Integrated Computer Technologies in Mechanical Engineering - 2022: Synergetic Engineering (Lecture Notes in Networks and Systems, 657) 3031362004, 9783031362002
The International Scientific and Technical Conference “Integrated Computer Technologies in Mechanical Engineering”―Syner
150 95 75MB
English Pages 790 [778] Year 2023
International Scientific and Technical Conference
Contents
Aerospace Engineering
Technology of Holes Strengthening by Pneumo-Impulse Hole Mandrelling
1 Introduction
2 Enhancing the Strength by Surface Plastic Deformation Methods
3 Technological Model of Pneumo-Impulse Hole Mandrelling
4 Results and Discussion
5 Conclusion
References
Experimental Setup with Stand-Alone Automatic Modular Logger of Parameters of Thermal Pulse Processing
1 Introduction
2 Development of a Stand-Alone Automatic Modular Parameter Logger
3 Hardware-Software Complex for Registration of Experimental Data
4 Experimental Setup with Stand-Alone Automatic Modular Parameter Logger
5 Conclusion
References
Estimation of the Minimum Uncut Chip Thickness Utilising Conventional Milling of S960QL and C45E Steels
1 Introduction
2 Design of Experiment and the Cutting Tests
3 Data Evaluation
4 Conclusion
References
Theoretical Foundations of Physical Modeling of the Descent and Landing Process of Controlled Precision Airborne Cargo Landing Systems
1 Introduction
2 Analysis of Modern Aviation Needs for Flight Researches of Critical Flight Modes and Special Flight Situations Arising During the Flight Operation of CPACLS
3 FDSM as a Design Object, a Technological Object and Flight Research Tool. Modern Concepts of Formation and Quality Control of a FDSM
4 Features of Application of Basic Provisions of Similarity Theory for Substantiation of Possibility of Creation a FDSM of a CPACLS and Carrying Out Flight Researches on Them. Criteria and Scale of Similarity in the Task of Forming the Required Quality of a FDSM of a CPACLS
5 Formalization of the System of Connections Imposed on the Similarity Scale of the Main Parameters of the Full-Scale CPACLS and Its FDSM
6 Similarity Scales of the Basic Parameters of a Full-Scale CPACLS and Its FDSM While Satisfying Possible Combinations of Similarity Criteria
7 Secondary Scales of Similarity of Parameters and Characteristics in Modeling the Flight of a Full-Scale CPACLS on Its FDSM
8 Conclusions
References
Algorithm of Computational Modeling the Self-heating Process of Pneumatic Tire in Operation
1 Introduction
2 Main Idea of Algorithm
3 Determination of Deformation Cycles
4 Calculation of Heat Load
4.1 Algorithm of Calculation Heat Generation Function
4.2 Strain Cycles Approximation
4.3 Determination of the Heat Generation Function
5 Solving of Stationary Heat Conduction Problem
6 Conclusions
References
Recent Advances in Development and Characterization of CVD Multilayer Composite Coatings—A Comprehensive Review
1 Introduction
2 A Survey of Trends in Development Processes of Multilayers Composite Coatings
3 Development of Composite Layers Coated Tools with Improved Performance
4 Coating Structure
4.1 Single Layer Coatings
4.2 Complex Multi-layers Coating
4.3 Gradient and Nanostructured Coatings
5 Conclusion
References
Adhesive Joint Optimization. Application of Bezier Curves for Structural Shape Description
1 Introduction
2 Problem Formulation
2.1 Mathematical Model of Stress State
2.2 Optimization Problem
2.3 Genetic Algorithm of Optimization
3 Numerical Realization and Results
4 Conclusion
References
Study of Drilling Cutting Modes Impact on Shrinkage and Surface Roughness of Holes in Composite Materials
1 Introduction
2 Literature Review and State-of-the-Art
3 Purpose and Objectives of the Study
4 Experimental Procedures
5 Results of Studying the Holes Shrinkage and Surface Roughness in Composite Laminates
5.1 Results of Holes Experimental Studying in Specimens Made of ELUR-P-1,0, T-10–14, and T42/1–76
5.2 Results of the Carbon Fiber Specimens Experimental Studies Made of UOL-300-1A Tape
6 Conclusion
References
Influence of Physico-Mechanical Factors on Residual Technological Stresses and Strains of Composite Panels
1 Introduction
2 Literature Review
3 Research Methodology
4 Results
5 Conclusions
References
Control of Stress Conditions in Growth of Copper Oxide Nanostructures
1 Introduction
2 Materials and Methods
3 Experimental Results and Discussion
4 Conclusions
References
Required Geometric Parameters of Inflatable Air Shock Absorbers of Parachute Platforms Bench Dynamically Similar Models
1 Introduction
2 Literature Review on Operating Conditions and Loads Acting on Inflatable Air Shock Absorbers of Parachute Platforms and Modeling Features on DSM
3 Criteria and Scale of Similarity in the Task a Creating BDSM and Execution Experimental Researches of NO Landing on It
4 Results of Using the Similarity Theory and Theory of Calculation of Thin-Walled Shells of Rotation to Determine the Wall Thicknesses of Air Shock Absorbers of BDSM
5 Conclusions
References
Shape of Part Edges with the Various Surface Finish of the Adjacent Surfaces
1 Introduction
2 The State-of-the-Art and the Study Purpose
3 Study of Relationship Between the Adjacent Surfaces Microrelief Orientation (Lay) and the Edge Shape
4 Study of Edge Shape for the Different Combinations of Adjacent Surfaces Lay
5 Study of the Edge Rounding Minimum Possible Radius for the Different Combinations of the Adjacent Surfaces Lay
6 Conclusion
References
Study of a Geometry Accuracy of the Bracket-Type Parts Using Reverse Engineering and Additive Manufacturing Technologies
1 Introduction
2 Literature Review and State-of-the-Art
3 Purpose and Objectives of the Study
4 Experimental Procedures
4.1 Creating a “Digital Portrait” (a Point Cloud) of the Aircraft Transmission Bracket and Its Digital Mock-Up (3D Model) of the Specified Geometry
4.2 The Aircraft Transmission Bracket 3D Printing at Selected Process Modes
4.3 Realization of the 3D Printed Aircraft Transmission Bracket Inspection by Comparing Its “Digital Portrait” and Digital Mock-Up
5 Conclusion
References
Design of High Fatigue Life Joints of Fuselage Structures Considering Fracture Mechanics
1 Introduction
2 Modern Principles of Designing Fuselage Joints
2.1 Residual Strength and Crack Growth
2.2 Inspection Program
2.3 Widespread Fatigue Damage
3 Fuselage Joints of Tail Section
4 Discussion
References
Analysis of Gluing Assembly Defects of Ultra-Thin Composite Face Sheets with Honeycomb Core
1 Introduction
2 Literature Review
3 Research Methodology
4 Experimental Research
5 Results
6 Conclusions
References
Information Technology for Determining the Flight Performance of a Paraglider Wing
1 Introduction
2 Classification
3 Paragliding Design Technique
4 Algorithm for Determining the Performance Characteristics of a Paraglider Wing
5 Software Architecture
6 Conclusion
References
Information Modeling
Research of Mathematical Models Based on Optimization Paraboloid and Alternative Method of Regression
1 Introduction
2 Literature Analysis and Problem Statement
3 Methods and Materials
3.1 Initial Dataset Preprocessing
3.2 Approximation Using a Parabola of the Second Degree
3.3 Approximation Using a Hyperbolic Curve
3.4 Approximation Using an Exponential Function
3.5 Approximation Using the Method of Least Absolute Deviations
4 Comparative Analysis of Alternative Mathematical Models
5 Conclusions
References
A Model of a Transport Multi-section Conveyor Based on a Neural Network
1 Introduction
2 Formal Problem Statement
3 Analytical Model of the Conveyor Section
4 Conveyor Section Model Using a Neural Network
5 Preparation of a Data Set for Training a Neural Network
6 Prediction Model Analysis
7 Conclusion
References
Simulation of the Impact of Migration Caused by the Russian War in Ukraine on the Dynamics of COVID-19 in Poland
1 Introduction
2 Materials and Methods
3 Results
4 Conclusions
References
Limiting Properties of Boundary Value Problems Solutions for Some Higher-Order Equations
1 Introduction
2 Problem Statement
3 Results
3.1 Theorem
3.2 Remark
3.3 Proof
4 Conclusions
References
Method for Correcting the Mathematical Model in Case of Empirical Data Asymmetry
1 Introduction
2 Literature Analysis and Problem Statement
3 Methods and Materials
4 Conclusions
References
Model for Determining the Ballistic-Temporal Characteristics of the Movement of a Controlled Parachute System
1 Introduction
2 Classification of Tasks for Motorized and Non-motorized Controlled Parachute Systems
3 Formalized Model Determining Time-Ballistics Characteristics of Controlled Parachute Systems
4 Conclusion
References
Design Improvements of an Industrial Centrifugal Fan Based on the Computer Mathematical Simulation
1 Introduction
1.1 Fan Design
1.2 The Modern Approaches to the Analysis of the Fan Structural Strength
2 Problem Formulation
2.1 Calculation Model
2.2 Finite-Element Model
2.3 Bearing Capacity
3 Results and Discussion
3.1 Static Strength Analysis
3.2 Modal Analysis
3.3 Bearing Capacity
4 Conclusions
References
Bagging Model for Heart Stroke Detection
1 Introduction
2 Materials and Methods
3 Results
4 Conclusions
References
Modification of the Goland-Reissner Overlapped Adhesive Joint Model to Describe Stress Values in Circular Joints
1 Introduction
2 Formulation of the Problem
3 Construction of the Solution
3.1 Gluing Area
3.2 Outside of the Gluing Area
3.3 Boundary Conditions
4 Model Problem
5 Conclusion
References
Stress State of the Adhesive Joint with Longitudinal Defect. Analytical and Finite Element Models
1 Introduction
2 Formulation and Solution of the Problem
2.1 Basis Equations
2.2 Constructing a Solution to the Problem
3 Model Problem
4 Conclusions
References
Modeling the Structure of an Aggregation of Toxic Cyanobacteria When Planning Their Elimination from the Air
1 Introduction
2 Problem Statement
3 Research Methodology
4 Results
5 Discussion
6 Conclusions
References
Information Technology in the Design and Manufacture of Engines
Thermal Analysis of Functionally Graded Dry Clutch Disc Subjected to Characteristic Parameters Effects
1 Introduction
2 Formulation of the Problem
2.1 Basic Equations
2.2 Materials and Methodology
2.3 Finite Element Model and Boundary Conditions
3 Numerical Results
4 Conclusions
References
Application of Markov Processes Theory for Computational Prediction of Turbine Blade Reliability
1 Introduction
2 The Current State of the Problem
2.1 Overview of Markov Processes
2.2 Definition of Characteristic Functions
3 Algorithm and Methods of Determination of Turbine Blade Reliability
4 Processing of Results
5 Summary
References
Calculation of Velocity and Temperature of Nickel Powder Particles in a Supersonic Nozzle During Low-Pressure Cold Spraying
1 Introduction
2 Brief Review of Key Cold Spaying Parameters
3 One-Dimension Isentropic Model
4 Calculation Results and Discussion
5 Conclusions
References
Numerical Simulation of Strength and Aerodynamic Characteristics of Small Wind Turbine Blades
1 Problem Statement
2 Materials and Methods
3 Results and Discussion
4 Conclusion and Further Research
References
Integrated Computer Modeling of Dynamic Processes in Rotor Machines and Systems with Elastic-Damper and Magnetic Bearings
1 Introduction
2 Literature Review
3 The Aim of the Research
4 The Subject of Research and Mathematical Formulation
4.1 Dynamic System for Analysis
4.2 Mathematical Formulation of Rotor Dynamics
5 Integrated Software Tool for Calculating Rotor Dynamics
6 Calculation Results and Program Verification
6.1 Dynamic Analysis of the Rotors of ECU
6.2 Dynamic Analysis of the Rotors in Passive-Active Magnetic Suspension
7 Discussion and Conclusion
References
Influence of Using Different Material Models of an Aircraft Gas Turbine Engine Fan Blade and a Bird when Simulating the Dynamics of a Collision Process in Flight
1 Introduction
2 Literature Review
3 Object and Aim of Research
4 Calculation Model, Preconditions and Analysis Algorithm
4.1 Finite Element Model, Constraints and Initial Conditions
4.2 Material Models
4.3 Preliminary Loading Conditions for Blades and Calculation Sequence
5 Impact Simulation of a Straight-Flying Bird and a Rotating Fan Wheel
6 Validation of Results According to Experimental Data
7 Discussion and Conclusion
References
Optimization of the Small Horizontal Axis Wind Turbine Blade Based on the Interactive Design Approach
1 Introduction
2 Structure and Blade Design
2.1 Design Wind Turbine
2.2 Finite Element Model
3 Results and Discussions
4 Conclusions and Future Works
References
Simulation of Aircraft Engine Dynamic Effect on Aircraft Wing Caused by a Fan Blade-Off
1 Introduction
2 Development of Models of Bearings and Bearing Supports
3 Development of Models of Engine Rotors and Stator
4 Development of a Finite Element Model of an Engine Mount
5 Numerical Analysis of Engine Vibration with a Fan Blade-Off
6 Conclusion
References
Nano-Modeling
Numerical Simulation of the Steady Creep of Single-Crystal Alloys
1 Introduction
2 Creep Theory of Monocrystalline Alloy
3 Method for Identification of Material Parameters for Creep Model
4 The Results of Approximation of the Creep Curves of the Monocrystalline Alloy ZhS-32
5 Conclusions
References
Substrate Thickness as a Control Factor in Growth of Copper Oxide Nanostructures
1 Introduction
2 Theoretical Model
3 Results and Discussion
4 Conclusions
References
Project Management and Business Informatics
Space Law: Current Situation and Development in Ukraine and Latvia
1 Introduction
2 Literature Review
3 Main Content Presentation
4 Conclusions
References
Legal Support of Economic Activity in the Conditions of the Pandemic
1 Introduction
2 Literature Review
3 Main Content Presentation
3.1 Special Legal Regime as a Means of Legal Regulation of Economic Relations
3.2 Special Regime of Economic Activity in a Pandemic as a Means of Overcoming the Pandemic
3.3 Special Legislation in the Field of Economic Activity in a Pandemic
4 Conclusions
References
Methodological Approaches to the Management of Ecological Safety of Acoustic Environment in the Territories of Urban Systems
1 Introduction
2 Formulation of the Problem for the Development of Architectural and Planning Solutions to Reduce Noise Pollution at the Level of Urban Systems
3 Results of Experimental Data on the Influence of Infrastructure Networks at the Level of Urban Systems
4 Conclusion
References
The Effect of Sanctions Against the Background of the Russian Occupation of Ukraine
1 Introduction
2 Literature Review
3 Main Content Presentation
3.1 The Role of Sanctions and Its Impact on the Russian Economy
3.2 How Are the Sanctions Affecting Russian Economy?
4 Conclusion
References
Use of Information Systems in Disclosure of Criminal Offenses
1 Introduction
2 Main Content Presentation
3 Conclusion
References
Artificial Intelligence in Labor Relations: Prospects for Evolution of Labor Laws
1 Introduction
2 Automation of Workplaces and Introduction of Artificial Intelligence as the Main Factors in Changing Nature and Labor Structure
3 Neurotechnologies and Labor
4 Conclusions
References
Scientific Approaches to Reforming Staff Management of State Forensic Science Institutions of the Ministry of Justice of Ukraine
1 Introduction
2 Literature Review
3 Main Content Presentation
3.1 Concept of HR Management as Part of Formation of Professionals Team. Tasks, Methods, Functions, Principles of HR Management
3.2 Analysis of Current Regulations of HR Management of State Forensic Science Institutions of the Ministry of Justice of Ukraine
3.3 HR Department of State Forensic Science Institution of Ministry of Justice of Ukraine and the Need to Reform It in Order to Create a HR Management Unit
3.4 Reforming Regulatory Framework for HR Management of State Forensic Science Institutions of the Ministry of Justice of Ukraine
3.5 Ways to Improve Training and Retraining of Forensic Staff of State Forensic Science Institutions of the Ministry of Justice of Ukraine
4 Research Results
5 Conclusion
References
Clustering of Patients with Diabetes Mellitus Using Expectation-Maximization Algorithm
1 Introduction
2 Materials and Methods
3 Results
4 Conclusions
References
Synergetic Approach and Issues of Criminological Science Methodology
1 Introduction
2 Main Content Presentation
3 Conclusions
References
Evaluation of Effectiveness the Innovative Diffusion of the Socio-economic Systems
1 Introduction
2 Authentication of Determinants of the Effectiveness of Innovative Diffusion of Socio-economic Systems
3 Forming of the Indexes System the Evaluation of Effectiveness the Innovative Diffusion of the Socio-economic Systems
4 An Evaluation of Influence the Effectiveness of Innovative Diffusion on Potential Development of the Socio-economic Systems
5 Conclusions
References
Particular Aspects of International Cooperation of Ukrainian Forensic Science Institutions with Foreign Specialists in Collecting, Studying and Processing Human Genomic Information and Conducting Molecular Genetic Analysis
1 Introduction
2 Literature Review
3 Main Content Presentation
4 Conclusions
References
Analysis of Project Implementation Methods and Models Throughout Risk Management
1 Introduction
2 The Importance of Risk Management Activities
3 Generalized Network Models
4 Probabilistic Network Models
4.1 Characteristics Field of Use of Probabilistic Models
5 Alternative Network Models
6 Deterministic Network Models
7 Accounting for Resource Constraints and Optimizing Their Use
8 Conclusions
References
Robotics and UAV
Analysis of the Architecture of Perceiving a Dynamic Environment for an Unmanned Aerial Vehicle
1 Introduction
2 Indicators for Assessing the Quality of Programs and Methods of Pattern Recognition
3 Information Processing Methods for Extracting Data
4 Software Analysis
5 Features of Perceiving a Dynamic Environment for an Unmanned Aerial Vehicle
6 Conclusions
References
Research of Human-Machine Interaction on the Example of the “Operator-UAV” System
1 Introduction
2 Literature Review
3 Materials and Methods
4 Results
4.1 Operator’s Stress Tolerance Study
4.2 Operator’s Resource Study
4.3 The Viability of the “Operator-UAV” System Study
5 Methods for Reducing Risks in the Work of UAV Operators
6 Conclusion
References
Signal and Image Processing
A Fast Procedure for Image Lossy Compression by ADCTC Using Prediction of Distortions’ MSE
1 Introduction
2 ADCTC and Test Images
3 Analysis of Distortion Statistics
4 Proposed Prediction Algorithm and Its Testing
5 Conclusions
References
Comparison of Noisy Medical Image Lossy Compression by BPG and AGU
1 Introduction
2 Lossy Compression Coders
2.1 BPG Encoder
2.2 AGU Encoder
3 Test Images for Simulation
3.1 Image Properties
3.2 Noise Models
4 BPG and AGU Compression Methods Analysis
5 Conclusions
References
Lossy Compression of Three-Channel Remote Sensing Images Based on Controlled Visual Quality
1 Introduction
2 Visual Quality Metrics
3 Method of Providing a Controlled Visual Quality
3.1 Lossy Compression Encoders
3.2 Basic of Providing a Controlled Visual Quality
4 Approaches to Lossy Compression of Remote Sensing Images
5 Visual Quality Control Method for 3D Version of AGU
6 Accuracy Analysis of Visual Quality Control
7 Conclusions
References
Smart Energy and Grids
Pump Assistant Ground-Source Two-Phase Thermosiphon for the Heating and Cooling of Households
1 Introduction
2 Theory
2.1 Concept
2.2 Power Consumption
2.3 Other Aspects
3 Experimental
4 Results and Discussion
5 Conclusions
References
Software Engineering and IT-Infrastructure
Development of Integrated CAD/CAE Systems Based on Parameterization of the Simulated Process
1 Introduction
2 Analysis of Functional Possibilities and Problems of Using CAD/CAE Systems in Modeling the Billet Stress-Strain State
3 The Method of Selecting the Scheme and Parameters of Billet Deformation Using Parametric Models
4 Development of Mathematical Models of SPD RS-180 and RS-60 Processes for Building a Parametric Model of Die Equipment
5 Patterns of Influence of the Scheme and Die Parameters on the Change in the Billet Shape During SPD
6 Conclusion
References
Cooperative Navigation of Personal Electronic Devices Formation Movement
1 Introduction
2 Formation of Personal Electronic Devices
3 Localization of PED Inside of the Formation
4 Data Time Synchronization
5 Collaborative Data Processing
6 Numerical Demonstration
7 Conclusion
References
Using Multilayer Neural Networks to Predict the Output Flow Parameters of Multi-section Conveyor
1 Introduction
2 Problem Statement
3 Literature Review
4 Main Material
5 Results
6 Conclusion
References
Architecting Autonomous Underwater Vehicles by Adapting Software Product Lines
1 Introduction
2 Software Product Lines for Cyber-Physical Systems
3 Extending the Feature-Model Metamodel for Cyber-Physical Attributes
4 The Life Cycle of Product Lines for Cyber-Physical Systems
5 Conclusions
References
Transport Systems and Logistics
Study of the Formation of Passenger Flows in Transport Hubs
1 Introduction
2 Literature Review
3 Research Methodology
4 Results and Discussion
5 Conclusions
References
Analysis of the Possibility of Using Analytical Methods to Model the Risks and Consequences of Transport Events in the Transport of Dangerous Goods by Railway Transport
1 Introduction
1.1 Problem Statement
2 The Purpose and Tasks of the Research
3 Presenting Main Material
4 Conclusions and Suggestions
References
An Air Traffic Safety Management Based on Probabilities
1 Introduction
2 Probability-Based Approach
3 Risk of Preplanned Trajectory Loss
4 Numerical Demonstration
5 Conclusion
References
Author Index
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Lecture Notes in Networks and Systems 657
Mykola Nechyporuk Vladimir Pavlikov Dmitriy Kritskiy Editors
Integrated Computer Technologies in Mechanical Engineering - 2022 Synergetic Engineering
Lecture Notes in Networks and Systems
657
Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
Advisory Editors Fernando Gomide, Department of Computer Engineering and Automation—DCA, School of Electrical and Computer Engineering—FEEC, University of Campinas—UNICAMP, São Paulo, Brazil Okyay Kaynak, Department of Electrical and Electronic Engineering, Bogazici University, Istanbul, Türkiye Derong Liu, Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, USA Institute of Automation, Chinese Academy of Sciences, Beijing, China Witold Pedrycz, Department of Electrical and Computer Engineering, University of Alberta, Alberta, Canada Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Marios M. Polycarpou, Department of Electrical and Computer Engineering, KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, Nicosia, Cyprus Imre J. Rudas, Óbuda University, Budapest, Hungary Jun Wang, Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong
The series “Lecture Notes in Networks and Systems” publishes the latest developments in Networks and Systems—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNNS. Volumes published in LNNS embrace all aspects and subfields of, as well as new challenges in, Networks and Systems. The series contains proceedings and edited volumes in systems and networks, spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. The series covers the theory, applications, and perspectives on the state of the art and future developments relevant to systems and networks, decision making, control, complex processes and related areas, as embedded in the fields of interdisciplinary and applied sciences, engineering, computer science, physics, economics, social, and life sciences, as well as the paradigms and methodologies behind them. Indexed by SCOPUS, INSPEC, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science. For proposals from Asia please contact Aninda Bose ([email protected]).
Mykola Nechyporuk · Vladimir Pavlikov · Dmitriy Kritskiy Editors
Integrated Computer Technologies in Mechanical Engineering - 2022 Synergetic Engineering
Editors Mykola Nechyporuk National Aerospace University “Kharkov Aviation Institute” Kharkov, Ukraine
Vladimir Pavlikov National Aerospace University “Kharkov Aviation Institute” Kharkov, Ukraine
Dmitriy Kritskiy National Aerospace University – Kharki Kharkov, Ukraine
ISSN 2367-3370 ISSN 2367-3389 (electronic) Lecture Notes in Networks and Systems ISBN 978-3-031-36200-2 ISBN 978-3-031-36201-9 (eBook) https://doi.org/10.1007/978-3-031-36201-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
International Scientific and Technical Conference
Integrated Computer Technologies in Mechanical Engineering—Synergetic Engineering Kharkiv, Ukraine November 18, 2022
Organized by: National Aerospace University “Kharkiv Aviation Institute”
Kharkiv Regional State Administration
Patronage: Ministry of Education and Science of Ukraine
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International Scientific and Technical Conference
General Schedule (GMT+2)
November 18, 2022 (Friday) 9:25
TESTING SESSION : https://meet.google.com/xvk-xhzi-fis REGISTRATION OPENS
9:45
OPENINGS CONFERENCE
10:00 PLENARY SESSION 10:50 TECHNICAL BREAK 11:00 Session 1—MECHANICAL Session 2—SOFTWARE ENGINEERING ENGINEERING https://meet.google.com/ AND PROJECT MANAGEMENT xvk-xhzi-fis https://meet.google.com/cry-xyej-pkm 13:50 TECHNICAL BREAK 14:10 Session 1—MECHANICAL Session 2—SOFTWARE ENGINEERING ENGINEERING https://meet.google.com/ AND PROJECT MANAGEMENT xvk-xhzi-fis https://meet.google.com/cry-xyej-pkm 15:40 CONCLUSIONS 15:45 CLOSING OF THE CONFERENCE : LOOKING TO THE FUTURE
ICTM-2022 Committees Steering Committee Honorary Chair Nechyporuk Mykola
Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine)
Executive Chair Pavlikov Vladimir
Doctor of Technical Sciences, Senior Research Associate (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine)
International Scientific and Technical Conference
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Program Committee (In Alphabetical Order) Bo An
Bodyanskiy Yevgeniy
Bychkov Sergiy Danylov Valeriy
Dolmatov Anatolii
Dorosh Mariia
Druzhinin Evgeniy
Fedorovych Oleg
Gł˛ebocki Robert Gorbenko Anatoliy Grebenikov Oleksandr
Hulianytskyi Leonid
Karatanov Oleksandr
Kashanov Olexandr Kharchenko Vyacheslav
Doctor of Philosophy, Associate Professor (Nanyang Technological University, Nanyang, Singapore) Doctor of Technical Sciences, Professor (Kharkiv National University of Radio Electronics, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (Antonov Company, Kyiv, Ukraine) Doctor of Technical Sciences, Professor (National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Associate Professor (Chernihiv National University of Technology, Chernihiv, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor Habilitatus, Professor (Warsaw University of Technology, Warsaw, Poland) Doctor of Technical Sciences, Professor (Leeds Beckett University, Leeds, UK) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Senior Research Associate (V. M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine) Doctor of Philosophy, Associate Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine); Doctor of Philosophy (Yuzhnoye State Design Office, Dnipro, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine)
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International Scientific and Technical Conference
Kiseleva Elena
Korostelev Oleg Kritskiy Dmitriy
Lobur Mykhaylo
Lukin Vladimir
Nikolaev Alexey
Plankovskyy Sergiy Pohudina Olha
Polosukhin Illia Ponomarenko Mykola
Ponomaryov Volodymyr Popov Viktor Przystalski Karol Sanin Anatoliy,
Shakhovska Nataliya
Shypul Olga
Sokolov Oleksandr
Corresponding Member of the National Academy of Science of Ukraine, Doctor of Physics and Mathematics, Professor (Oles Honchar Dnipro National University, Dnipro, Ukraine) Doctor of Technical Sciences (State Kyiv Design Bureau Luch, Kyiv, Ukraine) Doctor of Philosophy, Dean (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (Lviv Polytechnic National University, Lviv, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Physics and Mathematics, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Professor (O. M. Beketov National University of Urban Economy in Kharkiv, Kharkiv, Ukraine) Doctor of Philosophy, Associate Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) (NEAR.ai, San Francisco, CA, USA) Doctor of Technical Sciences, Senior Research Associate (Tampere University, Tampere, Finland) Doctor of Technical Sciences, Professor (Instituto Politécnico Nacional, Mexico City, Mexico) Doctor of Philosophy (Joint Stock Company FED, Kharkiv, Ukraine) Doctor of Philosophy (Jagiellonian University, Kraków, Poland) Doctor of Technical Sciences, Professor (Oles Honchar Dnipro National University, Dnipro, Ukraine) Doctor of Technical Sciences, Professor (Lviv Polytechnic National University, Lviv, Ukraine) Doctor of Philosophy, Associate Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (Nicolaus Copernicus University, Toru´n, Poland)
International Scientific and Technical Conference
Stoyan Yuriy
Szalay Tibor
Turkin Ihor
Ugryumov Mykhaylo
Volosyuk Valerii
Vozel Benoit Whitehead Charles K. Yakovlev Sergey
Yudelson Michael V. Zaslavskyi Volodymyr
Zavgorodniy Andrew
Corresponding Member of the National Academy of Science of Ukraine, Doctor of Technical Sciences, Professor (A. Pidgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine) Doctor of Philosophy, Associate Professor (Budapest University of Technology and Economics, Budapest, Hungary) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (V. N. Karazin Kharkiv National University, Kharkiv, Ukraine) Doctor of Technical Sciences, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Philosophy, Associate Professor (University of Rennes 1, Rennes, France) Doctor of Sciences, Professor (Cornell University, Ithaca, NY, USA) Doctor of Physics and Mathematics, Professor (National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine) Doctor of Philosophy (Carnegie Mellon University, Pittsburgh, PA, USA) Doctor of Technical Sciences, Professor (Taras Shevchenko National University of Kyiv, Kyiv, Ukraine) Doctor of Philosophy (LinkedIn, Milpitas, CA, USA)
Organizing Team National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine
Publication Chair Kritskiy Dmitriy
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International Scientific and Technical Conference
Secretary Starovoit Tetiana
Members Bykov Andrii Hryhorenko Tetiana Kalashnikova Vasylysa Karatanov Oleksandr Krytska Olha Morikova Anastasia Pohudina Olha Shypul Olga Tsegelnyk Yevgen
The Official Language of the ICTM-2022 is English Time for Presentations: Presentation at the plenary session is 20 minutes. Paper presentation and discussion at the regular session are 10 minutes.
