strategic environmental management used by mining companies in Finland. J Clean Prod 210:466–476 10. Wang G, Xu Y, Ren H (2019) Intelligent and ecological coal mining as well as clean utilization technology in China: Review and prospects. Int J Mining Sci Technol 29(2):161–169 11. Wei Z, Peng L, Dong-Sheng Z, Zhi Y (2018). A novel clean mining technology involving the underground disposal of waste rock in coal mines. Arch Mining Sci 63 12. Xu J, Gao W, Xie H, Dai J, Lv C, Li M (2018) Integrated tech-paradigm based innovative approach towards ecological coal mining. Energy 151:297–308 13. Yang L, Liang J, Yu L (2003) Clean coal technology—Study on the pilot project experiment of underground coal gasification. Energy 28(14):1445–1460 14. Zhengfu BIAN, Inyang HI, Daniels JL, Frank OTTO, Struthers S (2010) Environmental issues from coal mining and their solutions. Mining Sci Technol (China) 20(2):215–223
Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing with Partial and Fully Textured Surface Sanjay Sharma
Abstract In the present numerical-based study, the effect of triangular shape textured on the bearing steady-state performance has been investigated. The triangular shape texture having different values of depth size, number of textures and location has been used in the study to find the load carrying capacity and coefficient of friction and compared with untextured bearing. The pressure and fluid-film thickness in the lubricant flow domain having characteristics of iso-viscous and Newtonian fluid, which is governed with Reynold’s equation, have been solved by discretizing the domain into four-noded quadrilateral isoparametric elements by using finite element method. Four different cases of texture distribution, namely full textured region (0°–360°), first half textured region (0°–180°), second half textured region (180°360°) and increasing pressure textured region (144°–288°) on the bearing surface have been studied under low and average eccentricity ratios of 0.2, 0.4 and 0.6. The performance enhancement ratio has been also calculated in order to finalize optimum design parameters. The results indicate that surface texturing in an increasing pressure region has a positive effect on the bearing performance enhancement ratio, when the bearing operates at lower eccentricity ratio of 0.2 and texture depth of 1.0. Keywords Hydrodynamic journal bearing · Surface textures · Load carrying capacity · Coefficient of friction
Nomenclature Dimensional Parameters c D e
Geometrical clearance, Mm Diameter of journal, Mm Eccentricity of journal, Mm
S. Sharma (B) School of Mechanical Engineering, Shri Mata Vasihno Devi University, Katra 182320, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_158
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Q r RJ , Rb t W0 x y X J, Z J z
S. Sharma
Lubricant flow, Mm3 s−1 Radial coordinate Radius of journal and bearing, mm Time, s External load, N Circumferential coordinate Axial coordinate Journal center coordinate Coordinate along film thickness
Non-dimensional Parameters c¯ ¯ F¯0 F, F¯x , F¯z h¯ h¯ min p, ¯ p¯ c p¯ L , p¯ max t¯ W¯ 0
c/RJ (F, F0 ps RJ2 ) (Fx , Fz ps RJ2 ) h/c h min c ( p, pc ) ps ( pL , pmax ) ps t c2 ps μr RJ2
μ¯
μ μr ωJ μr RJ2 c2 ps , Speed parameter
W0 ps RJ2
Matrices and Vectors F¯ {p} ¯ Q¯ R¯ , R¯ x z R¯ H
Matrix (Fluidity) Vector (Nodal pressure) Vector (Nodal flow) Nodal RHS vectors (Journal center velocity) Column vector (Hydrodynamic terms
1 Introduction Application of surface texturing on mechanical components results in the improvement of various tribological aspects. The use of surface texture on a bearing inner surface is an effective technique to enhance the static performance of hydrodynamic
Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing …
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bearing. However, it would also result in an increase in friction losses and leads to more power requirement. In order to keep losses at a minimum level, the location, numbers and depth of texture is an important parameters to be optimal. Several researchers have studied different parameters and configuration of surface textures in different bearings in order to improve its performance. Qiu et al. [1] studied six different texture shapes to evaluate the friction and stiffness coefficient of gas lubricant parallel slider bearings. They found and finalized ellipsoidal shape to yield the maximum performance of bearing. Dadouche and Conlon [2] experimentally studied the effect of dimple shape texture on the steady-state performance characteristics of highly loaded journal bearing contaminated lubricant. They found that lightly textures bearing showed exceptional performance as compared to highly textured journal bearing. Su et al. [3] conducted parametric analysis by using finite difference method (FDM) to obtain the optimum surface textures parameters in terms of load carrying capacity of slider bearing with elastic deformation. Zheng et al. [4] studied the effect of hexagonal texture size on tribological properties under mixed lubrication. They found that friction coefficient decreases with increase in width within examined range. Feng and Peng [5] numerically studied the effect of rotary speed, texture depth, width and number on the static performance of water-lubricated thrust bearing by using JFO (Jakobsson–Floberg–Olson) cavitation model and found that well-designed groove texture can improve the performance of bearing. Shinde and Pawar [6] studied numerically by using COMSOL multiphysic 5.0 software to find static performance analysis of square-shape microtextured hydrodynamic journal bearing by placing texture in 90°–180° region results in improvement in static performance. Liang et al. [7] used CFD technique to investigate the effect of partial texture location on the performance of hydrodynamic bearing. They reported that texture located in inlet area and having shallow size can improve the performance of bearing. Yu et al. [8] used CFD technique to investigate the effect of surface texture on grease-lubricated journal bearing. They reported that texture shape has a more significant effect on its static performance parameters. Tala-Ighil et al. [9] studied textured hydrodynamic journal bearing with spherical dimple by using FDM to investigate the static performance and found that these values increase with the change in texture geometrical parameters. Cupillard et al. [10] studied circular dimple shape texture hydrodynamic journal bearing by using computational fluid dynamics to investigate the static performance parameters and found low friction coefficient for deep dimples than swallow dimples. Uddin et al. [11]studied texture hydrodynamic slider bearing with star-like structure by using finite difference method (FDM) to investigate the static performance parameters and found friction coefficient lower than the ellipse-, circle- and chevron-shaped texture. Yadav and Sharma [12] studied dimple textured hydrodynamic thrust bearing by FEM and JFO mass conservation algorithm to investigate the static performance parameters and found that for better performance, an optimum location of the lubricant supply hole and depth of dimple is required. Brizmer and Kligermam [13] studied microdimple texture hydrodynamic journal bearing by using laser surface texturing (LST) to investigate the load capacity and attitude angle and found that maximum load capacity for partial textures. Shen and Khonsari [14] studied different texture shapes in hydrodynamic slider bearing
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by using sequential programming method to investigate and optimized the static performance parameters. Tala-Ighil et al. [15] studied static performance parameters of micro cavities texture hydrodynamic journal bearing by using finite difference method (FDM) and found maximum at an appropriate arrangement of the textured area. Sharma and Yadav [16] studied dimple-shape textured hydrodynamic thrust bearing operating on non-Newtonian lubricant by using finite element method (FEM) to investigate the static performance parameters and found that these parameters are significantly affected by the behavior of lubricant. Gruetzmacher et al. [17] investigated the frictional and wear performance of multi-scale steel surfaces in comparison with laser-patterned and purely microcoined surfaces. They found that the COF decreases as the depth of the multi-scale pattern structures is increased. The best wear behavior was shown by the purely microcoined sample, whereas pronounced wear occurred in the other two cases. Gropper et al. [18] studied textured tilting pad thrust bearing and optimized the texture depth, circumferential and radial extent of the textured region. They also concluded that the optimum texture depth is significantly dependent on texture density as well as the operating conditions. Gropper et al. [19] studied textured tilting pad thrust bearings under hydrodynamic conditions, using a mass conserving cavitation algorithm. They compared three different discretization methods and significantly reduced the computational time for textured journal bearing. Some studies (Charitopoulos et al. [20]) Vl˘adescu et al. [21] related to investigation of both thermal and surface texturing effects have also been considered. Tala-Ighil and Fillon [22] reported that the average film temperature for full and first half textured region was observed to be higher than texturing in the second half and pressure build-up region of the fluid domain. After go through the literature, it is found that the effect of triangular-shaped texture on the static performance of hydrodynamic bearing has not been yet performed. Four cases of different texture distribution have been studied with different texture depths, and out of four cases considered, the most favorable in terms of static performance is finalized and finally the performance enhancement ratio has been calculated. The study has been carried out considering the bearing operation only under low and average eccentricity ratios of 0.2, 0.4 and 0.6. The objectives of the study are to find the value of optimum triangular texture depth and location in order to get maximum static characteristics performance and performance enhancement ratio. The obtained results have been expected to be valuable for bearing designers.
2 Methodology The triangular-shape textured journal bearing has been shown in Fig. 1. The nondimensional governing Reynold’s equation for a hydrodynamic bearing is given by Sharma et al. [23]:
Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing …
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Fig. 1 a Cross-sectional view bearing having textured surface; b Triangular-shape texture views; c Full triangular-shape textured distribution on the bearing surface
∂ ∂ ∂ ∂p ∂p F1 ∂h 3 3 + = h F2 h F2 h + 1− ∂α ∂α ∂β ∂β ∂α ∂t F0
(1)
where F 0 , F 1 and F 2 are viscosity functions and are given by the following expressions: 1 F0 = 0
1 dz, F 1 = μ
1 0
z dz and F 2 = μ
1 0
z F1 z− dz μ F0
2.1 Fluid Film Thickness The minimum fluid film thickness for triangular-shape textured hydrodynamic bearing is given by Sharma et al. [23]:
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S. Sharma
h = 1 − X J cos α − Z J sin α + h
(2)
where, X J and Z J = journal center coordinates at equilibrium position and h = Variation in fluid film thickness due to triangular shape textures on the bearing surface and its value is given by following equation given by Qiu et al. [24].
2.2 FEM Formulation After discretization of lubricant flow domain with four-noded quadrilateral isoparametric elements and by using Galerkin’s orthogonality criterion, the global system equation is obtained as Sharma et al. [23]: F { p} = Q + RH + x R X J + z R Z J
(3)
3 Results and Discussion 3.1 Effect of Textures on Load Carrying Capacity and Coefficient of Friction: See Figs. 2 and 3.
5
δt=0.0 δt=1.0 δt=2.0
25
δt=0.5 δt=1.5 COEFFICIENT OF FRICTION
LOAD CARRYING CAPACITY
6
4 3 2 1 0 0.2
0.4 ECCENTRICITY RATIO
(a)
0.6
20
δt=0.0 δt=1.0 δt=2.0
δt=0.5 δt=1.5
15 10 5 0 0.2
0.4 ECCENTRICITY RATIO
0.6
(b)
Fig. 2 a, b Variation of load carrying capacity and coefficient of friction with eccentricity ratios of textured hydrodynamic journal bearings having full texture region (0°–360°)
Lubrication Characteristics of Newtonian-Lubricated Hydrodynamic Bearing … 14
5
δt=0.0
δt=0.5
δt=1.0
δt=1.5
COEFFICIENT OF FRICTION
LOAD CARRYING CAPACITY
6
δt=2.0
4 3 2 1 0 0.2
0.4 ECCENTRICITY RATIO
0.6
δt=0.0 δt=1.0 δt=2.0
12
1641 δt=0.5 δt=1.5
10 8 6 4 2 0.2
0.4 ECCENTRICITY RATIO
(a)
0.6
(b)
Fig. 3 a, b Variation of load carrying capacity and coefficient of friction with eccentricity ratios of textured hydrodynamic journal bearings having pressure developing region (144°–288°)
3.2 Performance Enhancement Ratio (PER) In order to find the optimal values of texture depth and location, the performance enhancement ratio defined by Sharma et al. [23]: PER =
LCC of textured bearing LCC of plain bearing COF of textured bearing COF of plain bearing
Performance enhancement ra o
From Fig. 4 it is clear that at texture depth of 1.0 the performance enhancement ratio is maximum in pressure increasing region only, at eccentricity ratio of 0.2.
2 1.6 1.2 0.8 0.4 0
0-360
0-180
180-360
144-288
Texture region ε = 0.2
ε = 0.4
ε = 0.6
Fig. 4 Performance enhancement ratio of textured hydrodynamic journal bearings at texture depth of 1.0
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Table 1 Most optimistic results obtained in the present study Textured region
Effect on Percentage Percentage Percentage Percentage Eccentricity Textured bearing increase in decrease increase in decrease ratio depth performance LCC in LCC COF (%) in COF (%)
0°–360°
Negative
–
14.66
15.35
–
0.6
0.5
0°–180°
Negative
–
1.15
1.07
–
0.6
0.5
180°–360° Negative
–
13.7
14.00
–
0.6
0.5
144–288
34.48%
–
–
25.49
0.2 and 0.4
1.0
Positive
References 1. Qiu M, Delic A, Raeymaekers B (2012) The effect of texture shape on the load-carrying capacity of gas-lubricated parallel slider bearings. Tribol Lett 48:315–327 2. Dadouche A, Conlon MJ (2016) Operational performance of textured journal bearings lubricated with a contaminated fluid. Tribol Int 93:377–389 3. Su B, Huang L, Huang W, Wang X (2017) The load carrying capacity of textured sliding bearings with elastic deformation. Tribol Int 109:86–96 4. Zheng L, Gao Y, Zhong Y Lu GO Liu Z, Ren L (2018) The size effect of hexagonal texture on tribological properties under mixed lubrication. Ind Lubr Tribol 70:1798–1805 5. Feng H, Peng L (2018) Numerical analysis of water-lubricated thrust bearing with groove texture considering turbulence and cavitation. Ind Lubr Tribol 70:1127–1136 6. Shinde AB, Pawar PM (2017) Effect of partial grooving on the performance of hydrodynamic journal bearing. Ind Lubr Tribol 69: 574–584 7. Liang X, Liu Z, Wang H, Zhou X (2016) Hydrodynamic lubrication of partial textured sliding journal bearing based on three-dimensional CFD. Ind Lubr Tribol 68:106–115 8. Yu R, Li P, Chen W (2016) Study of grease lubricated journal bearing with partial surface texture. Ind Lubr Tribol 68:149–157 9. Tala-Ighil N, Maspeyrot P, Fillon M, Bounif A (2007) Effects of surface texture on journalbearing characteristics under steady-state operating conditions. Proc Inst Mech Eng Part J Eng Tribol 221:623–633 10. Cupillard S, Glavatskih S, Cervantes MJ (2008) Computational fluid dynamics analysis of a journal bearing with surface texturing. Proc Inst Mech Eng Part J Eng Tribol 222:97–107 11. Uddin.MS, Ibatan, Shankar TS (2016) In fluence of surface texture shape , geometry and orientation on hydrodynamic lubrication performance of plane-to-plane slider surfaces 12. Yadav SK, Sharma SC (2016) Performance of hydrostatic textured thrust bearing with supply holes operating with non-Newtonian lubricant. Tribol Trans 59:408–420 13. Brizmer V, Kligerman Y (2012) A laser surface textured journal bearing. J Tribol 134:031702 14. Shen C, Khonsari MM (2015) Tribology international numerical optimization of texture shape for parallel surfaces under unidirectional and bidirectional sliding. 82:1–11 15. Tala-Ighil N, Fillon M, Maspeyrot P, Effect of textured area on the performances of a hydrodynamic journal bearing. Tribol Int 44:211–219 16. Sharma SC, Yadav SK (2014) Tribology International performance analysis of a fully textured hybrid circular thrust pad bearing system operating with non-Newtonian lubricant. 77:50–64 17. Grützmacher PG, Rosenkranz A, Szurdak A, Gachot C, Hirt G, Mücklich F (2017) Effects of multi-scale patterning on the run-in behavior of steel—Alumina pairings under lubricated conditions. 1700521:1–8 18. Gropper D, Harvey TJ, Wang L (2018) Tribology international numerical analysis and optimization of surface textures for a tilting pad thrust bearing. Tribiol Int 124:134–144 19. Gropper D, Harvey TJ, Wang L (2018) A numerical model for design and optimization of surface textures for tilting pad thrust bearings. Tribol Int 119:190–207
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20. Charitopoulos A, Fillon M, Papadopoulos CI (2018) Tribology international numerical investigation of parallel and quasi-parallel slider bearings operating under Thermo Elasto Hydro Dynamic ( TEHD ) regime. Tribiol Int 1–11 21. Vl˘adescu S, Fowell M, Mattsson L, Reddyhoff T (2019) Tribology international the effects of laser surface texture applied to internal combustion engine journal bearing shells—An experimental study. Tribiol Int 134:317–327 22. Tala-Ighil N, Fillon M (2015) A numerical investigation of both thermal and texturing surface effects on the journal bearings static characteristics. Tribol Int 90:228–239 23. Sharma S, Jamwal G, Awasthi RK (2019) Enhancement of steady state performance of hydrodynamic journal bearing using chevron-shaped surface texture. 0:1–11 24. Khatri CB, Sharma SC (2016) Influence of textured surface on the performance of non-recessed hybrid journal bearing operating with non-Newtonian lubricant. Tribol Int 95:221–235 25. Raimondi AA, Boyd J (1958) A solution for the finite journal bearing and its application to analysis and design: Iii. ASLE Trans 1:194–209 26. Chandrawat HN, Sinhasan R (1988) A study of steady state and transient performance characteristics of a flexible shell journal bearing. 137–148
Big Turbo-Generator Shaft Vibrations Control Using Magnetorheological Fluid Damper Tarun Kumar, Rajeev Kumar, and Satish Chandra Jain
Abstract In this paper, a passive magnetorheological (MR) fluid damper is used to control vibrations generated in big turbo-generator shaft. The turbo-generator shaft is modeled using finite element method. The torque vibrations appeared in generator during different electrical faults are simulated using d-q-0 model using predictor– corrector method. The modified Bouc–Wen model is used to simulate the MR fluid damper. The complete dynamic system is modeled using MATLAB. The simulation results showed that MR fluid damper effectively reduces the rotor vibrations, but for better control of vibrations an active MR fluid damper can be more effective. Keywords Vibration control · FEM · MR fluid damper · Simulation
1 Introduction Turbo-generator coupled shaft subjected to torsional vibrations in power generation system due to various short circuit in network, it may be 3-phase, 2-phase, 1-phase to line or mal-synchronization [1]. These faults are very sensitive to turbogenerator rotor system because of its bulky size and long length. Various types of fault occurring due to grid interaction cause an impulsive electromagnetic torque on the coupled shaft system which generates torsional vibrations in it with amplitude 2–6 times of the nominal torque [2]. Sometimes these vibrations lead to severe damage to coupled shaft system or complete failure of it due to aggregation of stress. Low cycle fatigue wears of 70% of life of rotor and creep accounts for remaining 30% [3, 4]. The synchronous generator draws power from a prime mover (turbine) as shown in Fig. 1. The different researchers simulated generator vibrations during various electrical faults under loaded and unloaded conditions using different models [5–12]. Further, various researchers presented computational methods to measure torsional vibration produced on coupled shaft due to electrical faults on generator under unloaded condition and estimated the fatigue life of rotor under vibrations T. Kumar (B) · R. Kumar · S. C. Jain Indian Institute of Technology Mandi, Kamand Campus, VPO Kamand, Mandi, HP, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_159
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Fig. 1 Schematic of power generation and transmission system [10]
[13–23]. Although no complete analysis is available in the literature in which turbogenerator shaft vibrations are numerically simulated under loaded conditions for various electrical faults. Further, no comprehensive literature is available in which such turbo-generator rotor’s vibrations is controlled using piezoelectric magnetorheological fluid damper and vibrations produced are compared with the uncontrolled system. Therefore, in this paper a comparative result of vibrations under uncontrolled and controlled rotor using passive MR fluid damper is compared numerically.
2 Mathematical Formulation 2.1 Generator Modeling The synchronous generator is analyzed using dq0 model. The set of six coupled equations is used to calculate the currents, voltages and magnetic fluxes on d-q-0 reference frame as shown in differential eqs. (1)–(6). Wherein u, i, ψ denote voltages, currents and flux linkages, respectively. The subscripts d, q, 0 are associated with the d-axis components, q-axis components and zero sequence components, respectively, of voltages, currents and flux linkages as given in Eq. (7). The detailed dq0 model for synchronous generator modeling is given in [9, 10]. [u]dq0 = −[R][i]dq0 −
d[ψ]dq0 + ω[ψ]dq0 dt
(1–6)
where T [u]dq0 = u d u q u 0 u f d u D u Q T [i]dq0 = i d i q i 0 i f d i D i Q T [ψ]dq0 = ψd ψq ψ0 ψ f d ψ D ψ Q x0 =
xa + xb + xc ,x → u, i, ψ 3
(7)
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The subscripts a, b, c are associated with phase quantities on stator reference frame. The equation of mechanical motion is given as: J dω = −Tem + Tdrive p dt Tem =
3 p ψq i d − ψd i q 2 dγ = −ω dt
(8) (9) (10)
where T drive J p
is torque from turbines is polar moment of inertia is number of pole pairs.