November 18, 2022, Friday
(GMT+2) 9:25
TESTING SESSION: https://meet.google.com/xvk-xhzi-fis
9:30
REGISTRATION OPENS
09:45
OPENINGS CONFERENCE Dr. Andrii Humennyi, Acting Vise–Rector National Aerospace University “KhAI”, Ukraine
10:00 10:30
PLENARY SESSION Prof. Adam Wojciechowski, Prof. Piotr Lipi´nski – Keynote Speakers 1 Lodz University of Technology, Poland Dr. István Biró—Keynote Speaker 2 Budapest University of Technology and Economics, Hungary
10:50
TECHNICAL BREAK (continued)
International Scientific and Technical Conference
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(continued) (GMT+2) 11:00
SESSION 1—MECHANICAL ENGINEERING https://meet.google.com/xvk-xhz i-fis Dr. Olga Shypul – Chairman
11:00
Borys Lupkin, Kateryna Maiorova, Iurii Vorobiov, Viacheslav Nikichanov and Valeriy Sikulskyi. Study of Drilling Cutting Modes Impact on Shrinkage and Surface Roughness of Holes in Composite Materials
11:10
Oleh Pihnastyi and Olha Ivanovska. A model of a transport multi-section conveyor based on a neural network
11:20
Yevhen Nemanezhyn, Gennadiy Lvov and Yuriy Torba. Numerical Simulation of the Steady Creep of Single-Crystal Alloys
11:30
Oleksandr Tarasov, Liudmyla Vasylieva, Alexander Altukhov, Dmytro Pavlenko and Daria Tkach. Development of integrated CAD / CAE systems based on parameterization of the simulated process
11:40
Andrii Kondratiev, Viktor Kovalenko, Tetyana Nabokina and Igor Taranenko. Influence of Physico-Mechanical Factors on Residual Technological Stresses and Strains of Composite Panels
11:50
Nasr A.Jabbar, Ihsan Y. Hussain and Oday I. Abdullah. Thermal analysis of Functionally Graded dry clutch disc subjected to characteristic parameters effects
12:00
Vitalii Voronko, Yuri Dyachenko, Iryna Voronko, Oleksandr Zastela and Svitlana Myronova. Technology of Holes Strengthening by Pneumo-Impulse Hole Mandrelling
12:10
Olga Shypul, Oleksiy Pavlenko, Yevgen Aksonov and Sergiy Zaklinskyy. Experimental Setup with Stand-Alone Automatic Modular Logger of Parameters of Thermal Pulse Processing
12:20
Sergey Kurennov, Kostiantyn Barakhov, Igor Taranenko and Hanna Barakhova. Adhesive joint optimization. Application of Bezier curves for structural shape description
12:30
Andrii Breus, Sergey Abashin and Oleksii Serdiuk. Control of stress conditions in growth of copper oxide nanostructures
12:40
Oleksandr Shorinov, Serhii Polyvianyi and Anatolii Dolmatov. Calculation of Velocity and Temperature of Nickel Powder Particles in a Supersonic Nozzle During Low-Pressure Cold Spraying
12:50
Olena Sierikova, Elena Strelnikova and Kyryl Degtyariov. Numerical Simulation of Strength and Aerodynamic Characteristics of Small Wind Turbine Blades
13:00
Vasyl Ruzaikin. Pump assistant ground-source two-phase thermosiphon for the heating and cooling of households
13:10
Gennadii Martynenko and Lyudmyla Rozova. Integrated Computer Modeling of Dynamic Processes in Rotor Machines and Systems with Elastic-Damper and Magnetic Bearings (continued)
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International Scientific and Technical Conference
(continued) (GMT+2) 13:20
Oleg Baranov. Substrate thickness as a control factor in growth of copper oxide nanostructures
13:30
Valeriy Sikulskyi, Kateryna Maiorova, Ihor Bychkov, Svitlana Myronova and Stanislav Sikulskyi. Shape of Part Edges with the Various Surface Finish of the Adjacent Surfaces
13:40
Gennadii Martynenko, Natalia Smetankina, Volodymyr Martynenko, Vyacheslav Merculov and Mykola Kostin. Influence of Using Different Material Models of an Aircraft Gas Turbine Engine Fan Blade and a Bird When Simulating the Dynamics of a Collision Process in Flight
13:50
TECHNICAL BREAK
14:10
Vitaliy Gaidachuk, Tetyana Nabokina, Igor Taranenko, Oleksandr Gaidachuk and Andrii Kondratiev. Analysis of Gluing Assembly Defects of Ultra-Thin Composite Face Sheets with Honeycomb Core
14:20
Sergey Filipkovskij, Valentin Chigrin, Alexander Sobolev and Evgeny Vasilevskij. Simulation of Aircraft Engine Dynamic Effect on Aircraft Wing Caused by a Fan Blade-Off
14:30
Kateryna Maiorova, Valeriy Sikulskyi, Iurii Vorobiov, Oleksandra Kapinus and Anton Knyr. Study of a Geometry Accuracy of the Bracket-type Parts Using Reverse Engineering and Additive Manufacturing Technologies
14:40
Danylo Krasii and Oleksiy Larin. Application of Markov processes theory for computational prediction of turbine blade reliability
14:50
Dmytro Zhyriakov, Oleksandr Grebenikov, Andrii Humennyi and Dmytro Konyshev. Design of high fatigue life joints of fuselage structures considering fracture mechanics
15:00
Osamah Ihsan Ali and Istvan Gabor Gyurika. Recent Advances in Development and Characterization of CVD Multilayer Composite Coatings — A Comprehensive Review
15:10
Julia Viazovychenko and Oleksiy Larin. Algorithm of computational modeling the self-heating process of pneumatic tire in operation
15:20
Volodymyr Martynenko. Design Improvements of an Industrial Centrifugal Fan Based on the Computer Mathematical Simulation
15:30
Khalil Deghoum, Oday Ibraheem Abdullah, Mohammed Tahar Gherbi, Hakim S. Sultan and Adnan N. Jameel Al-Tamimi. Optimization of the Small Horizontal Axis Wind Turbine Blade Based on The Interactive Design Approach
15:40
Dmytro Kritskiy, Oleksandr Karatanov, Andrii Pohudin, Sergii Koba. Information Technology for Determining the Flight Performance of a Paraglider Wing
15:50
CONCLUSIONS
15:55
CLOSING OF THE CONFERENCE : LOOKING TO THE FUTURE
11:00
SESSION 2—SOFTWARE ENGINEERING AND PROJECT MANAGEMENT https://meet.google.com/cry-xyej-pkm Dr. Dmytro Chumachenko – Chairman (continued)
International Scientific and Technical Conference
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(continued) (GMT+2) 11:00
Kseniia Bazilevych, Mykola Butkevych, Halyna Padalko and Nataliia Dotsenko. Bagging Model for Heart Stroke Detection
11:10
Iryna Petrova, Denys Chekin and Kateryna Sylenok. Scientific Approaches to Evaluation of Social Efficiency of Forensic Examination
11:20
Victoriya Abramova, Sergii Kryvenko, Sergey Abramov, Viktor Makarichev and Vladimir Lukin. A Fast Procedure for Image Lossy Compression by ADCTC Using Prediction of Distortions’ MSE
11:30
Valeriyi Kuzmin, Maksym Zaliskyi, Yulia Petrova and Alexei Holubnychyi. Research of Mathematical Models Based on Optimization Paraboloid and Alternative Method of Regression
11:40
Oleksii Lytvynov, Mykola Nechyporuk and Hanna Spitsyna. Synergetic Approach and Issues of Criminological Science Methodology
11:50
Olena Agapova, Karina Palkova, Ilmars Blumbergs and Janis Grasis. Space Law: Current Situation and Development in Ukraine and Latvia ˙ ˙ Aleksandar Ivanovic, Valeriia Nataliia Filipenko, Gabriele JuodkaitE-Granskien E, Turska, Yuri Myroshnichenko and Serhii Lukashevych. Particular Aspects of International Cooperation of Ukrainian Forensic Science Institutions with Foreign Specialists in Collecting, Studying and Processing Human Genomic Information and Conducting Molecular Genetic Analysis
12:00
12:10
Oksana Yurchenko, Oleh Strelko, Halyna Vasilova, Maryna Rudiuk and Oleksii Goretskyi. Analysis of the Possibility of Using Analytical Methods to Model the Risks and Consequences of Transport Events in the Transport of Dangerous Goods by Railway Transport
12:20
Victoriia Naumenko, Vladimir Lukin and Sergey Krivenko. Comparison of noisy medical image lossy compression by BPG and AGU
12:30
Olha Pohudina, Mykhailo Kovalevskyi, David Naso and Rossella Bartolo. Analysis of the Architecture of Perceiving a Dynamic Environment for an Unmanned Aerial Vehicle
12:40
Fangfang Li, Sergey Krivenko and Vladimir Lukin. Lossy Compression of Three-channel Remote Sensing Images Based on Controlled Visual Quality
12:50
Olena Savchuk, Anastasiia Matvieieva and Dmytro Kritskiy. Legal Support of Economic Activity in The Conditions of the Pandemic
13:00
Dmytro Chumachenko, Adam Wojciechowski and Sergiy Yakovlev. Simulation of the Impact of Migration Caused by the Russian War in Ukraine on the Dynamics of COVID-19 in Poland
13:10
Galyna Mygal, Olga Protasenko and Natalia Kobrina. Research of human-machine interaction on the example of the “operator-UAV” system
13:20
Natalia Teliura, Alona Reshetchenko, Nataliia Tsapko, Oksana Myrgorod and Oleksandr Pirohov. Methods of Organizing Measures Directed to Environmental Safety and Noise Protection Management in Urban Areas (continued)
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(continued) (GMT+2) 13:30
Ivan Ostroumov and Nataliia Kuzmenko. An air traffic safety management based on probabilities
13:40
Tetiana Podzolkova, Ihor Shynkarenko and Sergii Perepelytsia. Use of Information Systems in Disclosure of Criminal Offenses
13:50
TECHNICAL BREAK
14:10
Nina Padalko, Anatolii Padalko and Halyna Padalko. Limiting Properties of Boundary Value Problems Solutions for Some Higher-Order Equations
14:20
Valeriyi Kuzmin, Yulia Petrova, Maksym Zaliskyi and Oleksandr Lavrynenko. Method for Correcting the Mathematical Model in case of Empirical Data Asymmetry
14:30
Andrii Pohudin, Olha Pohudina, Serhii Koba and Oleksii Dzvonyskyi. Model for determining the ballistic-temporal characteristics of the movement of a controlled parachute system
14:40
Svitlana Gutsu, Andriy Bublikov and Inna Alekseienko. Artificial Intelligence in Labor Relations: Prospects for Evolution of Labor Laws
14:50
Oleh Pihnastyi and Olha Ivanovska. Using multilayer neural networks to predict the output flow parameters of multi-section conveyor
15:00
Oleh Uhrovetskyi, Oleg Kurdes and Dariia Topal. Scientific Approaches to Reforming Staff Management of State Forensic Science Institutions of the Ministry of Justice of Ukraine
15:10
Ievgen Meniailov, Serhii Krivtsov and Tetyana Chumachenko. Clustering of Patients with Diabetes Mellitus using Expectation-Maximization Algorithm
15:20
Oleh Strelko, Tetiana Hrushevska, Borys Toropov, Yuliia Berdnychenko and Oleksandra Soloviova. Study of the Formation of Passenger Flows in Transport Hubs
15:30
Ivan Ostroumov and Nataliia Kuzmenko. Cooperative navigation of personal electronic devices formation movement
15:40
Grebenikova Olena, Lyba Vasyl. Evaluation of effectiveness the innovative diffusion of the socio-economic systems
15:50
CONCLUSIONS
15:55
CLOSING OF THE CONFERENCE: LOOKING TO THE FUTURE
Contents
Aerospace Engineering Technology of Holes Strengthening by Pneumo-Impulse Hole Mandrelling . . . . Vitalii Voronko, Yuri Dyachenko, Iryna Voronko, Oleksandr Zastela, and Svitlana Myronova
3
Experimental Setup with Stand-Alone Automatic Modular Logger of Parameters of Thermal Pulse Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olga Shypul, Oleksiy Pavlenko, Yevhen Aksonov, and Sergiy Zaklinskyy
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Estimation of the Minimum Uncut Chip Thickness Utilising Conventional Milling of S960QL and C45E Steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imre Palágyi, István Biró, and Tibor Szalay
29
Theoretical Foundations of Physical Modeling of the Descent and Landing Process of Controlled Precision Airborne Cargo Landing Systems . . . . . . . . . . . . Olexander V. Betin, Petro G. Kyriienko, Denys O. Betin, and Olena Y. Betina
39
Algorithm of Computational Modeling the Self-heating Process of Pneumatic Tire in Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuliia Viazovychenko and Oleksiy Larin
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Recent Advances in Development and Characterization of CVD Multilayer Composite Coatings—A Comprehensive Review . . . . . . . . . . . . . . . . . . . . . . . . . . . Osamah Ihsan Ali and Istvan Gabor Gyurika
63
Adhesive Joint Optimization. Application of Bezier Curves for Structural Shape Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergei S. Kurennov, Konstantin P. Barakhov, Igor M. Taranenko, and Hanna S. Barakhova Study of Drilling Cutting Modes Impact on Shrinkage and Surface Roughness of Holes in Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Borys Lupkin, Kateryna Maiorova, Iurii Vorobiov, Viacheslav Nikichanov, and Valeriy Sikulskyi
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Influence of Physico-Mechanical Factors on Residual Technological Stresses and Strains of Composite Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrii Kondratiev, Viktor Kovalenko, Tetyana Nabokina, and Igor Taranenko
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Control of Stress Conditions in Growth of Copper Oxide Nanostructures . . . . . . 111 Andrii Breus, Sergey Abashin, and Oleksii Serdiuk Required Geometric Parameters of Inflatable Air Shock Absorbers of Parachute Platforms Bench Dynamically Similar Models . . . . . . . . . . . . . . . . . 123 Olexander V. Betin, Igor V. Kaluzhynov, Denys O. Betin, and Olena Y. Betina Shape of Part Edges with the Various Surface Finish of the Adjacent Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Valeriy Sikulskyi, Kateryna Maiorova, Ihor Bychkov, Svitlana Myronova, and Stanislav Sikulskyi Study of a Geometry Accuracy of the Bracket-Type Parts Using Reverse Engineering and Additive Manufacturing Technologies . . . . . . . . . . . . . . . . . . . . . 146 Kateryna Maiorova, Valeriy Sikulskyi, Iurii Vorobiov, Oleksandra Kapinus, and Anton Knyr Design of High Fatigue Life Joints of Fuselage Structures Considering Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Dmytro Zhyriakov, Oleksandr Grebenikov, Andrig Humennyi, and Dmytro Konyshev Analysis of Gluing Assembly Defects of Ultra-Thin Composite Face Sheets with Honeycomb Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Vitaliy Gaidachuk, Tetyana Nabokina, Igor Taranenko, Oleksandr Gaidachuk, and Andrii Kondratiev Information Technology for Determining the Flight Performance of a Paraglider Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Dmytro Kritskiy, Oleksandr Karatanov, Serhii Koba, and Andrii Pohudin Information Modeling Research of Mathematical Models Based on Optimization Paraboloid and Alternative Method of Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Valeriyi Kuzmin, Maksym Zaliskyi, Yuliia Petrova, and Oleksii Holubnychyi
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A Model of a Transport Multi-section Conveyor Based on a Neural Network . . . 215 Oleh Pihnastyi, Anna Burduk, and Olha Ivanovska Simulation of the Impact of Migration Caused by the Russian War in Ukraine on the Dynamics of COVID-19 in Poland . . . . . . . . . . . . . . . . . . . . . . . 227 Dmytro Chumachenko, Adam Wojciechowski, and Sergiy Yakovlev Limiting Properties of Boundary Value Problems Solutions for Some Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Nina Padalko, Anatoliy Padalko, and Halyna Padalko Method for Correcting the Mathematical Model in Case of Empirical Data Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Valeriyi Kuzmin, Yuliia Petrova, Maksym Zaliskyi, and Oleksandr Lavrynenko Model for Determining the Ballistic-Temporal Characteristics of the Movement of a Controlled Parachute System . . . . . . . . . . . . . . . . . . . . . . . . . 261 Andrii Pohudin, Olha Pohudina, Serhii Koba, and Oleksii Dzvonyskyi Design Improvements of an Industrial Centrifugal Fan Based on the Computer Mathematical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Volodymyr Martynenko Bagging Model for Heart Stroke Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Kseniia Bazilevych, Mykola Butkevych, Nataliia Dotsenko, and Halyna Padalko Modification of the Goland-Reissner Overlapped Adhesive Joint Model to Describe Stress Values in Circular Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Sergei S. Kurennov, Konstantin P. Barakhov, and Olexandr G. Poliakov Stress State of the Adhesive Joint with Longitudinal Defect. Analytical and Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Sergei S. Kurennov, Konstantin P. Barakhov, and Olexandr G. Poliakov Modeling the Structure of an Aggregation of Toxic Cyanobacteria When Planning Their Elimination from the Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Kostiantyn V. Nosov, Yriy G. Bespalov, Tetiana O. Klochko, and Olena V. Vysotska
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Information Technology in the Design and Manufacture of Engines Thermal Analysis of Functionally Graded Dry Clutch Disc Subjected to Characteristic Parameters Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Nasr A. Jabbar, Ihsan Y. Hussain, and Oday I. Abdullah Application of Markov Processes Theory for Computational Prediction of Turbine Blade Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Danylo Krasii and Oleksiy Larin Calculation of Velocity and Temperature of Nickel Powder Particles in a Supersonic Nozzle During Low-Pressure Cold Spraying . . . . . . . . . . . . . . . . . 346 Oleksandr Shorinov, Sergii Polyvianyi, and Anatolii Dolmatov Numerical Simulation of Strength and Aerodynamic Characteristics of Small Wind Turbine Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Olena Sierikova, Elena Strelnikova, and Kyryl Degtyariov Integrated Computer Modeling of Dynamic Processes in Rotor Machines and Systems with Elastic-Damper and Magnetic Bearings . . . . . . . . . . . . . . . . . . . 371 Gennadii Martynenko and Lyudmyla Rozova Influence of Using Different Material Models of an Aircraft Gas Turbine Engine Fan Blade and a Bird when Simulating the Dynamics of a Collision Process in Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Gennadii Martynenko, Natalia Smetankina, Volodymyr Martynenko, Vyacheslav Merculov, and Mykola Kostin Optimization of the Small Horizontal Axis Wind Turbine Blade Based on the Interactive Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Khalil Deghoum, Oday I. Abdullah, Mohammed T. Gherbi, Hakim S. Sultan, and Adnan N. Jameel Al-Tamimi Simulation of Aircraft Engine Dynamic Effect on Aircraft Wing Caused by a Fan Blade-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 S. V. Filipkovskij, V. S. Chigrin, A. A. Sobolev, and E. T. Vasilevskij Nano-Modeling Numerical Simulation of the Steady Creep of Single-Crystal Alloys . . . . . . . . . . 421 Yevhen Nemanezhyn, Gennadiy Lvov, and Yuriy Torba Substrate Thickness as a Control Factor in Growth of Copper Oxide Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Oleg Baranov
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Project Management and Business Informatics Space Law: Current Situation and Development in Ukraine and Latvia . . . . . . . . 441 Olena Agapova, Ilm¯ars Blumbergs, Janis Grasis, and Karina Palkova Legal Support of Economic Activity in the Conditions of the Pandemic . . . . . . . 453 Olena Savchuk, Anastasiia Matvieieva, and Dmytro Kritskiy Methodological Approaches to the Management of Ecological Safety of Acoustic Environment in the Territories of Urban Systems . . . . . . . . . . . . . . . . 465 Natalia Teliura, Alona Reshetchenko, Nataliia Tsapko, Oksana Myrgorod, and Oleksandr Pirohov The Effect of Sanctions Against the Background of the Russian Occupation of Ukraine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Giga Abuseridze and Olena Agapova Use of Information Systems in Disclosure of Criminal Offenses . . . . . . . . . . . . . . 482 Tetiana Podzolkova, Ihor Shynkarenko, and Perepelytsia Sergii Artificial Intelligence in Labor Relations: Prospects for Evolution of Labor Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 Svitlana Gutsu, Andriy Bublikov, and Alekseienko Inna Scientific Approaches to Reforming Staff Management of State Forensic Science Institutions of the Ministry of Justice of Ukraine . . . . . . . . . . . . . . . . . . . . 508 Oleh Uhrovetskyi, Oleg Kurdes, and Dariia Topal Clustering of Patients with Diabetes Mellitus Using Expectation-Maximization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Ievgen Meniailov, Serhii Krivtsov, and Tetyana Chumachenko Synergetic Approach and Issues of Criminological Science Methodology . . . . . . 535 Oleksii Lytvynov, Mykola Nechyporuk, Oleksandr Kliuiev, and Hanna Spitsyna Evaluation of Effectiveness the Innovative Diffusion of the Socio-economic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Grebenikova Olena and Lyba Vasyl
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Particular Aspects of International Cooperation of Ukrainian Forensic Science Institutions with Foreign Specialists in Collecting, Studying and Processing Human Genomic Information and Conducting Molecular Genetic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Nataliia Filipenko, Gabriele Juodkait˙e-Granskien˙e, Aleksandar Ivanovic, Valeriia Turska, and Yu. Myroshnichenko Analysis of Project Implementation Methods and Models Throughout Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Evgeniy Druzhinin, Bohdan Haidabrus, Oleksandr Psarov, and Valeriy Cheranovskiy Robotics and UAV Analysis of the Architecture of Perceiving a Dynamic Environment for an Unmanned Aerial Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 Olha Pohudina, Mykhailo Kovalevskyi, David Naso, and Rossella Bartolo Research of Human-Machine Interaction on the Example of the “Operator-UAV” System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 Galyna Mygal, Olga Protasenko, and Natalia Kobrina Signal and Image Processing A Fast Procedure for Image Lossy Compression by ADCTC Using Prediction of Distortions’ MSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Victoriya Abramova, Sergii Kryvenko, Sergey Abramov, Viktor Makarichev, and Vladimir Lukin Comparison of Noisy Medical Image Lossy Compression by BPG and AGU . . . 636 Victoria Naumenko, Vladimir Lukin, and Sergii Kryvenko Lossy Compression of Three-Channel Remote Sensing Images Based on Controlled Visual Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Fangfang Li, Sergiy Kryvenko, and Vladimir Lukin Smart Energy and Grids Pump Assistant Ground-Source Two-Phase Thermosiphon for the Heating and Cooling of Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Vasyl Ruzaikin and Ivan Lukashov
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Software Engineering and IT-Infrastructure Development of Integrated CAD/CAE Systems Based on Parameterization of the Simulated Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 Oleksandr Tarasov, Liudmyla Vasylieva, Oleksandr Altuhov, Dmytro Pavlenko, and Daria Tkach Cooperative Navigation of Personal Electronic Devices Formation Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 Ivan Ostroumov and Nataliia Kuzmenko Using Multilayer Neural Networks to Predict the Output Flow Parameters of Multi-section Conveyor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Oleh Pihnastyi, Anna Burduk, and Olha Ivanovska Architecting Autonomous Underwater Vehicles by Adapting Software Product Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 Carlos Cares, Daniel Lühr, Sandro Mora, Claudio Navarro, Leonardo Olivares, Samuel Sepúlveda, and Gastón Vidal Transport Systems and Logistics Study of the Formation of Passenger Flows in Transport Hubs . . . . . . . . . . . . . . . 733 Oleh Strelko, Tetiana Hrushevska, Borys Toropov, Yuliia Berdnychenko, and Oleksandra Soloviova Analysis of the Possibility of Using Analytical Methods to Model the Risks and Consequences of Transport Events in the Transport of Dangerous Goods by Railway Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Yurchenko Oksana, Oleh Strelko, Halyna Vasilova, Maryna Rudiuk, and Oleksii Goretskyi An Air Traffic Safety Management Based on Probabilities . . . . . . . . . . . . . . . . . . . 755 Ivan Ostroumov and Nataliia Kuzmenko Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
Aerospace Engineering
Technology of Holes Strengthening by Pneumo-Impulse Hole Mandrelling Vitalii Voronko1
, Yuri Dyachenko2 , Iryna Voronko2(B) and Svitlana Myronova2
, Oleksandr Zastela2
1 O.M. Beketov National University of Urban Economy in Kharkiv, 17 Marshala Bazhanova
Street, Kharkiv 61002, Ukraine 2 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected]
Abstract. For increasing the aviation structural materials efficiency, above the all for ensuring the operational service life of airframes and engines, their parts, units, aggregates are specially processed to enhance their surface strength. Besides the heat treatment methods, the most widely used techniques are various methods of a surface plastic deformation (SPD) or burnishing. These processes are resulted in strain hardening of processed assembly component surfaces, providing a beneficial compressive residual stress. Altogether it’s resulted in the increase of fatigue strength and the durability of aircraft structures. It’s well known that the “thinnest places” of the airframe structural elements in terms of fatigue failures (75…80%) are the joints. Most of these joints are bolted or riveted ones that require the preliminary drilling of holes, which are concentrators of stress. Besides the holes for joints there are “free” empty holes for functional purposes are also in the aircraft structures (for fuel flowing, technological, drainage, assembly, weight-saving ones, others). The diameters of such holes are from 3…5 mm to 40 mm or more. Among bolted joints the most common are holes with a diameter of 5–12 mm (up to 95%). Increasing the service life of aircraft structures directly due to improving their joints fatigue characteristics is achieved by using the high-tensile fasteners (with radial tightness from 0.8 to 3% and higher, depending on the design, operating conditions, type of fastener, properties of the structural element material). The tightness in the joints resulted in appearing the beneficial compressive residual stresses. To increase the magnitude of tightness, provide the uniformity of its distribution in the joints and to increase joints leakproofness there are used the special types of fasteners (rivets with compensators, solid rivets, rivets made of high-strength steels and titanium alloys), as well as special techniques of joining (impulse riveting, riveting with increased pressure, riveting with forming a flush shop head, riveting in a closed volume, etc.), and also joining with tightness by bolts and rivet bolts and by conical bolts. Keywords: Fatigue Strength · Cold Bolt Hole Expansion (CBX) · Pneumo-Impulse Hole Mandrelling · Direct and Reverse Mandrelling Technological Scheme
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 3–15, 2023. https://doi.org/10.1007/978-3-031-36201-9_1
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1 Introduction The most common material used in aircraft manufacturing nowadays is aluminum. About 75–80% of a modern aircraft total weight are aluminum alloys [1, 2]. The most frequently used alloy is 7075, which contains aluminum, zinc, magnesium and copper. It is the strongest of all aluminum alloys. In terms of strength, it is close to steel, but has 3 times less density. Aluminium-magnesium alloys (with magnesium content from 3% to 6%) have special corrosion resistance that allows the aircraft successful using in the moist sea atmosphere. The corrosion resistance of such a material is 100 times better than of steel [3]. Designers are constantly searching for a material that can meet expectations regarding mechanical properties withal will reduce the weight of the aircraft. A promising candidate for this role, for example, is carbon fiber [4, 5] (threads with a diameter of 5 to 15 microns, formed mainly by carbon atoms). However, the manufacturing of the aircraft made of composite materials is much more expensive than when using aluminium alloys. Besides, carbon fiber has somewhat lower indices of the aircraft safety level. Since 2006, the American military has used Allite Super Magnesium material, which is now being used for civil industry [6]. The material is 50% lighter than titanium and 56% stronger than Grade 1 Titanium. Magnesium alloy dampens shocks 20% better than aluminium and is 21% stronger (6061 aluminium alloy). Supermagnesium is somewhat stronger than steel and 75% lighter than steel. Research is also underway to create brand new materials that would combine the best properties of alloys used in aviation [7]. An example is an intermetallic compound (B2) of the FeAl type. Intermetallics have high hardness and high chemical resistance and a higher melting point than the parent metals. In forementioned example an intermetallic B2 is obtained from an equal number of atoms of both nickel and aluminium. B2 crystals are only a couple of nanometers thick and penetrate the steel structure, giving it the strength of titanium. Titanium has good chemical resistance indices [8–10]. In humid air, in sea water, its corrosion resistance is no worse than stainless steel, and in hydrochloric acid it is many times better than stainless steel. And, like the stainless steel, titanium can be processed by cutting, forging and welding, and can be used for manufacturing the casted parts. The main advantages of titanium and its alloys, such as the combination of high specific strength and chemical resistance at normal and elevated temperatures (about 300–500 ºC), make them indispensable in the modern aircraft and spacecraft manufacturing. The especial importancy this material has for the supersonic aircraft manufacturing. In order to withstand the adverse phenomena of supersonic speeds, the common duralumin skins are changed to titanium ones, since the use of heavier, than titanium, steel for skins at such speeds and resulting temperatures of structure is not rational. Replacing the aluminium alloys or steel with titanium in the passenger aircraft is resulted in the parts weight savings of approximately 15–40%. Despite the higher cost of titanium compared to these metals, the additional costs pay off. It’s effective using of titanium in fire protection bulkheads, as the electrical and thermal conductivity of this metal is 5 times lower than steel, and 15 times lower
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than aluminium. In some aircraft kinds, which have the wing with a great load (about 1000 kg/m2 ), the wing was made of titanium and had an integral cross-section (except the control system wiring). The use of titanium alloys in the production of jet engines allows to reduce the mass by 100–150 kg [11]. The airframe also becomes less in weight (by 300 kg or more). In engines, titanium is used for the manufacturing of air intake elements, housings, compressor blades and discs, and not only. It is especially rational using of titanium in new turbofan engines. In the civil aircraft, titanium parts make up 1/7 of the total mass of the turbofan engine, in the military ones is 1/5 of the total mass. In multistage rockets, the second and third stage engines housings, cylinders and spherical tanks for compressed and liquefied gases, nozzles, etc. are made from titanium alloys. In space capsules “Mercury” and “Gemini”, the airframe, outer and inner skins are made of titanium alloys [12]. Titanium is also widely used for the casted workpieces, as such near-net-shaped blanks allow to reduce the scope of cutting work and reduce the waste of expensive metal [13]. As for the application of titanium in aviation electronics, this metal is very useful due to its gas-absorbing abilities. It absorbs the gases that remained after pumping out the device or got into the device while operation. Titanium applied to the device parts surface acts as a built-in pump capable of working throughout its life. Implementing the titanium alloys for the group of parts manufacturing in aerospace and mechanical engineering is complicated because of some problems that must be solved for the further successful using of this material [14–16]. The main disadvantages, in addition to the high cost of production, are caused by the properties of this alloy [17]: – high chemical activity of this metal at high temperatures which resulted in active reactions with all the atmospheric gases, especially with the liquefied ones, that is why Ti and its alloys can be melted only in a vacuum or in inert shield gases atmosphere; – problems of titanium waste recycling for further manufacturing; – poor antifriction properties caused by Ti adhesion to many materials; sliding contact of titanium with titanium barely works in terms of friction at all; – Ti and many of its alloys tend to hydrogen embrittlement and salt corrosion; – poor machinability by cutting, like the machinability of austenitic stainless steels; – aforementioned high chemical activity and tendency to grain growth at high temperature as well as phase transformations during the welding cycle cause difficulties when welding titanium and its alloys. Despite that titanium alloys are hard-to-process, the problem can be solved by creating certain conditions and applying special methods of this material processing. In this work, it is proposed to consider of enhancing the strength of aerospace components made of Ti alloys by methods of surface plastic deformation, as well as it is proposed the technological model of pneumatic impulse cold hole expansion while direct and inverse travels of the cold expansion mandrel.
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2 Enhancing the Strength by Surface Plastic Deformation Methods Long-term research in the field of mechanical engineering components service live extending has shown that it’s the preliminary holes strengthening by the surface plastic deformation (SPD) methods is the most effective. As a result of local plastic deformation caused by application of strengthening tool, a beneficial stress-strain state (SST) is created in the holes zones. Residual compressive stresses reduce the amplitude of local stresses and deformations while cyclic loading, which ensures an increase in the durability of the structure without increasing its mass. Strengthening the holes by the SPD methods [18, 19] is carried out as barrier crimping around the hole by stamping the segmental or ring “notches”, incusing the hole edges, by ball burnishing as well as burnishing with wedge-type and rolling mandrels. Application the holes burnishing method with multi-rolling mandrel faces a number of shortcomings. There are some of them: the complexity of the multi-rolling mandrel design, low reliability of this method, phenomenon of “flaking” the hole surface in case of the overstepping the deformation degree. At aircraft manufacturing enterprises the most common methods of holes strengthening now is the cold bolt holes expansion (CBX) by mandrel tool of specific shape, as it’s the method that can be implemented with ease [20–23]. Technically, such a “mandrelling” process can be accomplished in two ways: 1. Directly by a hard mandrel tool during its axial travel relative to the surface of the hole to be strengthened – due to the conical shape of the mandrel the hole radial deformations occur. 2. Through the intermediate elements inserted in the hole to be strengthened (for example, splitted or assembled elastic ones, highly conductive holders and bushings). The radial expansion of such elements can be carried out by a rigid mandrel, by a magnetic field or an elastic medium (for example, polyurethane or hydroplastic, which create a high specific pressure in a closed volume), which provides, by means of these intermediate elements, the strengthening deformation of the hole walls. The second variant is more interesting because it has advantages over the first one – the possibility of being implemented with a one-side access, in particular. In addition, among its prospects there are an increasing the accuracy of the hole shape and quality of its surface, as well as reducing the hole material overlaps. But excessive wearout of the cold expansion mandrel and the intermediate elements, structural weakening of the mandrel strength, inconveniencies with the use of electricity and elastic tooling significantly complicate the second approach application. The first variant is classic one and can be performed according to the schemes of either pressing (the mandrel works for compression) or retracting (with a tensile loading of the mandrel). Both schemes require a compulsory two-side access or free access to places of operations. The mandrel pressing scheme didn’t become widespread because it requires the use either of stationary press equipment, which is excluded when jig assembling, or bulky portable pneumatic, pneumohydraulic, or hydraulic presses (of C-type). Such squeezers are difficult to operate with due to their significant overall dimensions and weight and the limitations of the access in working zone.
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At enterprises, mostly, cold hole expansion is are carried out according to the first variant, with retraction of mandrel by means of broaching machines of the BM6-100, BM10-130, BM12-150, BM24-180 type, or by machines for installing and tightening the bolts of the MTB-1.5, MTB-4, MTB-7, MTB-2A (automatic), MTB-4A type. Typically, together with them there are used the pneumohydraulic pumps-multipliers of the model, for example, PHPM 1-250 or other models like pneumatic diaphragm pump (PDP) or pneumohydraulic device (PHD). The peculiarity of such machines is that they can be used only in places with free access to both sides of hole. This requires work of two employees in expansion operations, using of several mandrels for reducing the auxiliary time, and it is also accompanied by inconveniencies due to their regular connection and disconnection to the power unit. Moreover, those auxiliary pneumohydraulic drives create significant additional inconveniences for operators during their work, limiting their “maneuverability”. The cold expansion process itself with a small static (quasi-static) speed of mandrel movement is accompanied by excessive splashing out of material at the edges of the holes as well as with distortion of the hole shape (“corsetness” phenomenon). These disadvantages, and first of all the need for a two-side access to holes to be strengthened, significantly limit this variant of cold hole expansion processing (within 8…34% of the total number of bolted joints) application in manufacturing. The SPD methods have shown high effectiveness in the high-speed (impulse) variants of their implementation. Impulse processes are high-energy, their peculiarities are the high initial speed of the processes, exceeding 15…20 m/s, and short time of process, within 1‧10–4 …1‧10–3 of second. That is why they provide, above the all, the possibility of processes carrying out with an open circuit of loading, which in turn allows a one-side access processing. Impulse technologies are inherent the energy carriers to be used, which largely determine their prospects. Energy carriers should ensure the accumulation of the necessary energy and pressure with subsequent release of them within a short time [24]. For example, gunpowder and other explosives are a relatively cheap energy carrier, autonomous in application, but their use is associated with strict requirements for storaging and accounting, and their operation in shop conditions requires additional ventilation to remove spent poisonous gases and fumes [25]. High-pressure gas as an energy carrier requires additional equipment (compressor stations) to accumulate its potential energy. A classic example of mechanical energy “storage” is a compressed spring. But such an energy carrier, by analogy with the previous one, requires special drives, machines or jigs to compress the spring for its subsequent immediate release and the carrying out the useful work. Electric energy is a common energy carrier and it is widely used not only in everyday life, but also in industrial conditions. In impulse technologies, the electrical energy, which stored in special accumulators (for example, capacitor batteries), is released as an electric discharge in the liquid and creates a high-pressure shock wave that affects the surface being strengthened. The impact on the object of the electric energy carrier can also be carried out through the inductor, creating an electromagnetic force (an electromagnetic field that interacts with the working tool).
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A special place in impulse technologies is occupied by compressed air – it is widely used in shops of aviation enterprises as an energy carrier for various types of equipment. It is usually used a compressed air from a pneumatic network, with a pressure of 0.5 MPa, and that is due to a number of well-known advantages: the affinity of pneumatic manual guns, jigs and equipment, the simplicity of their designs, accessibility in maintenance, high productivity, reliability, safety, compact overall dimensions and weight acceptable for manual tooling. The potential energy that accumulated by the energy carrier, turning into kinetic energy, acts on the technological object – as a rule via transmission medium and (or) suitable in terms of properties and geometric shape working tools. And it is the transmission medium and working tools largely determine both the basic design features of impulse equipment and the quality of the technological processes to be performed. The working tool used in impulse hardening technologies is most often a solid body of a special geometric shape (cold expansion mandrel, tapered mandrel, balls, rollers, etc.); less often it is the mechanically compressed or rubber-like elastic medium or even magnetic field can be used. High-speed strengthening of holes by method of CBX by means of equipment using compressed air with a pressure of 0.5 MPa as an energy carrier has got a name pneumo-impulse hole mandrelling. As an example of its efficiency – pneumo-impulse mandrelling of holes in specimens made of D16T aluminium alloy showed a 2–3 times increase in their fatique life compared to ones with non-strengthened holes [26]. Pneumo-impulse hole mandrelling with it direct (“expanding”) and subsequent reverse (“smoothing”) working movement of the mandrel tool has the main advantage, which is inherent to the process with a one-sided access. Such a reverse motion of the mandrel tool herewith provides not only tool removal from the processed hole for further strengthening of the next hole, but also is accompanied by a smoothing effect, thus, improves a quality due to the localization of deformation zones, improving accuracy and roughness of the processed holes. Besides of this, pneumo-impulse equipment, as executive devices for high-speed CBX (for pneumo-impulse hole mandrelling), have minimal overall dimensions, provide reducing the number of employed workers, improve their working conditions and increase productivity [27]. In addition, for automated (humanless or robotic) variants of process such a non-closed loading schemes during hole expansion process allows a significant design simplification of the executive device’s holders and clamps, guiding elements, moving and positioning manipulators and significantly reduces their metal consumption. The examples of such impulse executive device that can implement the scheme of reversable cold bolt hole expansion with a one-sided access are the magnetic impulse machine MIM-P1 (portable) and the device for pneumo-impulse hole mandrelling PIDM-100, designed in National Aerospace University “Kharkiv Aviation Institute” [28, 29]. Magnetic pulse machines, regardless of the effectiveness the working mandrel tool reciprocating motion organization, also have some of the above-mentioned shortcomings, the main of which are related to labour safety while high-voltage operations, to using the additional bulky and expensive drives and low productivity of process (due to the regulation of work modes). These shortcomings significantly limit wide implementation of such machines in manufacturing.
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And vice versa, the practical experience of using in assembly processes the pneumoimpulse devices developed at the department of Aircraft Manufacturing Technologies of the National Aerospace University “Kharkiv Aviation Institute” confirmed their prospects for wide implementation. There are such devices as pneumatic impulse riveting guns PG-4 (pneumatic gun), BPG-80 (beat action pneumatic gun), PID-90 (pneumatic impulse device), PID-90M (modified) and pneumatic impulse hole mandrelling devices PIDM-90, PIDM-100. Assembly technological processes, for example, when accomplishing the bolted joints, include such basic steps for hole preparation as drilling, countersinking, reaming, strengthening, calibration followed by the installation of the fastening element (bolt) [30, 31]. Operations with a large number of holes in the structure are characterized by constant repetition and are mainly performed in the conditions of unit and aggregate assembly in jigs, without changing the assembly object position, with using the manual or portable mechanized executive devices. The objects of such jig assembly are groups of flat units and panels, long-sized parts and units, panels of single and double curvature, aggregates. Thus, the most common typical representatives of the sub-assemblies can be sections, panels of fuselage compartments, wings, centerplane, tail unit group as well as wing and empennage control surfaces.
3 Technological Model of Pneumo-Impulse Hole Mandrelling Cold expansion of bolt holes in aerospace components, as a SPD method, is an important technological factor of their strengthening, which ensures an increase of the joints and the whole structures service life. Pneumo-impulse reversable hole mandrelling, under conditions of sufficiently enstiffenning of assembly components in a jig, due to the it open dynamic loading scheme not only solves the problem of one-sided access to the places of operations, but has some additional benefits. The main of them is the process automatization option in the robotic complex environment. Reversable mandrelling involves direct and reverse working action of the tool. This is a conducive fact, because it ensures the tool removal from the processed hole after its direct action, and moreover, it resulted in improving the hole quality due to the “secondary” calibration (smoothing) effect of the reverse action. The quality parameters that ensure the enhancing of structures fatigue strength, above the all, as it was aforementioned, are: the accuracy of the strengthened hole diametrical dimensions, the geometric accuracy of its shape, the location of the axes relatively to the bearing surface and its flatness (are there any flushes near the edge of the hole or not). For cold expanded holes, a sign of their profile distortion is corsetness, that is, noncylindricity of hole along its length, which is characterized by the deviation of the hole generatrix from straightness in its longitudinal cross-section. It should be emphasized that the deviation of the generatrix is a general parameter of the hole quality, which reflects not only the type of its shape (corsetness), but also the accuracy of diametrical dimensions. Corsetness δ, the value of which is determined by the difference between the largest and smallest diameter of the hole, reflects only the accuracy of the geometric shape.
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It is no less important the surface layer conditions of the hole being strengthened (roughness, microhardness, magnitude of residual stresses, depth of the strain hardened layer, structure and parameters of the crystal lattice, density of dislocations, other metallographic parameters as well as physical and mechanical properties of material). The working tool for pneumo-impulse reversable hole mandrelling is a mandrel of classic design, the profile of its working part consists of a conical “intake” section, which directly deforms the material, with its front angle α, then a cylindrical strip with a diameter of dm , which ensures the hole final size and the quality of its surface, and a rear section with an angle β, for reduction of friction forces (Fig. 1). Mandrel diameter dm exceeds the diameter of the original hole Dh in the plate by the value , which determines the tightness value (interference of the tool and the hole) and is given in the input data and is called the nominal (or geometric) tightness = dm − Dh .
Fig. 1. Scheme of sizes for surfaces of holes and pneumo-impulse mandrel, subjected to elastic and plastic deformations in direct (a) and reverse (b) stages of pneumo-impulse hole mandrelling.
In the process of direct expansion, the hole in the plate, its material undergoes elastic and plastic deformation. At the same time, the tool – the mandrel – deforms elastically, decreasing in the radial direction to the size dm.el . The difference Cm = dm − dm.el is the elastic deformation of the mandrel in the process of direct expansion; the value of it depends on the properties of mandrel material, the material of the plate and its thickness, on the geometric parameters of the mandrel, the direction of the mandrel initial speed (Vm , Vm.rev ), the value of tightness, a friction contact conditions, and so on. After direct movement of mandrel, its diameter is restored to original size dm , and the diameter of the strengthened hole in the plate decreases as a result of the elastic deformations after unloading. The deformation Cmat.el = dm.el − Dh.el is called the material springback and it depends primarily on the properties of the plate material itself, as well as on the aforementioned factors. The value of m = Dh.el − Dh is the deformation of the hole after direct motion of mandrel relatively to its previous (input) value. Reverse hole expansion is the final stage of the hole strengthening and it is the result of the reverse working action of the mandrel with dm diameter. The angle β in this case will be the front, “intaking” one, and α angle will be a rear. For reverse movement of hole exapansion, the diameter of the hole after direct movement Dh.el will be incoming. By analogy with the direct movement of hole expansion, the reverse one has the following designations (Fig. 1) correspondingly: dm.rev , Cm.rev is the diameter and elastic deformation of the mandrel, rev is the deviation of the hole generatrix (calibration deformation) relatively to the hole after direct movement of expansion, c
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is the calibrating tightness, Cmat.rev is the elastic springback of material, Dh.el is the final “inspectable” (nominal) diameter of the hole to be strengthened. Deviations of the hole generatrix during direct and reverse movement of hole expansion due to the redistribution of stresses along the plate thickness will be uneven (non-linear), and they will determine the type and accuracy of the shape and diametrical sizes of the expanded strengthened hole. Based on practical expediency, the mandrel, during its reverse action, by its diameter dm should ensure the nominal (final, inspectable) diameter of the strengthened hole Dh.rev = Dnom , which is either specified in the drawing and another technical documentation or taken for technological considerations. At the same time, it is necessary to take into account not only the value of tightness , but also the elastic components of deformations arising in the process (coefficients as a percentage of dm or of Dnom ): – of mandrel Cm , arising during the expansion process; – of plate with a hole, Cmat , which occur after the hole expanding process is finished (material springback). Thus, in the general form, the diameter of the mandrel dm , which ensures the “inspectable” nominal diameter of the strengthened hole Dnom = Dh.rev during reverse expansion movement, will be determined as follows: +Tm Cm Cmat dm = Dnom + 100% 100%
(1)
The whole tolerance Tm of the mandrel tool considers the values of tolerances for the diameter of the strengthened hole (according to its drawing) and for the manufacturing and wearing out of the tool. If necessary, based on the requirements, regarding the conditions of operation or the purpose of the structure with strengthened holes (free, functional, for fastening elements, with appropriate fits, others), nominal diameters can be assigned considering allowances for subsequent calibration, and tolerances – considering the magnitude of corsetness. The inlet diameter of the initial hole Dh that is to be expanded with a given value of tightness , taking into account the hole surface roughness Rz , which is improved by filling its valleys with its peaks, is determined by the expression: Dh = dm /(1 +
) − Rz 100%
+TD
.
(2)
Considering the scheme in Fig. 1: Dh = dm − Cm.rev − Cmat.rev − rev − m . According to the definition of the nominal tightness as = dm − Dh , which value depends on design and technological considerations, the “geometric” calculation Dh = dm − would be rather approximate, since it is obvious that = Cm.rev − Cmat.rev − rev −m . Moreover, the actual (true) hole tightness, for example, with direct movement of mandrel, will be determined by the diameter corresponding to its largest expansion, i.e. real = (Dh.el + Cmat.el ) − Dh = − Cm.el .
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4 Results and Discussion Since to determine the main technological parameters of the mandrel reverse movement in pneumo-impulse hole mandrelling it is not enough to use only the data that available for immediate measurement (for example, the deviation of the strengthened holes generatrix, mandrel diameters, original or final diameters of holes), a technological model of pneumo-impulse hole mandrelling was proposed. This technological model for finding out parameters of the mandrel for strengthening the holes in structural materials by one of the methods of surface-plastic deformation, namely by pneumo-impulse reversable hole mandrelling, makes it possible to obtain the necessary parameters of the strengthened hole processing. Using this model, it is possible to find the value of original hole diameter that provides the nominal diametrical size of the strengthened hole with high calculating accuracy. The peculiarity of the presented sequence of calculations is the consideration of physical changes that occur in material of the strengthened hole. At the beginning of the process, while direct mandrel movement, the material is subjected to the strain hardening, but on the one hand it is only a portion of it, and on the other hand – as only the mandrel exits the hole, its diameter changes, due to elastic deformations, trying to return to its original shape. After the reverse movement of the mandrel tool, hole surface has the final strain hardening, and springback is minimized as much as possible. It is these changes that the proposed model, which ensures the calculations reliability, considers. This approach to the problem simplifies the procedure for selecting a working tool, that is mandrel ensuring the specified parameters of the ready part and the expected properties of the strengthened holes walls. That, in turn, gives opportunity to forecast the loads rate that the product can withstand during operating.
5 Conclusion Elastic deformations (Cm , Cmat.el ) that depend on the properties of mandrel and plate materials, tightness, geometric parameters of the mandrel, and some other aforedescribed factors, require research and clarification. Besides, for ensuring an inspectable diameter of the hole to be strengthened, it is also necessary to consider the tolerances for the original and strengthened holes, deviations of the strengthened hole from the correct shape (for example, corsetness) and the initial hole surface roughness, tolerances of the mandrel and its real wearout, since these parameters can be essential, in some cases even appear to be determinatives. It is obvious that the study of the influence of the dynamic process technological parameters, furthermore, under various conditions of mutual influence, on the quality indices of cold expanded holes for finding out the optimal parameters of process that accompanied by elastic deformations of both the tool and the plate with the hole faces not only with material and financial difficulties, but also with significant time expenses. Considering this and considering the state-of-the-art of special soft-ware and computer technology, it is proposed the further study of the pneumo-impulse reversable hole mandrelling by numerical simulating using the finite element method (FEM). It is also would be appropriate an exploratory research to confirm the efficiency of pneumo-impulse hole mandrelling implementing. When conducting the exploratory
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experiments and the subsequent field experimental studies, it is recommended to use the specimens of plates made of commonly used modern and promising structural materials, the mandrels made of 107WCR5 brand steel (the steel alloyed with chromium, manganese, vanadium), geometric parameters of which are recommended in the Manufacturing Instructions (kind of manufacturing regulatory documentation, compiled by results of scientific studies). Due to the need in technical inspection and measurements of holes quality parameters after direct movement of mandrel (before accomplishing the reverse movement of pneumo-impulse hole mandrelling), it is recommended to use a PID-90M pneumatic impulse gun, carrying out the process in two separate movements, thus simulating direct and reverse schemes of pneumo-impulse hole mandrelling (Fig. 1).