The above set of Eqs. (1)–(10) is used to model the synchronous generator. The Park’s transformation as given in Eq. (11) is used to express parameters u, i, ψ from a-b-c reference frame to d-q-0 reference frame. ⎤ ⎤ ⎡ ⎤ ⎡ 4π ⎤⎡ cos γ + x xa cos γ cos γ + 2π xd a 3 3 ⎣ xq ⎦ = 2 ⎣ sin γ sin γ + 2π sin γ + 4π ⎦⎣ xb ⎦=[ξ ]⎣ xb ⎦ 3 3 3 1 1 1 x0 xc xc 2 2 2 ⎡
(11)
where, x → u, i, ψ.
2.2 Fault Modeling of Loaded Generator The generator is connected to the infinite load through grid as shown in Fig. 2. ua , ub , uc are voltages at the terminal of the generator and u∞a , u∞b , u∞c are voltages at the point of fault on the transmission line. The generator is considered to transfer power
Fig. 2 Synchronous generator connected to infinite load through grid [10]
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through line with resistance Re and inductance L e for each phase. During occurrence of fault, the phase voltages at generator terminals are different in loaded condition [10] in contrary to unloaded case [9] as given by Eqs. (12) and (13).
d [u abc ] = [u ∞abc ] + Re [I ][i abc ] + L e [I ] [i abc ] dt
(12)
From (11) and (12), we get
d u dq0 = [ξ ][u ∞abc ] + Re [I ][ξ ][i abc ] + L e [I ] [ξ ][i abc ] dt
(13)
Line-to-Ground Short Circuit In line-to-ground short circuit, one phase of the transmission line comes in contact with the ground. The mathematical model for line to ground fault to evaluate electromagnetic torque is taken from [10] and described by Eqs. (14)–(17). Let us assume that phase a is grounded then boundary conditions will be as u ∞a = 0, i b = i c = 0 Equations (14)–(17) are directly obtained from [9, 10]. L f dσ
di f d di D − L Dσ = −R f d i f d + R D i D + u f d dt dt
(14)
di f d di d di D + L md + LD = −R D i D dt dt dt
(15)
L md
di q di Q + LQ = −R Q i Q dt dt ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡1 −R ia ia L 3 0 ⎢ L ⎥ ⎢ i ⎥ ⎢ −ωL ⎥ ⎢ i ⎥ d2 ⎥ ⎢ d ⎥ ⎢ d1 ⎥ ⎢ d ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ L q2 ⎥ d ⎢ i q ⎥ ⎢ ωL q1 ⎥ ⎢ i q ⎥ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ − ua ⎢ ⎢ L md1 ⎥ dt ⎢ i f d ⎥ ⎢ −ωL md2 ⎥ ⎢ i f d ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎣ L md1 ⎦ ⎣ i D ⎦ ⎣ −ωL ⎦ ⎣ i D ⎦ L mq2 iQ iQ ωL mq1 L mq
(16)
(17)
uA can be evaluated using (12) by applying boundary conditions. The electromagnetic torque of synchronous generator is given by (18) (Ref. [9]) ⎤T ⎡ ia (L d1 − L q1 ) sinγ ⎥ ⎢i fd ⎢ L md2 ⎥ ⎢ = − p⎢ ⎦ ⎣ iD ⎣ L md2 −L mq1 iQ ⎡2
3
Tem
⎤ ⎥ ⎥i a ⎦
(18)
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The description of various parameters is given in [10]. Line to Line Short Circuit In line-to-line short circuit, two phases of the transmission line come in contact with each other. The mathematical model for line to line fault to evaluate electromagnetic torque is taken from [10] and described by Eqs. (14)–(16) and (19). Let us assume that phase b and phase c are shorted then boundary conditions will be as u ∞b = u ∞c , i a = 0, i b = −i c ⎡
⎤T
⎡
⎤
⎡
√2 R 3
⎤T ⎡
L d2 id ⎢ ⎥ ⎢ −L ⎥ ⎢ i ⎥ ⎢ ωL d1 ⎥ ⎢ ⎥ ⎢ ⎢ q1 ⎥ q ⎥ ⎢ ⎥ d⎢ ⎥ ⎢ ωL q2 ⎥ ⎥ ⎢ L md2 ⎥ . ⎢ i f d ⎥ = ⎢ ⎢ ⎥ dt ⎢ ⎥ ⎢ ωL md1 ⎥ ⎥ ⎣ L md2 ⎦ ⎣ iD ⎦ ⎢ ⎣ ωL md1 ⎦ −L mq1 iQ ωL mq2
⎤ ib ⎢i ⎥ ⎢ d ⎥ ⎢ ⎥ ⎢ iq ⎥ ⎢ ⎥ − ub + uc ⎢i fd ⎥ ⎢ ⎥ ⎣ iD ⎦
(19)
iQ
uc and ub can be evaluated using (12) by applying boundary conditions. The electromagnetic torque of synchronous generator is given by (20) (Ref. [9]) ⎤T ⎡ ib − L q1 ) sinγ 2(L d1 √ ⎥ ⎢i fd ⎢ 3L − md1 ⎥ ⎢ √ = − p⎢ ⎦ ⎣ iD ⎣ −√3L md1 − 3L mq2 iQ ⎡
Tem
⎤ ⎥ ⎥i b ⎦
(20)
The description of various parameters is given in [10]. Three Phase Short Circuit In three-phase short circuit, all three phases of the transmission line come in contact with each other and ground. The mathematical model for three phase fault to evaluate electromagnetic torque is taken from [10] and described by Eqs. (1), (2), (4)–(6). The boundary conditions in this fault will be as u ∞a = u ∞b = u ∞c = 0 The electromagnetic torque of synchronous generator is given by (21) (Ref. [9]) Tem =
3 p( − L md i md i q + L mq i d i mq ) 2
(21)
The description of various parameters is given in [10]. Phase Synchronization Fault In mal-synchronization fault, the power delivered by the synchronous generator is different from required power at grid. For power plant to operate continuously the
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frequency of the phase at generator terminals should be same as frequency of phase at network. Out of phase synchronization results in excessive vibrations in the rotor and unit trip. Therefore, the mathematical model for mal-synchronization fault to evaluate electromagnetic torque is taken from [10] and described by Eqs. (1), (2), (4)–(6). The phase difference over transmission line at the time of fault is defined as δ. The torque equation for this type of fault is same as (21).
2.3 Dynamic Modeling The complete dynamic model for line-to-ground and line-to-line faults are represented by set of coupled Eqs. (22), (24), (25) and for three-phase and malsynchronization faults are represented by set of coupled Eqs. (23)–(25). The coupled turbine-generator equations are solved numerically using fourth-order Adams predictor–corrector scheme with startup by fourth-order Runge–Kutta method in MATLAB [10]. The elements of L and X matrices can be evaluated using equations describing the different faults. ⎤ ⎤ ⎡ ⎡ ⎤ i phase u fd i phase ⎥ ⎢ ⎢ ⎥ d ⎢ i fd ⎥ ⎥ + X⎢ i f d ⎥ = ⎢ 0 ⎥ L· ⎢ ⎦ ⎦ ⎣ ⎣ ⎣ iD iD 0 ⎦ dt iQ iQ 0 ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ id ud id ⎢i ⎥ ⎢u ⎥ ⎢i ⎥ ⎢ q ⎥ ⎢ q ⎥ q ⎥ d⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ L · ⎢ i f d ⎥ + X⎢ i f d ⎥ = ⎢ u f d ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ dt ⎢ ⎣ iD ⎦ ⎣ 0 ⎦ ⎣ iD ⎦ iQ iQ 0 ⎡
(22)
(23)
J dω = −Tem + Tdrive p dt
(24)
dγ = −ω dt
(25)
2.4 Rotor with MR Fluid Damper Modeling The turbo-generator shaft is modeled using finite element method. The rotor is divided into number of elements with two rotational dofs (single dof at each node). The Hamilton’s principal is used to drive elemental stiffness and mass matrices. Three types of sections, solid circular, hollow circular and tapered, are used to model
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Fig. 3 Modified Bouc–Wen model for MR fluid damper (Left) and force–velocity plot [25]
the rotor elements. The elemental matrices are combined to form the global matrices, and complete FEM equation is represented by Eq. (26). [I ]g θ¨ g + [C]g θ˙ g + [K ]g {θ }g = T p g + [Tuniform ]g
(26)
A magnetorheological damper or magnetorheological shock absorber is a damper filled with magnetorheological fluid, which is controlled by a magnetic field, usually using an electromagnet. The most extensively used model for modeling hysteretic systems is the modified Bouc–Wen model. The modified Bouc–Wen model predicts the force displacement behavior of the damper well [24]. The simple modified Bouc–Wen model is shown in Fig. 3. The Eq. (27) is used to represent the MR fluid damper. The hysteretic component z accounts for the time history of the response [24]. The spring k 1 and its initial displacement x 0 allow for both the additional stiffness and the force offset produced by the presence of an accumulator. F = c1 y˙ + k1 (x − x0 )
(27)
where 1 [αz + c0 x˙ + k0 (x − y)] c0 + c1 z˙ = −γ |x˙ − y˙ |z|z|n−1 − β(x˙ − y˙ )|z|n + δ(x˙ − y˙ )
y˙ =
3 Validation The dynamic model described in Sect. 2.3 is already validated by the author in ref [10]. The values of inductance L e and resistance Re are assumed to be zero for validation purpose only.
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Table 1 Comparison of first three natural frequencies from MATLAB and Ref. [19] S. No.
No. of elements considered in MATLAB
1
5
2
3
4
5
6
6
7
8
9
10
First 3 natural freqs. from MATLAB (×103 Hz)
First 3 natural freqs. of Ref. [19] (×103 Hz)
% Error
1.682
1.534
9.647
7.737
4.911
57.544
46.247
8.315
456.187
1.576
1.534
2.737
5.553
4.911
13.072
10.566
8.315
27.071
1.557
1.534
1.499
5.045
4.911
2.728
9.472
8.315
1.551
1.534
1.108
4.859
4.911
−1.058
8.786
8.315
5.664
1.547
1.534
0.847
4.774
4.911
−2.789
8.401
8.315
1.034
1.547
1.534
0.847
4.773
4.911
-2.810
8.487
8.315
2.068
Avg. % error
174.46
14.294
6.047
13.91 1.904
−0.302
0.0353
Further, the finite element model for the rotor is validated using reference [19]. The first three natural frequencies are calculated to validate the global mass and stiffness matrices. The first three natural frequencies are compared with the reference [19], and convergence of the code is established in Table 1. The first three natural frequencies of the tube as given in reference [19] show close resemblance to the simulated results. The MR fluid damper is validated for vehicle model given in ref [25]. Modified Bouc–Wen Model is used to model the damper. Various parameters are given in Reference [25]. From Fig. 4a, b, it is clear that there is close resemblance in results of ref [25] and simulated results.
4 Results and Discussion The generator is modeled using dq0 formulation. The complete system is connected with the infinite load through grid with resistance Re and inductance, L e . Per
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Fig. 4 a Comparison of mass m1 position as obtained from MATLAB simulations and reference [25] results. b Comparison of mass m2 position as obtained from MATLAB simulations and reference [25] results
unit system is used to calculate the peak values of torque. The coupled turbinegenerator equations are solved numerically using fourth-order Adams predictorcorrector scheme with startup by fourth order Runge–Kutta method in MATLAB. The peak values of per unit torque for line to ground, line to line, three-phase and mal-synchronization faults are 12 pu, 7.9 pu, 17.1 pu and 8.1 pu, respectively. The actual value of electromagnetic torque is calculated by multiplying per unit value by torque base 3 pUn In ωn−1 [9]. The rotor is modeled using finite element method. Complete rotor is divided in to 187 numbers of elements and 188 numbers of nodes. Each element has two degrees of freedom, i.e., one rotational degree of freedom at each node. Since torsional vibrations are independent from translational and axial vibrations so other DOFs are neglected to keep the formulation simple. The whole rotor is modeled using three type of elements (solid cylinder, hollow cylinder and tapered), and stiffness is calculated accordingly. The generator torque is applied on nodes associated with the generator position on rotor length. The complete dynamic system is solved using ODE15s in
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MATLAB. The diameter of various sections of coupled rotor along length is shown in Fig. 5. Further, MR fluid damper is used to damp the torsional vibration of turbogenerator shaft. The damper is placed at element No. 111. The vibrations at element No. 39 are simulated for various different faults. The various parameters of the MR fluid damper are shown in Table 2. The comparative results for control and uncontrolled system vibrations at element No 39 are shown in Fig. 6a–d. The peak torque, valley torque and average amplitude of torque for first cycle in control and uncontrolled condition during various electrical faults are given in Table 3. The amplitude reduction for controlled system during line to ground faults is 17.32%, for line-to-line fault is 8.57%, for three-phase fault is 4.57% and for malsynchronization fault is 40% at element No. 39. Further peak value of torque remains same for line-to-ground fault and three-phase fault for controlled and uncontrolled
Fig. 5 Diameter of various sections of rotor along the length
Table 2 Various parameters of the MR fluid damper
S. No.
Parameter
Value
1
c0
5300 Ns/m
2
c1
93,000 Ns/m
3
k0
51,400 N/m
4
k1
540 N/m
5
A
96,300 N/m
6
B
2,000,000 m−2
7
2,000,000 m−2
8
N
2
9
207
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Fig. 6 a Torque at element 39 under one-phase fault. b Torque at element 39 under two-phase fault. c Torque at element 39 under three-phase fault. d Torque at element 39 under mal-synchronizationphase fault Table 3 Various values of torque at element No 39 under controlled and uncontrolled conditions Fault type
Torque (×105 Nm)
Uncontrolled
MR fluid damper
Line to gound
Peak
3.38
3.24
Valley
−3.20
−2.19
Avg. torque amplitude
3.29
2.72
% Amplitude reduction
–
17.32
Peak
1.77
0.557
Line to line
Three phase
Mal-synchronization
Valley
– 1.73
−2.65
Avg. torque amplitude
1.75
1.60
% Amplitude reduction
–
8.57
Peak
4.79
4.89
Valley
−4.39
−3.88
Avg. torque amplitude
4.59
4.38
% Amplitude reduction
−
4.57
Peak
0.119
0.465
Valley
−2.29
−0.989
Avg. torque amplitude
1.20
0.72
% Amplitude reduction
–
40
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systems, but for other faults the peak value of torque increases extensively as shown in Fig. 6d.
5 Conclusions The dynamic models of loaded synchronous generator under four types of electric faults are analyzed in MATLAB. The rotor is modeled using FEM, and complete dynamic system is numerically simulated using ode15s in MATLAB. Further, magnetic rheological fluid damper is employed at element No. 111, and torsional vibrations at element 39 are simulated and compared with the uncontrolled system. The MR fluid damper leads to different reduction of torsional vibration for different faults. Passive MR fluid control effectively control line-to-ground and three-phase faults. In line-to-ground fault, effect of damper is minimum. In case of malsynchronization fault, the average amplitude of the vibrations remains same, but vibration peak values increase tremendously. Further, active MR fluid damper can be studied to control the vibrations come during various electrical faults on the line.
References 1. Wachel JC, Szenasi FR (1993) Analysis of torsional vibrations in rotating machinery. In 22nd Turbomachinery symposium, Dallas, TX, Sept, pp 14–16 2. Pasca N, Marsavina L, Muntean S, Negru R (2012) Failure analysis of a storage pump shaft 3. Shafts, transmitting. Design of power-transmitting shafts (1984) 4. Khangar VS, Jaju SB (2012) A review of various methodologies used for shaft failure analysis. Int J Emerging Technol Adv Eng ISSN 2250-2459 5. Asi O (2006) Fatigue failure of a rear axle shaft of an automobile. Eng Fail Anal 13(8):1293– 1302 6. Reid MJ (1988) Analysis of the causes of recent roll shaft failures in natal sugar mills. In: Proceedings of the annual congress-South African Sugar Technologists’ Association 7. Mitrovic, D. Stojanovic D. Petrovic N. “Analysis of Torsional Torques of Big Turbine-Generator Shafts.” 8. Cudworth CJ, Smith JR (1990) Steam turbine generator shaft torque transients: a comparison of simulated and test results. In: Generation, transmission and distribution, IEE Proceedings C, vol 137, no 5. IET, pp 327–334 9. Lupsa-Tataru L (2009) Comparative simulation study on synchronous generators sudden short circuits. Model Simul Eng 2009:8 10. Kumar T, Bangunde A, Kumar R, Jain SC (2018). The torque simulation of synchronous generator under loaded condition for different types of electrical disturbances. In: 2018 2nd International conference on power, energy and environment: towards smart technology (ICEPE). IEEE, pp 1–9. https://doi.org/10.1109/EPETSG.2018.8659271 11. Jackson MC, Umans SD (1980) Turbine-generator shaft torques and fatigue: Part IIIrefinements to fatigue model and test results. IEEE Trans Power Apparatus Syst 3:1259–1268 12. Szenasi FR, Von Nimitz W (1978) Transient analyses of synchronous motor trains. In: 7th Turbomachinery symposium, Houston, TX, vol 7, pp 111–117 13. Szolc T (2000) On the discrete-continuous modeling of rotor systems for the analysis of coupled lateral torsional vibrations. Int J Rotating Mach 6(2):135–149
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14. Taplak H, Parlak M (2012) Evaluation of gas turbine rotor dynamic analysis using the finite element method. Measurement 45(5):1089–1097 15. Sakamoto S, Hirata T, Kobayashi T, Kajiwara K (1999) Vibration analysis considering higher harmonics of electromagnetic forces for rotating electric machines. IEEE Trans Magnet 35(3):1662–1665 16. He Q, Du D (2010) Modeling and calculation analysis of torsional vibration for turbine generator shafts. J Inf Comput Sci 7(10):2174–2182. 17. Chan DSH (1996) The transient torsional vibration behaviour of a turbine-generator system under short circuit excitation. In: ASME 1996 international gas turbine and aeroengine congress and exhibition. American Society of Mechanical Engineers, pp V003T07A001-V003T07A001 18. Vaziri A, Nayeb-Hashemi H (2006) A theoretical investigation on the vibrational characteristics and torsional dynamic response of circumferentially cracked turbo-generator shafts. Int J Solids Struct 43(14):4063–4081 19. Sung C-C, Varadan VV, Bao X-Q, Varadan VK (1994) Active torsional vibration control experiments using shear-type piezoceramic sensors and actuators. J Intell Mater Syst Struct 5(3):436–442 20. Liu C, Jiang D, Chen J (2014) Coupled torsional vibration and fatigue damage of turbine generator due to grid disturbance. J Eng Gas Turbines Power 136(6):062501 21. Oliquino R Jr, Islam S, Eren H (2003) Effects of types of faults on generator vibration signatures. Curtin University of Technology, Western Australia 22. Tsai J-I, Lin C-H, Tsao T-P (2004) Assessment of long-term life expenditure for steam turbine shafts due to noncharacteristic subharmonic currents in asynchronous links. IEEE Trans Power Syst 19(1):507–516 23. Chan KS, Enright MP, Golden PJ, Naboulsi S, Chandra R, Pentz AC (2012) Probabilistic highcycle fretting fatigue assessment of gas turbine engine components. J Eng Gas Turbines Power 134(6):062502 24. Dyke SJ, Spencer BF Jr, Sain MK, Carlson JD (1997) On the efficacy of magnetorheological dampers for seismic response reduction. In: Proceedings of the ASME design engineering technical conferences 25. Butz T, Von Stryk O (2002) Modelling and simulation of electro-and magnetorheological fluid dampers. ZAMM-J Appl Math Mech/Zeitschrift Für Angewandte Mathematik Und Mechanik: Appl Math Mech 82(1):3–20
Antagonistic Actuation of Pneumatic Artificial Muscle (PAM) with Chain-Sprocket Mechanism Bhaben Kalita, Arunjyoti Borgohain, and Santosha K. Dwivedy
Abstract In this work, a system has been designed with the help of a chain-sprocket mechanism to lift a load for particular applications with the help of a pneumatic artificial muscle (PAM) actuator. The system will mimic the human arm where one PAM and spring are in antagonistic arrangement just like the human biceps and triceps. The PAM behaves as the biceps and the spring as the triceps muscle to provide additional support to the system. The PAM will contract when the air pressure is supplied and actuate the system to reach the desired position. With the help of this system, the experiment can be performed to obtain various system parameters for a wide range of the muscle. The system is designed focusing on its flexibility for multiple types of experiments with different dimensions of pneumatic muscles. Finally, the hysteresis present in the muscle has been plotted and compared to verify the developed system. The developed system with the chain-sprocket mechanism is very useful to understand the dynamics of the PAM, and as well as the designer and researcher can use this for various applications in the field of medical and robotic industries. Keywords Pneumatic artificial muscle · Chain-sprocket mechanism · Antagonistic arrangement · Hysteresis
1 Introduction Pneumatic artificial muscle has large applications in the field of medical robotics as well as in industries over the years. Due to the operational similarity with the natural muscles, PAM has attracted many researchers and designers for various applications which requires high force-to-weight ratios, high fatigue life and compliance [1, B. Kalita (B) · S. K. Dwivedy Indian Institute of Technology Guwahati, Guwahati 781039, India S. K. Dwivedy e-mail: [email protected] A. Borgohain Yantrabot Technologies Pvt. Ltd, Guwahati 781028, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_160
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2]. These artificial muscles are generally unidirectional which generate the pulling force by inducing compressed air pressure into it. PAM consists of a core of elastic tube cover with spiral network braid which limits the expansion of the muscle with an increase in the air pressure. Generally, the materials used for the core of the muscle are latex and silicon rubber where for the braid, fibers like nylon are used [3]. The operation of such kind of muscle basically depends on the shape, contraction behavior, generated tensile force along with the materials and geometry of the core and the braid of the muscle [4]. These factors of the artificial muscles for various applications have motivated an investigation for optimizing the performance of the operating principles of these PAMs. Most of the medical, robotic and industrial applications need the bidirectional linear or rotational motion. So, to obtain the required motion, there is a need for an antagonistic arrangement of two artificial muscles similar to the biceps–triceps muscle arrangement of the human arm. These biceps and triceps muscle can be classified as flexor and extensor muscles to move the human hand. Biceps is included in the flexor group, and it will bend the arm by decreasing the angle between the forearm and upper arm, whereas triceps muscle is included in the extensor muscle. In flexion condition, biceps muscle will contract while triceps muscle will relax, while in extension condition, the biceps will relax and triceps contracted. The length of muscle is shorter, and the contracting force becomes smaller [5]. PAMs consist of the core and the braid mimics the same principle like human muscles and the air pressure as its source of actuating force. This antagonistic arrangement is mostly employed with conventional actuators like hydraulic or pneumatic cylinders along with newer smart materials, such as shape-memory alloys [6] and electrostrictive materials [7]. Due to the advantages of PAM over the other actuators, in various applications like aerospace [8], 3D printing [9], rehabilitation [10] where the artificial muscle is used to obtain the antagonistic actuation. Balara and Tóthová [11] described about the static and dynamic properties of the PAM with the help of antagonistic arrangement experimental rig. Vocke et. al. [12] designed a mechanism with the help of the PAM to achieve the bidirectional motion. But due to the presence of the highly nonlinear force-contraction characteristic and compliance in PAM, there is a challenge to achieve the antagonistic actuation. Many researches have been done on various arrangements attached with the PAM to achieve antagonistic actuation with a wide range of actuator motion along with the control of the stiffness of the actuator. These arrangements involve simple pulleys [13, 14], variable complex radius pulleys [15] and offset lever mechanisms [16, 17] to modify the system kinematics. Due to the advantage of this antagonistic arrangement to achieve a wide range motion of the muscle actuator, a system has been designed and developed in this work. The system has been attached with a chain-sprocket mechanism with two artificial muscles which provided the required rotational as well as the linear motion for a particular application. A detailed study about the idea and flexibility of the system with its attachments has been done in the next section followed by the justification with typical hysteresis plot for different loads with the developed system.