References 1. Devezas, T.: Trends in aviation: rebound effect and the struggle composites x aluminum. Technol. Forecast. Soc. Chang. 160, 120241 (2020). https://doi.org/10.1016/j.techfore.2020. 120241 2. Kombarov, V., Sorokin, V., Tsegelnyk, Y., et al.: Numerical control of machining parts from aluminum alloys with sticking minimization. Int. J. Mechatron. Appl. Mech. 1(9), 209–216 (2021). https://doi.org/10.17683/ijomam/issue9.30 ´ 3. Sliwa, R.E., Pawłowska, B., Balawender, T., Zwolak, M.: Extrusion of lightweight aluminum and magnesium alloys structures for aviation applications. Key Eng. Mater. 926, 523–536 (2022). https://doi.org/10.4028/p-mmgjqk 4. Kondratiev, A., Píštˇek, V., Smovziuk, L., et al.: Stress-strain behaviour of reparable composite panel with step-variable thickness. Polymers 13(21), 3830 (2021). https://doi.org/10.3390/pol ym13213830 5. Ugrimov, S., Smetankina, N., Kravchenko, O., Yareshchenko, V.: Analysis of laminated composites subjected to impact. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) ICTM 2020. LNNS, vol. 188, pp. 234–246. Springer, Cham (2021). https://doi.org/10.1007/978-3-03066717-7_19 6. Al-Samman, T., Letzig, D., Yi, S.: Microstructure-mechanical properties and application of magnesium alloys. Metals 11(12), 1958 (2021). https://doi.org/10.3390/met11121958 7. Zagvozdkina, T.N., Karachevtsev, F.N., Dvoretskov, R.M., Mekhanik, E.A.: The use of optical methods of measurement to investigate the composition of new aviation materials. Meas. Tech. 58(3), 362–366 (2015). https://doi.org/10.1007/s11018-015-0717-z 8. Yanko, T., Dmytrenko, O.: Prospects for the implementation of new materials and technologies in the aerospace industry. Trans. Aerospace Res. 4, 1 (2021). https://doi.org/10.2478/tar-20210019 9. Xin, S., Zhang, J., Mao, X., et al.: Research and development of low-cost titanium alloys. J. Phys: Conf. Ser. 1347, 012022 (2019). https://doi.org/10.1088/1742-6596/1347/1/012022 10. Niknam, S.A., Khettabi, R., Songmene, V.: Machinability and machining of titanium alloys: a review. In: Davim, J.P. (ed.) Machining of Titanium Alloys. MFMT, pp. 1–30. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43902-9_1 11. Pavlenko, D., Dvirnyk, Y., Przysowa, R.: Advanced materials and technologies for compressor blades of small turbofan engines. Aerospace 8(1), 1 (2020). https://doi.org/10.3390/aerosp ace8010001 12. Okninski, A.: Solid rocket propulsion technology for de-orbiting spacecraft. Chin. J. Aeronaut. 35(3), 128–154 (2021). https://doi.org/10.1016/j.cja.2021.07.038
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13. Fuchs, C., Rodríguez, I., Baier, D., Zaeh, M.F.: Process planning for the machining of Ti6Al-4V near-net shaped components. Proc. CIRP 101, 58–61 (2021). https://doi.org/10.1016/ j.procir.2020.03.155 14. Zong, Y., Ye, J.: Research on the development of titanium alloy recovery technology in civil aviation industry. In: 2020 International Conference on Optoelectronic Materials and Devices, vol. 11767, pp. 304–310. SPIE (2021). https://doi.org/10.1117/12.2592369 15. Shapovalova, O.M., Ivanova, L.N.: Titanium alloys properties for aerospace engineering at the negative and cryogenic temperatures. Space Sci. Technol. S.1(11), 152–155 (2002). https:// doi.org/10.15407/knit2002.01s.152 [in Ukrainian] 16. Kritskiy, D., Pohudina, O., Kovalevskyi, M., et al.: Powder mixtures analysis for laser cladding using OpenCV library. In: Nechyporuk, M., et al. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2021. LNNS, vol. 367, pp. 924–937. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-94259-5_72 17. Peters, M., Kumpfert, J., Ward, C.H., Leyens, C.: Titanium alloys for aerospace applications. Adv. Eng. Mater. 5(6), 419–427 (2003). https://doi.org/10.1002/adem.200310095 18. Kang, L., Yang, C.: A review on high-strength titanium alloys: microstructure, strengthening, and properties. Adv. Eng. Mater. 21(8), 1801359 (2019). https://doi.org/10.1002/adem.201 801359 19. Sharif, S, Rahim, E., Sasahara, H.: Machinability of titanium alloys in drilling. In: Amin, A.N. (eds.) Titanium Alloys – Towards Achieving Enhanced Properties for Diversified Applications, pp. 117–137. IntechOpen, London (2012). https://doi.org/10.5772/35948 20. Skvortsov, V.F., Arlyapov, A.Y., Boznak, A.O., Kim, A.B.: Residual stresses in compression and tension mandrelling thick-walled cylinders. In: 2014 International Conference on Mechanical Engineering, Automation and Control Systems (MEACS), pp. 1–4. IEEE (2014). https://doi.org/10.1109/MEACS.2014.6986948 21. Fu, Y., Ge, E., Su, H., et al.: Cold expansion technology of connection holes in aircraft structures: a review and prospect. Chin. J. Aeronaut. 28(4), 961–973 (2015). https://doi.org/ 10.1016/j.cja.2015.05.006 22. Skvortsov, V.F., Boznak, A.O., Kim, A.B., et al.: Reduction of the residual stresses in cold expanded thick-walled cylinders by plastic compression. Defence Technol. 12(6), 473–479 (2016). https://doi.org/10.1016/j.dt.2016.08.002 23. Pichon, G., Daidie, A., Fau, A., Chirol, C., Benaben, A.: Cold working process on hard metal stacked assembly. In: Roucoules, L., Paredes, M., Eynard, B., Morer Camo, P., Rizzi, C. (eds.) JCM 2020. LNME, pp. 41–47. Springer, Cham (2021). https://doi.org/10.1007/978-3-03070566-4_8 24. Plankovskyy, S., Popov, V., Shypul, O., et al.: Advanced thermal energy method for finishing precision parts. In: Gupta, K., Pramanik, A. (eds.) Advanced Machining and Finishing, pp. 527–575. Elsevier, Amsterdam (2021). https://doi.org/10.1016/B978-0-12-817452-4.000 14-2 25. Zhivov, A.: Ventilation. In: Zhivov, A. (eds) Ventilation and Energy Efficiency in Welding Shops A Practical Guide SpringerBriefs in Applied Sciences and Technology SAST. Springer, Cham, vol. 29, p. 85 https://doi.org/10.1007/978-3-030-77295-6_6 (2022) 26. Voronko, V.V.: Designing of the process and tools for high-speed aperture burnishing in aluminum aircraft constructions. Dissertation, National Aerospace University “Kharkiv Aviation Institute” (2007) [in Russian] 27. Krivtsov, V.S., Vorob’ev, Y.A., Voron’ko, V.V.: Advanced devices for mandreling bores. Kuznechno-Shtampovochnoe Proizvodstvo (Obrabotka Metallov Davleniem) 12, 18–30 (2004)
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28. Voronko, V., Dyachenko, Y., Voronko, I., et al.: Automation of the pneumatic impulse mandreling technological. In: Arsenyeva, O., et al. (eds.) Smart Technologies in Urban Engineering – 2022. STUE 2022, LNNS, vol. 536, pp. 1–12, Springer, Cham (2023). https://doi.org/ 10.1007/978-3-031-20141-7_52 29. Vorobiov, I., Maiorova, K., Voronko, I., et al.: Creation and improvement principles of the pneumatic manual impulse devices. In: Nechyporuk, M., et al. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2021. ICTM 2021. LNNS, vol. 367, pp. 178–191. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-94259-5_17 30. Moravec, J., Blatnický, M., Dižo, J.: An application of a magnetic impulse for the bending of metal sheet specimens. Materials 15(10), 3558 (2022). https://doi.org/10.3390/ma15103558 31. Aksonov, Y., Kombarov, V., Tsegelnyk, Y., et al.: Visualization and analysis of technological systems experimental operating results. In: 2021 IEEE 16th International Conference on Computer Sciences and Information Technologies (CSIT), vol. 2, pp. 141–146. IEEE (2021). https://doi.org/10.1109/CSIT52700.2021.9648592
Experimental Setup with Stand-Alone Automatic Modular Logger of Parameters of Thermal Pulse Processing Olga Shypul1(B)
, Oleksiy Pavlenko1 , Yevhen Aksonov2 and Sergiy Zaklinskyy1
,
1 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected] 2 O.M. Beketov National University of Urban Economy in Kharkiv, 17 Marshala Bazhanova
Street, Kharkiv 61002, Ukraine
Abstract. The problem of creating a database of reliable experimental data on the technological parameters inherent in thermal pulse processing, considering the complexity of the processes occurring during the working cycle in a closed volume combustion chamber is investigated. The aim of the work is to develop a special experimental setup for studying the parameters of transient processes of interaction between the combustion products of a gas mixture and workpieces. The following results were obtained. Firstly, an be easily reconfigured stand-alone automatic modular logger of thermal impulse processing in a wide range of parameters and values, which, along with guaranteed protection of sensors and electronic components from the dangerous effects of high temperatures, pressure, and shock waves have been developed. Secondly, a specialized autonomous software and hardware complex that provides polling of sensors and storage of the experimental measurements for further analysis and processing has been developed. Thirdly, an experimental setup, consisting of the experimental chamber with a data logger inside, attached through an accelerating tube to a shock wave generator or a pulsecompression detonation system, depending on the problem being solved has been developed. Keywords: thermal pulse processing · shock wave generator · stand-alone data logger · automatic measurements
1 Introduction The modern approach to the existence of industrial technologies require not only the initial automatic adjustment of equipment operation modes but also the use of systems for their control and real-time adjustment according to the concept of digital twins [1, 2]. A digital twin is a virtual copy of a physical object and is used as a predictive model simulator and/or a diagnostic and equipment tuning tool. In process control, the digital twin allows testing and verification of various operation strategies, while the performance of the control system depends on the available sensors and actuators of the digital twin © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 16–28, 2023. https://doi.org/10.1007/978-3-031-36201-9_2
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[3]. A typical cycle for the development of the digital twin of equipment and its further use for modeling technological processes is considered in [4–8]. The task of digitalization of modern technologies, especially in the production of precision parts and mechanisms, is dictated, among other things, by the need to ensure such important quality indicators as service life and reliability, which in turn are largely determined by the quality of edge processing and surface cleaning. Unique advantages in terms of efficiency and technological capabilities, among other finishing methods, are provided by technologies for finishing with products of the fuel mixture combustion, which are represented by various types of heat and power methods, including thermal pulse one [9, 10]. The development of a digital twin of thermal pulse processing technology requires the use of a complex model of all equipment operation processes, including gas-dynamic processes of generating and filling the working chamber with a gas mixture, ignition, and combustion of fuel, heat exchange with processed objects, the exhaust of combustion products, etc. [11]. It is clear that the creation of this complex model, especially the implementation of calculations based on it, is unreasonably difficult. Therefore, it is rational to develop digital twins of individual physical and chemical processes inherent in thermal pulse processing and further general digitalization of the entire technology. There are several ways to build digital twins of processes, but the common principle for them is the principle of processing and comparing information from real sensors that measure process parameters with the readings of virtual sensors of the digital twin [12, 13]. Of course, the results received from digital twin will only be reliable if the input data used to describe the process being modeled is complete and comprehensive. Due to the complexity of the processes that occur during the working cycle of processing parts with a detonation mixture in the chamber of thermal impulse equipment and several factors affecting the quality of processing, the main problem in developing a digital twin is to create a database of reliable initial data on pressure parameters, shock wave velocity and temperature on the surface parts directly during the work cycle [14–16]. This task is complicated because even the most modern methods of experimental studies of heat transfer under shock waves are either unsuitable for the conditions of high-temperature detonation processes or require the use of additional equipment to transmit the signal from the sensors outside the working chamber. This not only introduces errors caused by the interference of the wave pattern when shock waves are reflected from the equipment elements but also requires a violation of the integrity of the walls of the working chamber, which is unacceptable due to the reduction of its strength characteristics. Concerning the above, the issue of developing an autonomous system for recording experimental data synchronized with the time control system of thermal pulse installation is topical. The use of stand-alone data loggers in research and industry is well known. They are used where the presence of a person is impossible due to the risk of injury, exposure to harmful environmental factors, the long duration of technological processes, etc. Various systems for automatic registration of parameters of dangerous processes, meteorological probes, hydrological, hydrographic buoys, etc. are widely used [17–21]. An autonomous gauge for blast impulse determination close to explosive charges [22]
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works under conditions similar to those that arise during thermal impulse action. Unfortunately, the passive method of measuring parameters, underlying its functioning, cannot be used in the study of processes in the combustion chamber of thermal pulse machines. In any case, the design of such equipment is based on the general idea – all electronic equipment is isolated from the external environment by a special closed capsule, and only special sensors have contact with the environment. Depending on the conditions of use, the recorders can be powered by a centralized power supply network or equipped with autonomous power sources. Based on the foregoing, the purpose of this work is to develop a special experimental setup that includes a shock wave generator with a wide energy range and an Stand-alone Automatic Modular Parameter Logger (hereinafter referred to as the data logger) with the ability to measure, write, read, convert and store information, a sufficient level of protection sensors, electronic components, and power supplies from high temperatures and pressures for its further use in the study of the parameters of transient processes of the interaction of combustion products of the fuel mixture with machined parts.
2 Development of a Stand-Alone Automatic Modular Parameter Logger There are several requirements for the data logger housing. First of all, it must be strong enough and reliably protect the experimental equipment from the effects of high temperatures and pressure from the outside. The geometric dimensions of the data logger depending on the maximum energy that the process equipment can develop and the volume of its working chamber. Its internal volume should be sufficient to accommodate sensors, electronic components, and power supplies. The housing should be easy to assemble and disassemble and allow quick and flexible replacement of sensors. Considering all the requirements, the authors of this work analyzed various shapes and chose the most suitable for use octagonal shape of the data logger housing (Fig. 1) [23]. The basic design of the data logger housing consists of ten parts: two octagonal end caps and eight rectangular side walls of a trapezoidal cross-section. The shape of the logger parts, as the use of rubber seals, makes it possible to increase the strength of the structure during assembly and prevents the penetration of high temperature and pressure gases into it. Depending on the study objectives, the data logger can be placed in the working chamber of the thermal-pulse installation both horizontally and vertically. The horizontal arrangement (Fig. 1) is most typical for studying the parameters of the part walls oriented at an angle to the direction of propagation of the combustion front or detonation wave. The octagonal shape of the logger body together with a special cradle (Fig. 2) allows you to change the inclination angle of the face with sensors in 15° increments. When conducting experimental studies, it may be necessary to measure the speed of propagation of the combustion front, temperature gradient, etc. In this case, the sensors must be oriented in the direction of propagation on the segment of the measured length. To do this, the logger housing must be oriented vertically (Fig. 3).
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Fig. 1. The Stand-alone Automatic Modular Parameter Logger (Housing Structure).
Fig. 2. Profile of the cradle.
Additionally, it should be noted that the logger housing can be equipped with fairings of various shapes, reflectors, reflectors, and visors to bring its shape closer to the real part. Also, to study the behavior of burrs under the effect of high temperatures, their models, made from foil of the corresponding grades of materials of different thicknesses, can be attached to the logger housing with special holders.
Fig. 3. Vertically located data logger with spherical fairing.
The location of the sensors on the data logger housing. The sensors used in the data logger are of similar design and size. A high-speed pressure sensor of model 113B [24] manufactured by PCB Piezotronics Inc., USA (Fig. 4) is typical. The length of the sensor is 1.48 inches (37.4 mm), the diameter of the sensitive part is 0.218 in. (5.54 mm), and
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the mounting thread size is 5/16–24 UNF (inch analog of the metric M7 × 0.75), wrench size – 6 mm.
Fig. 4. High-speed pressure sensor of model 113B manufactured by PCB Piezotronics Inc.
Sensors can be installed on both end caps and side walls in any combination and any place. It depends only on the aim of the experiment. An example of the sensors location on the end caps and side walls of the logger housing is shown in Fig. 5.
Fig. 5. An example of the location of sensors (pos. SR) on the data logger housing end caps and side walls.
3 Hardware-Software Complex for Registration of Experimental Data To log the results of experiments, a specialized autonomous hardware and software complex was developed. It consists of several analog-to-digital (ADC) conversion blocks and a data logging block. The overall dimensions of the hardware part of the complex are determined by the internal volume of the logger housing. This imposes certain restrictions on the functionality of the complex. At the same time, it must comply with the list of mandatory requirements that ensure the fulfillment of tasks as part of the experimental setup.
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The following requirements are put forward for ADC blocks. 1. Compatibility of the input interface with PCB 113B22 type pressure sensors. 2. Operational amplifier with digital gain control and zero offset. 3. An ADC chip that converts a 16-bit signal with a sampling frequency of at least 100 kHz. 4. Minimal power consumption for longer battery life. The following requirements are put forward for the data logger. 1. Autonomous operation from the rechargeable battery. 2. Operating modes: “Data logging” and “Standby”. Ensuring the lowest possible power consumption in standby mode. 3. Selecting configuration parameters and entering the “Data logging” mode without physical contact with the equipment by the command from automatic sensor. 4. Data logging within a specified time from a start signal is received, followed by an automatic transition to standby mode. The duration of operation in the “Data logging” mode is determined from 0.1 s to 1 s by the setup settings. 5. Storage of registered data in read-only memory (ROM). The ROM storage capacity should provide data recording from eight ADC blocks with a total duration of up to 5 min. 6. Synchronization of the results of experiments in time at the moment the command to start the data recording process is received. 7. Fault tolerance and work stability are provided by a programmable logic integrated circuit (FPGA). 8. Timing is carried out using a high-precision clock generator. 9. Reading the registered data occurs after the experiments by connecting the complex to an external data interface. By these requirements, a software and hardware complex for recording experimental data, which block diagram is shown in Fig. 6 was developed. A block diagram of the configuration of the software part of the data logger is shown in Fig. 7. Data Logger Hardware. Due to the limited internal dimensions, the use of complex filters for the analog signal coming from the sensors is not possible. Therefore, the simplest filtering schemes are used, and noise suppression is provided by reducing the length of the analog signal transmission lines. Therefore, the ADC blocks are made as separate devices placed next to the pressure sensors. Setting parameters, reading and saving data from ADC blocks, their further transfer to a PC is provided by the data logger. A feature of the input stage of the ADC block is the digital control of the gain and zero-level offset. This makes it possible to store their values in non-volatile memory and load them after power is applied to the device. In addition, this approach allows you to create different settings profiles and change them as needed when an external command arrives without the need for physical contact with the data logger. To reduce power consumption, the power supply of individual interfaces and ADC blocks is activated separately, depending on the needs of a particular experimental study, determined by the corresponding configuration parameters.
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Fig. 6. Block diagram of the hardware-software complex for logging the experimental data: SR – sensor; OA – operational amplifier; ADC – analog-to-digital converter; ADC Int. – ADC interface; FPGA – field-programmable logic integrated circuit; ROM – Read-Only memory; CG – clock generator; ST-SP – start/stop sensor; EI – external interface; IC – interface converter; PC – personal computer.
Fig. 7. Block diagram of the internal configuration of the software part of the complex: RAM – random access memory; RTEI – receiver-transmitter of external interface; EI – external interface; RTC – real-time counter; S/SP – start/stop sensor signal processor; CF – clock frequency.
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The start of the registration process and the setting of the logger configuration profiles are controlled by a pressure change in the working chamber of the machine. When filling with a fuel mixture, the pressure in it increases, and after the end of the working cycle returns to atmospheric. Thus, when the start/stop sensor located inside the logger responds to an increase in pressure to a certain level relative to atmospheric pressure, the polling of the sensors and the recording of the obtained values into memory start automatically. At the end of the working cycle, when the pressure in the working chamber drops to atmospheric pressure, the start/stop sensor stops the data logger. The Software Part of the Complex. Is built on a field-programmable logic integrated circuit (FPGA), which internal configuration is shown in Fig. 7. Unlike microcontrollers, the FPGA is programmed with a logic circuit that performs parallel signal processing. This ensures high speed, stability, and fault tolerance of the developed device. An external high-speed serial interface is provided for setting parameters and transferring saved data. Communication with a PC is carried out via USB protocol by an interface converter. Synchronization of the time of each experiment is carried out by a command to start data recording coming to the FPGA from the start/stop sensor. The memory management module provides synchronous writing of data received from the ADC to the RAM and their subsequent serial transfer to the ROM. After a series of experimental studies, the logger is connected to a PC to transfer data stored in ROM. Data Processing System Software. Reading the experimental data and setting the parameters of the data logger is carried out by a specially developed utility for a PC. It is possible to save the current settings of the complex in profile files and their further loading. The analysis and processing of the obtained data are carried out by the software for visualization and data processing of the CNC system [25, 26]. This software has the following features. 1. Graphical presentation of data arrays as time-synchronized graphs. 2. The possibility of convenient interactive navigation (zooming, moving) in the coordinate field of the program window. 3. Ability to perform mathematical calculations on data sets, including filtering and spectral analysis functions. 4. Automatic calculation and display of graph extrema and average value on a given interval. 5. Data arrays export and import. 6. Sorting in the list of arrays and searching by specified parameters. Thus, the developed hardware and software complex provides autonomous registration of experimental results, storage in the ROM, and further transfer to a PC for analysis and processing.
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4 Experimental Setup with Stand-Alone Automatic Modular Parameter Logger The creation of technical support for experimental studies of processes in the chamber of thermal impulse equipment during the working cycle is aimed at providing reliable information about their parameters. The sensors installed in the data logger make this task possible. Work on their verification and, if necessary, calibration should be carried out under normal conditions by the regulations established by the manufacturer. Therefore, for this task, placing the logger directly in the combustion chamber of a thermal pulse machine is not advisable due to unavoidable measurement errors due to heating of its structure, reflection of detonation waves, etc. The experimental setup for shock wave generation developed by the authors of this work (Fig. 8), along with the above purpose, is used, among other things, to calibrate the sensors of the data logger under conditions as close as possible to the real picture of the processes inherent in thermal pulse processing.
Fig. 8. Experimental setup for shock wave generation: 1 – “Exen-3” pneumopulse generator; 2 – accelerating tube; 3 – exemplary sensors; 4 – experimental chamber.
The setup consists of an “Exen-3” pneumopulse generator (1) designed to create air shock waves by converting compressed air energy into shock wave energy, an accelerating tube (2) of about two meters long with built-in sensors (3), and an experimental chamber (4) where the data logger is accommodated. The experimental chamber has the same shape and internal dimensions as the working chamber of the real equipment. Since the combustion of the fuel mixture does not occur in it, the requirements for its strength are significantly reduced. Therefore, exemplary sensors can be installed directly on its wall, as close as possible to the sensors of the data logger. The pneumatic circuit diagram of the experimental setup is shown in Fig. 9.
Fig. 9. Pneumatic circuit diagram of the experimental setup: 1 – shut-off valve; 2 – reducer; 3 – filter; 4 – oil-moisture separator; 5 – check valve; 6 – solenoid valve; 7 – receiver; 8, 10, 11, 12 – exemplary sensors; 9 – speed valve; 13 – experimental chamber.
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When the solenoid valve 6 opens, receiver 7 is filled with compressed air to a predetermined pressure controlled by the pressure sensor 8. In several milliseconds after the opening of high-speed valve 9, the receiver is discharged. The flow of compressed air forms a shock wave at the end of accelerating tube 2 (according to “Exen-3” device specifications). The experimental chamber (Fig. 10) is equipped with brackets, by which the position of the logger along the x and y axes is changed. The design of the data logger makes it possible to change the angle of inclination α of the surface with sensors to the direction of propagation of the generated shock wave.
Fig. 10. Location of the stand-alone parameter logger in the chamber of the experimental setup.
The “Exen-3” pneumopulse generator provides high repeatability of shock waves in a narrow energy range. Exemplary sensors installed on the accelerating tube show the parameters at the entrance to the experimental chamber. Calibration of the logger sensors is carried out by comparing the parameters at the entrance to the chamber with those measured by the logger directly inside it and compiling calibration tables. To calibrate the data logger sensors in a wide range of detonation energies, the “Exen-3” pneumopulse generator should be replaced with a pulse-compression detonation (PCD) system (Fig. 11). It allows to control the distribution of pressure, temperature, and concentration of combustible substances before detonation initiation and, as a result, can provide both gradient and linear distribution of gas pressure in the detonation tube for various application strategies [27].
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Fig. 11. Experimental setup based on a pulse-compression detonation system: 1 – drive motor; 2 – pulse-compression generator of detonation wave; 3 – accelerating tube with exemplary sensors and universal interface.
5 Conclusion As a result of the work carried out, a stand-alone automatic logger of thermal pulse processing parameters was developed. It is located inside the chamber during the work cycle. The data logger housing protects the sensors and electronic components from dangerously high temperatures, pressures, and shock waves. The design of the data logger housing makes it easy to replace panels with sensors. This provides a quick and flexible reconfiguration of the data logger for measurements with sensors for various purposes in a wide range of parameters and values. A specialized stand-alone hardware-software complex has been developed. It provides registration of experimental results in a limited volume of the internal space of the logger, their storage in ROM, and subsequent transfer to a PC for analysis and processing, as well as for further use in the chain of the digital twin of the process. The electronic components of the hardware-software complex are isolated from the external environment by the data logger housing and are powered by autonomous energy sources. The storage capacity of the memory device is sufficient for a cycle of experiments, after which the logger is removed from the working chamber, opened, and connected to a personal computer to transfer the stored information. After charging the batteries, the data logger is ready for a new cycle of experiments. The design of an experimental setup, consisting of a shock wave generator and experimental chamber with a data logger fixed inside it is proposed. The generator and chamber are interconnected by an accelerating tube. Exemplary sensors installed on the tube near the chamber register the parameters of pressure and temperature at its inlet. Based on the comparison of these parameters with those measured by the logger inside the chamber, calibration tables are compiled. This ensures the reliability of data measured by the logger sensors. To check the measurement accuracy of the logger and the limits of its application the use of a pulse-compression detonation system for the generation of detonation waves in a wide energy range is proposed.
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Estimation of the Minimum Uncut Chip Thickness Utilising Conventional Milling of S960QL and C45E Steels Imre Palágyi, István Biró(B) , and Tibor Szalay Department of Manufacturing Science and Engineering, Budapest University of Technology and Economics, M˝uegyetem Rakpart 3, 1111 Budapest, Hungary {biro.istvan,szalay.tibor}@gpk.bme.hu
Abstract. Today’s technological advancements aim to make small-sized, compact products. This trend creates requirements for higher accuracy of cutting processes, where the minimum uncut chip thickness presents itself as a technical limit of removing chip in a controllable way. The following research aims to estimate the minimum uncut chip thickness and how it is affected by the cutting speed and the feed per cutting edge utilising conventional milling with a conventional-sized end mill. Keywords: minimum uncut chip thickness · ploughing · conventional milling · cutting force measurement
1 Introduction Surfaces with the highest accuracy are usually made with machining processes since this is the most cost-efficient solution for mass production in the current state of technology. However, when the layer of material we wish to remove is too thin, the resulting surface’s microgeometry becomes deteriorated by roughness with visible cracks. In these conditions, the feed per cutting edge ( f z ) is in the scale of the cutting edge radius (r β ), which results in the uncut chip thickness (h), which is the thickness of the removed material layer, being in the scale of the minimum uncut chip thickness (hmin ) (see Fig. 1) [1]. The material moves on the cutting edge in two directions: one portion stays on the workpiece and suffers plastic deformation, and the other peels off as a chip [2]. The size effect aligns with the before-mentioned effect as the specific cutting force drastically increases by decreasing the uncut chip thickness [3]. The cutting edge radius affects the minimum uncut chip thickness: hmin increases as the r β radius increases, and so does the effective rake angle (γ eff ) at the same applied feed [4]. Regarding the machining parameters, previous researches show that the minimum uncut chip thickness increases as the feed is increased [5]. The force components of cutting indicate the effect of hmin as well since the normal force (F n ) is relatively higher compared to the cutting force (F c ) at an uncut chip thickness smaller than hmin due to the intensified ploughing [6]. The following research aimed to examine the circumstances © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 29–38, 2023. https://doi.org/10.1007/978-3-031-36201-9_3
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of how the minimum uncut chip thickness and the corresponding cutting conditions could be estimeted by utilising simply realisable conventional milling tests for S960QL high-strength structural steel and C45E normalised carbon steel.
Fig. 1. The transition between the ploughing-dominated plastic deformation and the chip formation
2 Design of Experiment and the Cutting Tests The cutting tests have been designed as conventional milling tests with constant depth (ap ) and width (ae ) of cut, as presented in Fig. 2. Conventional milling was chosen as the applied cutting process because, in the case of side milling, the uncut chip thickness varies between the theoretical zero and a maximum value (hmax ), which is determined by the f z feed per cutting edge due to the kinematics of milling. This circumstance provides the opportunity to examine the conditions of chip removal at many different uncut chip thicknesses without altering any cutting parameter. During conventional side milling, the uncut chip thickness is zero when the cutting edge meets the workpiece and then gradually increases until it reaches hmax , which is approximately in the position where the cutting edge left the point of entry into the material by 90° of the tool’s rotation [7]. Since the cutting edge enters the material with an uncut chip thickness smaller than hmin , the impact of ploughing was expected to be more significant than it would be during climb milling. (In the case of climb milling, the cutting edge is already separating the material in the chip’s root before the uncut chip thickness gets below hmin .) Therefore, ae has been determined to provide side milling with half of the tool’s nominal diameter. Only a single cutting edge was applied to avoid the force components per cutting edge being summed up during the measurement, therefore being able to observe the conditions of a single chip’s removal directly. A Sumitomo WEX2016E end milling cutter with a Sumitomo AXMT123504PEERG insert was used to carry out the cutting tests. The tool’s nominal diameter is d = 16 (mm), the axial immersion angle is κ r = 90°, the tip radius of the insert is r ε = 0.4 (mm), and the cutting edge radius is in the range of r β = 30…70 (µm).