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2 Design of the Setup A system has been designed to enable the antagonistic arrangement of PAM mimic like the biceps and triceps of the human arm as shown in Fig. 1. The system can be efficiently used to test the PAM with different lengths and types. Figure 2 describes the schematic diagram of a system where one PAM which mimics like the triceps of the human arm can be replaced with a spring to simplify the system. In the developed system, these both types of experiment can be performed to understand the working of the PAM and its effect on the application. Aluminum slotted extrusions are used primarily to build a lightweight and stable frame. To fix the muscles as the arrangement of a human arm as shown in Fig. 1, four snap hooks (6) have been attached as shown in Fig. 3. A chain-sprocket mechanism is designed as shown in Fig. 4 which is adjustable with the required dimension of the PAM. To actuate the muscles mounted on the frame, compressed air is supplied through the 4 mm tube (9). The frame also has two manual airflow regulators (3, 4) which enables controlled air supply to individual muscles.
2.1 Features of the Setup The system is a bench-top design with size Length × Breadth × Height (L × B × H) = 17 × 15 × 35 inch. The frame is built with aluminum T-slot 2020 extrusions (2) fixed with the help of M5 bolts and T-nuts. The base is a mild steel plate (1) of size L × B = 14 × 11 inch and 8 mm thick. The frame also has an adjustable platform (11) with extrusion links on which a shaft (13) of 20 mm diameter is installed with the help of two pillow block bearings (12) as shown in Fig. 4. This helps to fit the PAM of different sizes to perform the experiments. The shaft (13) has a through hole of 8 mm diameter at one end for assembling the arm (14). The arm is a threaded rod of 8 mm diameter. It has two M8 wing nuts (19) for easy adjustment of the position of the loads mounted. On the other end of the shaft, a hole with M6 thread is provided to fix a spur gear of Module = 1 which is driving a rack (25) sliding on a slotted body Fig. 1 Human hand muscles
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Fig. 2 An arrangement of spring-muscle system
(24) made of acrylic sheet. An idle gear is also mounted for smooth movement of the rack. For easy mounting of the muscles and springs, four snap hooks (6) of steel material are used. Two hooks are mounted on the top portion of the frame with two M5 Allen bolts (5) as shown in Fig. 3. Other two hooks are tied to two ends of a chain (16). The chain roles over a sprocket (17) of teeth = 36 nos. which is mounted on the shaft (13) and fixed with grub screw. To measure angular deflection while inflating the muscles, the value can be recorded from the protractor (15) as shown in Fig. 3. The rear end of the frame also has a scale (22) with which the linear movement of the rack (25) can be observed and recorded simultaneously. This linear displacement corresponds to the change in muscle length while extension/contraction. The CAD models are shown in Fig. 5a where only one PAM is in the antagonistic arrangement with a spring, and Fig. 5b depicts that two PAMs are attached in the system. In the case of using only one muscle, a nonlinear tension spring is used which will play role of a secondary muscle as shown in Fig. 5a. In Fig. 6a, the PAM of length L 1 is mounted between the hooks of the frame and the chain. To the other end of the chain, a nonlinear spring of length S 1 is mounted. Then, the movable platform is adjusted by distance, H x and set to an initial tensioning value T 1 . A mass of weight, W, is loaded to the arm and placed at fix distance, R, with the help of wing nuts. Now, while supplying the compressed air to the muscle L 1 , it will make the muscle to inflate to a length L 2 which in turn rotates the sprocket by an angle, δ and increase the length of spring to length, S 2 as shown in Fig. 6b. The new tension, T 2 , on the chain will depend on the value of weight, W, radius, R, and the deflection angle, δ.
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Fig. 3 Schematic of the proposed system
The pressure on the muscle while deflecting can be recorded from the dial indicator corresponding to the muscle. The same motion can be achieved in both directions with the help of two PAMs attached to the system as shown in Fig. 5b for the required application. The developed system has been shown in Fig. 7 with front and rear view to conducting various experiments to understand the behavior of the PAM. The typical hysteresis for the PAM has been observed which is provided proper justification to the system. The hysteresis present in the muscle has been plotted with the help of contraction in the muscle with the variation in pressure for a particular load. In
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Fig. 4 Chain-sprocket mechanism attached to the system
Fig. 8a, b, the typical hysteresis has been shown in 15 N and 30 N load ,respectively, subjected to the system. These plots are almost similar to the hysteresis plots which is found in the previous literature [18, 19]. With the help of these plots, different system parameters present in the PAM can be obtained and used for the study of nonlinear nature exhibiting in the muscle. The safe operating range of different system parameters can be obtained from the mathematical work proposed by Kalita and Dwivedy [20, 21] for various resonance conditions.
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Fig. 5 CAD model of the system with a one PAM and a spring b two PAMs in the antagonistic arrangement
3 Conclusion The developed system with the chain-sprocket mechanism can be used for testing a wide range of muscle available in the market before use in the applications. The system has been designed keeping in mind some major factors like the flexibility, robustness, weight and uses. The principle of the antagonistic arrangement can be easily understood with the help of this design. Various nonlinear behaviors present in the muscle can be studied by performing experiments on this system. The system can provide the angular motion along with the linear motion simultaneously by attaching an arrangement of rack and pinion in the system. The configuration of the designed chain-sprocket mechanism with the PAM requires less control to achieve the demands like desired load and position along with the fewer implementation costs. The behavior of the muscles is completely different for different values of the
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Fig. 6 Experimental observation of the antagonistic arrangement with one PAM and and the spring
system load. So, to justify the developed system, hysteresis present in the muscle has been plotted for different loads and shows the typical behavior. Hence, the system is very useful for various applications in the field of health care and industries.
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Fig. 7 Actual developed system with a front view and b rear view
(a)
(b)
12
10
10
8
8
6
6
4 4
2
2 0
0 0
50
100
150
200
250
300
0
50
100
150
200
250
300
Fig. 8 Typical hysteresis present in the used PAM in the developed system for a 15 N and b 30 N
References 1. Tondu B, Lopez P (1997) The McKibben muscle and its use in actuating robot-arms showing similarities with human arm behaviour. Ind Robot 24(6):432–439
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2. Klute G, Czerniecki J, Hannaford B (2002) Artificial muscles: actuators for biorobotic systems. Int J Robot Res 21(4):295–309 3. Daerden F, Lefeber D (2002) Pneumatic artificial muscles: actuators for robotics and automation. Eur J Mech Environ Eng 47(1):11–21 4. Tóthová M, Pitel J (2013) Dynamic model of pneumatic actuator based on advanced geometric muscle model. In: 9th International conference on computational cybernetics (ICCC). IEEE, Tihany, Hungary, pp 8–87 5. Lightner S, Lincoln R (2002) The fluidic muscle: a ‘new’ development. Int J Mod Eng 2(2):1–8 6. Hirose S, Ikuta K, Sato K (1989) Development of shape memory alloy actuator. improvement of output performance by the introduction of a σ-mechanism. J Ad Robot 3(2):89108 7. Szufnarowski F, Schneider A (2010) Compliant piezo-flexdrives for muscle-like, antagonistic actuation of robot joints. In: International conference on biomedical robotics and biomechatronics. IEEE, Tokyo, Japan, pp 381–388 8. Wereley N, Kothera C, Bubert E, Woods B, Gentry M, Vocke R (2009) Pneumatic artificial muscles for aerospace applications. In: 50th structures, structural dynamics, and materials conference 17th adaptive structures conference, p. 2140. AIAA, Palm Springs, California 9. Peele BN, Wallin TJ, Zhao H, Shepherd RF (2015) 3D printing antagonistic systems of artificial muscle using projection stereolithography. Bioinspir Biomim 10(5):055003 10. Knestel M, Hofer EP, Barillas SK, Rupp R (2008) The artificial muscle as an innovative actuator in rehabilitation robotics. IFAC Proc Vol 41(2):773–778 11. Balara M, Tóthová M (2012) Static and dynamic properties of the pneumatic actuator with artificial muscles. In: 10th Jubilee international symposium on intelligent systems and informatics. IEEE, Subotica, Serbia, pp 577–581 12. Vocke RD III, Kothera CS, Wereley NM (2014) Mechanism and bias considerations for design of a bi-directional pneumatic artificial muscle actuator. Smart Mater Struct 23(12):125039 13. Tondu B, Ippolito S, Guiochet J, Daidie A (2005) A Seven-degrees-of-freedom robot-arm driven by pneumatic artificial muscles for humanoid robots. Int J Robot Res 24(4):257–274 14. Shin D, Sardellitti I, Khatib O (2008) A hybrid actuation approach for human-friendly robot design. In: International conference on robotics and automation. IEEE, Kobe, Japan, pp 1747– 1752 15. Shin D, Yeh X, Khatib O (2011) Variable radius pulley design methodology for pneumatic artificial muscle-based antagonistic actuation systems. In: International conference on intelligent robots and systems. IEEE, San Francisco, CA, USA, pp 1830–1835 16. Woods BKS, Kothera CS, Wereley NM (2011) Wind tunnel testing of a helicopter rotor trailing edge flap actuated via pneumatic artificial muscles. J Intell Mater Syst Struct 22(13):1513–1528 17. Vocke III RD, Kothera CS, Chaudhuri A, Woods BKS, Wereley NM Design and testing of a high-specific work actuator using miniature pneumatic artificial muscles. J Intell Mater Syst Struct 23(3):365–378 18. Zang X, Liu Y, Heng S, Lin Z, Zhao J (2017) Position control of a single pneumatic artificial muscle with hysteresis compesnsation based on modified Prandtl-Ishlinskii model. Bio-Med Mater Eng 28(2):131–140 19. Li H, Kawashima K, Tadano K, Ganguly S, Nakano S (2011) Achieving haptic perception in forceps’ manipulator using pneumatic artificial muscle. IEEE/ASME Trans Mechatron 18(1):74–85 20. Kalita B, Dwivedy SK (2019) Nonlinear dynamics of a parametrically excited pneumatic artificial muscle (PAM) actuator with simultaneous resonance condition. Mech Mach Theory 135:281–297 21. Kalita B, Dwivedy SK (2019) Dynamic analysis of pneumatic artificial muscle (PAM) actuator for rehabilitation with principal parametric resonance condition. Nonlinear Dyn 97(4):2271– 2289
Deep Neural Network Approach for the Prediction of Journal Bearing Static Performance Characteristics Sunil Kumar, Vijay Kumar, and Anoop Kumar Singh
Abstract Deep neural network approach is an excellent way of performance predictions of mechanical systems due to the advancement in computational technologies. In the present document, the predictions of static performance characteristics are made for the hole-entry hybrid journal bearing. Maximum pressure and minimum fluid film thickness values are obtained using FEM and used as target output for feedforward backpropagation neural network model. In this model, hidden layers and number of neurons in these layers are decided heuristically. Logistic activation function is used for hidden and output layer neurons. Using the developed model, predictions for journal bearing performance are made within and out of the prescribed range of input parameters. The percentage error obtained for neural network training, testing and predictions is very small (−1.0% < error < 1.0%). It is concluded that a lot of time is saved in predictions using deep neural network approach compared to the mathematical analysis the journal bearing performance. The use of multiple hidden layers for journal bearing performance predictions and multiple data sets for input and output neurons is the novelty of the present work. Keywords Feedforward backpropagation · Logistic activation function · Multiple hidden layers · Hybrid journal bearing · Hole entry · FEM
1 Introduction Deep neural network is used to solve complex problems and has been proved as a powerful tool for most precise and accurate predictions of system behaviour [1, 2]. This approach can be used in journal bearing performance predictions. There is no available literature which shows the mapping of FEM results with neural network model to predict the performance of journal bearings. Theoretical investigation for analysing performance characteristics of journal bearing is more time-consuming process. Many calculations are required to obtain the desired characteristics. By S. Kumar (B) · V. Kumar · A. K. Singh Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_161
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using the literature data, predictions can be made for the performance characteristics by neural network approach. Sinano˘glu investigated the pressure variations at the circumference of journal bearing experimentally. The results obtained by neural network model were found very close to experimental values [3]. Neural network model was successfully used for the fault detection of bearings and their prevention [4, 5]. Only one hidden layer was considered by researchers in the previous work for the predictions of journal bearing performance using neural network [1, 3]. But, in this work, multiple hidden layers along with multiple input and output data sets are considered for more accurate predictions. Hybrid journal bearing is considered for FEM analysis in the present work due to its vast applications in heavy load and high-speed machines [6, 7]. Static and dynamic performance characteristics of journal bearings using different restrictors can be analysed theoretically and experimentally [8–10]. Their results can be used as target output for the performance predictions using neural network. Orifice, capillary and constant flow valve restrictors are used to control the flow through journal bearings. Many researchers analysed the hybrid journal bearings using orifice restrictor as a flow control device [11, 12]. FEM was used for theoretical analysis [13, 14]. In the present work, finite element method (FEM) is used to obtain journal bearing performance characteristics. Maximum pressure p max and minimum fluid film thickness h min are used as target output for feedforward backpropagation neural network model. Hidden layers and number of neurons in these layers are decided heuristically. Logistic activation function is used in hidden and output layers. Predictions for journal bearing performance are made within and out of the prescribed range of input parameters. The use of multiple hidden layers for journal bearing performance predictions and multiple data sets for input and output neurons is the novelty of the present work.
2 FEM Analysis of Journal Bearing Journal bearings have been analysed by many researchers to determine their static and dynamic performance characteristics [8, 9]. In the present FEM analysis, the flow field is discretised as shown in Fig. 1. The number of nodes is taken as 48, and the number of four-noded elements is 36. The number of holes is considered as 24 (12 per row) in hole-entry hybrid journal bearing in the present work. The governing Reynolds equation in non-dimensional form for lubricant flow field in journal bearing is given as Eq. (1) [15–17]; ∂ ∂ ∂p ∂p F1 ∂h ∂ 3 3 + = h F2 h F2 h + 1− ∂α ∂α ∂β ∂β ∂α ∂t F0
(1)
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Fig. 1 Discretised domain of flow field
In this equation, α and β show circumferential and longitudinal coordinates, respectively. F 0 , F 1 and F 2 are the viscosity functions. For Newtonian isothermal case, μ = 1 which is constant and F 0 = 1/μ, F 1 = 1/2μ and F 2 = 1/12μ. p, and h are the pressure, speed parameter and film thickness, respectively. This governing equation can be modified depending upon the fluid model used. Finally, the performance characteristics can be determined by using appropriate relations. The pressure values are obtained by Eq. (2); e
p=
nl
pj Nj
(2)
j=1
Fluid film thickness can be obtained by the expression given in Eq. (3); h = 1 − X J cos α − Z J sin α
(3)
where X J and Z J are the journal centre coordinates. The lubricant flow through orifice restrictor can be calculated by Eq. (4); 1/2 Q R = C s2 1 − p c
(4)
The solution of modified Reynolds equation is obtained by Galerkin’s approach of FEM and given as Eq. (5);
F P = Q + R H + X˙ J R x + Z˙ J R z Extended form of modified Reynolds equation is given as Eq. (6);
(5)
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Table 1 Dimensionless operating and geometric parameters for FEM analysis
Parameters
Value/range
External load (w)
0.2–1.4
Speed parameter ()
1.0, 1.2, 1.4
L/D ratio
1.0
Land width ratio (ab )
0.25
Restrictor
Orifice
Restrictor design parameter C s2
⎡
F 11 ⎢ . ⎢ .. ⎢ ⎢ ⎢ F i1 ⎢ . ⎢ . ⎢ . ⎢ ⎢ F j1 ⎢ ⎢ .. ⎣ .
F 12 .. . F i2 .. . F j2 .. .
··· .. . ··· .. . ··· .. .
F1j .. . Fi j .. . F jj .. .
··· .. . ··· .. . ··· .. .
F n1 F n2 · · · F n j · · ·
0.1
⎧ ⎫ ⎫ ⎤⎧ ⎫ ⎧ ⎪ ⎪ p1 ⎪ F 1n ⎪ Q1 ⎪ R H1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ .. ⎥ .. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ . ⎥⎪ . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ pi ⎪ ⎪ ⎪ F in ⎥⎪ Q R ⎪ ⎪ ⎪ ⎪ ⎪ H i i ⎬ ⎨ ⎨ ⎨ ⎬ ⎬ ⎥ .. ⎥ .. . . . . = + . ⎥⎪ . ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p F jn ⎥⎪ Q R Hj ⎪ j j⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎥⎪ ⎪ .. ⎪ ⎪ .. ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ . ⎪ ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ ⎭ ⎭ ⎭ F nn Qn R Hn pn ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ R x1 ⎪ R z1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R R ⎪ ⎪ ⎪ x z ⎨ i⎬ ⎨ i⎪ ⎬ .. .. ˙J + + X˙ J Z . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rx j ⎪ ⎪ ⎪ Rz j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ R xn R zn
(6)
The boundary conditions for the elastic deformation flow field are as follows: i. ii. iii. iv. v.