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31
Fig. 2. The setting of the milling tests
The factors (i.e. parameters) of the milling tests are listed in Table 1. The cutting speed (vc ) and feed per cutting edge ( f z ) intervals were chosen according to earlier research [3, 5] and taking the recommended parameters by the tool manufacturer into account. The research in [3, 5] implies an exponential correspondence of the minimum uncut chip thickness to the cutting parameters. Therefore, five levels (i.e. values) of both the cutting speed and the feed rate per cutting edge have been determined to form a geometric progression. This design of experiment resulted in a set of nine different parameter combinations (as presented in Table 1), of which cross-sections with the same factor levels were regarded as control measurements (see Tests 3 and 8). The s sampling rate of the force measurement was set to acquire data as the milling cutter turns ϕ = 0.159155 = 0.5 °, similarly to how it was executed in [3], and expressed by Eq. (1), where ω stands for the tool’s angular speed. Figure 2, where vf shows the direction of the tool’s feed rate, interprets the cutting parameters included in Table 1. s=
100 · vc ω = ϕ 3 · d · ϕ
(1)
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The experiments were carried out in the C45E normalised carbon steel and the S960QL high-strength structural steel by a KONDIA B640 3-axis CNC machining centre in the laboratory of the Department of Manufacturing Science and Engineering, Budapest University of Technology and Economics. During the tests, a KISTLER 9257B piezoelectric force sensor and a KISTLER 5070A charge amplifier detected the force components affecting the workpiece in the static X-Y-Z coordinate system of the machining centre. An OMRON E3FDS10B4 proximity sensor registered the cutting edge’s arbitrarily chosen position by producing a high signal value when the cutting edge arrived in that position in each period of the milling process (i.e. each time the milling cutter made a full rotation). Data of the force measurement and the proximity signal were acquired by a National Instruments USB-4431 and a National Instruments BNC-2110 data collectors. Table 1. Parameter settings of the milling tests No
Feed per edge f z (mm)
Cutting speed vc (m/min)
Depth of cut ap (mm)
Width of cut ae (mm)
Sampling rate s (Hz)
1
0.057
50
2
8
11937
2
71
16950
3
100
23873
4
141
33661
5
200
47746
6
0.020
7
0.034
8
0.057
9
0.095
10
0.160
100
2
8
23873
Each cutting test was composed of 50 full periods of milling when both the ap and ae parameters were stabilised at their designated values determined in the design of experiment. The data acquisition frequency has been set to register the same amount of data per period during each milling test, similarly to previous research [5].
3 Data Evaluation First, the average trends of the X-Y-Z force components for the time interval of a single period have been determined from the 50 measured periods. Second, the averaged X-Y-Z force components have been transformed into cutting force (F c ) and normal force (F n ) components. With regard to the force measurement’s coordinate system in Fig. 2, the force components have been transformed following Eq. (2). Figure 3 shows an example of the average characteristics of the force components, where U represents the proximity
Estimation of the Minimum Uncut Chip Thickness
signal.
⎧ ⎫ ⎨ Fc (ϕ) = Fx · cos ϕ + Fy · sin ϕ ⎬ F (ϕ) = Fx · sin ϕ − Fy · cos ϕ ⎩ n ⎭ Fp (ϕ) = Fz (ϕ)
600 400
(2)
Fx
Fy
Fz
Fc
Fn
U
200
600 400 200
0
0
-200
-200
-400
-400
-600
U (V/100)
F (N)
33
-600 -30
0
30
60
90
120 150 180 210 240 270 300 330 (°)
Fig. 3. The measured and transformed force components of cutting (example by Test 9 in C45E)
The F c and F n components form the F active force of cutting (see Fig. 2) and determine the Θ kinetic angle as Eq. (3). Concerning that the axial immersion angle is κ r = 90°, the uncut chip thickness (h) is approximated by the f z feed rate per cutting edge and the ϕ angular position of the cutting edge according to Eq. (4). The ϕ position is derived from the proximity sensor’s signal. Fn (3) θ = arctan Fc h(ϕ) = fz · sin ϕ
(4)
Two distinctive uncut chip thickness values have been determined to indicate the conditions of increased ploughing indirectly (see the graphical explanation in Fig. 4): a. On the plots of kinetic angle per uncut chip thickness, a maximum value is identifiable as marked by the points hΘmax . Since a trait of ploughing is that the F n normal force increases relatively to the F c cutting force [6], the uncut chip thickness at the maximal kinetic angle may indicate the condition where ploughing is still the dominant processing mechanism over chip removal.
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b. A general condition of a cutting process dominated by ploughing is that the measure of the F n force exceeds that of the F c , resulting in the kinetic angle being Θ > 45° according to Eq. (3). The h45° marks the uncut chip thickness where the fitted exponential trend of Θ rises above 45° for the first time as the uncut chip thickness decreases.
55
C45E S960QL C45E trend S960QL trend
hΘmax
Θ (°)
50
45
hΘmax
40
35 h45°
h45°
30 0
0.02
0.04
0.06
0.08
0.1
h (mm)
Fig. 4. Determination of the distinctive uncut chip thicknesses based on the kinetic angle (example by Test 9)
Figures 5 and 6 represent the trend of h45° and hΘmax as a function of the cutting speed, respectively. It is observable that values of both distinctive uncut chip thicknesses increase with the cutting speed, though no more sophisticated trend can be ascertained from the currently measured data. However, an exponential trend of both h45° and hΘmax as a function of the feed rate is determinable, as presented in Figs. 7 and 8, respectively. The cause of this effect may be that as the feed rate increases, the cutting edge is pushed into the workpiece creating an increasing slide-off on the surface-to-be-machined before the cutting edge can cleave into the material. The cutting edge’s intense slide-off increases the material surface’s elastic limit mechanically, making the workpiece’s local surface layer more resilient against plastic deformation (a crucial mechanism of chip removal). Due to the increased resiliency, more considerable deformation provided by a larger uncut chip thickness is required to start the material separation. The effect of the feed rate on h45° and hΘmax bears characteristical similarities to the results of [3] in terms of the minimum uncut chip thickness. The explained trends of h45° and hΘmax have been formalised: Eq. (5) represents the trend considering the cutting speed, and Eq. (6) shows the exponential characteristic per the feed rate. Both models have been determined using the linear or linearised
Estimation of the Minimum Uncut Chip Thickness
35
least-squares regression method, and the resulting coefficients are collected in Table 2 alongside the R2 determination coefficient. h(vc ) = cv,1 · vc + cv,2
(5)
h(fz ) = cf ,1 · ecf ,2 ·fz
(6)
Table 2. Coefficients of the distinctive uncut chip thickness models by Eqs. (5) and (6) Model
h(vc )
Material h45° (µm)
hΘmax (µm)
h( f z ) S960QL
C45E
S960QL
C45E
cv,1
0.558
0.104
cf,1
0.508
0.289
cv,2
−20.545
−2.916
cf,2
38.383
21.845
R2
0.642
0.665
R2
0.950
0.881
cv,1
0.124
0.047
cf,1
0.396
0.602
cv,2
0.426
2.513
cf,2
32.634
30.333
R2
0.753
0.585
R2
0.991
0.914
0.05
C45E S960QL C45E model S960QL model
h45° (mm)
0.04 0.03 0.02 0.01 0.00 20
60
100
140 vc (m/min)
Fig. 5. Effect of the cutting speed on h45°
180
220
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C45E S960QL C45E model S960QL model
hΘmax (mm)
0.04 0.03 0.02 0.01 0.00 20
60
100
140
180
220
vc (m/min)
Fig. 6. Effect of the cutting speed on hΘmax
0.20
C45E S960QL C45E model S960QL model
h45° (mm)
0.15 0.10 0.05 0.00 0
0.03
0.06
0.09
0.12
fz (mm)
Fig. 7. Effect of the feed rate per on h45°
0.15
0.18
Estimation of the Minimum Uncut Chip Thickness
0.20
C45E S960QL C45E model S960QL model
0.15 hΘmax (mm)
37
0.10 0.05 0.00 0
0.03
0.06
0.09
0.12
0.15
0.18
fz (mm)
Fig. 8. Effect of the feed rate on hΘmax
4 Conclusion The research presented in this article concerns how the minimum uncut chip thickness could be determined by simply realisable side milling tests using a macroscopic-sized and single-point end milling cutter with commercial tool geometry. During the conventional milling tests, the force components affecting the workpiece were measured, and the uncut thickness was approximated with the help of a geometric model and the signal of a proximity sensor. Conclusions were made from the ratio of said force components as the function of two cutting parameters: the cutting speed and the feed rate per cutting edge. Two indicators have indirectly represented the minimum uncut chip thickness: a. the h45° addresses the uncut chip thickness where the conditions of ploughing become dominant (the Θ kinetic angle derived from the ratio of the normal force and cutting force rises above 45°), and b. the hΘmax represents the conditions of the most intense ploughing by determining the maximum of the Θ kinetic angle. The values of h45° and hΘmax range between 5…75 (µm), which are in the scale of the applied cutting edge’s r β radius, meeting the results of prior researches on the minimum uncut chip thickness. The cutting speed caused both indicators to increase; however, the currently available measured data do not allow any more specific conclusions to be drawn. On the other hand, a more distinguishable impact of the feed rate was determined as it exponentially increases both indicators. The effect of the cutting speed and the feed rate have been formalised as functions. Additional milling tests with a wider range of cutting parameters have to be carried out to determine the said trends more reliably. Further examination of the measured data is required to acquire more precise results, including applying a more sophisticated approximation of the uncut chip thickness by using a model considering the cutting edge’s cycloidal tool path.
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Acknowledgement. The research reported in this paper is part of project no. BME-NVA-02, implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021 funding scheme. The author would also like to acknowledge the support provided by the CEEPUS III HR 0108 project.
References 1. Aramcharoen, A., Mativenga, P.T.: Size effect and tool geometry in micromilling of tool steel. Precis. Eng. 33(4), 402–407 (2009). https://doi.org/10.1016/j.precisioneng.2008.11.002 2. Agmell, M., Johansson, D., Laakso, S.V.A., Ahadi, A., Ståhl, J.-E.: The influence the uncut chip thickness has on the stagnation point in orthogonal cutting. Procedia CIRP 58, 13–18 (2017). https://doi.org/10.1016/j.procir.2017.03.183 3. Biró, I., Szalay, T.: Extension of empirical specific cutting force model for the process of fine chip-removing milling. Int. J. Adv. Manuf. Technol. 88(9–12), 2735–2743 (2016). https://doi. org/10.1007/s00170-016-8957-x 4. Mikołajczyk, T., Latos, H., Pimenov, D.Y., Paczkowski, T., Gupta, M.K., Krolczyk, G.: Influence of the main cutting edge angle value on minimum uncut chip thickness during turning of C45 steel. J. Manuf. Processes 57, 354–362 (2020). https://doi.org/10.1016/j.jmapro.2020. 06.040 5. Biró, I., Szalay, T., Geier, N.: Effect of cutting parameters on section borders of the empirical specific cutting force model for cutting with micro-sized uncut chip thickness. Procedia CIRP 77, 279–282 (2018). https://doi.org/10.1016/j.procir.2018.09.015 6. Wojciechowski, S., Matuszak, M., Powałka, B., Madajewski, M., Maruda, R.W., Królczyk, G.M.: Prediction of cutting forces during micro end milling considering chip thickness accumulation. Int. J. Mach. Tools Manuf. 147, 103466 (2019). https://doi.org/10.1016/j.ijmach tools.2019.103466 7. Lee, J.-H., Sodemann, A.A.: Geometrical simulation of chip production rate in microendmilling. Procedia Manuf. 26, 209–216 (2018). https://doi.org/10.1016/j.promfg.2018. 07.029
Theoretical Foundations of Physical Modeling of the Descent and Landing Process of Controlled Precision Airborne Cargo Landing Systems Olexander V. Betin , Petro G. Kyriienko , Denys O. Betin(B) and Olena Y. Betina
,
National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine {o.betin,p.kirienko,d.betin,o.betina}@khai.edu Abstract. Research has been carried out aimed at developing the theoretical foundations of physical modelling of the descent and landing process of controlled precision airborne cargo landing systems (CPACLS). The conditions, features of application and operation of CPACLS are revealed. Analyzed the needs of modern aviation to conduct flight researches of critical modes and special flight situations arising during the flight operation of CPACLS. Features of application of the basic provisions of similarity theory to substantiate the possibility of creating of a free flying dynamically similar model (FDSM) of a CPACLS and conducting flight research on them are considered. The criteria and scales of similarity in the task of formation of the required quality of a FDSM of a CPACLS have been established. The system of connections imposed on the scale of similarity of the basic parameters of a full-scale CPACLS and its FDSM has been formalized. Similarity scales are obtained for the basic parameters of a full-scale CPACLS and its FDSM when a possible combination of similarity criteria is satisfied. Secondary scales of similarity of parameters and characteristics are obtained by modeling the flight of a full-scale CPACLS on its FDSM. Keywords: Theoretical Foundations · Physical Modeling · Descent and Landing Process · Controlled Precision Airborne Cargo Landing Systems
1 Introduction Controlled precision airborne cargo landing systems (CPACLS) refer to aviation equipment, in particular, to controlled parachute systems with platforms for delivery of various cargoes to hard-to-reach areas of natural disasters, accidents, geological rescue and exploration works, and also have military applications (Fig. 1). The second name of these systems is controlled gliding cargo systems, which determines the appearance of a gliding parachute in the design. But whatever you call it, during operation (descent or flight, as well as landing) CPACLS is an aircraft that has certain aerodynamic and dynamic parameters and characteristics. CPACLS provide accurate cargo landing and reduced cargo loss, and allows the system to be used at different times of the day and under different weather conditions [1–3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 39–50, 2023. https://doi.org/10.1007/978-3-031-36201-9_4
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Fig. 1. Appearance of CPACLS
Minimum configuration of CPACLS: stabilizing parachute, main gliding parachute (with aerodynamic quality 4…5), suspension system, cargo platform with shock absorber system, flight computer (or command unit) in the main gliding parachute sling control box (with aerodynamic quality 4…5). In modern CPACLS the accuracy of landing 5…150 m. Operating temperature range from −50 to +85 C. To shorten the landing time at the final stage of landing, a round classical landing parachute is often used. The operating altitudes of the landing of modern CPACLS are mainly H = 150...5000 m and rarely exceed them when the flight speed of the aircraft (helicopter) – carrier, and, consequently, the beginning of parachute system deployment V = 250...400 km/h. Generally, the mass of the landing cargo does not exceed mc = 1000 kg. The use of a parachute system, practically homing to the beacon, allows, depending on the conditions of use, to reduce cargo losses by up to 20% [1–3]. Reducing cargo loss and improving landing accuracy are the main tasks facing the designers of CPACLS.
2 Analysis of Modern Aviation Needs for Flight Researches of Critical Flight Modes and Special Flight Situations Arising During the Flight Operation of CPACLS When creating a CPACLS, the piloting technique of which, it would seem, should not be particularly difficult, it is necessary to investigate following problems [1–6]: 1) features of the behavior of the carrier aircraft (helicopter) and the CPACLS in the landing modes (in case of a sudden change in the aircraft’s center of gravity); 2) features of flight and landing of CPACLS in normal and complicated conditions (e.g. high altitude conditions), as well as improving landing accuracy; 3) features of the behavior of CPACLS in case of possible piloting errors; 4) features of CPACLS piloting in difficult meteorological conditions (e.g. in conditions of intense icing, rain, snow, wind gusts, etc.);
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5) the possibility of increasing controllability at high angles of attack by using energy means of increasing lift (blowing compressed air jets onto the steering surfaces in order to stabilize their vortex systems at high angles of attack) and other modern means of improving the load-bearing properties of units and control surfaces; 6) motion dynamics, optimal trajectories and piloting methods when separating two objects with aerodynamic quality and comparable dimensions; 7) features of CPACLS landing, when using various types of shock absorber systems. In the practice of flight operation of the CPACLS, there may be special flight situations related with damage to the structure. The causes of these situations can be divided inti three main group [1–3]: 1) human factor – going into uncalculated modes due to piloting errors, collision with another aircraft due to air traffic control errors, etc.; 2) aircraft failures – fatigue failure of aircraft structural components, failures of control system units, etc.; 3) uncalculated environmental effects – destruction of units and systems due to exposure to difficult weather conditions (increased atmospheric turbulence, lightning discharges, vertical wind gusts near the ground, heavy icing), magnetic anomalies, collisions with birds, etc. In modern conditions, the causes of special flight situations can also serve as illegal interventions, as well as unauthorized use of weapons on a CPACLS. To solve these problems, it is necessary to have advanced highly accurate experimental information and proven mathematical software that allows obtaining data on the dynamics of flight under failure and structural damage of CPACLS and their on-board systems.
3 FDSM as a Design Object, a Technological Object and Flight Research Tool. Modern Concepts of Formation and Quality Control of a FDSM The need to study the flight modes and special situations discussed above has led to creation of a method that uses a reusable unmanned aerial vehicle (UAV) for research purposes as a flight research tool, capable of performing a remotely piloted or automatic flight according to a given program (Fig. 2) – a free flying dynamically similar model (FDSM) [7–9]. The most effective is the use of a FDSM for the study of critical flight modes related with complex spatial motion of the aircraft, phenomena with a high probability of leading to aircraft destruction, as well as for testing design concepts characterized by a high degree of technical risk. It should be noted here that for a FDSM, not only do the experimental conditions must satisfy certain laws, but it must also be created according to these laws, i.e. have the appropriate quality. Quality – a complex or totality of essential features, characteristics, indicators, properties and features that distinguish an object or phenomenon from others and give it
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certainty. The quality of the aircraft as an object of production is a complex of tactical and technical characteristics and indicators, that determine its operational reliability [10]. The required quality of FDSM is determined by its similarity to a full-scale CPACLS, and is formed by design, manufacture, ground and flight tests.
Fig. 2. Full-scale aircraft and its FDSM
The required quality of a FDSM can be achieved only as a result of very complex and integrated process, including the development of technical task, design, preparation of model manufacturing, manufacturing, ground tests, adjustment of parameters to the required similarity values, numerical modelling, flight tests, as well as partially the process of operating a FDSM.
4 Features of Application of Basic Provisions of Similarity Theory for Substantiation of Possibility of Creation a FDSM of a CPACLS and Carrying Out Flight Researches on Them. Criteria and Scale of Similarity in the Task of Forming the Required Quality of a FDSM of a CPACLS Modelling is the replacement of the study of a phenomenon in full-scale conditions with the study of a similar phenomenon on a model [7, 11]. Flight researches of flight modes of a full-scale CPACLS on a FDSM belong to physical modelling, which differs from other modelling types (mathematical, analogue, etc.) by the fact that the model has the same physical nature as the full-scale CPACLS. At the same time, a physical experiment is carried out in full-scale flight conditions and the study, based on the results of the experiment, of the laws of interaction of the FDSM with the external environment, with gravitational and other fields of the Earth and (or)
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the effect of this environment on the units and systems of the FDSM with the subsequent transfer of physical modelling results to the full-scale CPACLS. A formalized mathematical record of similarity conditions forms a system of connections (system of similarity criteria) that must be superimposed on environmental parameters, parameters and characteristics of a FDSM. The system of similarity criteria id the theoretical basis for solving applied and methodological problems of creating FDSM and conducting flight research on them, and the method of forming this system is the theoretical basis for modelling the flight dynamics of a full-scale CPACLS on a FDSM. The Standard Atmosphere (SA) is currently used as the calculation basis [12]. Transfer of the results of research flights of a FDSM to a CPACLS is only possible if geometric, kinematic and dynamic (aerodynamic and overall) similarity conditions are fulfilled. A FDSM must have the same shape, required by the similarity of elasticgeometric characteristics, mass inertial parameters and similar control laws of the automatic control system as the CPACLS. As a result, a FDSM will behave in flight in the same way as a full-scale CPACLS in the appropriate conditions. Touching upon the questions of similarity theory, we will briefly dwell on those described more than once, but necessary for modelling the similarity scales [7, 9, 11]: geometric n φn Sn Wn = const1 ; kφ = = 1; ks = = k2 ; kw = = k3 , m φm Sm Wm
(1)
tn Vn k an k = const2 ; kv = = = const3 ; ka = = 2 = const4 , tm Vm kt am kt
(2)
k = kinematic kt = dynamic
Rn = kR = const5 , Rm
(3)
where k – scale of linear size; kφ – sce of angles; ks – scale of areas; kw – scale of volumes; kt – scale of times; kv – scale of linear velocities; ka – scale of linear acceleration; kR – scale of forces; n , m – randomly chosen analogous sizes (for example, characteristic sizes) of the comparable bodies (natural object and model); φn , φm – angles between respective lines of geometrically similar bodies; Sn , Sm , Wn , Wm – areas and volumes of geometrically similar bodies; tn , tm – similar time periods of movement of geometrically similar bodies; Vn , Vm . , an , am – linear velocities and accelerations of the movement of geometrically similar bodies in similar periods of time; Rn , Rm – equally directed forces of the same nature acting on geometrically similar bodies in similar time periods. Hereinafter code “n” defines the relation of criterion or indicator to parameters or characteristics of full-scale CPACLS, and code “m” – to the corresponding parameters or characteristics of BDSM. The geometrical similarity of CPACLS and FDSM is understood not as specified similarity of structures (though it is possible too), but as similarity of exterior surfaces of bodies of identical form. Let’s pay attention, that the requirement of geometrical similarity extends not only to shapes of streamline bodies (including roughness, waviness,
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gaps, ledges and other distortions of the specified shape) when studying the aerodynamic phenomena, but also onto borders of flows (if those are available) [7, 9]. Aerodynamic similarity includes the similarity of force interactions of flows with streamlined bodies and also expresses certain requirements for the physical properties of the medium (gases or liquids) under consideration. By studying the similarity of the air flows around the full-scale CPACLS and its model, one can judge both the aerodynamic similarity and the conditions achieving a degree completeness of the similarity. The equations of motion, continuity and energy of gases are a mathematical description that allows you to obtain the basic similarity criteria of gas flows [7, 9, 11]. Thus, two gas flows are similar if the criterion conditions are met [11] Frn = Frm , Ren = Rem , Mn = Mm , Shn = Shm , Pr = Prm , χn = χm , n
(4)
where Fr – Froude criterion; Re – Reynolds criterion; M – Mach criterion; Sh – Strouhal criterion; Pr – Prandtl criterion; χ – adiabatic index. The similarity criteria included in condition eq. (4) have the following form: Fr =
μcp cp V V Vt V2 ; Re = ; M = ; Sh = ; Pr = ; χ= , g ν c λ cϑ
(5)
where V – flow velocity; g – acceleration of gravity; – characteristic √ linear size of the streamlined body; ν – coefficient of kinematic air viscosity; c = χ RgT – speed of sound in the incoming stream; t – time; μ – coefficient of dynamic air viscosity; cp – specific heat capacity of air at constant pressure; cυ – specific heat capacity of air at constant volume; λ – air thermal conductivity coefficient; R – absolute gas constant; T – incoming flow temperature. All criteria are written for unperturbed flow. They form the basis, i.e. the system of dimensionless quantities that determine all other flow parameters. Similarity criteria have a certain physical meaning and characterize real factors affecting the aerodynamic force. Since the flights of a full-scale CPACLS and flight researches on a FDSM are carried out in the Earth’s atmosphere, it makes sense to limit the similarity consideration only atmospheric air flows, thus reducing the number of similarity conditions, which take the following form according to the criteria [7, 11]: Fr:
V2 1 νn Vn Vm Vn tn Vm tm Vn2 = m ; Re: kv = · ; M: = ; Sh: = . gn n gm m k νm cn cm n m
The same conditions in the scale ratios by criteria gn 1 νn cn k Fr: kv = k ; Re: kv = · ; M : kv = ; Sh: kv = . gm k νm cm kt
(6)
(7)
When studying unsteady flow modes, condition of kinematic similarity of flows by Sh criterion in the form of (6) necessarily be fulfilled (at dynamic similarity of flight modes this condition is automatically fulfilled). However, the same condition in the form (7) is an identity an cannot be used when considering aerodynamic similarity conditions together in a scale form.
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5 Formalization of the System of Connections Imposed on the Similarity Scale of the Main Parameters of the Full-Scale CPACLS and Its FDSM According to definition, at dynamic similarity the similarity of polygons of forces operating onto full-scale CPACLS and its FDSM. In particular, Fn Gn mn an mn gn = or = , Fm Gm mm am mm gm
(8)
where FFmn = mmmn aanm – the relation of inertial forces; GGmn = mmmn ggnm – the relation of forces of weight; mH , mM – masses of a full-scale CPACLS and its FDSM. After simplification by mmmn , we obtain that the scale of linear acceleration ka =
an gn = , am gm
a, taking into account kinematic scales of similarity (2) in the form ka = we receive formulas for the scales of times and linear velocities gM gH ; kv = k . kt = k gH gM
(9) k kt2
i kv =
k kt ,
(10)
According to SA, the acceleration of gravity at altitudes in range of 0…5000 m does not change considerably. Therefore, it is possible to consider that for CPACLS and its √ FDSM gn = gm , and, therefore, kt = kv = k and ka = 1. It is easy to see that writing of the relation (10) with respect to kv coincides with writing of the air flow similarity condition according to the Fr in form (7). The necessity of satisfying relation (10) in dynamic similarity leads to the necessity of analyzing only those combinations in which Fr criterion is present. Thus, apart from the trivial general case of joint satisfaction of the criteria Fr, Re and M, and the variant where only the similarity condition of Fr is satisfied, it is necessary to consider two more options for the joint pairwise satisfaction of the similarity criteria: (Fr and Re), (Fr and M). One of the main objectives of similarity analysis is to obtain similarity scales that allow finding the relationship between the relevant parameters and characteristics of a full-scale CPACLS and its FDSM as well as the parameters and characteristics of modeling phenomena. The scales of similarity of key parameters of full-scale CPACLS and FDSM are [9] k =
Ijn n mn ; km = ; kI = m mm Ijm
(11)
(j = x, y, z, xy, xz, yz), where kI – scale of moments of inertia; Ijn , Ijm – the inertia moments (axial and centrifugal) of full-scale CPACLS and FDSM concerning axes of analogous coordinate systems.
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Fulfillment of the similarity condition of the Fr criterion means compliance with the similarity taking into account the gravity forces for the compared flows. The force of gravity G ∗ for some elementary volume of air Δ3 and mass Δm∗ is determined by the expression ΔG ∗ = Δm∗ g = ρ 3 g,
(12)
where ρ – air density at the height of the experiment. For finite air volumes in the case of dynamic similarity Gn G∗ mn gn ρn 3n gn = ∗n or = , Gm Gm mm gm ρm 3m gm
(13)
where ρn , ρm – air density at flight altitudes Hn of a full-scale CPACLS and aerodynamic similarity Hm its FDSM. From where it is not difficult to find that mn ρn 3n = or km = kρ k3 , mm ρm 3m
(14)
where kρ = ρρmn – scale of air densities. According to Steiner’s theorem, moment of inertia of system of material points I relatively any axis is equal to the moment of inertia of system Ic relatively parallel axis through the center of mass (CM), plus the product of mass of system m onto squared distance between these axes L, i.e. [9, 13] I = Ic + mL2 .
(15)
Therefore, the ratio of moments of inertia relatively any analogous axes of a full-scale CPACLS and its FDSM will have form In Icn + mn L2n = , Im Icm + mm L2m
(16)
where Icn , Icm – the own moment of inertia of CPACLS and its FDSM. cn = kI (by similarity definition), then after the corresponding Since IImn = IIcm transformations (16) taking into account (14), we will receive kI =
In mn L2n = = km k2 = kρ k5 . Im mm L2m
(17)
The same ratio can be received also for the centrifugal moments of inertia, using concept of the main axes of inertia [9].