There is an ambient pressure at bearing edge. At internal nodes, flow is observed to be zero. Flow is nonzero on holes and external boundaries. Restrictor flow and bearing input flow are identical. At trailing edge of the positive region, p = ∂∂αp = 0.
Non-dimensional operating and geometric parameters used in FEM analysis are presented in Table 1:
3 FEM Solution Procedure The journal bearing solution procedure using FEM is shown as a flowchart in Fig. 2.
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Fig. 2 Overall solution scheme for FEM analysis
The input data is computed to achieve fluid film thickness for initial journal centre coordinate position. Fluidity matrix is generated and solved for the Newtonian lubricant. Pressure values are obtained using boundary conditions for lubricant flow field. Iterative procedure is done to achieve journal equilibrium for applied radial load (w). Increments in journal centre coordinates are computed to achieve the convergence criteria, i.e. PERR < 0.001. Finally, static performance characteristics are obtained for the journal bearing. Scilab software is used in the present work for FEM analysis and DNN predictions. It provides the computational environment for matrices manipulation and algorithm implementation.
4 Neural Network Model Development The values of p max and h min are obtained using FEM and considered as target output for training and testing a feedforward neural network model. Backpropagation algorithm is used for minimising the error and updating the weights [1, 3]. Weights are selected initially as random numbers. The deep neural network is multilayer neural network which contains more than one hidden layers. This approach is used for the solution of complex problems. Large number of input data is required for the processing. The number of neurons in each hidden layer can be decided heuristically whereas the neurons in input and output layers are fixed depending upon the problem. Feedforward deep neural network
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adopts trial-and-error approach to solve complex problems associated with complex relationship in input and output data set. Backpropagation algorithm is employed for learning of deep neural network because of its better implementation in complex structure of hidden layers. Feedforward backpropagation algorithm uses the gradient decent approach for the adjustment of connected weights and finally to minimise the error. The best possible results can be predicted by using this approach [1, 3]. Logistic activation function is mostly used in deep neural networks. It is used to get the nonlinear output from the hidden layers and output layer. This function gives the output value in the range 0–1. If the target output is not in this range, then some standardisation or normalisation methods should be used to convert the data in the range 0–1. Schematic of feedforward deep neural network is shown in Fig. 3. Input to the first hidden layer (h i1 ) is given as Eq. (7) where xi0 represents the data set for input layer. The first hidden layer response vi1 is given by logistic activation function and represented by Eq. (8); h i1 =
n0
wi1 i0 xi0
(7)
i 0 =1
vi1 =
1 1 + e−h i1
Fig. 3 Schematic of feedforward deep neural network
(8)
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Input to the second hidden layer h i2 is given as Eq. Using logistic activation (9). function, the response of the second hidden layer vi2 is obtained and given as Eq. (10): n1
h i2 =
wi2 i1 vi1
(9)
i 1 =1
1 1 + e−h i2
vi2 =
(10)
Input to the output layer h i3 is given as Eq. (11). Using logistic activation function, the response of output layer yin is obtained and given as Eq. (12): n2
h i3 =
win i2 vi2
(11)
i 2 =1
yin =
1 1 + e−h i3
(12)
The associated weights between layers are updated using the expression (13) to minimise the error. Mean square error (E) is obtained by using Eq. (14): w(new) = w(old) − η
∂E + αw ∂w
2 T 2 1 yin − yin 2 i =1
(13)
n
E=
(14)
n
where yiTn is the target output and yin is the neural network model output. (η) is the learning rate, and its value should be in the range 0–1. (α) is the momentum term used to improve training speed and accuracy. The value should be in the range 0–0.9999. A bias which is a positive value can be added in hidden and output layer neurons. Deep neural network for journal bearing analysis is shown in Fig. 4. In this figure, whereas radial load (w) and rotational speed (Ω) are considered as inputparameters, maximum pressure p max and minimum fluid film thickness h min are the output parameters. A bias (+1) is connected with all the neurons of hidden layers to get the nonzero value during the iteration process. Both weights and bias are updated in each run. The parameters for the neural network predictor are shown in Table 2. In this table, (η) is the learning rate which shows the speed of learning for the error convergence. (n i ), (n h1 ), (n h2 ) and (n o ) represent neurons in input layer, first hidden layer, second hidden layer and output layer, respectively. (α) shows the momentum term and (N) shows the number of epochs. Epochs are the training cycles performed in the program
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Fig. 4 Feedforward deep neural network for journal bearing
Table 2 Parameters for neural network predictor Neural network
η
α
ni
n h1
n h2
no
N
Activation function
0.9
0.8
2
18
18
2
15,000
logistic
to achieve the convergence criteria. The initial weights are chosen randomly. Hence, the number of iterations to converge the error varies each time when running the programme because, initially, the weights are chosen as random values. One pattern each time is computed during iteration process.
5 Results and Discussion The minimum fluid film thickness h min versus restrictor design parameter (Cs2 ) for FEM model validation is shown in Fig. 5. The FEM results are validated with literature data [18]. This graph shows the decrease in h min with the increase in Cs2 . The FEM results for h min are very close to the reference values for Cs2 as 0.08–0.13. Slight variations in h min values are observed due to the consideration of thermal effects in the literature. Results obtained by FEM are considered as target output parameter for neural network training. The range for radial load (w)is (0.2–1.4), and the values of speed parameter (Ω) are 1, 1.2 and 1.4. FEM results p max versus w are shown in Fig. 6.
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Fig. 5 h min versus Cs2 (FEM model validation)
Fig. 6 p max versus w (FEM results for Ω = 1, 1.2 and 1.4)
It is observed that p max values linearly increase with increase in w, but due to the increase in Ω, the p max values decrease for the same values of w. The graph for minimum fluid film thickness h min versus radial load (w) is shown in Fig. 7. The range for radial load (w) is 0.2–1.4, and the values of speed parameter (Ω) are 1, 1.2 and 1.4. It is observed that h min values linearly decrease with increase in w, but due to the increase in Ω, the h min values increase for the same values of w.
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Fig. 7 h min versus w (FEM results for Ω = 1, 1.2 and 1.4)
Fifteen sets of input parameters (radial load w and speed Ω) are considered for deep neural network model training as listed in Table 3.The ranges for w and Ω are 0.2–1.4 and 1.0–1.4, respectively. Maximum pressure p max and minimum fluid Table 3 Deep neural network training results S. No.
w
Ω
p max (Target)
p max (NN)
1
0.2
1.0
0.5783
0.5766
2
0.5
1.0
0.6483
3
0.8
1.0
0.7224
4
1.1
1.0
5
1.4
6
0.2
7
Error (%)
h min (Target)
h min (NN)
Error (%)
0.29
0.9602
0.9542
0.62
0.6458
0.39
0.9005
0.9026
−0.23
0.7217
0.10
0.8407
0.8396
0.13
0.800
0.8045
−0.51
0.7806
0.7783
0.29
1.0
0.8822
0.8796
0.29
0.7202
0.7229
−0.37
1.2
0.5782
0.5787
−0.09
0.9649
0.9610
0.40
0.5
1.2
0.6472
0.6464
0.12
0.9121
0.9164
−0.47
8
0.8
1.2
0.7195
0.7188
0.10
0.8594
0.8606
−0.14
9
1.1
1.2
0.7949
0.7977
−0.35
0.8065
0.8051
0.17
10
1.4
1.2
0.8732
0.8724
0.09
0.7536
0.7536
0.00
11
0.2
1.4
0.5780
0.5805
−0.43
0.9688
0.9661
0.28
12
0.5
1.4
0.6463
0.6471
−0.12
0.9221
0.9271
−0.54
13
0.8
1.4
0.7172
0.7164
0.11
0.8753
0.8775
−0.25
14
1.1
1.4
0.7906
0.7912
−0.08
0.8285
0.8274
0.13
15
1.4
1.4
0.8663
0.8643
0.23
0.7817
0.7803
0.18
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Table 4 Deep neural network testing results S. No.
w
Ω
p max (Target)
p max (NN)
1
0.7
1.0
0.6972
0.6954
2
1.3
1.0
0.8545
0.8567
3
0.7
1.2
0.6950
0.6939
4
1.3
1.2
0.8468
5
0.7
1.4
0.6933
6
1.3
1.4
0.8409
0.84107
Error (%)
h min (Target)
h min (NN)
Error (%)
0.26
0.8606
0.8609
−0.03
−0.26
0.7404
0.7406
−0.03
0.16
0.8770
0.8796
−0.30
0.8491
−0.27
0.7713
0.7703
0.13
0.6927
0.09
0.8909
0.8945
−0.40
−0.02
0.7973
0.7957
0.20
Table 5 Deep neural network predictions within range S. No.
w
Ω
p max (Fig. 6)
p max (NN)
Error (%)
h min (Fig. 7)
h min (NN)
Error (%)
1
0.4
1.0
0.62
0.6229
−0.47
0.92
0.9218
−0.20
2
0.6
1.0
0.67
0.6702
−0.03
0.88
0.8820
−0.23
3
1.0
1.0
0.77
0.7767
−0.87
0.80
0.7982
0.23
4
1.2
1.0
0.83
0.8314
−0.17
0.76
0.7591
0.12
5
0.4
1.2
0.62
0.6236
−0.58
0.93
0.9332
−0.34
6
0.6
1.2
0.67
0.6698
0.03
0.89
0.8984
−0.94
7
1.0
1.2
0.77
0.7710
−0.13
0.82
0.8232
−0.39
8
1.2
1.2
0.82
0.8240
−0.49
0.79
0.7875
0.32
9
0.4
1.4
0.62
0.6248
−0.77
0.94
0.9419
−0.20
10
0.6
1.4
0.67
0.6697
0.04
0.91
0.9112
−0.13
11
1.0
1.4
0.77
0.7658
0.55
0.84
0.8438
−0.45
12
1.2
1.4
0.82
0.8113
1.06
0.81
0.8113
−0.16
film thickness h min are considered as output parameters. The target output and neural network model output are compared, and error is calculated. The calculated sum of squares error is obtained as 0.0000934. The percentage error observed for p max is −0.51 to 0.39 and for h min is −0.47 to 0.62 during training of the model. Normalisation of input data set and output target values is not required because all the values are already in the range of 0–1. Neural network model is tested for six input data sets, and the results are presented in Table 4. The error observed for maximum pressure as −0.27 to 0.26 and for minimum fluid film thickness as −0.40 to 0.20 during testing the neural network model. Predictions are made for the journal bearing performance within the prescribed range of load (0.2–1.4), and the percentage error is presented in Table 5. The error is calculated and found in the range of −0.87 to 0.03 for maximum pressure and − 0.94 to 0.23 for minimum fluid film thickness. The neural network predictions out of prescribed range of load are presented in Table 6.
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Table 6 Deep neural network predictions out of range S. No.
w
Ω
p max
h min
1
1.5
1.0
0.8996
0.7058
2
1.6
1.0
0.9166
0.6895
3
1.7
1.0
0.9305
0.6740
4
1.5
1.2
0.8932
0.7373
5
1.6
1.2
0.9112
0.7215
6
1.7
1.2
0.9264
0.7062
7
1.5
1.4
0.8856
0.7652
8
1.6
1.4
0.9044
0.7504
9
1.7
1.4
0.9206
0.7359
This deep neural network model is suitable for predictions of journal bearing performance within and out of the prescribed range. By using feedforward backpropagation neural network model, very precise and accurate results are obtained for all the predictions. DNN approach was used extensively in recent years due to its excellence in detecting the faults and improving the life span of journal bearings [5, 19, 20]. It gives more accurate results compared to ANN. The number of iterations used to converge the error should be less to save the convergence time. In the past, 100,000– 500,000 number of iterations were required to achieve the convergence criteria [1, 3, 21], but now, the same problem can be solved within 20,000 iterations for the same value of percentage error due to the advancement in computational technology. Hence, the time saving means getting the solution converged in comparatively less number of iterations.
6 Conclusion Mapping of the patterns in deep neural network found to be excellent for journal bearing performance predictions. Feedforward backpropagation algorithm handles the complexity of the problem in 15,000 iterations and gives the more accurate and precise results. The neural network model predictions are found to be very near to target values. The error is calculated for deep neural network and found to be very small (−1.0% < error < 1.0%) during training and testing of the model. The predictions are made within and out of the prescribed range with very high accuracy. This deep neural network approach is perfect for performance predictions within and out of the prescribed range of input parameters. It is concluded that lot of time is saved to achieve the convergence criteria using DNN compared to the mathematical analysis of the journal bearing performance.
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Appendix Journal Bearing notations (non-dimensional) α, β
Circumferential and longitudinal coordinates
X J, Z J
Journal centre position
p
Hydrodynamic pressure
Journal rotational speed
C s2
Restrictor design parameter
pc
Pressure at holes
F 0, F 1, F 2
Viscosity functions
μ
Lubricant dynamic viscosity
h
Fluid film thickness
w
Radial load
QR
F P Q RH Rx , Rz
Flow through restrictor Fluidity matrix Nodal pressure vector Nodal flow vector Vector for hydrodynamic terms Vectors due to journal centre velocity
References 1. Sinano˘glu C (2006) A neural predictor to analyse the effects of metal matrix composite structure (6063 Al/SiCp MMC) on journal bearing. Industr Lubr Tribol 58(2):95–109 2. Patel PM, Prajapati JM (2011) A review on artificial intelligent system for bearing condition monitoring. Int J Eng Sci Technol 3(2):1520–1525 3. Sinano˘glu C, Kurban AO, Yildirim S¸ (2004) Analysis of pressure variations on journal bearing system using artificial neural network. Industr Lubr Tribol 56(2):74–87 4. Samanta B, Al-Balushi KR, Al-Araimi SA (2003) Artificial neural networks and support vector machines with genetic algorithm for bearing fault detection. Eng Appl Artif Intell 16(7–8):657– 665 5. Liu R, Yang B, Zio E, Chen X (2018) Artificial intelligence for fault diagnosis of rotating machinery: a review. Mech Syst Sign Process 108:33–47 6. Garg HC, Sharda HB, Kumar V (2006) On the design and development of hybrid journal bearings: a review. Tribotest 12(1):1–19 7. Garg HC, Kumar V (2013) Static performance characteristics of hybrid journal bearings with plugged entry holes. Industr Lubr Tribol 65(5):333–340 8. Dhawan R, Verma S (2014) Analyzing micropolar lubrication in noncircular hybrid journal bearings. Tribol Trans 57(2):182–189 9. Kumar R, Verma S (2016) Effect of micropolar lubrication in non-circular hole-entry hybrid journal bearing with constant flow valve restrictor. Industr Lubr Tribol 68(6):737–751
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10. Khatak P, Garg HC (2018) Performance comparison of hole-entry and slot entry hybrid journal bearings considering combined influence of thermal effects and micropolar lubricant. Industr Lubr Tribol 70(6):1037–1050 11. Nicodemus ER, Sharma SC (2011) Orifice compensated multirecess hydrostatic/hybrid journal bearing system of various geometric shapes of recess operating with micropolar lubricant. Tribol Int 44(3):284–296 12. Ram N, Sharma SC (2012) Analysis of orifice compensated non-recessed hole-entry hybrid journal bearing operating with micropolar lubricants. Tribol Int 52:132–143 13. Kucinschi BR, Fillon M, Fre J, Pascovici MD (2000) A transient thermo elasto hydrodynamic study of steadily loaded plain journal bearings using finite element method analysis. J Tribol 122(1):219–226 14. Khatri CB, Sharma SC (2017) Behaviour of two-lobe hole-entry hybrid journal bearing system under the combined influence of textured surface and micropolar lubricant. Industr Lubr Tribol 69(6):844–862 15. Dowson D (1962) A generalized Reynolds equation for fluid-film lubrication. Int J Mech Sci 4(2):159–170 16. Fowles PE (1970) A simpler form of the general Reynolds equation. J Lubr Technol 92(4):661– 662 17. Crosby WA, Chetti B (2009) The static and dynamic characteristics of a two-lobe journal bearing lubricated with couple-stress fluid. Tribol Trans 53(2):262–268 18. Sharma SC, Kumar V, Jain SC, Nagaraju T (2003) Study of hole-entry hybrid journal bearing system considering combined influence of thermal and elastic effects. Tribol Int 36(12):903– 920 19. Thamba NB, Aravind A, Rakesh A, Jahzan M, Duraiswamy RP, Mangalaraja RV (2018) Automatic fault classification for journal bearings using ANN and DNN. Arch Acoust 43(4):727–738 20. Ren L, Sun Y, Cui J, Zhang L (2018) Bearing remaining useful life prediction based on deep autoencoder and deep neural networks. J Manuf Syst 48:71–77 21. Sinano˘glu C (2009) Design of neural model for analysing journal bearings considering effects of transverse and longitudinal profile. Industr Lubr Tribol 61(3):132–139
Kinematics and Foldability Analysis of Bennett Mechanisms and Its Networks Tony Punnoose Valayil
Abstract Bennett 4R mechanism is a one degree-of-freedom spatial mechanism having four revolute (R) joints and four kinematic links. In this research, two types of Bennett 4R mechanisms were taken to check their range of motion or working range using MATLAB SimMechanics toolbox. It was found that working range for type 2 Bennett 4R mechanism was better than type 1 Bennett 4R mechanism. Kinematic characteristics of both Bennett 4R mechanisms were also discussed. Then, range of motions for both the types of Bennett 4R mechanisms was also found by connecting it in a network. Each network is an interconnection of nine Bennett 4R mechanisms of same type. Coupler curves of Bennett 4R mechanisms were also plotted in MATLAB. Type 2 Bennett 4R mechanism has better foldability than type 1 Bennett 4R mechanism, and therefore, it is more compact after folding. Range of motion was then found out for Bennett RRRS mechanism. Bennett RRRS mechanism was obtained by replacing the last revolute joint in a 4R Bennett mechanism by a spherical joint. It was observed that working range for Bennett RRRS mechanisms and Bennett 4R mechanisms was found to be same. Keywords Foldability · Strength · Stiffness
1 Introduction In 1903, G. T. Bennett invented a spatial mechanism with four links and four joints [1]. All four links are twisted to make it a spatial mechanism. It has a closed-loop kinematic structure. Due to its special geometrical properties, researchers have tried using it in applications like deployable structures, foldable tends, liquid mixing machines, etc. Two types of Bennett 4R mechanisms were identified. In this research, networks were made using these two types of mechanisms using MATLAB SimMechanics
T. Punnoose Valayil (B) Coimbatore Institute of Technology, Coimbatore 641014, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_162
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toolbox. Kinematics of two types of Bennett 4R mechanisms is discussed and application where Bennett 4R mechanism could be used as a rigid structure is also discussed.
2 Kinematics of Type 1 Bennett 4R Mechanism Type 1 Bennett 4R mechanism has four links such that their opposite links are of equal length. It also has four joints. Its graphic diagram is shown in Fig. 1 [2]. As shown in Fig. 1, ABCD is a type 1 Bennett 4R mechanism having non-parallel and non-intersecting revolute joint axes. A, B, C and D are the revolute joints in the mechanism. The two opposite links of type 1 Bennett 4R mechanism have same twist angle and length [2]. DC = l3 = AB = l1 , AD = l4 = BC = l2 , DC = AB = α, AD = BC = β, where l1 , l 2 , l 3, l 4 represent lengths of the links AB, BC, CD and AD. ‘α’ represents twist angles in the links AB and DC and ‘β’ represents twist angles in the links AD and BC. Type 1 Bennett 4R mechanism exists only if one more condition is satisfied. That is, the opposite link lengths and twist angles should satisfy Eq. (1) [2] sin α sin β = l1 l2
Fig. 1 Type 1 Bennett 4R mechanism
(1)
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3 Kinematics of Type 2 Bennett 4R Mechanism It is also known by the name of equilateral Bennett mechanism. Equations for its existence are given by Eq. (2) [3] α+β =π l1 = l2 = l3 = l4
(2)
4 Coupler Curves The link which connects input link (AD) and output link (BC) of the Bennett 4R mechanism is known as coupler link (CD). The path traced by a point in the coupler link during its motion is known as coupler curve. It is very important to find this path, because this path denotes the output for the application used by the mechanism. It also helps to solve rigid-body guidance and path generation problems [4]. The revolute joint coordinates for the Bennett 4R mechanisms are obtained as [2]. A(0, 0, 0), B(l1 , 0, 0), D(l2 cos θ, l2 sin θ, 0), C((l1 + l2 cos η), (l2 cos α sin η), (l2 sin α sin η)). To plot the coupler curve for type 1 and type 2 Bennett 4R mechanisms, the revolute joint coordinates for joint C are considered. Now, the input angle ‘θ ’ and the output angle ‘η’ are obtained using Eq. (3) [2].
z (m)
50
0
-50 100 50 0
y (m)
-50 -100 50
100
Fig. 2 Coupler curve for type 1 Bennett 4R mechanism
150
200
x (m)
250
300
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z (m)
50
0 -50 100
50
0 y (m)
-50
-100 0
50
100 x (m)
150
200
Fig. 3 Coupler curve for type 2 Bennett 4R mechanism
⎫ ⎧ θ ⎪ ⎪ ⎪ ⎬ ⎨ 2π + 2 arctan K tan 2 , θ ∈ [0, π ]⎪ η= ⎪ ⎪ θ ⎪ ⎪ ⎭ ⎩ 2 arctan K tan , θ ∈ [π, 2π ] 2
(3)
where K =−
sin(β ± α) sin(α) − sin(β)
(4)
For plotting the above coupler curve (Fig. 2) for type 1 Bennett 4R mechanism, the following values are considered. α = 30◦ ∗3.14/180 rad, β = 17.46◦ ∗3.14/180 rad, K = +3.68 (K+ve taken), l1 = 167.7 mm and l2 = 100 mm. Coupler curve for type 2 Bennett 4R mechanism is shown in Fig. 3. For plotting the above coupler curve for type 2 Bennett 4R mechanism, the following values were considered α = 30◦ ∗3.14/180 rad, β = 150◦ ∗3.14/180 rad, K = 1.15 (K+ve taken), l1 = 100 mm and l2 = 100 mm.
5 Bennett Network A single Bennett 4R mechanism was taken, and its links were extended to connect eight other Bennett 4R mechanisms to form a network as shown in Fig. 4. Black colored lines in Fig. 4 represent extended links. Red color represents the links of eight Bennett 4R mechanisms which are connected to it. The arrangement of network is made in such a manner that actuation needs to be given to the input link of the Bennett 4R mechanism placed at the center (numbered 1 in Fig. 4) so that the entire network gets actuated. The four extended links of the mechanism 1 divide the whole network into four regions, I, II, III and IV. Type 1 and type 2 Bennett 4R mechanisms are used to construct two different networks.
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Fig. 4 Bennett network developed in MATLAB
6 Replacement of R (Revolute) by S (Spherical) A lower-over-constrained type of Bennett mechanism was introduced by Morgan V. Brown and Paul Milenkovic [5]. In this Bennett RRRS mechanism, the last revolute joint in Bennett 4R mechanism was replaced by a spherical joint. Spherical joint has three degree-of-freedom, whereas revolute joint has only one degree-of-freedom. This Bennett RRRS mechanism with spherical joint also has one degree-of-freedom like the Bennett 4R mechanism. Hence, spherical joint was introduced to type 1 and type 2 Bennett 4R mechanisms, and their range of motion was calculated. Minimum and maximum foldability was analyzed using MATLAB SimMechanics. Revolute joint at D of Bennett mechanism ABCD (shown in Fig. 1) was replaced by spherical joint.
7 Results Foldability test was conducted on type 1 and type 2 Bennett 4R mechanisms. The coordinates of Bennett 4R mechanism at various input angles (θ ) were calculated. The configuration of the type 1 Bennett 4R mechanism at θ = 12◦ and θ = 131◦ is given below in Fig. 5a, b. There is singular configuration before and after the specified minimum and maximum input angle. The configuration at which the mechanism losses its degree-of-freedom, and there is no motion possible to the next position.
Fig. 5 a Type 1 at input angle θ = 12◦ , b Type 1 at input angle θ = 131◦
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Fig. 6 a Type 2 at input angle θ = 12◦ , b type 2 at input angle θ = 147◦
The coordinates of Bennett 4R mechanism at various input angles (θ ) were calculated. The configuration of the type 2 Bennett 4R mechanism at θ = 12◦ and θ = 147◦ is given in Fig. 6a, b. There is singular configuration before and after the specified minimum and maximum input angle. The configuration at which the mechanism losses its degree-of-freedom, there is no motion possible to the next position. Foldability test using MATLAB SimMechanics provided the range of motion of Bennett 4R mechanisms. Input angle working range for all equal linkages (Type 2) is from 1° to 147° and for opposite equal linkages (Type 1) is from 12° to 131°. For type 2 Bennett 4R mechanism, the minimum folding angle is 1°, and maximum folding angle is 147°, but for type 1, the minimum angle is 12° and maximum angle is 131°. Hence, the foldability of using type 2 Bennett 4R mechanism is high, and hence, it is more compact than type 1 Bennett 4R mechanism after folding and expanding. There is singular configuration before and after the specified minimum and maximum input angle. The configuration at which the mechanism losses its degree-of-freedom, there is no motion possible to the next position. Foldability test was now performed in MATLAB SimMechanics to find the range of motion for Bennett mechanisms with spherical joint (RRRS) and is shown in Figs. 7 and 8. Input angle working range for all equal linkages (type 2 Bennett RRRS mechanism) is from 1° to 147° and for opposite equal linkages (type 1 Bennett RRRS mechanism) is from 12° to 131°. For type 2 Bennett RRRS mechanism, the minimum folding angle is 1° and maximum folding angle is 147°, but for type 1 Bennett RRRS mechanism, the minimum is 12° and maximum is 131°. Hence, the foldability of using type 2 Bennett RRRS mechanism is high, and hence, it is more compact than
Fig. 7 a Input angle (θ) at 12° for type 1 Bennett RRRS mechanism, b input angle (θ) at 131° for type 1 Bennett RRRS mechanism
Fig. 8 a Input angle (θ) at 1° for type 2 Bennett RRRS mechanism, b input angle (θ) at 147° for type 2 Bennett RRRS mechanism
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Fig. 9 a Network at θ = 30◦ for type 1 Bennett 4R mechanism, b network at θ = 130◦ for type 1 Bennett 4R mechanism
type 1 Bennett RRRS mechanism after folding and expanding. There is singular configuration before and after the specified minimum and maximum input angle. The configuration at which the mechanism losses its degree-of-freedom, there is no motion possible to the next position. It is to be noted that the Bennett mechanism with spherical joints (RRRS) has the same foldability like the Bennett 4R mechanism. Hence, type 2 Bennett RRRS mechanisms can also be used in place of type 2 Bennett 4R mechanism where compact folding is required. The results are same because singularity happens on revolute joint which is not replaced by spherical joint. Once this is also replaced, a better result can be obtained due to three degree-of-freedom motion of spherical joint. This also indicates that the Bennett RRRS mechanism has the same properties like Bennett 4R mechanism as mentioned by Morgan V. Brown and Paul Milenkovic [5].The coordinates of all the joints in the network are calculated for type 1 Bennett 4R mechanisms for two different input angles of θ (θ = 30◦ , 130°). The network was modeled for these configurations in MATLAB SimMechanics, and different shapes of the network are obtained as shown in Fig. 9a, b. There is singular configuration before and after the specified minimum and maximum input angle. The configuration at which the mechanism losses its degree-of-freedom, there is no motion possible to the next position. The coordinates of all the joints are calculated for different input angles of θ (θ = 30◦ , 121°) for type 2 Bennett mechanism. The network was modeled for these configurations in MATLAB SimMechanics, and different shapes of the network are obtained as shown in Fig. 10a, b. From Figs. 9 and 10, it is found that the input angle working range differs for both type 1 and type 2 Bennett 4R mechanisms. Input angle working range for type 1 is from 30° to 130°, and the input angle working range for type 2 is from 30° to 121°. Even though type 1 and type 2 have same minimum folding angle, type 2 is better when considering its shorter links. Therefore, working with type 2 is more compatible than working with type 1 when it comes to network. Singularity occurs
Fig. 10 a Network at θ = 30◦ for type 2 Bennett 4R mechanism, b network at θ = 121◦ for type 2 Bennett 4R mechanism
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at θ = 29◦ and θ = 122◦ for the type 2 network and at θ = 29◦ and θ = 131◦ for type 1 network.
8 Different Cases of Twist Angles Twist angles present in the links of a Bennett mechanism play an important role in kinematic analysis. Twist angles are the main reasons for the formation of saddle surfaces in a deployable Bennett network. Selection of twist angles is based on actual needs [3]. By changing the twist angle, the shape of the network is analyzed for type 1 and type 2 Bennett 4R mechanisms. Case 1: With α = 20◦ , l1 = 167.7 mm and l2 = 100 mm. Case 2: With α = 30◦ , l1 = 167.7 mm and l1 = 100 mm. From above Fig. 11a, b, it can be observed that when twist angle changes, the shape of the networks changes. When twist angle is increased from 20° to 30°, more saddle like shape is formed. Thus, twist angle is a governing factor for formation of saddle surface in Bennett mechanism. By choosing the appropriate link length and twist angle, a perfect and expected saddle shape can be obtained. Saddle surfaces are most commonly used in building roofs in structures. These saddle roofs have a hyperbolic paraboloid shape. This is a doubly curved surface having a concave surface on one side and a convex on the other. Unlike other structures, these roofs gain their strength through their shape and not by their mass. These roofs are lightweight and have high stiffness. They can withstand unequal loading, both dead load and live load. An example of saddle roof is shown in Fig. 12. It is a railway station at Warszawa Ochota, Poland. Such type of roofs if constructed using Bennett mechanisms can be folded. A physical model of Bennett 4R mechanism which was constructed using PVC pipe is shown in Fig. 13. Their links were connected using nut and bolt, acting as revolute joint. In Fig. 13 (left), a single Bennett 4R mechanism can be observed, and to the right, a network made of Bennett 4R mechanism can be observed. A saddle shape can be observed in the network. It can also be constructed using steel for roofs in buildings.
Fig. 11 a Case 1, b Case 2
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Fig. 12 Saddle roof
Fig. 13 Physical model
9 Conclusion Two types of Bennett 4R mechanisms having different kinematic characteristics were analyzed. Foldability analysis was done using these two types to find their working range. Results indicated that type 2 Bennett 4R mechanism has better working range than type 1 Bennett 4R mechanism. Then, further analysis was done by replacing the last revolute joint in a Bennett 4R mechanism by a spherical joint. Working range during foldability test for Bennett 4R mechanism and Bennett RRRS mechanism provided similar results. Hence, it can be concluded that either 4R or RRRS can be used for making deployable structures. Then, a network using Bennett 4R mechanisms was constructed. In the case of networks, working range for type 2 Bennett 4R mechanism was better than type 1. While analyzing the shape of the Bennett
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network, a saddle shape was observed. It was observed that twist angles present in the links of the Bennett mechanism were the reason for obtaining perfect saddle surface. The saddle structures gain their strength through their shape. Thus, Bennett networks provide a simple way for constructing deployable saddle roofs.
References 1. Bennett GT (1903) A new mechanism. Engineering 76:777–778 2. Changjian Z, Sanmin W, Yuantao S, Jianfeng L (2015) Kinematic and dynamic characteristics analysis of Bennett’s linkage. J Harbin Inst Technol 95–100 3. Fufu Y, Jianmin L, Yan C (2015) Deployable Bennett network in saddle surface. In: The 14th IFToMM world congress, Taipei 4. Soni AH (1974) Mechanism synthesis and analysis. McGraw-Hill Book Co, New York 5. Morgan VB, Paul M (2011) Properties of the Bennett mechanism derived from the RRRS closure ellipse. J Mech Rob 021012–1–021012–8
Nanofibers for Sustainable Filtration: A Waste to Energy Approach Prakash Giri, Ashish Kakoria, Sahil Verma, and Sumit Sinha-Ray
Abstract Air, a crucial requirement of human life, has been increasingly polluted day by day with rapid growth of industries, especially fossil fuel-driven ones, deforestation and urbanization. It has been severely affecting our lives and environment. One active as well as passive source of air pollution is cigarette smoke which has various health concerns. This smoke on the one hand releases complex aerosols like vapor oxides, tar and particulate pollutants in the surroundings, and on the other hand, cigarette buds get accumulated in land and ocean, thus affecting the ecological aspects of the environment, which demands for the requirement of efficient smoke filtration with proper disposal of used filters. This study analyzed the use of PAN nanofibers in cigarette smoke filtration with reusability in energy applications. PAN nanofibers showed 42.61% porosity with increased tortuosity; yet permeability was at par with commercial cigarette filters. The nanofiber filters gained weight up to 115% because of absorbed pollutants after a complete cigarette smoking test. Positive electrical conductivity upon carbonization and catalytic contribution in electrochemical tests was observed for the nanofiber filters. Keywords PAN · Nanofibers · Smoke · Filtration · Energy
1 Introduction Smoke consists of various gaseous and particulate pollutants. It is observed that cigarette smoke consists of complex aerosols including elements like Al, As, Cd, Cr, Cs, Cu, Hg, Pb, Mn, Na, Ni, Se, Zn, etc.; compounds like CO, CO2, hydrogen cyanide, formaldehyde, acrolein, acetaldehyde, ammonia, nitrogen dioxide, sulfur dioxide, aromatics amines, etc., and particulate pollutants including PM2.5. Pb, Cd and Cr which are highly carcinogenic, and many aromatic compounds in the smoke Ashish Kakoria and Sahil Verma contributed equally for this chapter. P. Giri · A. Kakoria (Deceased) · S. Verma (Deceased) · S. Sinha-Ray (B) School of Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175005, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_163
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are found to be cacogenic [1–6]. Particulate pollutants can cause diseases like lung cancer, chronic obstructive pulmonary disease and asthma [7]. PM2.5 is small enough to enter lungs and interfere with oxygen exchange in the arteries [7]. Cigarette buds when released in atmosphere can get accumulated in water bodies and severely affect aquatic ecology. Thus, efficient filtration of smoke constituents and proper disposal/utilization of filter buds is necessary. Air filters made up of fiber materials are used for most of the smoke filtration applications. Filtration of PM in the flow stream is subjected to combined effect of diffusion, inertial impaction, interception, electrostatics and earth’s gravitational force [8, 9]. Filtration contribution of 40% and 60% by diffusion and interception was observed by Overton, showing the contribution of inertial impaction to be negligible and interception to be important process for the filtration of sub-micron particles [10, 11]. Traditional filtration media including spun-bonded, melt blown and glass fibers has low particulate matter filtration efficiency because of large pore size; on the other hand, preparing thicker media causes high pressure drop and energy costs [6]. Thus, electrospinning process was carried out for the formation of nanofibers.
2 Methodology 2.1 Materials Polyacrylonitrile (PAN) (mol. wt. = 150 kDa) was purchased from Sigma-Aldrich. N, N-Dimethylformamide (DMF) was purchased from Alfa Aesar.
2.2 Electrospinning and Membrane Preparation 9 wt. % solution of PAN in DMF was prepared by stirring the mixture at 40 °C and 700 rpm, for 14 h. The solution was electrospun with a conventional electrospinning apparatus whose parameters are detailed in Table 1. The schematic of electrospinning setup is shown in Fig. 1. Table 1 Electrospinning parameters
S. No.
Parameter
Value
1
Syringe capacity
10 mL
2
Syringe diameter
18 mm
3
Needle size
18 gauge
4
Needle to collector distance
12 cm
5
Solution flow rate
1.6 mL/h
6
High voltage
10 kV
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Fig. 1 Schematics of electrospinning setup (left) and PAN fiber mat prepared by electrospinning (right)
Thus, the produced fiber mat was compressed in polymer press at 50 °C and 392 kPa for 60 s. These mats were rolled into cigarette filters of 7.6 mm diameter and 15 mm length.
2.3 Characterization and Testing Scanning electron microscopy (SEM) and ImageJ software were used for architectural characterization of the nanofiber mat. Pressure and permeability were determined with the help of manometer and anemometer. Cigarette smoking test was performed on a laboratory-made bottled setup that mimics human smoking process with its squeezing and expanding action (Fig. 2).
2.4 Carbonization First, the nanofiber mat was stabilized in air at 300 °C followed by carbonization at 900 °C in N2 atmosphere. A ramp rate of 10 °C/min. and soaking time of 1 h. were used for the purpose. Cooling to room temperature was also performed in N2 atmosphere [12].
2.5 Electrochemistry Electrical conductivity was tested by using DC voltage source. Cyclic voltammetry (CV) and linear sweep voltammetry (LSV) tests were performed by Auto Lab electrochemical workstation (302 N) to determine the activities of the carbonized nanofibers.
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Fig. 2 Pressure drop and flow measurement (left), schematics of setup for cigarette smoking test (right)
To prevent the formation of bubbles, rotating disk electrode (RDE, PINE Instruments) setup was used with rotation ranging from 100 to 1600 rpm. The electrochemical experiments were performed in three-electrode setup, using Pt wire as a counter electrode and Ag/AgCl (3M) as reference electrode. The potential against Ag/AgCl was converted into potential against reversible hydrogen electrode (RHE) by using the equation E(RHE) = E(Ag/AgCl) + (0.197 + 0.059 × pH)
(1)
3 Results 3.1 Fiber Morphology Fiber mat with average fiber diameter of 430 nm was produced with the electrospinning process. Porosity of the fiber mat was determined to be 42.61%, with a mean pore diameter of 2.60 µm (Fig. 3). The diameter was reduced to 335 nm after stabilization and to 275 nm after carbonization.
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Fig. 3 SEM of PAN nanofibers mat (left) and carbonized mat (right)
3.2 Filtration Properties The tortuosity of prepared filter was determined to be 1.078 as suggested by Pisani [13] τk =
1 1 − α(1 − ∅)
(2)
where α is the shape factor and ∅ is the porosity. The Knudsen diffusivity was calculated based upon the semi-empirical formulation as [14, 15] DKeff
εp dp × = × τk 3
8RT πM
(3)
where εp is the volumetric porosity, dp is the pore diameter, R is the specific gas constant, T is temperature in absolute scale and M is molecular wt. of the fluid used. The permeability was determined as [16–18] −2μ × k= Pa + Pb
RTLN − DKeff Pb − Pa
(4)
where μ is the dynamic viscosity of the flowing medium (air), Pa and Pb are pressures at inlet and outlet, L is the length of filter medium and N is the molar flux through the medium. Measuring the required values of porosity, pore diameter, temperature, pressure drop and size of the filter, the value of Knudsen diffusivity was calculated to be 159.98 × 106 µm2 /s and permeability to be 0.0349 µm2 . The normal time taken by smoke to pass through the filter section would be t=
15 × 10−3 L = = 0.0115 s v 1.3
(5)
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where v is taken as the exit velocity of air. However, the diffusion time would be 2 15 × 10−3 L2 tD = = = 1.4 s Deff 159 × 10−6
(6)
Therefore, the apparent length would be extended to 1.82 m, which is nearly 121 times more than the filter length. These results show good filtration capabilities with increased hydraulic retention time or increased effective length of the filter, allowing for efficient diffusion across the filter section.
3.3 Reusability Analysis PAN nanofiber filters were used for cigarette smoke filtration where the filter gained up to 115% of its initial weight. Compressed polymer mat on simple rolling gained 15% of initial wt., and the absorption capacity was seen to increase with fluffiness of the filter. A section of the filter was washed in ethanol and the absorbed particles leeched out in the solution. A similar study is demonstrated in Ref [19]. This elucidates the reusability of the filters. PAN nanofibers are porous and rich in carbon content. Their carbonization showed a positive electrical conductivity. The conductivity of the specimen was measured to be 98.47 −1 m−1 (c.f. Fig. 4). For CV analysis, the carbonized sample was purged with oxygen in 0.1M KOH electrolyte. The presence of reduction peak in CV confirmed that the sample was oxygen reduction reaction (ORR) active. Furthermore, LSV studies carried out in 0.1M KOH in the potential range of 1.164 to −0.036 V (vs. RHE) under the scan rate of 10 mV/sec showed an onset potential of 0.78 V (vs. RHE) with maximum current density of 0.94 mA/cm2 , at 1600 rpm (Fig. 5).