6 Similarity Scales of the Basic Parameters of a Full-Scale CPACLS and Its FDSM While Satisfying Possible Combinations of Similarity Criteria To analyze the system of relations imposed on the basic parameters of a FDSM by similarity conditions, consider possible combinations of similarity criteria. First option: At the same time the similarity conditions according to the criteria Fr, Re and M are satisfied, which, according to conditions (7), requires the satisfaction of
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equality [9] gn νn gm νm = 3 . 3 cn cm
(18)
Under SA conditions, such equality is possible only when HH = HM and, therefore, scale of linear dimensions [9] cn n gm k = · = 1. (19) gn cm νm Therefore, in this variant kρ = 1, k = 1, km = 1 and kI = 1, which means that the external shape, masses, moments of inertia and position of the center of mass of a full-scale CPACLS and a FDSM are identical. However, they can be made of different materials, have different structural layouts and airborne hardware, and allow investigation almost all flight modes of a full-scale CPACLS. Second option: The similarity condition for the Fr criterion only is satisfied, with auto-modelity for the criteria Re and M [9]. This is only option where the choice of scale of linear size k does not depend of the aerodynamic similarity heights Hn and Hm . However, after selected the scale k and assigning the aerodynamic similarity heights, the scales km and kI are uniquely defined by relations (14) and (17). The decision to build such models requires a particularly careful analysis of the issues related with the reliability of the information obtained when modelling the flight modes of a full-scale CPACLS on its FDSM. Third option: Similarity conditions according to the Fr and Re are satisfied with the auto-modelity according to the Mach M criterion [9]. In this variant, according to conditions (7), as well as ratios (14) and (17), similarity scale formulas for basic parameters of a full-scale CPACLS and its FDSM 5 2 2 g υn2 · ρn gm υn gm 3 ρn m 3 νn · ; km = · 2 ; kl = · · (20) k = gn νm2 gn υm · ρm gn υm2 ρm implicitly relate the scales k , km and kI to the aerodynamic similarity heights Hn and Hm . If there is a pair of heights (Hn , Hm ), for which the scales k , km and kI have acceptable values and a FDSM with such scales is physically feasible, then such a FDSM allows to investigate a certain flight mode or maneuver of a full-scale CPACLS. Fourth option: Similarity conditions according to the Fr and M are satisfied with the auto-modelity according to the Re criterion [9]. In this variant, according to conditions (7), as well as ratios (14) and (17), similarity scale formulas for basic parameters of a full-scale CPACLS and its FDSM k =
cn2 gm Tn T 3 · ρn T 5 · ρn · 2 = ; km = n3 ; kI = n5 , gn cm Tm Tm · ρm Tm · ρm
(21)
which, as in the previous version, implicitly relate the scales k , km and kI to the aerodynamic similarity heights Hn and Hm . The strong analogy of k , km and kI scales with aerodynamic similarity heights determines that when the Fr and M criteria are satisfied together, it is possible to modelling only a single maneuver or flight mode of a full-scale CPACLS.
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7 Secondary Scales of Similarity of Parameters and Characteristics in Modeling the Flight of a Full-Scale CPACLS on Its FDSM In order to adequacy in modelling of flight modes of a full-scale CPACLS on its FDSM, it is necessary that at any similar points of similar trajectories, forces of the same nature must be equally directed and proportional. With respect to the inertial forces acting on a full-scale CPACLS and its FDSM, according to (3), (9) and (14), it is possible to write gn mn an mn an Fn = = = km ka = kρ k3 . Fm mm am mm am gm
(22)
Then, in general terms, the scale of forces kR =
gn Rn Fn = = km ka = kρ k3 , Rm Fm gm
(23)
and when gn = gm , the same scale of forces kR = kρ k3 .
(24)
In addition, the condition [9, 11] must be fulfilled F ·n F ·n F ·m Gn = or = Gm F ·m Gn Gm
(25)
where F ·n , F ·m – the equalities of external forces (without gravity forces) acting on a full-scale CPACLS and its FDSM respectively. The last entry is the equality of vertical overloads ny.n = ny.m . A similar conclusion can be drawn for the longitudinal nx.n , nx.m , as well as side nz.n , nz.m overloads. From where, in similar time periods tn and tm the scale of operating overloads when modelling the flight of a full-scale CPACLS on its FDSM kol =
ny.n nx.n nz.n = = = 1. nx.m ny.m nz.m
(26)
In the rotational movement of a full-scale CPACLS, angular movements φ, velocities ω and accelerations α occur due to the action of the moments of forces M around geometrically similarly spaced axes. To obtain the scales of moments of forces (kM ), angular movements kφ , velocities (kω ) and accelerations (kα ) we will use the positions and ratios available in mechanics of the rotational movement of the body [13]. Thus, the moment of force relative to some point is the product of force by the shortest distance from this point to the line of action of the force, i.e. M = RR ,
(27)
where R – force applied to a solid body; R – the length of the perpendicular dropped from the center of rotation to the line of force action.
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Then, for similar moments of forces applied to the CPACLS and its FDSM in similar time periods tn and tm , according to (1) and (23) it is possible to write kM =
gn gn Mn Rn R.n = = kρ k3 k = kρ k4 , Mm Rm R.m gm gm
(28)
and when gn = gm the same scale of moments of forces. kM = kρ k4 ·
(29)
As for the angular movements of the full-scale CPACLS ( φn ) and its FDSM ( φm ), in order to provide geometric and kinematic similarity in modelling of flight dynamics, it is necessary that angular movements be in the same ratio as the angular parameters of geometrically similar objects. Therefore, according to (1), it can be written that φn φn = = kφ = 1. φm φm
(30)
The angular velocities and accelerations (ω and α) are related to similar linear parameters (V and a) when moving along a trajectory with radius r by the ratios ω=
a V and α = , r r
(31)
that are valid for constants, instantaneous and average values, in all cases of body movement along the circle. Then for similar moments of forces applied to the CPACLS and its FDSM in similar time periods tn and tm , according to (1), (9) and (10), gn 1 ωn Vn rm αn an rm gn 1 kω = = = = = (32) √ ; kα = ωm Vm rn gm k αm am rn gm k and at gn = gm the same scales 1 1 kω = √ ; kα = . k k
(33)
All other required parameters, characteristics and scales can be obtained similarly from the similarity scale formulas obtained above.
8 Conclusions Research has been carried out aimed at developing the theoretical foundations of physical modelling of the descent and landing process of CPACLS. The conditions, features of application and operation of CPACLS are revealed. Critical flight modes and special flight situations arising during the flight operation of the CPACLS are analyzed. The features of application of physical modelling methods to solve this problem are considered. The features of a FDSM of a CPACLS are determined and modern concepts of formation and control of its quality are presented. The features of application of the basic provisions of similarity theory are considered, the criteria are established and similarity scales are obtained in the task of formation of required quality of FDSM of CPACLS.
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References 1. Kuyanov, A.Yu.: Cargo controlled parachute system sherpa. Precise airborne ejection. Freefall navigation system. http://gidgrowth.com/gruzovaya-upravlyaemaya-parashyutnaya-sis tema-sherpa-tochnaya..html (in Russian) 2. Foreign parachute systems for cargo landing – InVoenInfo. http://invoen.ru/vvt/zarubezhnieparashyutnie-sistemi-desantirovanie-gruzov. (in Russian) 3. Ploskov, S.Yu.: Modern parachute systems for high-precision cargo delivery. Engineering Journal: Science and Innovation # 3·2021. http://engjournal.article/2066/2066.pdf. (in Russian) 4. Betin, A., Ryzhenko, A., Sadovnychiy, S., Ricardo, P.-F.: Simulation of aircraft flight dynamics by means of dynamically similar models. In: Modeling and Simulation Technologies, pp. 64–69. American Institute of Aeronautics and Astronautics, Boston, USA (1998) 5. Betin A., Sadovnychiy S., Ryshenko A.: Simulation of aircraft emergencies by means of dynamically similar models. In: Modeling and Simulation Technologies, pp. 328–334. American Institute of Aeronautics and Astronautics, Portland, USA (1999) 6. Sadovnychiy, S., Ryzhenko, A., Betin, A.: Flight control system damage simulation using freely flying models. The Aeronaut. J 109(1091), 45–50 (2005) 7. Povkh, I.P. Aerodynamic experiment in mechanical engineering. In: Povkh, I.P. (ed.) L.: Mechanical engineering, 364 p. (1979). (in Russian) 8. Use of free-flying models to research flight dynamics: review of foreign press for 1941–1970. Reviews. Translations. Abstracts. – M.: CAHI. – 1971. – Vol. 352. – 92 pages. (in Russian) 9. Betin, A.V., Ryzhenko, A.I., Ryabkov, V.I., Cheranovsky, O.R.: Determination of the Sizes and Mass and Inertial Parameters of Free-Flying Dynamically Similar Aircraft Models: Schoolbook, 101 p. Ministry of education of Ukraine, Kharkov aviation institute. Kharkov aviation institute, Kh 1992. (in Russian) 10. Gzhirov, R.I.: Designer’s quick reference guide: handbook. In: Gzhirov, R.I. (ed.) L.: Mechanical Engineering, 464 p. (1983). (in Russian) 11. Krasnov, N.F.: Aerodynamic. In: N.F. Krasnov. – M.: Science, 614 p. (1971). (in Russian) 12. GOST 4401-81. Standard atmosphere. Parameters. – M.: State committee of the USSR on standards, 107 p. (1981). (in Russian) 13. Kukhling, H.: Reference book on physics. In: Kukhling, H. (ed.) M.: World, 520 p. (1982). (in Russian)
Algorithm of Computational Modeling the Self-heating Process of Pneumatic Tire in Operation Yuliia Viazovychenko(B)
and Oleksiy Larin
National Technical University «Kharkiv Polytechnic Institute», Kharkiv, Ukraine {yuliia.viazovychenko,Oleksiy.Larin}@khpi.edu.ua
Abstract. The paper deals with development of computational algorithm for modeling of complex processes in elastomer composites. It was implemented as example for investigation of self-heating processes. The calculation approach was purposed and applied for calculation of heat process of tire in operation. The approach deals with computational modeling of stress-strain state processes and solving thermal conductivity problem. It includes obtaining data from one calculation, its processing and implementation to another. The several APDL-macros and sub routines has been written in frame of work. Keywords: Algorithm · Engineering Computing · Data Analysis · FE-method · Approximation · Self-Heating · Strains Cycles · Approach
1 Introduction In engineering practice, to analyze the processes flow and the operation of systems, due to their complex nature, it is sometimes impossible to implement an integrated approach that takes into account all the features. This leads to developing necessitate of step-bystep approaches and computational algorithms for their reliable modeling. An example of one of these processes is the self-heating process of a rubber-cord composite material under cyclic loading. The nature of this process has a complex character and many different components. Taking into account them in a complex is an incredibly difficult engineering task. One of such component is energy dissipation phenomenon caused by a combination of viscoelastic properties and cyclic loading. It should be noted that heating of such materials causes a decrease in strength characteristics [1], delamination [2] of the cord reinforcement from the rubber matrix, changes in their damping properties [3], accelerates the course of fatigue processes [4], and significantly reduces the fatigue life. Also, temperature change affects stiffness [5, 6] and elongation at break [7]. Thus, determining the self-heating characteristics of such materials is an important task for understanding their behavior in operation. Thus, the paper proposes a computational algorithm for calculating the heating temperature of a structure caused by self-heating, using a rolling pneumatic tire as an example. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 51–62, 2023. https://doi.org/10.1007/978-3-031-36201-9_5
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2 Main Idea of Algorithm Self-heating process in elastomer is determined by the amount of heat, which is released per deformation cycle. Thus, this phenomenon simulation requires a consistent formulation in the calculation problems. For this, it is proposed a numerical algorithm, which consists of 3 main stages (Fig. 1).
Fig. 1. General scheme of algorithm
1. The first stage represents the static problem solution of the tire interaction with the road surface, which is loaded with an internal pressure and a vertical load formed by the car weight. The results of this stage are deformation cycles for one revolution in each of the layers. 2. The second stage is contained in determining the heat generation rate in each element. For this, the amount of released energy during one cycle of deformation is determined. Corresponding calculations were carried out for the schematic cycle. In previous works, the loss modules for all tire materials were determined experimentally, which, in turn, determine the amount of dissipated energy [8]. 3. The third stage consists of solving the of stationary thermal conductivity problem. In this case, as a load in each element, the heat generation rate function, which was determined at the 2nd stage, is applied.
3 Determination of Deformation Cycles The deformed state problem as solved on the base of FE-method [9]. So the corresponding matrix equation is [K(U)] · U = F,
(1)
where K(U) is the stiffness matrix, U – is the nodal displacement vector, F – is the nodal force vector. It can be seen that Eq. (1) is nonlinear. It is due to availability of physical relation nonlinearity. For this material it was taken into account by using Neo-Hookean model.
Algorithm of Computational Modeling
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An 8-node finite element with 3 degrees of freedom at the node was used to build the FE model (Fig. 2).
Fig. 2. Representation of FE – model, a – complete model of a tire, b – segment of a tire
The initial simulation and calculation of deformed state was carried out within the framework of a static and the obtained results correspond to wheel rotation. A multi-scale approach (details in [10]) was used for reliable stress and strain levels identification. The analysis of the received data of deformed state of the tire elements allows us to determine the four zones of the largest strains of the cord layers (3 in the tire carcass layer and 1 in the breaker layer, see Fig. 3). These zones correspond to the following areas: shoulder (carcass and breaker), bead (carcass) and the place where the carcass is doubled. The bead zone has the largest strains in the sector (0–15). And the rest of the regions, respectively, have them in the sector (30–45).
Fig. 3. The largest strains zones of the frame: 1 – sub-breaker,2 – shoulder zone; 3 – carcass doubling; 4 – bead zone.
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The strain cycles definition: # Definition of the typical tire zones with the most deformed zones over the structural 1: for each zone k in range [1 .. 4] 2: define NodeZone[k] sets for typical nodes with the maximum strains 3: save node coordinates X, Y, Z for each NodeZones[k] to external file # Identification of the deformation cycles 4: for each zone k in range [1 .. 4] 5: define the node Nmax with max strains value in NodeZones[k] 6: define X, Y, Z coordinates of the node Nmax 7: change the coordinate system to cylindrical (X, Y, Z) -> (R, Phi, Z) # Z – axis of tire rolling 8: define (RNmax, PhiNmax, ZNmax) for the node Nmax 9: define NumCirc of nodes in circumferential coordinate 10: (R, Phi, Z) = (RNmax, PhiNmax, ZNmax) 11: dfi=2*Pi/NumCirc 12: for i in range [0.. NumCirc]: 13: select node with coordinates (R, Phi, Z) in NodeZones[k] 14: get strain Eps[i] for selected node 15: Phi=Phi+dfi 16: save data {Phi, Eps[i]} to the external file
A description of the macro associated with the definition of zones and strains cycles is represented above. The cycle construction procedure involved the approximation of strains by the values of these zones nodes, which are located along the lines, where two coordinates are fixed, and only the circumferential coordinate changes. Figure 4 represents cycles of the carcass deformed state in the control zones. Analysis of these graphs shows that the carcass deformation cycles are different depending on the zone and load level. For easer perception, the deformed state cycles are shown in polar coordinates (0° corresponds to the contact area of the tire with the road surface) Fig. 4 (b). Thus, the strains cycles are actually shown for four characteristic points. Building such dependencies for all nodes of the model, using them for further calculations, is a rather difficult and computationally expensive task.
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Fig. 4. Strain cycles representation in Cartesian (a) and polar (b) coordinates
Therefore, the entire model was conventionally divided into the four zones mentioned above, and it was considered that the general nature of the deformation cycle for all nodes within one zone is the same. But the deformation values in the cycle were scaled relative to the deformation value in each specific node. For obtaining data about strains in each node and scaling the values for each cycle several APDL-macros were wrote.
4 Calculation of Heat Load 4.1 Algorithm of Calculation Heat Generation Function The calculation of heat generation functions contains also with several steps. The main of them are shown in Fig. 5.
Fig. 5. General scheme of heat generation rate calculation
The first step includes importing of strain cycles from previous paragraph. The second one is involved to get suitable approximation of these cycles, for example by piecewise quadratic functions. On the next step the refinement of the obtained approximations is held. It was carried out by using of Fourier’s transformation to provide cyclic character of strain-time dependence. The last one deals with calculation the heat generation rate. It was held according to 2-parameters viscoelasticity model with using real operation cycles from previous step.
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4.2 Strain Cycles Approximation Based on the fact that deformation cycles were built from discrete values of strains along the circumferential coordinate, they were initially approximated by piecewise quadratic functions. The view of such approximation for one cycle as example is represented by (1): ⎧ ⎪ −0.3079x2 − 0.0336x + 0.1192 0 < x ≤ 0.4027 ⎪ ⎪ ⎪ ⎪ ⎨ 0.1974x2 − 0.1793x + 0.1009 0.4027 < x ≤ 0.6041 (2) f = −0.0361x2 + 0.0803x + 0.029 0.6041 < x ≤ 1.1277 ⎪ ⎪ ⎪ −0.0024x2 + 0.0011x + 0.029 1.1277 < x ≤ 2.2152 ⎪ ⎪ ⎩ 0.0036x2 − 0.0229x + 0.0997 2.2152 < x ≤ 3.1415 Since the tire deformation in one revolution is obtained from the static analysis, in order to proceed to the cyclic deformation, it is suggested to decompose the deformation cycle into a Fourier series, which will ensure the periodicity of the deformation. The graph of function (1) is shown on Fig. 6 (a).
Fig. 6. Approximation of the half cycle of shoulder zone
Thus, to reduce the length of the series while maintaining the necessary convergence, it is proposed using method of finding the discrepancy between the approximation of the cycle and the Fourier series, limiting itself to keeping only three of its terms. Fourier series expansion looks like this y(x) = a0 + a1 cos x + a2 cos 2x + a3 cos 3x.
(3)
At the same time, the non-viscous can be shown as ζ = ∫π0 (f (x) − y(x))2 dx.
(4)
At the same time, the approximation of the cycle repeats the form of the Fourier series, but with other coefficients. For some cycles, frequency components were selected, which in the Fourier series gave a greater contribution, compared to the first three: z(x) = g0 + g1 cos x + g2 cos 2x + g3 cos 3x.
(5)
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The coefficients of the obtained deformation cycle in this case are found as solutions of the following equations: gi :
dζ = 0, i = 0..3. dai
(6)
Fig. 7. Visualization of strain cycles approximation for zone 1 (a) and zone 2 (b) at different load modes
It was found that the cycles corresponding to the nature deformation of the side zone and the carcass connection zone for the nominal value of the internal pressure, are sufficiently well approximated by the first three terms of the Fourier series. At the same time, the deformation cycle of the sub-breaker zone in the overloaded mode includes the sixth frequency component instead of the third, and for the shoulder zone-from the fourth to the sixth.
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Figure 7 shows examples of the deformation cycles of different zones approximated by Fourier series, which were refined by searching for the discrepancy, for the standard loading mode. At the same time, the error of approximation by the Fourier series, which estimates the integral characteristic, is 0.003% in the standard mode, and 0.007 in the overloaded mode. With refinement by the method of finding the discrepancy, the error in the standard mode decreased to 0.2·10−5 , while in the overloaded mode, the approximation inaccuracy decreased to the error value of 0.4·10−5 . The calculation of the approximation integral error for the series without refinement was 0.009% for the normative mode, with refinement it decreased to 0.7·10−6 . 4.3 Determination of the Heat Generation Function At the second stage, the heat generation rate is determined per unit volume as follows d εij (τ ) , dt
˙ = ∫t0 σij (τ ) · Q
(7)
where εij (τ ) – strain cycles obtained when solving the problem of the stress-strain state of tire determination and approximated above, σij (τ ) – strain tensor, which depends on the material parameters. In this paper, the self-heating process is modeled on the basis of structural deformation cycles in operation. From stress-strain state calculations without taking into account viscoelasticity, we get only the amplitude strains values per cycle. Since the structure under study works under cyclic load, the resulting cycle was decomposed into a Fourier series, which will provide a dynamic (periodic) component. The resulting series in the general case has the following form ε(t) = ε0 +
N k=1
εk cos kω t.
(8)
The corresponding time dependences of the stress, taking into account the viscoelastic behavior [11], have the form σ (t) = ∫t−∞ G(t − τ )
∂ε d τ, ∂τ
(9)
where G(t) is the relaxation kernel. The work was limited to the two-parameter model of linear viscoelasticity. ˆ G(t) = E + Ee
ˆ
− Eη t
t
= E + E e T0 ,
(10)
ˆ η – are material constants obtained from the experiment [12]; T0 – relaxation where E, ˆ time (T0 = η/E). Then, on the basis of (10) and (8), the stress is determined from (8) and will be written as follows N N σ = Ae−iλt + B bi cos ωi t + C ci sin ωi t + D (11) i=1
i=1
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where A, B, C, D, bi , ci are integration constants that depend on the material characterˆ T0 , as well as the strain amplitudes values. They can be found analytically and istics E, have a complex algebraic form. Then, for the stationary case, the released energy amount is equal to AD = ∫T0 σ
dε ˆ T0 , ωi , εi ), dt = AD (E, dt
(12)
where ωi is the load frequency, T0 – one deformation cycle period. A brief description of the macros construction needed to the calculation and application of the heat generation rate function is shown below. Heat generation rate calculation: 1: for each zone k in range [1..4] 2: load “Cycle approximation parameters” – Fourier’s fitting parameters 3: for each p layer in range [1..7] 4: for each node i 5: load from file “Strains sets” strain value 6: scaling the cycle of this zone with respect to obtained # , – heat dissipation material parameters (strain amplitude dependent value) , with respect to amplitude value 7: load – amplitude of released energy 8: definition amount, with respect to amplitude value 9: definition of heat generation rate function speover time, using cifically for this node
5 Solving of Stationary Heat Conduction Problem This problem was also solved with usage of FE method. The matrix equation for stationary case can be represented as [P] = F,
(13)
where [P] – thermal conductivity matrix, – nodal temperature value, F – thermal ˙ nodal forces vector, which include the internal heat source Q.
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The second FE model was built to solve the problem of heat conduction problem. The 20-node FE with 1 degree of freedom at the node is used for it. FE-mesh of the models for the 1st and 3rd stages spatially completely coincides. Figure 8 illustrated surfaces with different boundary conditions.
Fig. 8. Boundary conditions
The temperature (T0 , C°) and the convective coefficient (hc , W/m2 C°) are set as boundary conditions on the external (G1), internal (G2) surface and the point of interaction with the tire rim (G3) [13]. Thus, on the surface of G1: T0 = 25, hc1 = 16,18; on the surface of G2: T0 = 25, hc2 = 5,9; on the surface of G3: T0 = 25, hc3 = 88000; ˙ is given, which As a load in each node, the function of the heat generation rate Q, is calculated by formula (6) for each zone according to the deformation cycle. Figure 9 shows the temperature distribution formed during rolling of a tire at a speed of 50 km/h (a, c) and 240 km/h (b, d) loaded with an internal pressure of 210 kPa and with a vehicle weight of 1 ton (a, b) and 2.5 tons (c, d). According to Fig. 9, the carcass zone is the most heated, due to the presence of larger deformations in it, compared to other layers. From the obtained results, it can be seen that this temperature distribution is close to the operational one. The obtained temperatures correspond to the actual heating levels of the rolling tire at the given operational modes, which proves the feasibility of using the proposed algorithm for further calculations. Figure 9 shows that the zone of maximum temperature was moving to the board zone with an increase of the vertical load value. Thus, the results of this study provide not only quantitative, but also qualitative characteristic of temperature state of tire. The identification of high temperature location is essential for further research of the mechanical behavior of the tire.
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Fig. 9. Temperature distribution field along the tire profile at an internal pressure of 210 kPa and a vertical load of 1 t (a, b), 2.5t (c, d)
6 Conclusions The work deals with the development of algorithm for investigation of complex processes on the example of self-heating phenomena in elastomers. The two types of problem of were solved on the base of FE method. Also, the data processing approach was purposed for preparing to usage deformed state calculation results in solving of heat conductivity problem. The series of specific APDL-macros has been written in frame of work. The proposed algorithm was tested by applying it for heat state calculation of tire in operation which was caused by self-heating process. As the results temperature distribution on tire profile were obtained. Their analysis showed good correlation with real temperature distribution is appeared in tire in operation.
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References 1. Li, Z., Wan, J., Zhang, L., et al.: Effects of heat and moisture on characteristics, tensile properties of RFL-coated rayon cords, and their adhesion with NR/SBR matrix. J. Appl. Polym. Sci. 134(48), 147–165 (2017) 2. Shim, W., Jang, J., Choi, J.H., et al.: Simulating rate- and temperature-dependent behaviors of adhesives using a nonlinear viscoelastic model. Mech. Mater. 147, 103–446 (2020) 3. Kondé, A., Rosu, I., Lebon, F., et al.: Thermomechanical analysis of an aircraft tire in cornering using coupled ale and Lagrangian formulations. Open Eng. 3(2), 191–205 (2013) 4. Luo, W., Li, M., Huang, Y., et al.: Effect of temperature on the tear fracture and fatigue life of carbon-black-filled rubber. Polymers 11(5), 201–214 (2019) 5. Johlitz, M., Diercks, N., Lion, A.: Thermo-oxidative ageing of elastomers: a modelling approach based on a finite strain theory. Int. J. Plast. 63, 138–151 (2014) 6. Charrier, P., Marco, Y., Le Saux, V., et al.: On the influence of heat ageing on filled NR for automotive AVS applications: constitutive models for rubber. In: VII – Proceedings of the 7th European Conference on Constitutive Models for Rubber, ECCMR, pp. 381–388 (2012) 7. Mott, P.H., Roland, C.M.: Aging of natural rubber in air and seawater. Rubber Chem. Technol. 74(1), 79–88 (2001) 8. Viazovychenko, Y., Larin, O.: Stochastic optimization algorithms for data processing in experimental self-heating process. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2020: Synergetic Engineering, pp. 644–653. Springer International Publishing, Cham (2021). https://doi.org/10.1007/9783-030-66717-7_55 9. Thompson, M.K.: ANSYS Mechanical APDL for Finite Element Analysis, p. 889. Butterworth-Heinemann (2017) 10. Larin, A.A., Petrova, Y.A.:“Determination of stress cycles of pneumatic tire elements using the submodeling procedure NTU “KhPI” Bull.: Series “Dyn. Strength Mach. 57, 37–49 (2014) 11. Haupt, P.: Continuum Mechanics and Theory of Materials, p. 643. Springer, Berlin Heidelberg (2002) 12. Larin, A.A., Vyazovichenko, Y.A., Barkanov, E., Itskov, M.: Investigation of viscoelastic characteristics of rubber-cord composites considering the process of their self-heating. Strength Mater. 50, 841–851 (2018) 13. Cho, J.R., Lee, H.W., Jeong, W.B., et al.: Numerical estimation of rolling resistance and temperature distribution of 3-D periodic patterned tire. Int. J. Solids Struct. 50(1), 86–96 (2013)
Recent Advances in Development and Characterization of CVD Multilayer Composite Coatings—A Comprehensive Review Osamah Ihsan Ali(B)
and Istvan Gabor Gyurika
Research Centre of Engineering Sciences, University of Pannonia, 10. Egyetem Street, 8200 Veszprem, Hungary [email protected]
Abstract. Recent years have seen significant progress in the improvements of surfaces properties of various materials, as well as efficiency and reliability in a wide ranges of engineering applications such as chemical vapor deposition (CVD technology) on cutting tools, forming tools, dies, and moulds, with perfected equipment and processes to synthesize the composite multilayer-coatings. The present paper has focused on concepts in the impact of modified surface on a set of properties that define the composite coating-structures and microstructures. Rated among this are coating structure, heat resistance, adhesion to a workpiece material, corrosion and oxidation resistance at elevated temperatures. The results of literatures survey showed that, the anti-corrosion and wear performance of the coatings was greatly improved by the addition of several compounds and elements such as aluminum oxide, silicon carbide, nitrides, borides or their mixtures. Also, the properties of substrates significate effected by the chemical composition, thickness, and number of layers of the coatings. As well, the present article provides some theoretical insights into the CVD process for further understanding and improving the coating structures and microstructure, as well as shedding light on some a survey of trends in development of the properties and characterization of composite-multilayered coatings. Keywords: Chemical vapor deposition · Composite-multilayered coatings · CVD process · Composite materials · Multi thin layers · Coating-structure and microstructures
1 Introduction The process known as chemicals vapor depositions (CVD), which involves the depositions of a solids through a chemical’s reactions involving ones or more gaseous chemicals species and is typically thermally activated, has been used for many years in a variety of applications (such as oxidations or/and wears resistant coatings for cementers carbides, steels or alloys, preforms for drawings graded-index optical fibres, thins films for integrated circuits, coatings for nuclear fuels, etc.) [1]. The substrates taken into consideration here often have a straightforward form and are constructed of non-porous © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 63–75, 2023. https://doi.org/10.1007/978-3-031-36201-9_6
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materials. More recently, porous materials have been densified using the same chemical processes, with the solid phase being deposited inside the substrate’s pores. Consequently, the so-called (CVD) technology, originally developed for processing composite coatings materials. Composite has been the most popular material treated using the CVD technology (i.e. a materials made of a carbons fibre preforms up to almost completes densifications, by a carbons matrix resulting forms in-situs pyrolysis of method, or refractory materials: silicon carbide, titanium carbide and even oxides such as alumina). So, it would seem that CVD should be regarded as a potential process method for the production of several composites, assuming the availability of sufficient gaseous precursors and infiltration conditions [2]. Furthermore, several surface treatments have been applied in various tribological applications during the recent 20 years. For e.g., ceramic materials like Tic, TiN, Cr2 O3 , Al2 03 , and Si3 N have started to be employed in hostile conditions like space and vacuum. Ceramic hard coatings have several applications due to their great wear resistance, erosion resistance, and heat resistance, as well as their strong bonding between the coating and the substrate. The two coatings that are utilized the most commonly are CVD-TiC and TiN. Their major benefits are high hardness and melting point, strong substrate adhesion, and low friction coefficients with metallic materials [3, 4]. The CVD multilayer composite coating was researched extensively as a novel coating type based on materials like Tic, TiN, Cr2 O3 , Al2 03 , etc. The tribological characteristics of CVD multilayer composite coatings are excellent, mainly in vacuum. Due to the CVD coating, it has a ceramic’s wear resistance, but because it is also a composite ceramic, it has a special quality that sets it apart from other ceramics. After rigorous, reliable, in-depth testing over an extended period of time, CVD multilayer composite coating may be employed in astronautic bearings [5]. As well, the effectiveness and dependability of a variety of high-speed steel carbide and cutting tools, in addition to the surface qualities of tool materials, have improved in recent years significantly due to different techniques for depositing wear-resistant coatings. It is now feasible to establish theories regarding the effects of modify surface on a variety of parameters that determine the operational performance of cutting tools due to new equipment and techniques for coating synthesis. Hardness, heat resistance, adherence to a workpiece’s material, resistances to corrosions and oxidations at highs temperatures, and resistances to mechanical and thermal cycling are rated among these. The serviceability of cuttings tools with wear-resistant coatings is dependent on the compositions, thickness, and number of layers of the coating, as well as the synthesis technique [6]. Today’s methods may deposit multi-layered coatings formed of single, dual, and triple compounds from periodic system groups such as carbides, nitrides, borides, oxides, and their combinations. Each layer may be produced as thin as several nano-meters, allowing it to be practically defect-free in concept [7]. The current article aims to conduct a comprehensive review on the basic concepts of surface modification of different engineering components, which used in various engineering applications via chemical vapor deposition coatings technique and study the extent effect on the most important mechanical, physical and chemical properties, especially effect the chemical composition of coatings, structures of coating and coating materials of CVD thin film layers and the substrate on resistance of corrosion and
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oxidation. Also, the present paper aims to study the types of CVD coatings structures and which the best for advanced modern engineering applications.