Fig. 4 Conductivity tests of carbonized PAN
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Fig. 5 a CV, b OER, c HER, d ORR of carbonized sample in 0.1M KOH under scan rate of 10 mV/s. using three-electrode setup
After ORR analysis, electrochemical cell was purged with N2 , and LSV procedures were carried out between 0 and −1.4 V (vs. RHE) in 0.1M KOH concentration at room temperature which confirmed the presence of hydrogen evolution reaction (HER) activity in the sample, revealing an onset potential of 0.6 V (vs. RHE) with the current density of 25.7 mA/cm2 . Furthermore, LSV test was again performed for the sample in the potential range of 0.9–1.9 V (vs. RHE) under scan rate of 10 mV/s with rotation ranging from 0 to 1600 rpm in 0.1M KOH, and it was observed that the carbonized sample was oxygen evolution reaction (OER) active, showing an onset potential of 1.58 V (vs. RHE) with the current density of 6.5 mA/cm2 , which is just 350 mV additional compared to IrO2 .
4 Conclusion Nanofibers of PAN in DMF made with electrospinning process showed good filtration capabilities and excellent reusability applications. Besides, commercial cigarette
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filters can also be tested for replacement with alternative carbon rich fiber filters, which have electrochemical reusability because of their porosity and carbon content. This experiment has already demonstrated that the fibers such as of PAN can have their application as a current collector, electrode material, catalyst support material or a catalyst itself. Moreover, this research opens further analysis about the effect of heavy metal deposition from cigarette smoke on these properties. This research provides good insight about filtration capabilities of PAN and its reusability applications. These fibers can be excellent alternatives for commercially available air/smoke filters, preventing themselves to be exposed in our surroundings by finding their reusability in energy applications. Acknowledgements Advanced Material Research Center (AMRC), IIT Mandi, is appreciatively acknowledged for their advanced instrument facility. P. G., A. K. and S. V. acknowledge scholarship from Ministry of Human Resource Development (MHRD), India.
References 1. Chiba M, Masironi R (1992) Toxic and trace elements in tobacco and tobacco smoke. Bull World Health Organ 70(2):269 2. Kensler CJ, Battista S (1963) Components of cigarette smoke with ciliary-depressant activity: their selective removal by filters containing activated charcoal granules. N Engl J Med 269(22):1161–1166 3. Patrianakos C, Hoffmann D (1979) Chemical studies on tobacco smoke LXIV. On the analysis of aromatic amines in cigarette smoke. J Anal Toxicol 3(4):150–154 4. Hossain MT, Hassi U, Huq SI (2018) Assessment of concentration and toxicological (Cancer) risk of lead, cadmium and chromium in tobacco products commonly available in Bangladesh. Toxicol Rep 1(5):897–902 5. Burnett RT, Smith-Doiron M, Stieb D, Cakmak S, Brook JR (1999) Effects of particulate and gaseous air pollution on cardiorespiratory hospitalizations. Arch Environ Health Int J 54(2):130–139 6. Morabet RE (2018) Effects of outdoor air pollution on human health. Elsevier Inc. 7. Mannucci PM, Harari S, Martinelli I, Franchini M (2015) Effects on health of air pollution: a narrative review. Intern Emerg Med 10(6):657–662 8. Zhu M, Han J, Wang F, Shao W, Xiong R, Zhang Q, Pan H, Yang Y, Samal SK, Zhang F, Huang C (2017) Electrospun nanofibers membranes for effective air filtration. Macromol Mater Eng 302(1):1600353 9. Wen D, Jianhui W, Bing P, Xiaobin Z, Fuwei X, Huimin L, Kejun Z (2015) An improved theoretical model of cigarette smoke filtration across mono-segment cellulose acetate filters. Beiträge zur Tabakforschung/Contrib Tobacco Res 26(5):232–240 10. Siddiqi K, Shah S, Abbas SM, Vidyasagaran A, Jawad M, Dogar O, Sheikh A (2015) Global burden of disease due to smokeless tobacco consumption in adults: analysis of data from 113 countries. BMC Med 13(1):194 11. Overton JR (1973) Filtration of cigarette smoke: relative contributions of inertial impaction, diffusional deposition, and direct interception. Beiträge zur Tabakforschung/Contrib Tobacco Res 7(3):117–120 12. Rahaman MS, Ismail AF, Mustafa A (2007) A review of heat treatment on polyacrylonitrile fiber. Polym Degrad Stab 92(8):1421–1432
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13. Pisani L (2011) Simple expression for the tortuosity of porous media. Transp Porous Media 88(2):193–203 14. Arya G, Chang HC, Maginn EJ (2003) Knudsen diffusivity of a hard sphere in a rough slit pore. Phys Rev Lett 91(2):026102 15. Knudsen M (1909) Effusion and the molecular flow of gases through openings. Ann Phys 29:179 16. Cunningham RE, Williams RJ (1980) Diffusion in gases and porous media. Plenum Press, New York 17. Carrigy NB, Pant LM, Mitra S, Secanell M (2013) Knudsen diffusivity and permeability of PEMFC microporous coated gas diffusion layers for different polytetrafluoroethylene loadings. J Electrochem Soc 160(2):F81–F89 18. Kerkhof PJ (1996) A modified Maxwell-Stefan model for transport through inert membranes: the binary friction model. Chem Eng J Biochem Eng J 64(3):319–343 19. Giri P, Kakoria A, Verma S, Sinha-Ray S (2021) Reuse of cigarette filters for energy applications. Function Text Cloth 2020:161–168. Springer, Singapore
Effect of Heat Treatment on Wear Behaviour of Austenitic Stainless Steel Waris Nawaz Khan, Furkan, and Rahul Chhibber
Abstract Stainless steel is one of the most widely used steels for structural applications. This paper quantifies the effect of heat treatment on the wear resistance behaviour of stainless steel 304 grades using pin on disc method. Ageing at different temperature ranges imparts hardness to the SS304 specimen which in turn affects the wear resistance characteristic. Different wear parameters have been calculated using Archard’s equation and wear scar diameter method. Microstructure has been studied using optical microscopy. Scanning electron microscopy has been used to investigate the wear mechanism. Keywords Wear · Stainless steel · Archard’s equation · Wear mechanism
1 Introduction Austenitic stainless steel with its superior mechanical and tribological properties is now used extensively in strategic applications. Stainless steel 304 grade is a preferred material for structures with high structural integrity and aggressive service environments. It cannot be usually heat treated, but can be annealed at different temperature ranges to bring it into application of refrigeration, chemical, paper and food processing industry. Some applications such as conveyor belt, screws and bolts have relative sliding motion against different components. This leads to wear of the mating surface and can ultimately lead to component’s failure [1]. Stainless steel 304 possesses adequate toughness and sufficient ductility to resist failure for a considerable long duration of time. They can also be readily welded with similar and dissimilar metals for various applications [2]. 304 grade austenitic stainless steel is W. N. Khan (B) · Furkan · R. Chhibber Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, India e-mail: [email protected] Furkan e-mail: [email protected] R. Chhibber e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_164
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Table 1 Chemical composition of SS 304 steel (%wt) Element
C
Cr
Si
Mn
Ni
P
Fe
Composition
0.025
18.2
0.5
1.6
8.5
0.045
Balance
Table 2 Chemical composition of EN 31 steel Element
C
Cr
Si
Mn
S
P
Fe
Composition
1.1
1.3
0.25
0.55
0.042
0.040
Balance
a high alloy Cr–Ni–Mo steel. It has high pitting resistance equivalent number which makes it significantly corrosion resistant. The literature reveals that the previous researchers have worked in the field of estimating wear characteristic of austenitic stainless family [3–7]. But very limited literature is available regarding the effect of heat treatment on austenitic stainless steel 304 grade’s wear characteristics [8]. This paper aims to investigate the effect of heat treatment on hardness, its correlation with microstructure and related wear behaviour. Pin on disc technique has been used to study the wear behaviour in laboratory. Various wear property parameters have also been calculated using Archard’s equation and scar diameter method. Wear mechanism has been studied using scanning electron microscopy (SEM).
2 Material Stainless steel 304 plate was received in the dimension of 250 × 250 mm. The chemical composition of stainless steel 304 is given in Table 1. The pins made out of SS304 were made to slide against the disc made of EN 31 bearing steel. Table 2 gives the chemical composition of EN31 bearing steel used.
3 Experiment 3.1 Specimen Preparation Pins were prepared from the SS304 plate in the dimensions shown in Fig. 1.
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Fig. 1 Geometry of pin specimen
Table 3 Heat treatment of pin specimen Specimen nomenclature
Solid solution annealing
Sensitization
Temperature (°C)
Soak time
Cooling medium
Temperature (°C)
Soak time
HT 1
1050
60
Water
500
60
Air
HT 2
1050
60
Water
620
270
Air
HT 3
1050
60
Water
650
270
Air
NHT
As received specimen, no heat treatment
Cooling medium
3.2 Heat Treatment The pin specimens were subjected to the following heat treatment as shown in Table 3.
3.3 Pin on Disc Wear Test The pin on disc test was conducted on Ducom wear testing machine with a load of 5 N at a rotating speed of 500 rpm for 15 min duration. The volume loss in material was calculated using two methods: (i) change in scar diameter (ii) weight loss of pin due to material removal. Figure 2 represents the schematic of wear testing machine used in this experiment.
4 Results and Discussion 4.1 Microstructural Examination: The specimens in as-received and heat-treated condition were polished with emery paper up to a grade of 2000 and then with a diamond paste. The mirror polished
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Fig. 2 Pin on disc wear test setup
specimens were then etched with Villela reagent to reveal the microstructure. Optical microscope was used to observe the developed microstructure owing to the heat treatment. The microstructure images in Fig. 3 clearly indicate that heat treatment has lead to grain coarsening. This in turn will affect the hardness of the specimen, thereby influencing the material loss due to wear.
4.2 Microhardness Measurement Microhardness measurement carried out with a load of 10gf with a dwell time of 10 s is shown in Fig. 4.
4.3 Dry Sliding Wear Test Dry sliding wear test was performed for specimens with a load of 5 N. The wear track radius was kept 60 mm with a rotational speed of 500 rpm for 15 min. Sliding distance is represented mathematically as per Eq. 1. SD =
π DNT 1000
(1)
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Fig. 3 Microstructure of SS 304 specimen a non-heat-treated NHT a HT 1, b HT 2, c HT 3, d HT 4 Fig. 4 Hardness measurement
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Table 4 Wear based on loss in weight Specimen
Initial weight (W i )
Final weight (W f )
Loss in weight (W )
Volume loss (mm3 )
Wear (mm3 /m)
HT 1
8.0837
8.0790
0.0047
0.05853
4.14 × 10–4
HT 2
7.7866
7.7632
0.0234
2.9140
20.6 × 10–4
HT 3
7.9982
7.9654
0.0278
3.4620
24.5 × 10–4
NHT
7.4296
7.3938
0.0308
3.8536
27.1 × 10–4
where SD is the sliding distance in metre, D is the wear track diameter in mm, N is the rotation of disc measured in rpm, and T is the time of pin sliding over disc in minutes. Here, D is 60 mm, N is 500 and T is 15. This makes the sliding distance to be 1413 m in total. Before and after each run of 15 min, the pin specimens were weighed on an electronic weigh balance with and accuracy of ±0.1 mg. Table 4 represents the initial and final weight of pin specimens. This weight loss data is used to calculate the volume loss as per Eq. 2. Volume loss is further used for calculating the volume loss per unit sliding distance which is generally referred as average wear in mm3 /m. W ρ
V =
(2)
where V is the volume loss in mm3 , W is the weight loss in grams occurring due to wear and ρ is the density of pin material in g/cm3 . Q=
V SD
(3)
Here, Q is the average wear occurring in mm3 /m, SD is the sliding distance in metres which is 1413 m in this case. A comparative plot of all specimens for wear occurring measured mm3 /m is shown in Fig. 5. The data in Table 4 and Fig. 5 clearly indicate that the highest loss in weight due to wear occurs in non-heat-treated specimen. Heat treatment improves the wear resistance characteristics, specifically in HT 1 condition where weight loss reduces multiple folds as compared to others. Wear coefficient and wear resistance are another important parameters which are used to define the wear resistance characteristics of a material. Equations 4 and 5 represent the mathematical expression for these parameters as per Archard’s equation. K =
Q∗H Fn
(4)
1 K
(5)
Rw =
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Fig. 5 Comparison of wear among specimens
Table 5 Wear coefficient and wear resistance
Specimen
Q (mm3 /m)
Wear coefficient (K)
Wear resistance (Rw )
HT 1
4.14 × 10–4
13.60 × 10–3
73.53
HT 2
20.6 ×
67.22 ×
10–3
14.88
HT 3
24.5 × 10–4
75.46 × 10–3
13.25
NHT
27.1 × 10–4
88.48 × 10–3
11.30
10–4
where K is the wear coefficient, Q is the average wear in mm3 /m, H is the hardness of pin specimen and Rw is the wear resistance. Table 5 represents the wear coefficient and wear resistance value for all the specimens calculated using the above-mentioned equations. Figure 6 shows comparative plot of wear coefficient and wear resistance.
4.4 Wear Calculation on the Basis of Wear Scar Diameter Each pin specimen after undergoing the rotationary sliding motion on disc developed scar on its surface. This scar is mainly due to the material loss at the interface of pin and disc. The scar is measured precisely using vernier calipers, and then based on mathemtical formular given in Eq. 6, the volume loss is calculated. Volume Loss = π ∗ (wear scar diameter in mm)4 /64 ∗ (sphere radius in mm) (6)
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Fig. 6 Comparative plot of wear coefficient and wear resistance
Table 6 Wear scar diameter and volume loss
Specimen
Wear scar diameter (mm)
Volume loss (mm3 )
HT 1
2
0.261
HT 2
3
1.325
HT 3
3.5
2.454
NHT
4
4.186
Table 6 represents the data of wear performance calculated as per the wear scar diameter method.
4.5 Wear Mechanism Scanning electron microscopy was used to estimate the wear mechanism occurring. SEM images shown in Fig. 7 were taken at the face of pin’s sliding surface. The
Fig. 7 Wear mechanism
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material removal is suggested to occur by ploughing mechanism. The cuts developed at the pin’s face suggest ductile abrasive wear, whereas the SEM images taken at the end of sliding shows the spalling and flacking.
5 Conclusion • Austenitic stainless steel of 304 grade has been given suitable heat treatments. • Pin on disc technique has been used to estimate the dry sliding wear behaviour in three heat-treated and one non-heat-treated as-received condition of specimens. • Wear characteristic parameters namely; loss in weight, volume loss, average wear, wear coefficient and wear resistance have been calculated using Archard’s equation. • Wear scar diameter has also been used to calculate the volume loss occurring. • Heat treatments improve the wear resistance of SS 304 steel. • Heat treatment 1 specimen exhibits the best wear resistance, with its resistance value turning out to be almost 6.5 times better than that of material in as-received condition. • Heat treatment 2 and Heat treatment 3 show an improvement of marginal 1.13 and 1.1 times in wear resistance as compared to as-received non-heat-treated specimen. • Applications involving rigorous wear due to sliding, SS304 can be suggested to be used in heat treatment 1 condition for better performance.
References 1. Kumar S, Mukhopadhyay A (2016) Effect of microstructure on the wear behavior of heat treated SS-304 stainless steel. Tribol Ind 38(4):445–453 2. Abudaia FB, Bull SJ, Oila A (2012) Surface wear resistance of austenitic stainless steels modified by colossal carbon supersaturation and TiN coating. J Mater Sci Eng B, 2(2):103–111 3. Hoier P, Malakizadi A, Friebe S, Klement U, Krajnik P (2019) Microstructural variations in 316L austenitic stainless steel and their influence on tool wear in machining. Wear 428:315–327 4. Mello CB, Ueda M, Lapienski CM, Reuther H (2009) Tribological changes on SS304 stainless steel induced by nitrogen plasma immersion ion implantation with and without auxiliary heating. Appl Surf Sci 256:1461–1465 5. Yang ZY, Naylor MGS, Rigney DA (1985) Sliding wear of 304 and 310 stainless steels. Wear 105(1):73–86 6. Bregliozzi G, Ahmed SIU, Schino AD, Kenny JM, Haefke H (2004) Friction and wear behavior of austenitic stainless steel: influence of atmospheric humidity, load range, and grain size. Tribol Lett 17(4):697–704 7. Straffelini G, Trabucco D, Molinari A (2002) Sliding wear of austenitic and austenitic-ferritic stainless steels. Metal Mater Trans A 33A:613–624 8. Bressan JD, Daros DP, Sokolowski A, Mesquita RA, Barbosa CA (2008) Influence of hardness on the wear resistance of 17–4 PH stainless steel evaluated by the pin on disc testing. J Mater Process Technol 205:353–359
Design and Development of Intelligent Moving Machine Using LabVIEW Amit Yadav, Ajeet Gaur, D. K. Chaturvedi, A. K. Saxena, and Dharvendra P. Yadav
Abstract The intelligent moving machine is designed for fast response, cost effectiveness and it avoids the obstacle in its path so it is called safe intelligent moving machine (SIMM). An introduction of intelligent behavior of moving machine that can adapt the change as like environment. Also describes the application of intelligent machines. This work presents design and development of an automatic intelligent system for industrial environment using LabVIEW and its real-time hardware implementation. The machine moves forward, backward, left and right through the signals supplied by the control station. LabVIEW programming has been proved to be more efficient than microcontroller to track this intelligent moving machine with the help of DAQ. The additional feature is that our vehicle is also identifying the obstacle in its path when it moving and generates necessary control commands to avoid it. Keywords Motor driver circuit · LabVIEW code · Hardware implementation
1 Introduction 1.1 Intelligent Moving Machine The intelligent moving machine which may avoid the obstacle move intelligently and reach to the destination safely, accurately in minimum time and minimum cost. Earlier many moving machines were developed using different techniques for a real-time path finder system. Man intelligent moving machine has become an important topic in the robots community. An advanced intelligent moving machine must integrate capabilities to detect human’s presence in their vicinity and interpret their motion for an active interaction; the machine must also be able to track the static and dynamic A. Yadav (B) · A. Gaur · D. K. Chaturvedi · A. K. Saxena · D. P. Yadav Dayalbagh Educational Institute, Dayalbagh, Agra, India ADRDE-DRDO, Agra, India D. P. Yadav ISTRAC, ISRO, Bengaluru, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_165
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obstacle in its path. The humans and moving machines coexist, closely interact and perform a certain amount of work. This research work is divided into two parts. The first part deals with the development of safe intelligent moving machine (SIMM), which maintains a certain relative positional relationship between the human and moving machine. The moving machine can carry loads that are required by people working in hospitals, airports, and industries. These machines are considered as an intelligent moving machine because they are able to interact with its environment and learn from the surroundings automatically. Interacting with nature and indoor situation involves learning and adaptation from changing environment. The intelligent machine can work as an assistant for humans in various situations. The second part of work related to the intelligent moving machine is obstacle avoidance in real-time situation. The real-time obstacle avoidance system of intelligent moving machines will be developed with help of soft computing techniques.
1.2 Motivation Human intelligence, speed of thinking, capability of adaptation and the way obstacle avoidance in real-time dynamic environment inspired the researcher to develop an intelligent moving vehicle which is safe, intelligent and optimally find its path in the static and dynamic, real-time environment.
1.3 Steps for Design and Development of Intelligent Machine First we have to design a block diagram panel for giving the commands to our automatic vehicle. The commands will be forward, backward, left and right. The command will be transmit through the DAQ, card to the receiver station. The functional block is shown below in Fig. 1. In this section, our receiver antenna gets the information from the DAQ card and starts doing task. The vehicle moves forward, backward, left and right. And also avoids obstacle in his path. The UV sensor is used to detect the objects and gives commands to control motor and speed of front wheels to take turns of automatic vehicle. The speed of automatic vehicle is 10 km/h [1]. It takes turn very fast in motion. The intelligent moving machine is safe running motor which takes action according to the situations. The control programming plays very important role in this type situation which delivers the correct action to the motor drives (Figs. 2 and 3).
CODE
DAQ
Fig. 1 The functional block
RF Signal
Trans Receive
Moving Machine
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DAQ Signal
Wireless System
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Fig. 2 Block diagram of intelligent machine
Start
Check start signal
Initialing I/O port of DAQ
Set direction of vehicle forward, back, left and right
Is Obstacle Detect N Vehicle goes on forward
Stop
Fig. 3 Flow chart
Y
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1.4 Proposed Work On the basis of historical evidences, microcontroller, embedded systems and appropriate communication systems work to know engineering and understanding of obstacle detection and their avoidance. The state of the art is both microcontroller and moving machine took advance steps in developing intelligent moving vehicle. The entire work focuses on the design and development of safe intelligent moving vehicle which has the capability to avoid obstacle in real time. These obstacles may be known or unknown, static or dynamic, however, development presented in this paper, only static and known obstacles are considered.