2 A Survey of Trends in Development Processes of Multilayers Composite Coatings In the production of multilayer composite coatings for tools, chemical (CVD) synthesis of coatings is currently commonplace worldwide. Utilizing heterogeneous thermochemical reactions that result in adsorption and chemosorption, CVD methods generate the coatcomprising chemicals in both the vapor-gas environment and on the working faces of the tool. The starting materials are gaseous metal halogens, MeG, whose interaction with other gas mixture compounds (H2 , N2 , CH3 , Ar) accounts for the synthesis of a coating. The properties, structure, and quality of coatings are affected by CVD processing different factors, the most important of which are deposition time and temperature, composition and reactant content in a vapor-gas technique, as well as pressure and flow rate. Structure, phase composition, the existence of flaws in a coating, its adhesion to a substrate, and hence its fundamental qualities are all affected by these variables. The crystallochemical similitude of a coating being applied and a substrate also has a substantial influence on adhesion strength. As a result, a coating is created on the working faces of a tool that has great strength and homogeneity; strong adherence to the substrate; and consistent thickness, even on complex-shaped faces. But CVD procedures frequently take a long time, operate at high temperatures (800 °C to 1500 °C), and do not comply to environment safeties. The other causes the carbide coating-substrate and grain-binder interfaces to form the brittle phase W3 Co3 C, which reduces the material’s strength by 20% to 30%. In the case of WC-Co substrates, this is caused by the reactions (x − 2)Me + xWC + yCo - > (x − 2)MeC + W3 Co3 C. Furthermore, because condensation sites form and coatings grow with prominent epitaxy, CVD methods show a considerable reliance of coating quality on substrate substructure. In particular, condensation nuclei of a coating only occur on cobalt grains because those show the best crystallographic fit when coatings of the TiC-type are being produced on carbide substrates (for example, WCCo). As a result, when the coat coalescence has been complete, straight above carbide grain pores might occur, resulting in qualitydegrading defect in coated carbide. As either a result, substantial attention is being occupied to the grain structure of carbide in the fabrication of these, which is demonstrated in particular by the use of fine and ultrafine structure to reduce of porosity. Prior to coating, a substrate’s surface is saturated with carbon in order to prevent embrittlement caused by produced phases on the coat-substrate interfaces. The issue of the diminished strength of carbide inserts after CVD-coating still has no adequate solution. Finally, it’s important to note that. The substrate and deposit are led to intense bombardment by ions of vaporized metal during the process, which causes the substrate’s surface layers to partly sputter and increases the temperature in the coating region. As a consequence, atomic transportation on the surfaces of a substrate is increased, a deposited condensate has been activated farther, and significant adhesive connections are generated between a coating and substrates. Also, CVD processes which is achieved by the highly ionized beam being accelerated by setting a negative (relative to a chamber’s
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case) potential on a substrate, or by the stream density and homogeneity being enhanced with plasma-optical magnetic systems. To summarize, CVD procedures still require additional research and refinement, and might be pursued in the creation of innovative processes that should incorporate all of the benefits of multilayer composite coatings [8, 9].
3 Development of Composite Layers Coated Tools with Improved Performance Many researches have been conducted to evaluate the performance of composite coatings, and two methods of developing composite coated carbides with wear-resistant coatings have been studied. The first discussed the selection of the optimum coating composition and “architecture” as well as process variables of a deposition technology that incorporates chemical methods, while the second addressed carbides whose structure and properties should comply with the optimum coating conditions, resulting in improved performance of a carbide-coating composite. To reduce the probability of carbide strength, decrease during coating due to decarburising, it has been proposed that a primary barrier layer be deposited first, followed by the major wear resistant coatings coated using specially developed CVD-processes. As a result of this method, a coating with a thin underlayer of complicated composition and a wear-resistant external surface was created. The barrier was designed to prevent interdiffusion between a substrate and a coating, assuring decarburising, excellent phase formation, and substrate embrittlement. Furthermore, the primary barrier layer, which has been used to deposit an underlayer, also included coating/substrate transitional zone with fewer defect. The outer coating was intended to achieve the highest level of wear resistance. Another set of processes was used to reduce the probability of surface defect in CVD coatings. After CVD layers were deposited, these were treated with a 1.5 keV metal (e.g. Ti+) plasma beam (ion bombardment) to remove stresses concentrator. Another challenge to solve was the structure and “architecture” of a coating. On thermodynamic basis, the properties of many refractory substances that qualify as wear-resistant coatings were analyzed. A coating was proposed as a “technical intermediate” whose purpose is to enhance the wear resistance of the contact surfaces while lowering the wear-inducing thermomechanical stress. Finally, based on the findings of the study, a single-layer coating does not tend to achieve the requirements of the properties; consequently, efforts were largely focused on the development of many-layered coatings with different composition and properties, in order to achieve the best operational performance of hard metal tools, particularly those intended for severe and intermittent machining [10, 11].
4 Coating Structure Composite coatings are a series of protective layers applied to a substrate for enhance mechanicals and chemical-properties and characterization of the substrates. There are many types of the structures as the following:
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4.1 Single Layer Coatings CVD coatings are sometimes composed of a single layer that contains one or more phases. Cost-effectiveness most probably explains the existence of this initial generations of composite coatings in many industrials processes. TiC, TiN, CrN, CrC, Al2 O3 , DLC, W2C, WC/C, MoS2 , diamond, softs metals, and some polymers are cases of commercially viable single-layer coatings. To fulfil the functionals demands, an adhesions interlayer may be necessary. Carbonaceous materials, especially DLC films, have received considerable attention. The hydrogen concentration of these amorphous formations with hybridizations ranges between 1 and 50% at%, forms which the majority of the structure’s characteristics are derived [12]. They most probably display the widest variety of frictions and wears behaviors among solid lubricants. Single carbide films are exceedingly hard and have a variety of outstanding features, including highs mechanicals strengths, chemicals inertness, and appealing frictions characteristics. There are several CVD processes available for producing high quality films withs micro and nanocrystalline structures on a wide range of substrates. CVD coatings of excellent quality provide good mechanical and tribological features. However, the surfaces of film coatings are often roughs, with sharps facets that must be polished using costly procedures, see Fig. 1 [13].
Fig. 1. (a) Single layer of TiAlN coating microstructure (b) single layer of CrAlN coating microstructure.
4.2 Complex Multi-layers Coating Multi-layered coatings are mainly composed of lamellae of two or more different materials that are regularly repeated and have a thickness of ups to a few tens of micrometers (if the thickness of each lamella is less than manometer, these films mays develop into superlattices) [14]. There are two types of multilayer coatings: isostructural (each layer has the same structure) and non-isostructural (individual layers have different structure). The magnetron-sputtered systems, such as TiB2/TiC, might be presented as a firsts example of hard coatings multilayers. TiC films develop epistatic layers structures in (111)-orientations on (001)-TiB2 s initially, and only after a particular transition
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zone produce their ‘own’ (200)-TiC. In contrast, TiB2 develops untextured on (200)-TiC initially and then forms the (001)-texture. This allows the creation of a multilayers structures with a (001)-TiB2 /(111)-TiC sequence. Only untextured TiC/TiB2 multilayers are observable in very thin individual layers [15]. In addition, depending on the deposition conditions, TiB(N)/TiN variations in the texturing sequences are observed in the system with peridicties ranging from 2.8 to 17 nm. Hardness is greatest for periodicities of 10 nm or fewer. Another form of multilayer coating involves the alternation of transitions metal compounds with carbon-based films, such as TiC/DLCs (on a Ti/TiC interlayers) or TiN/CNx (Fig. 2) [16]. Furthermore, nano-multilayer is the multilayers concepts modified to a thickness of individuals layers in the 100 nm ranges. The main purpose of these structures is to considerably improve the hardness, fractures toughness, and adhesions of the coatings, hence improving their wear resistance. A periodic structure comprising layers of twos (or mores) materials is known as a superlattices when the thickness of one layer is several nanometers or less. Because the material is organized so that the reciprocal spaces cell is smaller and the actual real space unit cell is layered, superlattice reflections are present. Here are some superlattices that have significantly improved tribological characteristics, including TiAlN/TiAlCrN, TiCN/ZrCN, CrAlYN/CrN, and TiHfN/CrN. The characteristics of multis-layers coatings dependent on the selections of the material, the kinetics of growths and deposition-process and the unique architecture of nanoscales thins films. Combining different structures and compositions within one coatings has so many benefits, including the achieve of unique individuals physicals properties (such as diffusions barriers + lows frictions), decrease of the mismatch in mechanicals and chemicals properties between the substrates and the coatings (chiefly to enhances adhesions), controlling of the residuals strains and consequently the stress within the coating, the ability to prevent cracks during operation under difficult conditions, and enhancements of adhesion. The following a list of mechanisms for toughening multilayer coatings. Among these mechanisms are: (1) (2) (3) (4)
the splitting of cracks at boundaries of the smalls-grains. Deflections of cracks at layer interfaces. lowering or concentrating stress by opening the interface, and. “Nano-plasticity” is the plastics deformations at the interfaces for stress relaxation and energy dissipations. This generation of coatings has seen significant progress as a result of the versatility of vacuums technology and the development of multiplex/hybrids processes, which have lately found significant industrial uses. [17, 18].
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Fig. 2. Example of a complex CVD-obtained multilayer coating microstructure.
So, these coatings improve the mechanical, physical and chemical properties of different substrate materials, as well as enhance their oxidation resistance. Actually, there are many studies focus on the oxidation behavior of the CVD-composite multilayer-coatings for C/C composites in airs has been investigated in details by many researchers, below will take studies of SiC-coated C/C composites as clear example. S. Kobayashi et al. [19] by directly chemically reacting SiCl4 gas and carbons at 1473 K, CVD-SiC coating on C/C composites was formed. The experiment results showed that CVD-SiC coatings successfully reduced the rates of oxidation of C/C composites. Pengfei, Z. et al. [20] The SiC layers were created via using a pressure-pulses CVD (pulse-CVD) process using methyl trichlorosilane (MTS) as the precursor. After 1 h of air exposure, the weights loss of as-prepared SiCs-coated C/C composites was just 1.3 wt%. Akihito, S. et al. [21] investigated the anti-oxidative properties of multilayered SiC-coating on C/C composite via low pressures CVD technique. The results showed that the failures of the multilayered coating have always been generated by preparations faults insides the coatings, and that the weights loss decrease as the depositions periods increase. In summary, previous studies haves shown that CVD-SiCs coatings could well protects C/C composites against oxidations at highs temperatures. Actually, various oxidizing environments may cause diverse oxidation behaviors in SiC ceramics. Water vapor, being a commons gas in the atmosphere and a gaseous combustions product, has a major effect on the oxidation behavior of CVD-SiC. Jingguo, F [22] studied oxidation-kinetics of bulks CVD-SiC in dry-oxygens and wets-oxygens in 1473 and 1673 K. The experiment results demonstrated that waters vapors improved the oxidation-kineticss of SiC when compared to dry-oxygen rates. As well, Mandal, D. [23] A paralinear-oxidation model was also presented to explains the longs-terms deterioration of CVD SiC. The results demonstrate that SiC suffers paralinear oxidations/volatilizations processes at greater partials waters pressures and
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higher gaseous flows rates. K.L. More [24] investigated the oxidation-behavior of SiCSiC composite coat through CVD SiC at higher waters vapors-pressures. The results showed that at greater water vapor pressures, SiC recession rates might be accelerated. Hui, Y. [25] investigated the oxidations behavior of CVD-SiC-coated C/SiC composites in dry O2 and H2O/O2 environments. The results of the experiments showed that the weights loss of C/SiC composites in H2O/O2 was less than in dry-oxygens with the same partials pressures of oxygens, as shown in Fig. 3. This is because the oxidations rates of SiC was significantly increased in oxygens, and the SiO2 scales formed in H2O acted as an efficient oxidations barrier. It clear that waters vapors were essential to the oxidations of the CVD SiC composites. In summary, previous studies have shown that CVD-SiC coatings could well protects C/C composites against oxidations at highs temperatures. Actually, various oxidizing environments may cause diverse oxidation behaviors in SiC ceramics. Waters vapor, being a commons gas in the atmosphere and a gaseous combustion product, has a major effect on the oxidations behavior of CVD-SiC.
Fig. 3. Schematic of oxidation behavior of C/C composite coated samples with CVD-SiC coating in dry oxygen and wet oxygen.
According of microstructural-features and elucidation in P.F. Zhang studies [26], a schematic model illustrating the corrosion of the Al/Al2O3 coating is proposed (Fig. 4). The coating’s characteristic alumina splat distribution causes an alumina skeleton to develop, which resists the corrosion test practically intact. The SEM analysis revealed that the corrosion at the top surfaces of the coatings is most extensive. 4.3 Gradient and Nanostructured Coatings Recent years have seen a development in the literatures and theoretical knowledge of this news generations of composite films coatings, particularly with related of coatings that are nanocrystallized and nanocomposite. The concept behind this generations of coatings is to maximize hardness (H) whiles making sure that the elastic modulus be as low as possible (E). The mechanism inhibits the growth of stable ones, which results
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Fig. 4. Cross sectionals morphologies of the Al/Al2 O3 composites coatings after the accelerate corrosion-testing; a) the skeletons resist effectively furthers progressions of the corrosion, (b) schematics depictions showing mechanisms as to how the Al2 O3 skeletons in the coatings resist advancements of corrosion.
in better resistance to dislocation movement at the nanoscale scale. And, by controlling the presences of interfaces between nanocrystallized (nc) metal nitrides/metals, or between amorphous and nanocrystallized phases, as with nc-MnNs/-Si3N4s (where M is Ti, W, V, or another transitions metal), or in TiB2, one can reduce the grains sizes in accordance with the wells-knowns Hall-Petchs relationships, see Fig. 5. The most recent advancements in the application of nanometer-scale coatings are presumably focused on improving their stability at highs temperatures in order to prevent diffusion, grain expansion, and phase transition. This appears possible if the layers materials are stables to create low-energy coherent interfaces and are thermodynamically stable with respects to one another [27]. Controlling the structures and chemical-compositions of composite coating at the nanoscales both a unique scientific subject and an industrials issue. This controls is typically achieved throughs threes structural configuration. The firsts configurations functionally graded coatings, which are seen to be the primal next step after multilayered coating. Grading the composition has been accomplished by severe controls of the depositions process to combines relatively hard TiAlN phase with a softer MoS2 phases, with an increase towards the latter phases to the tops surface. Another advantage of gradings the mixture is that’s it increases the adherence of DLC-based coatings with metal addition, such as Ti/TiC/DLC, as seen in Fig. 6. Ti and TiC(H) graded underlayers have achieved such increases in wears resistances of DLC films. Second, nanostructured coatings include nano-crystallized films (with nanometer grains sizes) and nanocomposite coatings. According to Veprek, nanocomposites are structures that combines amorphous and crystalline phases [28, 29].
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Fig. 5. CrN/TiAlN nanolayer coating-structure.
Fig. 6. Ti/TiC/DLC functionally gradient coating system that shows the gradual composition variation across the thickness of the layers.
5 Conclusion The current paper shows clearly that composite multilayer hard coatings are commonly utilized in several engineering applications, such as on cuttings and forming tools to enhance productivity, extend tool life, and enable a variety of unique engineering applications. Composite hard coatings depositions have now become a routine processing step in the tools industry. Currently, a wide ranges of CVD hard coating are available for a variety of applications. Also, the survey shows that, the increase most mechanical and chemical properties and characterizations of many components and tools coated via CVD-multilayered coatings. There are many materials had been successfully applied as multi-layer-coated the metals forming and plastics moldings dies and wears components such CrN-based multi
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component nitride, TiC, TiN, CrC, Al2 O3 , DLC, W2C, WC/C, MoS2 , TiCN/ZrCN, CrAlYN/CrN and TiHfN/CrN and the majors’ functions of these structures is to significant enhanceshardness, toughness, friction behavior, oxidation and corrosion resistance, and adhesions to a workpiece material coated. Regarding to the best coating-structures, the literature survey exhibits that, the composite multilayer coating-structure and the nanostructured-coatings are the best. So, the advantages of combine several structures and compositions within one coating includes achievements of various individuals mechanical and physicals properties. As well, increase the ability to stops cracks during operations under service conditions, and reductions of the mismatch in mechanicals and chemicals properties between the substrate and the coatings. Acknowledgment. This work was supported by the TKP2020-NKA-10 project financed under the 2020-4.1.1-TKP2020 Thematic Excellence Programme by the National Research, Development and Innovation Fund of Hungary.
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Adhesive Joint Optimization. Application of Bezier Curves for Structural Shape Description Sergei S. Kurennov , Konstantin P. Barakhov(B) and Hanna S. Barakhova
, Igor M. Taranenko ,
National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine [email protected]
Abstract. The goal of paper is solution of the problem of topological optimization of symmetrical double-shear adhesive joint. Suggested mathematical model of joint with variable thickness is generalizing of classical model of HollandReissner. The shape of doubler is described by means of Bezier curve. Seeking parameters in the problem of optimization are coordinates of reference points of Bezier curve. As objective function both joint length and doubler cross-section area can be considered. Restriction is applied on stress in adhesive film and in doubler. Direct problem of estimation joint stress state at given geometrical parameters is solved by means of method of finite differences. To solve the problem of optimization genetic algorithm was used. To improve convergence of genetic algorithm island model of evolution is suggested. Distinctive feature of suggested model of algorithm is in following: on one of «islands» mutations occur more frequently and more dispersion comparing with two other «islands». Such combination ensures quick evolution selection and stability of obtained results. Model problem is solved. Keywords: Adhesive Joint · Optimization · Genetic Algorithm
1 Introduction Lap adhesive joints are inherent part of up-to-date structures made of composites. Wide spreading of adhesive joints in composite structures is stipulated by their high manufacturability, gas tightness, low weight, high aerodynamic efficiency. Lap adhesive joints don’t disturb structure of composites and allow to realize high strength and other mechanical properties in structures. However, known drawback of lap joints is stress concentration near edges of joining [1, 2]. To reduce stress concentration and increase joint strength following different structural solutions are used: increasing thickness of adhesive film near joint edge [3], reduction thickness of plates near joint length (“scarf” joint) [4], application of different types of adhesives [5], embedding of transversal joining elements inside of joint [6], and other methods [7, 8]. Application of symmetrical doubleside joints allows to escape of structural bending and reduce peeling-off stress inside © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 76–87, 2023. https://doi.org/10.1007/978-3-031-36201-9_7
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adhesive film [9, 10]. Usually, to describe stress state of adhesive joints one-dimensional mathematical model of sandwich rods and beams with quite compliant filler are used. If parameters of layers are constant along joint length, then stress-state state of joint can be described in analytical form [1, 2]. However, if thickness of joining layer changes along joint length, then the problem has no analytical solution. Therefore, numerical methods are used for determination of stress state of joint with variable thickness along length. The problem of topological optimization is qualitatively more complicated comparing with classical problem of parametric optimization. The reason of this is in following: seeking value is a function of material distribution over structure but not the selection of set of small number of unknown parameters. Generally, topological optimization of lap joints consists in finding optimal joint length and dependence of joining plates thickness along joining length. One of the possible ways of this problem solution is discretization of seeking function. In this case the problem can be reduced to finding thickness of structural elements at system of points [11]. Transition from continuous functions to discrete one allows to optimize joint with stepped thickness variation [12]. If seeking function is assumed to be continuous then it can be described by known values in system of points with splines [13], Bezier functions [14, 15] or trigonometric series. Since number of parameters is big and solution of reversed problem of optimization even by numerical methods is impossible, then to solve optimization problem other methods can be used, for example, genetic algorithms. This method is concluded in finding optimal parameters of a problem by means of solving consequence of direct problems. Usually, direct problem of structural stress state estimation at given parameters is solved by means of method of finite elements [13–16]. Optimization can be conducted also for adhesive film thickness near joining edge [17], dimension and shape of squeezed recess of adhesive substance at joint edge [18], and also the structure of composite [19, 20]. General drawback of finite elements method application at solving problems of topologic optimization is relatively slow rate of algorithm operation. It can be related with two-dimensional formulation of considered problem. Due to this it is necessary to compose finite-element grid on each stage of iteration, moreover, adhesive film is divided on relatively fine elements. The goal of the paper is solution of the problem of topological optimization of adhesive joint in one-dimensional formulation. Application of well-known mathematical models of joints [1, 21], which are used for description of stress state of joints in analytical form, allow to reduce problem dimension without significant losing of precision and, as a consequence, to increase rate of calculation. One-dimensional problem formulation allows to use quite simple and fast method of finite differences [6, 11] for estimation of joints stress state. To increase rate of genetic algorithm convergence improved island model of genetic algorithm (Island Model GA) [22, 23] is used in the paper. The difference of suggested model of genetic algorithm from known models consists in following: conditions on «islands», i.e. in subpopulation, in which evolution selection occurs in parallel; way with other, are not the same like in other islands. On one of islands mutagenesis happens more frequently and with more dispersion than in on other islands. Such combination of high variability on one island in combination with regular migration of the best individuals between islands ensures good operation rate of evolution algorithm and stability of results.
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2 Problem Formulation 2.1 Mathematical Model of Stress State Let’s consider structure consisting of two plates, which jointed with each other by means of symmetrical doublers, Fig. 1 (a). Such structure doesn’t withstand bending at tensioncompression and widely used in engineering due to this. Since the structure geometry is symmetrical one can consider its quarter only. Transversal translations of central layer of main plate are equal to zero. If structure deformation is considered in frames of the theory of rods, then it is possible to analyze the zone of adhesive layer only. The structure is loaded with longitudinal forces 2F. Thickness of adhesive film is considered to be constant along joint length and uniform on each zone. The length of adhesive joint is designated as L. Differential element of adhesive zone and force factors applied to it are shown on Fig. 1 (b).
(a) (b) Fig. 1. Adhesive joint: (a) – structural scheme, (b) – differential element
Equilibrium equations of doubler and main plate are following dN1 /dx = −τ ; dN2 /dx = τ ; dQ1 /dx = σ ; dM1 /dx − s1 (x)τ − N1 ds1 /dx + Q1 = 0,
(1) where N1 , N2 – longitudinal forces in load-carrying layers; Q1 , M1 – lateral force and bending moment inside of doubler; τ , σ – shear and normal stress in adhesive film; s1 – distance from neutral axis of doubler to adhesive film in case of symmetrical structure of doubler s1 (x) = 0.5 δ1 (x), where δ1 (x) – doubler thickness. Equations of rod deformations have following view N1 = B1 dU1 /dx, N2 = B2 dU2 /dx, D1 d 2 W1 /dx2 = M1 ,
(2)
where U1 and U2 – longitudinal translations of load-carrying layers; W1 – transversal translations of doubler; B1 (x) and B2 – rigidity of layers at tension-compression, if layers are uniform by thickness, then B1 (x) = δ1 (x)E1 , B2 = δ2 E2 , where E1 and E2 – elasticity moduli of correspondent layer; D1 (x) – doubler bending rigidity, D1 (x) = δ13 (x)E1 /12.
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Stress in adhesive film [1, 10] σ = K · W1 , τ = P(U1 − U2 + s1 (x)dW1 /dx),
(3)
where K, P – rigidity of adhesive film at tension compression and at shear, which can be calculated as following, for example, K = E0 δ0−1 , P = G0 δ0−1 , where δ0 – adhesive film thickness, E0 i G0 – elasticity modulus and shear modulus of adhesive substance. Boundary conditions have following view N2 (0) = F, N2 (L) = 0, N1 (0) = 0, Q1 (0) = 0,M1 (0) = 0, U1 (L) = 0, Q1 (L) = 0,
dW1 /dx|x=L = 0.
System of Eqs. (1)–(3) can be reduced to the system of following differential equations with respect to U1 , U2 and W1 W1 1 dB1 dU1 B2 d 2 U2 B1 d 2 U1 dW1 dW1 + − U2 + s1 − U1 + U2 − s1 = 0, U1 + =0 2 P dx P dx dx dx P dx2 dx 2 2 dD1 d 3 w1 1 d 2 D1 K ds1 dw1 D1 d 4 w1 2 d w1 + + − s − 2s1 + w1 1 4 3 2 2 P dx P dx dx P dx dx dx dx P ds1 B1 ds1 d 2 U1 ds1 B1 d 2 s1 dU1 1 ds1 dB1 dU2 U2 − + s1 + − U1 + s1 = 0. − + 2 2 P dx dx dx P dx dx P dx dx dx dx (4)
Boundary conditions also can be reduced to the same variables U1 , U2 and W1 . 2.2 Optimization Problem The problem of optimization can be formulated as follwing – it is necessary to find joint length L and dependence of doubler thickness δ1 (x) on longitudinal coordinate; these parameters ensure extremal value of definite criterion of optimality at strength restrictions fulfilling. As optimality criterion doubler mass of joint cost can be selected. Doubler mass with precision up to arbitrary multiplier is proportional to its cross-sectional area: M = ∫L0 δ1 (x)dx → min .