2 Objective of Paper The primary objective of the work to develop a real-time obstacle avoidance machine to complete the task that requires distributed functionalities. In this context, the core research objectives of the thesis are given by: 1. 2. 3.
To design and develop the machine in lab. To design and develop an obstacle avoidance algorithm for moving machine. To implement in laboratory a framework for vehicle and the obstacle avoidance system.
2.1 Methodology The graphical approach also allows non-programmers to build programs by dragging and dropping virtual representations of laboratory equipment with which they are already familiar. The LabVIEW programming environment, with the included examples and the documentation, makes it simple to create small applications. This is a benefit on one side, but there is also a certain danger of underestimating the expertise needed for good quality of programming. For complex algorithms or largescale code, it is important that the programmer possess an extensive knowledge of the special LabVIEW syntax and the topology of its memory management. The most advanced LabVIEW development systems offer the possibility of building stand-alone applications. Furthermore, it is possible to create distributed applications, which communicate by a server scheme and are therefore easier to implement due to the inherently parallel nature of code.
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Fig. 4 Block diagram panel
2.2 Block Diagram Panel for IMM The LabVIEW is a graphical-based approach, so we start from the block diagram panel. In this section, we design a code programming for our moving machine which generates the signal in true/false conditions. If the condition is true, then our machine will move forward and if the condition is false our machine will stop. In this panel, the control output depends on five outcomes which are known as no action, forward, backward, left and right. These signals generate output which we can see at the front panel and from here we can send the signals to the DAQ card. This work is totally radio frequency based in this experimental setup no use of any type of chords. In programming code, four keys have been used which gives the signal for moving to the machine. This machine can be changing its position according to the path. So, it is called safe intelligent moving machine (SIMM) (Figs. 4 and 5).
3 Results Intelligent moving machine has been developed which is moving forward, backward, left and right. This is a smooth running machine controlled through softwarebased controlling technique. This is a highly useful intelligent machine for industrial purpose which can work more efficiently in real time industrial circumstances. This is completely controlled through a control station. This moving machine also effectively avoids the obstacles coming in its path, sensing them through the ultrasonic sensor. It changes its movement very fast. When we provide a signal, it changes its directions
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Fig. 5 Front panel result of LabVIEW code
in 0.3 s. The trajectory angle with obstacle is kept as 45° when machine is moving forward and backward [2]. At the same time, the direction preference for turning has been also taken as left turn for both, forward and backward movement.
4 Conclusion The intelligent moving machine is developed in this project is smooth running and response is also fast enough for utlization in lab. The developed automatic intelligent machine also senses and avoids the static obstacles in its path. So, the developed moving machine is very useful in industrial and airport as a load vehicle with necessary adoptation for the respective field of operation.
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5 Experimental Setup of Intelligent Moving Machine
References 1. Alonso-Mora J, Breitenmoser B, Rufli R, Beardsley P, Siegwart R (2013) Optimal reciprocal collision avoidance for multiple non-holonomic robots. J Distrib Autonom Rob Syst 203–216 2. Borenstein J, Koren Y (1988) Obstacle avoidance with ultrasonic sensors. IEEE J Rob Autom RA-4(2):213–218
Effect of Poling Orientation in Performance of Piezoelectric Materials Jitendra Adhikari, Rajeev Kumar, Vikas Narain, and Satish Chandra Jain
Abstract The present study proposes enhancement of harvested power and voltage by tuning the poling orientation in piezoelectric materials. The dependency of piezoelectric strain coefficients on performance is presented mathematically and to demonstrate the effect, a cantilever-based energy harvester having platinum substrate is considered with seven different materials. It is observed that PZT-2 shows an improvement of 598% in harvested power and 165% in voltage by poling tuning to 45°. Similar poling tuning helps PZT-7A to improve 325 and 106% in power and voltage generation, respectively. Huge improvement of 1425% in power and 290% for voltage is observed for PMN-0.35PT. PbTi03 shows a minimal improvement at poling angle of 30°. The performance of materials like Ba2 NaNb5 O15 and PVDF gets deteriorated with an increase in poling orientation. The peak values of power and voltage are observed at different poling angles for different piezoelectric materials. The least magnitudes of power and voltage generation occur at poling angle of 90° for any material system. Keywords Energy harvesting · Poling tuning · Piezoelectric material
1 Introduction Piezoelectric materials are high potential smart materials which gained importance because of its development in future self-powered microsystems. Technological advancement increases researchers’ interest in improving the efficiency and performance of piezoelectric energy harvesting devices. Studies in mechanical aspect of the piezoelectric energy harvester focus on materials, geometrical entities (cantilever, diaphragm, and spherical, etc.) and working mode of energy harvester. The working mode deals with piezoelectric strain coefficients: d 31 (transverse), d 33 (longitudinal) and d 15 (shear). Ng et al. [1] used simple geometrical configuration suitable for J. Adhikari (B) · R. Kumar · S. C. Jain School of Engineering, Indian Institute of Technology Mandi, Mandi 175005, India V. Narain Shri Bhawani Niketan Institute of Technology and Management, Jaipur 302039, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_166
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harvesting the energy using transverse mode (d 31 mode) when mechanical vibrational frequencies are low. Shen et al. [2] harvested energy based on longitudinal mode (d 33 mode). But this configuration is less effective due to polarization issues, and large mechanical force is required to produce the strain as compared to the transverse mode. Wang et al. [3] investigated energy harvesting for shear mode (d 15 mode) configuration. This arrangement appears to be most effective for output power, but it involves a complex fabrication process. Even though piezoelectric constant for d 31 is low as compared to the d 15 and d 33 modes still d 31 is most accepted due to its real-time applications when compared against its counterparts. Performance in the d 31 mode for a piezoelectric material can be escalated using dipole engineering and through alteration of crystallographic orientation [4–6]. Davis et al. analyzed the eff for various piezoelectrics using dipole engineering response of poling change on d31 [7]. The study primarily focusses on enhancement of performance of energy harvesters using various piezoelectric materials in cantilever configuration by changing the poling direction operating in d 31 mode.
2 Materials and Methodology This research deals with the effect of poling in the performance of energy harvesters of the piezoelectric materials. Different piezoelectric materials are considered for analysis including PMN-0.35PT(Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 ) [8], Barium sodium niobate (Ba2 NaNb5 O15 ) [9], PZT-5A and PZT-7A (Pb[ZrxTi1-x]O3 ), [10] Lead Titanate (PbTiO3 ) [11], PCR-1 (Piezoelectric Ceramics of Rostov) [8], polyvinylidene difluoride (PVDF) [9]. Table 1 presents the material properties of piezoelectric materials. Here, we have mathematically illustrated the dependency of alteration of poling direction over performance of piezoelectric materials. Suppose stress (T ) be a second-order tensor, d is the coupling piezoelectric coefficient and polarization (P) is a vector related by these two variables as Table 1 Physical properties of materials E S66
d31
d15
d33
T ε11 ε0
T ε33 ε0
29.9
−60
440
152
990
450
−7
52
37
235
51
−3.96 −6.05 14.7 33.4 34.32 −133
936
270
4610 3270
Material
E S11
E S12
E S13
E S33
PZT2
11.6
−3.3
−4.9
14.8 45
Ba2 NaNb5 O15
5.3
PMN-0.35PT
13.2
−1.98 −1.2 −5.8
E S44
8.33 15.4 13.2
PCR-1
12.5
−4.4
420
220
1130 650
PZT-7A
10.7
−3.22 −4.62 13.9 39.5 27.84 −60.2 364
151
843
427
PVDF
4.43 −9.91 −3.1
PbTiO3
7.7
−1.7
−1.2
15.9 38.8 33.8
−95
1.14 1.82 1.45
1.36
1.93 2.97 6.8
7.3
8.2
−6.8
68
190
19
31.4
56
240
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Pi = d i jk T jk
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(1)
Polarization vector (P) is transformed from old co-ordinate system to new coordinate system which can be written as: P = a P
(2)
where P and P are the polarization vectors in old and new co-ordinate system respectively and a is the direction cosines matrix for rotation about any axis that is given as: ⎤ l1 m 1 n 1 a = ⎣ l2 m 2 n 2 ⎦ l3 m 23 n 3 ⎡
(3)
where l1 = cos x , x , m1 = cos y , x ,n 1 = cos z , x , l2 = cos x , y , m 2 = cos y , y , n 2 = cos z , y , l3 = cos x , z , m 3 = cos y , z , n 3 = cos z , z . Substitution of Eqs. (1) into (2) leads to: Pi = α d i jk T jk
(4)
Stress tensor T jk is transformed from old co-ordinate system to new co-ordinate system as below: T jk = αT jk
(5)
Matrix α is derived from the direction cosines of x, y, z co-ordinate axes given as: ⎡
l12 ⎢ l2 ⎢ 2 ⎢ 2 ⎢ l α=⎢ 3 ⎢ l2 l3 ⎢ ⎣ l3 l1 l1 l2
m 21 m 22 m 23 m2m3 m3m1 m1m2
⎤ n 21 2m 1 n 1 2n 1l1 2l1 m 1 n 22 2m 2 n 2 2n 2 l2 2l2 m 2 ⎥ ⎥ ⎥ 2 n3 2m 3 n 3 2n 3l3 2l3 m 3 ⎥ ⎥ n 2 n 3 m 2 n 3 + n 2 m 3 l2 n 3 + n 2 l3 m 2 l3 + l2 m 3 ⎥ ⎥ n 3 n 1 m 1 n 3 + n 1 m 3 l2 n 3 + n 1 l3 m 1 l3 + l1 m 3 ⎦ n 1 n 2 m 1 n 2 + n 1 m 2 l1 n 2 + n 1 l2 m 1 l2 + l1 m 2
(6)
Substitution of Eqs. (5) in (4) and then simplifying it we have: Pi = di jk T jk
(7)
Here, poling vector is considered to be rotated with angle θ along 2nd direction eff depends on d31 , d33 , d15 and as shown in Fig. 1. Upon rotation, it is noted that d31 angle θ as mentioned below: eff = cos θ d31 cos2 θ + d33 sin θ − d15 sin2 θ d31
(8)
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Fig. 1 Schematic of functionally graded piezoelectric harvester
eff It is noticed from Eq. (8) that magnitude of d31 increases with surge in magnitudes of d31 and d33 . It also gets negatively affected with an increase in d15 . The voltage in d31 mode is related to piezoelectric coefficients and electrical permittivity as
V = C
eff d31 eff ε31
(9)
where V and C are the voltage and numerical constant that depends on stress and thickness of the piezoelectric layer, respectively: On further expansion [12] V =C
cos θ d31 cos2 θ + d33 sin θ − d15 sin2 θ ε11 sin2 θ + ε33 cos2 θ
(10)
The research investigates the reaction of poling orientation on energy harvester performance in transverse mode. A cantilever-shaped harvester with substrate layer made of platinum having dimensions 100 mm × 5 mm × 1 mm and piezoelectric patch of 20 mm × 5 mm × 1 mm dimensions is adhered with substrate material at a distance 10 mm from fixed end, as shown in Fig. 1. The harvester base tied to support vibrates with 1 g amplitude in transverse direction with frequency of the harmonic load varying from 20 to 100 Hz and resistance of magnitude 10 M was maintained across the piezoelectric patch. Finite element formulation was done using MATLAB to numerically simulate the performance of piezoelectric energy harvester. All S ij and d ij in Table 1 are reported as 10−12 m2 /N and pm/V, respectively.
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3 Results and Discussion eff The effect of poling orientation on voltage and power due to change in d31 is explored for different piezoelectric materials. It is observed that different materials have discrete response for change in poling angle. It is also noted from Eq. 8 that eff , a significant difference has to be there in between the to have higher values of d31 magnitudes of d31 and d15 . Sensing voltage has a proportional relation with piezoelectric strain coefficients and shares an inverse relation with dielectric constants as mentioned in Eq. 20 [12]. PZT-2 shows an improvement of 598% in harvested power and 165% in voltage when poling orientation changes from 0° to 45°. Huge improvement in power of around 1425% was observed for PMN-0.35PT along with a voltage gain of 290%. As the poling angle of PCR-1 increases, the harvested power also surges reaching the maximum value at 30° and further improvement leads to fall with least value at 90°. Here, power improvement of 33% and voltage increment of 17% were observed. PZT-7A indicates rise in power and voltage with increase in poling orientation touching the peak at 45° along with showing voltage and power enhancement of 106% and 325%, respectively. Material system like PbTiO3 shows minimal increase of 3.5 and 2.12% in power and voltage with peak orientation at 30°. On the contrary, a material system such as Ba2 NaNb5 O15 and PVDF experiences decrease in performance with increase in poling orientation. It is noticed that for any material system (Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15) the least value of harvested power and voltage occurs at 90°.
Fig. 2 PZT2: Power versus frequency
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Fig. 3 PZT2: Voltage versus frequency
Fig. 4 Barium sodium niobate: power versus frequency
4 Conclusion The effect of poling orientation on performance of harvested power and sensing voltage for different materials was identified in the present study. It is observed that
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Fig. 5 Barium sodium niobate: voltage versus frequency
Fig. 6 PMN-0.35 PT: power versus frequency
for PMN-0.35PT, PZT-2 and PZT-7A, harvested power get enhanced by 1425%, 598% and 325% due to large difference in magnitudes of piezoelectric coefficients. On the contrary, Ba2 NaNb5 O15 and PVDF show a negative trend with increase in poling angle. An interesting observation was that the poling angle for peak sensing voltage and generated power for different materials need not to be same.
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Fig. 7 PMN-0.35 PT: voltage versus frequency
Fig. 8 PCR-1: power versus frequency
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Fig. 9 PCR-1: voltage versus frequency
Fig. 10 PZT-7A: power versus frequency
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Fig. 11 PZT-7A: voltage versus frequency
Fig. 12 PVDF: power versus frequency
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Fig. 13 PVDF: voltage versus frequency
Fig. 14 PbTiO3 : power versus frequency
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Fig. 15 PbTiO3 : voltage versus frequency
References 1. Ng TH, Liao WH (2005) Sensitivity analysis and energy harvesting for a self-powered piezoelectric sensor. J Intell Mater Syst Struct 16(10):785–797 2. Shen Z, Liu S, Miao J, Woh LS, Wang Z (2013) Proof mass effects on spiral electrode D33 mode piezoelectric diaphragm-based energy harvester. In: 26th international conference on micro electro mechanical systems (MEMS), pp 821–824 3. Wang DA, Liu NZ (2011) A shear mode piezoelectric energy harvester based on a pressurized water flow. Sens Actuators A Phys 167(2):449–458 4. Yin J, Cao W (2000) Domain configurations in domain engineered 0.955 Pb (Zn 1/3 Nb 2/3) O 3–0.045 PbTiO3 single crystals. J Appl Phys 87(10):7438–7441 5. Zhang R, Cao W (2004) Transformed material coefficients for single-domain 0.67 Pb (Mg 1/ 3 Nb 2/ 3) O 3–0.33 PbTiO3 single crystals under differently defined coordinate systems. Appl Phys Lett 85(26):6380–2 6. Nakanishi R, Kanda K, Fujita T, Kanno I, Maenaka K (2019) Multilayer piezoelectric MEMS energy harvester based on longitudinal effect. J Phys Conf 1052(1) 7. Davis M, Damjanovic D, Hayem D, Setter N (2005) Domain engineering of the transverse piezoelectric coefficient in perovskite ferroelectrics. J Appl Phys 98(1):014102 8. Dantsiger AY, Razumovskaya ON, Reznitchenko LA, Grineva LD, Devlikanova RU, Dudkina SI, Gavrilyatchenko SV, Dergunova NV, Klevtsov AN (1994) Highly effective piezoceramic materials (Handbook). Kniga, Rostov-on-Don. (in Russian) 9. Comsol Multiphysics, Materials Database v4.3 in Materials Database (2012) 10. Dunn ML, Taya M (1993) Electromechanical properties of porous piezoelectric ceramics. J Am Ceram Soc 76:1697–1706 11. Xu Y (1991) Ferroelectric materials and their applications. North-Holland, Amsterdam 12. Kiran R, Kumar A, Kumar R, Vaish R (2018) Poling direction driven large enhancement in piezoelectric performance. Scripta Materialia 151:76–81
Parametric Analysis of Vertical Contact Mode Triboelectric Energy Harvester Satish Kumar, Rajeev Kumar, Vikas Narain, and Satish Chandra Jain
Abstract Triboelectric energy harvesters (TEHs) are a fast growing, recently presented mechanical energy harvesting technology. Because of their versatility, they can be manufactured in various configurations, and hence have a broad number of applications. The TEH is capable of harvesting mechanical vibrational energy. In this paper, simplified model of TEH system using simple coupling of commercial polytetrafluoroethylene (PTFE) with a thin copper (Cu) is presented. Based on the simplified model, real-time output characteristics of TEH at fixed value of resistance are derived using Simulink. The parametric study of the design parameters has been carried out, and the effects of these parameters on harvested power are illustrated for sinusoidal motion cycles. The thoroughly validated theoretical model is a very powerful tool for directing the design of the system structure and material selection, and performance optimization with regard to TEH’s application conditions of TEH. The output characteristics of triboelectric energy harvester are validated with the existing literature. Keywords Triboelectric mechanism · Energy harvesting · Contact electrification · Electrostatic induction
1 Introduction Nowadays, the tremendous increase of popularity of handhold small electronics like cell phones, robots, navigation, motion, and biological sensors requires very small S. Kumar (B) · R. Kumar · S. C. Jain School of Engineering, Indian Institute of Technology Mandi, Mandi 175005, India R. Kumar e-mail: [email protected] S. C. Jain e-mail: [email protected] V. Narain Shri Bhawani Niketan Institute of Technology and Management, Jaipur 302039, India © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_167
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power. These small electronics need a battery to operate, but sometimes it is difficult to replace/charge the battery. Energy storage systems have emerged as one of the most promising technologies for meeting smaller energy needs, such as running electronic devices with low power and wireless sensors. While several other methods based on various mechanisms for harvesting mechanical energy have been demonstrated, such as piezoelectric harvesters, electro-magnetic harvesters, and electrostatic harvesters, triboelectric energy harvesters (TEHs) are more influential in power output and energy conversion efficiency. That makes TEH the most common source of energy for portable electronics and also possibly to use for large-scale power production in the near future. The triboelectric mechanism is developed on the basis of contact electrification and electrostatic induction. Fan and Wang [1] published the first paper on triboelectric energy harvesting. Through mounting two polymer sheets made of materials with significantly different triboelectric features, they produced the TEH with metal films coated on upper and lower layers of the assembled structure. When subjected to mechanical deformation, due to the nano-scale surface roughness, a friction between the two films produces equal but opposite charging signs on two sides. Therefore, at the interface zone, a triboelectric potential layer is created, which serves as a charge “pump” to drive the electrons flow in the external load when there is a difference in the device capacitance. Such a versatile polymer TEH provides up to 3.3 V of output voltage at a output power density of 313 W/m2 [2]. Surface topography plays a significant role in deciding the effective area of contact between the triboelectric layers. Fan et al. [3] showed that patterned films exhibit substantially better performance than unpattern films. Triboelectric layer patterning is known to increase the efficiency of the triboelectric energy harvester. Zhu et al. [4] developed a triboelectric energy harvester that not only has quite a-simplified structure but also significantly higher output power allowed by nanoparticular surface treatments. Niu et al. [5] developed a mathematical model for TEH contact-mode and conducted experiments to validate those theoretical findings. Kia et al. [6] performed on systematic simulations of the macro/nanostructure adhesive contact behavior at the TEH interface; an interaction potential was used to describe the adhesive interactions while using Green’s half-space function, surface deformations are coupled. Niu et al. [7] demonstrated the first equivalent circuit model and the first integrated simulator for TEH system and validated through comparison with the analytical results. Rasel et al. [8] exhibited a human-skin triboelectric generator based on human skin, a microstructured surface PDMS sheet, a ground-connected Cu electrode, and polyurethane (PU) as substrate material. In this paper, a triboelectric tile is proposed to harvest the energy from walking. A simplified model of tile has been analyzed, and mathematical model of the tile has been developed to predict the harvested power.
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2 Mathematical Modeling Physical model of TEH is shown in Fig. 1. When a force is applied on the TEH, then it vibrates. During the vibration, PDMS layer makes intermittent contact with a bottom copper layer that results in an electric voltage difference between both layers. A simplified model of the TEH (see Fig. 2). As a result of contact electrification, the inner surfaces of top and bottom layers should have opposite static charges (tribocharges) of equal quantity and opposite charging density (σ ). As the two layers begin to separate from each other, a potential difference (V ) between the two electrodes would be triggered with increasing separation. It is appropriate for the insulator (layer 2) to presume that the tribocharges are distributed uniformly with marginal decay at the surface. So there are two sections to the total charge in layer1: One is the triboelectric (Sσ ) charge, another is the
Fig. 1 Physical model of TEH
Fig. 2 Simplified model of TEH
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transferred charge between the two electrodes (−Q). This charge flows through an external resistance (R). Thus, the total charge in layer 1 is (Sσ − Q). From the Gauss theorem, the electric field strength in the dielectric layer and the air gap are given by: Inside dielectric: E air = −
Q Sε0 εr 1
(1)
Inside the air gap: E air =
− QS + σ (t) ε0
(2)
The voltage between the two electrodes can be calculated as: V = E 1 d + E air y(t)
(3)
Substituting Eqs. (1) and (2) into (3) V =−
Q σ y(t) (d0 + y(t)) + Sε0 ε0
(4)
Combining Eq. (4) with Ohm’s law will calculate the performance of output properties: V = IR = R
dQ dt
(5)
Merging Eqs. (4) and (5) R
dQ Q σ y(t) =− (d0 + y(t)) + dt Sε0 ε0
(6)
Then Eq. (6) can be solved in Simulink where, Q = 0 at t = 0 A sinusoidal motion of the top electrode, defined in Eq. (7), was selected because this type of motion was fitted with the experimental testing rig (Keyboard Life Tester by Shenzhen ZXD Testing Equipment Co. Ltd.) [9]. Also a motion of constant velocity was found. Once t = 0, the top electrode’s two contact surfaces and the dielectric layer have been undergoing contact for a sufficiently long time to achieve their saturation by the contact electrification. The top electrode then began to isolate itself from the dielectric.
Parametric Analysis of Vertical Contact Mode Triboelectric … Table 1 Parameters used in theoretical estimation of constant velocity [7]
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Total external resistance in the circuit (R)
R = 10 M
Area size of the dielectrics (S)
S = 36 × 10–4 m2
Tribo-charge surface density (σ )
σ = 15 µCm−2
Thickness of the dielectric layer (d)
d = 250 µm
Relative permittivity (εr )
εr = 3.4
Effective thickness of dielectric layer (d 0 )
d 0 = 73.53 µm
Vacuum permittivity (ε0 )
ε0 = 8.854 × 10–12 Fm−1
Amplitude of the motion (A0 )
A0 = 5 mm
Frequency (f )
f = 3.2 Hz
Angular velocity (ω)
2π f (rad s−1 )
Initial phase angle θ
3π /2 rad
y(t) = A0 sin(ωt + θ ) + A0
(7)
where A0 and ω are the amplitude and the angular velocity of the motion, respectively. θ is the initial phase angle. The parameters used in the simulation matched those used in the experiments which are described in Table 1.
3 Result and Discussion Based on the analytical modeling, a Simulink model is developed in MATLAB (see Figs. 3 and 4). The top plate of the TEH is vibrated with a sinusoidal motion and generated current, voltage and power are calculated from Eq. (6) using Simulink at a fixed value of resistance. Geometric and material properties are given in Table 1. Using sinusoidal motion, the calculated charge has been compared with existing results [7] as shown in (see Fig. 5). It depicts that calculated charge is matched well with experimental published results. Figure 6 represented TEH-determined output voltage, current, and power during the compressive load cycles. The TEH developed a positive and negative peak voltage of 42 and −30 V, respectively, from the touch and release operation. The change in the separation distance between the top electrode and the dielectric layer added the difference in voltage between the two electrodes, while it was diminished by the flow of electric charges It would be a continuous and instantaneous operation. As the separation displacement of the top electrode decreases to zero, the number of electrical charges did not return to its original value, which makes the next cycle operation different from the previous one. Significant variations in amplitude at the peaks showed that the first peak voltage was slightly higher when in contact with the next one (see Fig. 6).
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Fig. 3 Simulink model
4 Parametric Study It is of high significance to increase TEH output power and to reduce device size. Here, implement a numerical analysis of the different parameters under which the power produced is increased and miniaturizes the size. Especially, the first scheme uses a predefined geometry with the exact value of the resistance circuit. The second scheme is to define TEH design parameters when the power output is either unchanged or increased.
4.1 Triboelectric Material In the construction of triboelectric energy harvester (TEH), the choice of triboelectric materials is the main factors behind the high performance. For this purpose, one uses the triboelectric series, where the triboelectric materials are ordered according to their polarity (ability of a material to gain/lose electrons). Materials such as glass
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Fig. 4 Subsystem of the Simulink mod
Fig. 5 Real-time transferred charge–time relationship at a fixed load resistance
or Nylon are positive tribomaterials and tend to lose electrons when coming into contact with negative-charge tendency materials [e.g., PTFE] that have a tendency to gain electrons. For this study, choose PTFE and nylon as the two triboelectrical materials, due to their opposite position in the triboelectric series. PTFE is a widely used polymer, due to its flexibility, manufacturing ease, transparency, biocompatibility, and super-hydrophobicity. It is also widely used as triboelectric material in the construction of TEH for a broad range of applications. On the other hand, nylon has
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Fig. 6 Calculated output performance of contact-mode TEH. a Output charge–time relationship at fixed load resistance; b output current–time relationship at fixed load resistance; c output voltage– time relationship at fixed load resistance; d the relationship of output power on fixed load resistance
good mechanical properties (strength and stiffness), high impact resistance, which is easy to fabricate and maintains its properties over a large temperature range.
4.2 Effect of Dielectric Thickness on the Output Power Aiming the improvement of the triboelectric effect, we studied the influence of the PDMS layer thickness on the generated electrical outputs. PDMS was the triboelectric material chosen to vary the thickness (between 10 and 250 µm), while Cu was used in film form. Here, the relation between the thickness of the friction layer and the process of transporting triboelectric charges in a PDMS film is investigated in this study. The power output is inversely proportional to the dielectric layer thickness
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Fig. 7 Relationship of output power with dielectric layer thickness (d)
(d), as shown in Fig. 7. They decrease rapidly as the dielectric layer thickness (d) increases from 0.1 to 0.2 mm which is usually used in experiments for TEH.
4.3 Investigation of Optimum Load Resistance Here the density of the surface charge, geometries including area, dielectric thickness and frequency are the same as in the previous section. The goal is to calculate the load resistance for different dielectric layer thicknesses in order to obtain maximum value of average power output. The simulation results are presented in (see Fig. 8), from which it is clear that the output power meets its limit at a certain load resistance. Fig. 8 Relationship of output power with load resistance (d)
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Fig. 9 Relationship of output power with motion parameter (frequency)
4.4 Effect of Motion Parameter (Frequency) on Output Power The nature of the profile of input motion influences the magnitude and rate of induced TEH outputs. Sinusoidal and constant velocity periodic motion profiles are commonly used to describe TEH, of which key parameters are the magnitude of the motion (amplitude) and the motion rate (frequency). Increasing the frequency of movement of the TEH layer increases the power output (Fig. 9). The overall peak power increases directly proportional to frequency.
4.5 Effect of Dielectric Surface Area on Output Power The output power is dependent on the size of a TEH surface area (S). This study used a pair of square-shaped TEH layers consisting of PDMS and Cu surface contacts, and power output was evaluated at various surface areas. As the surface area increases, the output power is increased (see Fig. 10). As a result, the power output was higher for the larger surface area unit, which was observed at a relatively lower load resistance, both favorable for practical TEH applications in terms of outputs [10].
4.6 Effect of Surface Charge Density on Output Power Triboelectric charge density (σ ) depends on the relative position of triboelectric pairs in the triboelectric series, structuring of triboelectric surfaces, contact area influenced by the applied force, and the surrounding environmental factors. The output power is proportional to the surface charge density (σ ) (Fig. 11) under sinusoidal movement,
Parametric Analysis of Vertical Contact Mode Triboelectric …
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Fig. 10 Relationship of output power with surface area
Fig. 11 Relationship of output power on surface density
where device parameters are the same as described in Sect. 3. As shown in Fig. 11, output power increases exponentially as the surface charge density (σ ) increase.
5 Conclusion In summary, a simplified model for TEH for sinusoidal motion is presented in this work and the influence of various design parameters on the triboelectric energy harvester (TEH) has been studied. TEH has been modeled under the vertical contact mode of the triboelectric harvester. Simulink model is developed for explicit expressions, and charge, current, voltage, and power have been calculated for an externally
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charged TEH with a fixed resistance of R = 10 M. Based on a simplified model, the electrical output achieved a peak voltage of 40 V, current of 4.1 µA and peak power of 170 µW. The simplistic model would be a very useful tool for directing the design of the triboelectric tile device structure and selection of materials. The output power is proportional to the density of surface charge, the area size, and the frequency. Output power gradually increases by increasing the density of the surface charge but has less effect on area size and frequency. Future research should concentrate on modeling and simulating of an ideal triboelectric tile by selecting ideal pairs of materials from the triboelectric series and selecting the correct design parameters. It will provide guidance for TEH’s rational design and could significantly promote flexible electronics growth.
References 1. Fan FR, Tian ZQ, Wang ZL (2012) Flexible triboelectric generator. Nano Energy 1(2):328–334 2. Jiang T, Zhang LM, Chen XY et al (2016) Structural optimization of triboelectric nanogenerator for harvesting water wave energy. ACS Nano 9:12562–12572 3. Fan FR, Lin L, Zhu G, Wu W, Zhang R, Wang ZL (2012) Transparent triboelectric nanogenerators and self-powered pressure sensors based on micropatterned plastic films. Nano Lett 12(6):3109–3114 4. Niu S, Wang S, Lin L, Liu Y, Zhou YS, Hu Y, Wang ZL (2013) Theoretical study of contact-mode triboelectric nanogenerators as an effective power source. Energy Environ Sci 6(12):3576–3583 5. Yang Y, Zhou YS, Zhang HL et al (2013) A single-electrode based triboelectric nanogenerator as self-powered tracking system. Adv Mater 25:6594–6601 6. Kia DS, Towfighian S, Jin C (2016) Predicting the output of a triboelectric energy harvester undergoing mechanical pressure. In: ASME 2016 conference on smart materials, adaptive structures and intelligent systems. American Society of Mechanical Engineers, pp V002T07A007–V002T07A007 7. Rasel MS, Halim MA, Park JY (2015) A PDMS based triboelectric energy harvester as selfpowered, active tactile sensor system for human skin. In: Sensors. IEEE pp 1–4 (2015) 8. Niu S, Zhou YS, Wang S, Liu Y, Lin L, Bando Y, Wang ZL (2014) A simulation method for optimizing the performance of an integrated triboelectric nanogenerator energy harvesting system. Nano Energy 8:150–156 9. Yang B, Zeng W, Peng Z, Liu S, Chen K, Tao X (2016) A fully verified theoretical analysis of contact-mode triboelectric nanogenerators as a wearable power source. Adv Energy Mater 1600505 10. Wang J, Li SM, Yi F et al (2017) Sustainably powering wearable electronics solely by biomechanical energy. Nat Commun 7:12744
Mathematical Model of Sliding Mode Triboelectric Energy Harvester Tarun Pratap Singh, Satish Kumar, and Rajeev Kumar
Abstract A triboelectric energy harvester converts mechanical energy to electrical energy, which is then collected and used to charge a rechargeable battery. This battery may be used to power small electronics devices for a myriad of applications such as temperature and humidity sensors, accelerometer or GPS tracking devices. In this paper, mathematical model of sliding mode triboelectric energy harvester has been developed to predict the harvested energy under mechanical motion, constant speed and accelerated motion. Numerical results reveal that harvested energy depends upon the type of motion, dielectric constant and surface charge density of triboelectric material. Further, experimental study has been conducted to validate the numerical result, and there is significant error between numerical and experimental result. Experimental setup will be further improved. Keywords Energy harvesting · Triboelectric · Electrostatic
1 Introduction As the energy requirement is growing day by day so it is important to search for alternative energy sources. Harvesting energy from friction and vibration is an effective approach because it is clean and widely available. Triboelectric energy harvester converts the mechanical energy into electrical energy. It can utilize the waste energy from human motion, ocean waves, car motion, etc. It works on triboelectric effect and electrostatic effect [1, 2]. There are two basic modes to extract energy by triboelectric energy harvester: (1) contact mode and (2) sliding mode [3]. It has been observed that contact mode has better output compared to sliding mode [4], but sliding mode triboelectric energy T. P. Singh (B) · S. Kumar · R. Kumar Department of Mechanical Engineering, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India e-mail: [email protected] R. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 R. Kumar et al. (eds.), Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-16-0550-5_168
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harvester is much easier for packaging and more effective for static charge generation [5]. Basic structure of contact mode and sliding mode energy harvester is same, basic difference between these two is direction of motion of top plate, in sliding it moves in lateral direction, whereas in case of contact mode, it moves in vertical direction. Triboelectric effect is known from decades, but first triboelectric nanogenerator (TENG) was invented by Wang group in 2012 [6]. The first TENG device consists of Kapton film and polyester (PET) as triboelectric materials, and working principle of this device was on vertical contact separation mode. Due to triboelectric effect, charges are generated on the surface of two different triboelectric materials and because of the electrostatic induction charge is transferred from triboelectric material to electrode. A continuous AC output was achieved if it repeated. Initially, most of the TENG were performed on the vertically contact mode. Sliding mode triboelectric nanogenerator was demonstrated by Wang et al. by using PTFE (Polytetrafluroethylene) and nylon (Polyamide6, 6) as triboelectric materials, glass as substrate and Cu as electrode [7]. This TENG has same working principle as vertical contact separation mode. Maximum output obtained as V OC = ∼1300 V and short-circuit current density of 4.1 mA/m2 with a peak power density of 5.3 W/m2 . Zhu et al. fabricated a TENG and obtained output by lightening two rows of 80 LED’s [8]. To harvest energy from distilled water, Wang group fabricated a sliding mode TENG [9]. A U-tube as triboelectric material and distilled water slides over it. This TENG works on triboelectric effect and Pascal’s law. This TENG is also used as multifunctional sensor which could measure displacement, pressure, torsion, etc. In sliding mode, TENG along with triboelectric effect, other effects are considering for energy harvesting. Han et al. designed a TENG based upon printed circuit board (PCB) technology by using PTFE and Kapton as triboelectric materials [7]. It is found that as the frequency changes from 600 Hz to 3 kHz, the power output changes from 1.3 to 25.7 W. This TENG had been used for charging a phone by using a small transformer. Park et al. fabricated a small-size rotating TENG. PTFE and fiber are taken as triboelectric material. Wind energy was used for providing rotation motion. As the wind speed increased from 10 to 30 m/s, the voltage output also increases from 5.1 to 23.8 V [10]. In this paper, a sliding mode triboelectric energy harvester is modeled to predict the harvested energy and numerical results were compared with the experimental result.
2 Basic Mechanism of Working of Sliding Mode TENG Basic mechanism of sliding mode consists sliding motion of one triboelectric material over other; generally, this motion can be achieved by two ways—to and fro motion of slider over the base and rotating motion of one plate over another plate.
Mathematical Model of Sliding Mode Triboelectric Energy Harvester
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Fig. 1 a Initial position of TENG, b sliding between plates, c maximum sliding position, d sliding back to initial position
2.1 Initial Position As shown in Fig. 1a, it has two dielectric layers and two metal electrodes. In this position, electric charge generates because of electronegativity difference between two triboelectric materials, but in this position there is no electron flow between them (I = 0).
2.2 Sliding (X < L) Let L is the Length of the dielectric material and x is the displacement from initial position. Bottom plate is fixed, and upper layer is moving with a constant velocity in parallel direction of the bottom plate as shown in Fig. 1b. To balance the charge generated in the dielectric, charges start generating on the electrode and due to the potential difference between two electrode electrons start moving from bottom electrode to upper electrode and current flows from top electrode to bottom electrode.
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2.3 Maximum Sliding Position (x = xmax = l) This is the position when upper plates slide to the maximum position of sliding. In this situation, there is no current flow (I = 0) because both the electrodes and the dielectric are balanced as shown in Fig. 1c.
2.4 Sliding Back to Initial Position In this position, top electrode starts sliding back to initial position, now upper plate has more electron so to balance these electrons, electrons start moving from top plate to bottom plate, so current direction is reverse, and it flows from bottom to top electrode as shown in Fig. 1d.
3 Mathematical Modelling Dielectric-to-dielectric sliding mode triboelectric energy harvester is considered to model as shown in Fig. 2. It has two dielectric layers and two metal electrodes. Let d 1 and d 2 are the thickness of dielectric layer, εr1 and εr2 are the relative permittivity of dielectric layers, ε0 is vacuum permittivity. R is the total external resistance in the circuit. Voltage–charge–displacement (V-Q-x) relationship shows a linear relationship between V and Q for sliding mode TENG, like in a capacitor. This linear V-Q relationship is because of capacitance between two electrodes. It does not intercept because of the triboelectric charge. So, this relationship can be shown as V =−
1 × Q + VOC (x) C(x)
(1)
Let σ is the charge distribution in non-overlapped region of the bottom electrode so −σ will be the charge distribution in non-overlapped region of the top electrode. Charge distribution in overlapped region can be expressed as – Fig. 2 Dielectric-to-dielectric sliding mode energy harvester
Mathematical Model of Sliding Mode Triboelectric Energy Harvester
For bottom electrode = For top electrode =
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−σ x (l − x)
σx (l − x)
By applying the Gauss theorem, electric field in overlapped region of dielectric layer 1 and dielectric layer 2 can be calculated asE1 =
σx ε0 εr1 (l − x)
E2 =
σx ε0 εr2 (l − x)
VOC = E 1 d1 + E 2 d2 VOC =
d1 σx d2 + ε0 (l − x) εr1 εr2
(2)
As dielectric thickness is very less compared to length, the capacitor between the overlapped region will dominate the total capacitance. The total capacitance C can be calculated as parallel plate capacitor model ε0 w(l − x) C= d1 + εdr22 εr1
(3)
From Eqs. 1, 2 and 3 V =−
σx d1 d1 d2 d2 1 Q+ + + wε0 (l − x) εr1 εr2 ε0 (l − x) εr1 εr2
(4)
In Eq. (4) Voltage can be replaced by Ohm’s law V =R
dQ dt
(5)
From Eqs. (4) and (5), generalized equation can be obtained dQ σx d1 d1 d2 d2 1 R Q+ + + =− dt wε0 (l − x) εr1 εr2 ε0 (l − x) εr1 εr2
(6)
This is the generalized equation for dielectric-to-dielectric sliding mode which has been solved with the help of fourth-order Runge–Kutta numerical method by applying boundary condition.
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xmax , x = v.t For t < v xmax , x = xmax For t ≥ v
4 Numerical Results A MATLAB code is developed based on the mathematical model. Obtained results are validated with the published results Niu et al. [5]. Figure 3 shows present results matches well with published results. Further, numerical results are predicted for current and voltage shown in Fig. 4a, b respectively. As the external resistance increases, the current and charge decrease but voltage output increases. After reaching the maximum separation distance (x max ), charge gets stabilized, but the values of current and voltage decrease exponentially. As the value of the resistance increases, less charge can transfer from bottom electrode to top electrode, so number of charge transfer is decreasing for higher load, for small load it shows a linear relationship for charge same as short-circuit condition as shown in Fig. 3 (Table 1). Further results were analyzed considering the motion of the slider as accelerating and deaccelerating. For analyzing this, following condition is imposed on the movement of top plate, generated charge voltage and current are presented in Fig. 5a–c, respectively (Table 2).
Fig. 3 Charge–time relationship comparison between present result and published result
Mathematical Model of Sliding Mode Triboelectric Energy Harvester
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Fig. 4 Theoretical calculated results. a Current–time relationship with different resistance, b voltage–time relationship with different resistance
Table 1 Parameters for constant velocity mathematical model [5]
Dielectric 1
εr1 = 4, d1 = 220 µm
Dielectric 2
εr2 = 2, d2 = 220 µm
Width of dielectric (w)
0.1 m
Length of dielectric (l)
0.1 m
Surface charge density (σ )
7 µcm−2
Maximum separation distance (x max )
0.08 m
Velocity (v)
1 m/s
xmax xmax xmax 1 2 ; For time range of ≤t