(5)
Restrictions can be applied to maximum stress in adhesive film. Here different strength criteria can be used, for example, criterion of maximum principal stress [21] (6) σg∗ (x) = |σ (x)| + σ 2 (x) + 4τ 2 (x) /2 ≤ σgmax , ∗|
where x ∈ [0; L]; σg (x) – absolute value of the first principal stress in adhesive film; σgmax – adhesive film strength. Kpome togo, pazpyxenie coedineni moet ppoicxodit v fopme pazpyva nakladki. B cily togo neobxodimo vvecti ogpaniqeni na makcimalnye nappeni v nakladke σp∗ = N1 (x)/δ1 (x) + δ1 (x)M1 (x)/2D1 (x) ≤ σpmax ,
(7)
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where x ∈ [0; L]; σp∗ (x) – maximum stress in doubler cross-section; σpmax – strength of doubler material. Moreover, upper and lower structural or technological restrictions can be applied on function δ1 (x). In the first case doubler thickness has not be less than definite predefined value δ1 (x) ≥ δmin ,
(8)
where δmin – definite minimal possible doubler thickness, which can be equal to, for example, to the thickness of composite monolayer. Restriction on maximum doubler thickness can be given by considerations of aerodynamic efficiency, possibility of quality control, manufacturing abilities etc. 2.3 Genetic Algorithm of Optimization Solution of the problem of topologic optimization in analytical form faces significant difficulties. Therefore, to solve the problem it is suggested to use genetic algorithm. For this (1) authors suggest to select joint length L and doubler thickness δi at node (reference) points as seeking variables, and find their optimal values which ensure, for example, minimal mass of doubler (5) at fulfilled restrictions on strength (6) and (7). However, in comparison with the problem about finding optimal material distribution along beam (1) length [11], if values of thickness δi in neighboring points differ significantly (that can happen due to crossbreeding or mutations in process of genetic algorithm operation), then stress in adhesive film (3), calculated by finite-difference method, will have (1) unrealistic surges. Therefore, optimal dependence for δi is sought between functions having property of smoothness a priory. The same conclusion can be done from intuitive considerations about following: more probably seeking function δ1 (x) is smooth, has no tearing, angular points and surges. It is suggested in this paper to seek function δ1 (x) as Bezier curve, which has a large amount of reference points. In this case the problem of optimization is reduced to finding of optimal coordinates of reference points. I.e. the shape and length of a doubler are defined by arranged set of coordinates of reference points of Bezier curve (x1 , y1 ), (x2 , y2 ), ..., (xn , yn ) , where n – quantity of reference points. In this case joint length is equal to L = xn . Genetic algorithm requires solution of direct problem of joint stress state estimation at known joint length and function δ1 (x). For numerical solution of the system of differential equations with variable coefficients (4) the direct method of finite differences is used. This method is the simplest and fast in software realization. If function δ1 (x) is assigned then also functions s1 (x), B1 (x) and D1 (x) are known too. To realize finite differences method let’s divide zone of adhesive joining x ∈ [0; L] on the system of node points with numbers from zero to N . The increment of division is (1) h = L/N . Translations of load-carrying layers are designated as following U1 (xi ) = ui , (2) (1) U2 (xi ) = ui and W1 (xi ) = wi . (1) (2) Solution of the system of differential Eqs. (4) is reduced to finding ui , ui and (1) wi . Presence of boundary conditions allows to use as unknown variables translations at outer nodes beyond of joining zone. Function δ1 (x) and functions s1 (x), B1 (x) and
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D1 (x) related to it are defined inside of joining zone only x ∈ [0; L] I.e. values of (1) (1) (1) (1) (1) (1) δ0 , .., δN , B0 , .., BN , D0 , .., DN are known. Therefore, at edge points x0 and xN derivatives from translations in (4) and in boundary conditions can be written in the form of differences by symmetrical finite-difference template. However, to write derivatives from s1 (x), B1 (x) and D1 (x) in difference form at boundary points one has to use single-side right and left templates. Writing differential Eqs. (4) in finite-difference form for points from 0, 1, ..., N , and also boundary conditions one can get the sys(1) (1) (1) (2) (2) tem of linear equations with respect to unknown u−1 , u0 , ..., uN +1 , u−1 , ..., uN +1 , i (1)
(1)
(1)
w−2 , w−1 , ..., wN +2 , which contains 3N + 11 equations. After solving obtained system of equations one can get transitions of load-carrying layers at node points. This allows to find stress in adhesive film (3) (set of stress values σi and τi at node points), longitudinal forces in load-carrying layers and also all other force factors in joint elements. To realize genetic algorithm it is necessary to involvefitness-function, which allows to arrange by quality different sets of seeking parameters (x1 , y1 ), (x2 , y2 ), ..., (xn , yn ) . If mass of joint (5) is selected as optimality criterion and restrictions on maximum first principal stress in adhesive film (6) and minimal thickness of doubler (8) are applied, then fitness-function can have following view: = ∫L0 δ1 (x)dx + fj , (9) j
where fj – penalty functions, which possess following general property – they are more than zero is restrictions are violated and they are equal to zero if restrictions are satisfied. Dependence of penalty functions on deviation of correspondent parameter from restriction can be linear or power, for example ⎧ 2 ⎪ ⎨ Z1 max σg∗ /σgmax − 1 , max σg∗ > σgmax f1 = ⎪ ⎩ 0, max σ ∗ ≤ σ max g g where Z1 – definite big number. By analogous way functions f2 , f3 etc., which correspond to restrictions (7), (8) and other possible restrictions, can be selected. Genetic algorithms possess some drawbacks, the most significant of them is the complexity of adjusting. At high changeability convergence is disturbed. And even good found values of seeking parameters face with risk to be lost due to mutations. At low changeability approximate solution can be found quickly, but then convergence become slow and population expression occurs (stabilization near definite non-optimal values of parameters). One of the possible exits from this contradiction is application of island model of evolution algorithm. In this case general population is divided by several insulated sub-populations (islands). On each of these islands evolution process happens independently. The best individuals can migrate from island to island with given period. The model with three islands, on one of which probability of mutations and mutation dispersion is higher than on other two ones, is considered in the paper. Such combination of two relatively stable islands with one island having higher level of mutagenesis allows to combine quick search of good solutions with results stability and with saving the best
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solutions in general population. Operation of evolution algorithm on a single island consists of further stages: 1) 2) 3) 4) 5) 6) 7)
Creation of original population; Selection – i.e. separation of the best individuals for crossbreeding; Separation of selected individuals in pairs; Crossbreeding and mutation; Returning of newborn individuals to the population; Extinction of the worst individuals; Checking-up the stop criterion. If criterion is fulfilled then algorithm is stopped, if no – return to the item 2 occurs.
All stages of the algorithm allow different methods of realization. For example, selection can be conducted in the form of selection of given quantity of the best individuals or in the form of probability selection. Moreover, probability can depend on both value of objective function (9) and on place of individual in arranged list. Separation of individuals on pairs can be arbitrary or can depend on degree of closeness or difference of individuals. Such approach requires introduction of degree of individual closeness. In this paper for this purpose the sum of squares of distances between reference points of Bezier curve of each two individuals is used as degree of individual closeness. I.e. following sum is used as closeness criterion for individuals with numbers i and j n (j) 2 (j) 2 (i) (i) i,j = xk − xk + yk − yk . k=1
Closer (similar) individuals i and j less value of i,j . The strategy of outbreeding was used in the paper, i.e. more difference between individuals more probably they will be pair for crossbreeding. Crossbreeding is realized in the form of random interchange of reference points by two parental individuals. I.e. reference point with number k of descendent of individuals i and j is the result from selection two correspondent of random (j) (j) (i) (i) reference points of parental individuals xk , yk and xk , yk . Mutations in this case consist of in changing of coordinates of some reference points of descendent on definite arbitrary values, which has zero mathematical expectation. Quantity of mutation reference points is also random value. Both extinction of the worst individuals and selection of individuals for crossbreeding can be realized by several methods. Fulfilling of given number of iterations or reaching of given level of homogeneity by population can be used as a criterion of stop. Since in island model of evolution algorithm several populations are considered, then above-mentioned algorithm is applied to each population separately. Island model of genetic algorithm consists of following stages: 1) 2) 3) 4)
Creation of original populations on each island; Fulfilling of K cycles of evolution selection on each islands; Migration from island to island of definite quantity of the best individuals; Checking-up the stop criterion. If criterion is reached – algorithm has to be stopped. If criterion is not reached – return to the item 2.
In suggested variant of island model of genetic algorithm three islands are considered. On one of islands mutations occur with higher probability and higher dispersion
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comparing with two other islands. After K cycles of generation changing two islands are selected arbitrary. These two islands interchange with the best m individuals (migrants). This ensures income of novel genetic information inside of population. Quantity of migrants has to be significantly less than entire quantity of individuals on island. As criterion of stop fulfilment of definite quantity of migrations can serve. After finishing of algorithm operation, it is necessary to select optimal solution from entire combination of individuals. Since parameters of a single, even the best individual, are results of arbitrary mutations and crossbreeding, they can differ from each other at different realization of algorithm. Average value of parameters in population is more stable to arbitrary deviations. Therefore, as the solution of problem optimization, it is recommended to take truncated selected mean value of parameters of the best individuals of all populations (or from all three populations). For calculation of truncated selected mean value one has to use, for example, a half of the best individuals of population.
3 Numerical Realization and Results Let’s consider results of solution got by application of suggested algorithm of topological optimization of adhesive joint for exact case. Joint with following parameters is considered: E1 = 100 GPa, E2 = 70 GPa, δ2 = 3 mm, δ0 = 0.1 mm, E0 = 2.274 GPa, G0 = 0.54 GPa, σgmax = 30 MPa, δmin = 0.5 mm. Joint is loaded with longitudinal force per unit length F = 300 kN/m. Maximum stress in doublet are restricted with the limit of σpmax = 115 MPa. Six reference point of Bezier curve were used in the paper. Calculation shown that optimal joint length is L = 133.85 mm. Figure 2 demonstrates diagram of doubler thickness changing along joint length and positions of Bezier curve.
Fig. 2. Doubler thickness and reference points of Bezier curve
It can be seen that at the left side of joint minimal allowable doubler thickness is equal to 0.5 mm. Figure 3 shows following parameters: (a) stress in adhesive film (dimensionless view), (b) – maximum stress in doubler and load-carrying plate. Maximum principal stress is equal to maximum allowable at joint edges. Maximum allowable stress in doubler σpmax is designates with black dotted line on Fig. 3 (b). One can observe that restriction (7) is also fulfilled as equality.
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Fig. 3. Stress in adhesive film
Analysis of stress state of joint by means of finite elements method was done to verify suggested model. Adhesive film was divided on elements with maximum dimension 0.1 δ0 . Figure 4 shows diagrams of stress components (3) in adhesive film near joint edges (i.e. at the most stressed zones), which were calculated by means of suggested model and by means of finite-element modelling (stress in mid plane of adhesive film are shown).
(a)
(b)
Fig. 4. Stress in adhesive film near joint edges; AM – analytical model, FEM – finite-element model
4 Conclusion Following conclusions were obtain due to solution of series of problems and results analysis: 1. Dependence of doubler length and shape has non-linear character as function of load. 2. Presence of restriction on minimal allowable doubler thickness leads to following result: found optimal doubler shape contains horizontal zone with minimal allowable thickness near non-loaded doubler edge. As far as authors know, such structural solution was not suggested previously by anybody.
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3. It is impossible to reach uniform stress distribution in joint at given original conditions. Apparently, key restriction, is constant thickness of main plate along joint length. Due to this situation joint load-carrying ability is restricted. 4. Application of Bezier curves in comparison with expansion of doubler section in trigonometric series possesses some advantages – less quantity of seeking parameters, possibility to automatically considering such known effects as, for example, horizontal position of tangent to diagram of doubler thickness at origin of coordinates. Suggested approach can be developed and generalized in further directions: 1. Restrictions on translations value are added, other strength criteria of adhesive film are used etc. 2. Solving of the problem of joints topological optimization, in which more compliant adhesive is used near joint edges and more rigid one – at central zones of adhesive contact. 3. Application of genetic algorithm of optimization for solving problems of topological optimization of coaxial tubes joining [26] and joints having circular symmetry [10]. 4. Application of more precise model of adhesive film stress state [21]. 5. Developing and more complicated analysis of island model of genetic algorithm. Application, for example, different objective functions on each of islands, and also combinations of genetic algorithms with other up-to-date methods of optimization.
References 1. da Silva, L.F.M., das Neves, P.J.C., Adams, R.D., Spelt, J.K.: Analytical models of adhesively bonded joints. Part I: Literature survey. Int. J. Adhes. Adhes. 29, 319–330 (2009). https://doi.org/10.1016/j.ijadhadh.2008.06.005 2. Shishesaz, M., Hosseini, M.: Effects of joint geometry and material on stress distribution, strength and failure of bonded composite joints: an overview. J. Adhes. 96, 1–69 (2018). https://doi.org/10.1080/00218464.2018.1554483 3. Barakhov, K.P., Taranenko, I.M.: Influence of joint edge shape on stress distribution in adhesive film. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2021: Synergetic Engineering, pp. 123–132. Springer International Publishing, Cham (2022). https://doi.org/10.1007/978-3-030-94259-5_12 4. Haghani, R., Al-Emrani, M., Kliger, R.: Effect of laminate tapering on strain distribution in adhesive joints: experimental investigation. J. Reinf. Plast. Compos. 29(7), 972–985 (2009). https://doi.org/10.1177/0731684408102698 5. Ramezani, F., Ayatollahi, M.R., Akhavan-Safar, A., da Silva, L.F.M.: A comprehensive experimental study on bi-adhesive single lap joints using DIC technique. Int. J. Adhes. Adhes. 102, 102674 (2020). https://doi.org/10.1016/j.ijadhadh.2020.102674 6. Karpov, Y.S.: Jointing of high-loaded composite structural components. Part 2. Modeling of stress-strain state. Strength Mater. 38(5), 481–491 (2006). https://doi.org/10.1007/s11223006-0067-9 7. Shang, X., Marques, E.A.S., Machado, J.J.M., Carbas, R.J.C., Jiang, D., da Silva, L.F.M.: Review on techniques to improve the strength of adhesive joints with composite adherends. Compos. B Eng. 177, 107363 (2019). https://doi.org/10.1016/j.compositesb.2019.107363 8. Kupski, J., Teixeira de Freitas, S.: Design of adhesively bonded lap joints with laminated CFRP adherends: review, challenges and new opportunities for aerospace structures. Compos. Struct. 268, 113923 (2021). https://doi.org/10.1016/j.compstruct.2021.113923
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9. Amidi, S., Wang, J.: An analytical model for interfacial stresses in double-lap bonded joints. The J. Adhes. 95(11), 1031–1055 (2018). https://doi.org/10.1080/00218464.2018.1464917 10. Kurennov, S., Smetankina, N.: Stress-strain state of a double lap joint of circular form. Axisymmetric model. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2021: Synergetic Engineering, pp. 36– 46. Springer International Publishing, Cham (2022). https://doi.org/10.1007/978-3-030-942 59-5_4 11. Kurennov, S., Barakhov, K., Taranenko, I., Stepanenko, V.: A genetic algorithm of optimal design of beam at restricted sagging. Radioelectron. Comput. Syst. 2, 83–91 (2022). https:// doi.org/10.32620/reks.2022.1.06 12. Veisytabar, M., Reza, A., Shekari, Y.: Stress analysis of adhesively-bonded single stepped-lap joints based on three-parameter fractional viscoelastic foundation model. Proc. Instit. Mech. Eng., Part L: J. Mater.: Des. Appl. 236(5), 933–949 (2022). https://doi.org/10.1177/146442 07211062497 13. Groth, H.L., Nordlund, P.: Shape optimization of bonded joints. Int. J. Adhes. Adhes. 11(4), 204–212 (1991). https://doi.org/10.1016/0143-7496(91)90002-y 14. Ayaz Ümütlü, H.C., Kiral, Z.: Airfoil shape optimization using Bézier curve and genetic algorithm. Aviation 26(1), 32–40 (2022). https://doi.org/10.3846/aviation.2022.16471 15. Zou, P., Bricker, J., Uijttewaal, W.: Optimization of submerged floating tunnel cross section based on parametric Bézier curves and hybrid backpropagation – genetic algorithm. Mar. Struct. 74, 102807 (2020). https://doi.org/10.1016/j.marstruc.2020.102807 16. Ejaz, H., Mubashar, A., Ashcroft, I.A., Uddin, E., Khan, M.: Topology optimisation of adhesive joints using non-parametric methods. Int. J. Adhes. Adhes. 81, 1–10 (2018). https://doi. org/10.1016/j.ijadhadh.2017.11.003 17. Rispler, A.R., Tong, L., Steven, G.P., Wisnom, M.R.: Shape optimisation of adhesive fillets. Int. J. Adhes. Adhes. 20(3), 221–231 (2000). https://doi.org/10.1016/S0143-7496(99)000 47-0 18. Belingardi, G., Goglio, L., Tarditi, A.: On the optimization of single lap metal/plastics adhesive joints. Key Eng. Mater. 221–222, 161–172 (2001). https://doi.org/10.4028/www.scientific. net/kem.221-222.161 19. Symonov, V.S., Karpov, I.S., Juraˇcka, J.: Optimization of a panelled smooth composite shell with a closed cross-sectional contour by using a genetic algorithm. Mech. Compos. Mater. 49(5), 563–570 (2013). https://doi.org/10.1007/s11029-013-9372-0 20. Hassan Vand, M., Abbaszadeh, H., Shishesaz, M.: Optimization of adhesive single-lap joints under bending moment. J. Adhes. 98, 1–26 (2021). https://doi.org/10.1080/00218464.2021. 1932485 21. Kurennov, S.S.: Refined mathematical model of the stress state of adhesive lap joint: experimental determination of the adhesive layer strength criterion. Strength Mater. 52(5), 779–789 (2020). https://doi.org/10.1007/s11223-020-00231-5 22. Gozali, A.A., Fujimura, S.: DM-LIMGA: dual migration localized island model genetic algorithm—a better diversity preserver island model. Evol. Intel. 12(4), 527–539 (2019). https:// doi.org/10.1007/s12065-019-00253-2 23. Palomo-Romero, J.M., Salas-Morera, L., García-Hernández, L.: An island model genetic algorithm for unequal area facility layout problems. Expert Syst. Appl. 68, 151–162 (2017). https://doi.org/10.1016/j.eswa.2016.10.004 24. Kumar, S., Pandey, P.C.: Behaviour of Bi-adhesive Joints. J. Adhes. Sci. Technol. 24(7), 1251–1281 (2010). https://doi.org/10.1163/016942409x12561252291982
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25. Öz, Ö., Özer, H.: On the von Mises elastic stress evaluations in the bi-adhesive single-lap joint: a numerical and analytical study. J. Adhes. Sci. Technol. 28(21), 2133–2153 (2014). https://doi.org/10.1080/01694243.2014.948110 26. Kurennov, S.S., Barakhov, K.P., Poliakov, A.G.: Stressed state of the axisymmetric adhesive joint of two cylindrical shells under axial tension. Mater. Sci. Forum 968, 519–527 (2019). https://doi.org/10.4028/www.scientific.net/msf.968.519
Study of Drilling Cutting Modes Impact on Shrinkage and Surface Roughness of Holes in Composite Materials Borys Lupkin , Kateryna Maiorova(B) , Iurii Vorobiov , Viacheslav Nikichanov , and Valeriy Sikulskyi National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine [email protected]
Abstract. Studying the influence of the cutting modes for holes drilled in carbon fibers ELUR-P-1,0 and UOL-300-1A, fiberglass T-10–14, organic fiber T42/1–76 composites on shrinkage and surface roughness was carried out. It was found that the surface roughness of the ELUR-P-1,0 carbon fiber specimens and the T-10–14 fiberglass specimens at both studied cutting modes was provided in a range Ra 3.2…Ra 6.3 – sufficient for the holes in composites. The opposite effect showed the same specimens with 50 m/min cutting speed and 0.1 mm/rev automatic feed rate. With ELUR-P-1,0, T-10–14 and T42/1–76 specimens were studied dependence of drilled holes shrinkage on cutting wedge angle range. Also, the study results showed that the cutting speed increase leads to the average shrinkage increasing. Studies of the hole surface structure after drilling the UOL-300-1A carbon fiber at 50 m/min cutting speed and 0.1 mm/rev automatic feed rate showed: splintering - up to 0.05 mm; slapping from 1.0 to 1.5 mm; delamination up to 1.5 mm; shrinkage 0.001 mm at the drilled hole entry. By results of experimental studies it’s concluded: an increase in cutting speed up to 50 m/min showed admissible holes shrinkage in the all tested composites, but there is a risk of the holes surface quality reducing (surface roughness increasing), which indicates the need for further research with drills of other types and finding out a range of appropriate cutting speeds. Keywords: Composite Laminate · Composites Material · Drilling · Cutting Mode · Shrinkage · Surface Roughness
1 Introduction For reducing the weight of aerospace technology and declining the labor input of its production, there are widely used structures and units made of various composite materials (hereafter, composites) [1–3]. Joining of such structures and units to other assembly components, as a rule, is carried out via holes with mechanical punctual fasteners (rivets, bolts, bolt-rivets), and is performed using either stationary press equipment or manual multi-impact and impulse tools [4–6]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 88–98, 2023. https://doi.org/10.1007/978-3-031-36201-9_8
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The most common method of making the holes for fasteners in composites is drilling, that is using the cutting tools. The drilling process is one of the most complex types of mechanical processing of composites, because when carrying out it is an oblique cutting scheme is implemented and the process itself is in a closed space [7, 8]. During the drilling process, the edges of the cutting tool are alternately contacting with the composite, causing thermomechanical effects that resulted in the serious defects, including delamination, burrs, fibers breakages, surface cavities and such defects as debonding and voids, which significantly reduce the quality of the machined holes in parts made of composites. Moreover, during machining the part made of composite, as a result of external loads application and due to the aforementioned temperature factors, the surface roughness of the material is changed and even its structure may got changes [9, 10]. One of the most common factors that affects the processed holes accuracy is the shrinkage of the composites resulted from the influence of temperature, which occurs due to the cutting tool friction during drilling [11–13]. The amount of heat accumulated in the hole depends on the cutting tool shape and the speed of cutting (thus, on spindle rotation rate) [14–16]. These are the key factors when drilling the composites, as they not only reduce the cleanliness of the holes surface and the assembly tolerances, but also affect the product fatigue strength and are the causes of parts spoilage. Therefore, finding out the cutting mode optimal parameters for each type of composites drilling using the existing equipment (considering properties of composite and its structural peculiarities) in order to process holes of high quality still remains a relevant issue.
2 Literature Review and State-of-the-Art The accuracy and the surfaces quality of the holes for fasteners that processed in composites are extremely important in ensuring the product manufacturing and operational performances. During machining the parts made of composites, because of the external loads and the temperature factors that influence on their surfaces, such phenomena as shrinkage and surface roughness worsening occur, and also changes in composite material structure and its chemical composition [17, 18]. All this is resulted in changes in the physico-mechanical and operational performances of the composite material [19]. There are some factors that have significant influence on the quality formation of the holes surfaces in composites: the cutting tool geometry, the tool cutting wedge orientation with respect to the direction of the composite fibers, and the cutting mode parameters. In the process of rotation and feeding of the cutting wedge while drilling, the dynamic characteristics on certain areas of the hole surface, which depend on the pattern of the drill bit contact with the composite, change cyclically [20]. The anisotropy of the composite laminates performances is caused by the differences in the physical and mechanical properties of their constituent materials. These constituents in different ways withstand the external load and temperature in the deformation zone of cutting. The orientation of the tool cutting wedge relative to the orientation of the reinforcing fibers is decisive both for the processing quality parameters and for the wear out of the cutting wedge itself. In Fig. 1 there are shown three typical cutting areas along the composite thickness during drilling with the force P: 1 – zone with fiber and matrix delamination; 2 – sliding zone; 3 – the zone where it’s assumed the absence of defects, with preservation of the fiber-matrix adhesive bonding.
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Fig. 1. Scheme of holes areas along composite thickness while its drilling: zone with fiber and matrix delamination (1), sliding zone (2), defectless zone with preservation of the fiber-matrix adhesive bonding (3).
Zone 1 is the entry zone of the tool into composite, where it is formed a diameter (dentry ), which is larger than ones in other zones of composite and larger than the drill diameter (ddrill ). This is explained as follows. During rotational and translational movements of the drill in the hole, the cutting wedge contacts the reinforcing fibers of one layer of the composite laminate at different angles ϕ while one rotation. Depending on the contact angle ϕ, the cutting force dynamic components change [20, Fig. 3]. The cutting speed, the tool geometrical parameters and the properties of the composite itself affect the intensity of heat generation. Heat accumulation in the cutting zone depends on the heat absorption and heat dissipation capabilities of the composite. The higher the thermal conductivity, the less the thermal expansion. Such a contact scheme is cyclically repeated and allows to identify typical areas in the hole, where the quality parameters formation and the tool wearing out occur in different ways. A similar physical and mechanical process occurs while conventional machining of fiber-reinforced composites (FRCs) [10–12, 16]. During such a process, the tool cutting wedge, contacting the composite, transfers the external loads to the material constituent components. Typical zones in the cutting plane across the cutting wedge when composites drilling can be categorized by the contact angles ϕ into the follows [20]: • from 0º to 90º, which is similar to the zone with angles ϕ from 180º to 270º; • from 90º to 180º, which is similar to the zone with angles ϕ from 270º to 360º; • 0º (180º) and 90º (270º). In hole areas with cutting wedge contact angles from 0º to 90º, as well as from 180º to 270º, the various external compressive forces act on the reinforcing fibers, for example, in the thermoset composites they are 5…6 times higher than in thermoplastics. When it is reached the limit value of compression forces, the fibers break down and turn into sawdust-like chips (fuzz). The temperature factor and the cutting force are the causes of the microcracks formation in the “fiber-matrix” contact zone, as well as of the destruction (pyrolysis) of the matrix material. Figure 2 shows the layers that are formed during drilling of composites and their sizes, in thickness. The first layer is an absorbed film, which is 3…5 μm; the second – a modified layer of 15…150 μm thickness; the third one is a dense destructured
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layer of 20…150 μm; the fourth—a layer of 150…250 μm with fibers fibrilation and “brushy” fracture; the fifth – a transition layer of 250…300 μm thickness; and the sixth, the last one, has the composite initial structure.
Fig. 2. Scheme of composite destructed superficial ply, which consists of an absorbed film of 3…5 μm thickness (1); a modified layer of 15…150 μm thickness (2); a dense destructured layer of 20…150 μm (3); a layer of 150…250 μm thickness with fibers fibrilation and “brushy” fracture (4); a transition layer of 250…300 μm thickness (5); composite initial structure (6).
Due to the fact that the specific heat capacity and thermal conductivity of composites are lower than those of steel, up to 80% of the heat released due to deformations, loadings and friction is “removed” through the tool. In carbon fiber and fiberglass composites at the recommended cutting speed of 50…100 m/min. Under the influence of deformation and frictional forces, the temperature in the closed processing zone reaches 200…250 °C [21]. Such temperatures affect the quality and service life of the composite matrix, namely, in the near-surface contact layers with a depth up to 300…500 microns, destruction of the composite matrix constituents is developed due to the temperature factor. At the same time, along the depth of the hole, the nominal size of the hole decreases (shrinkages) [22]. The composite destruction is a significant impact on the strength characteristics of riveted and bolted joints, especially in hybrid composites that common in the modern aircraft design [23, 24]. It is zones along the holes depth with a contact angle ϕ from 90° to 180° and from 270° to 360° are tended to destructive changes in the composites, where the tensile forces, which acting along the fiber, increase [20]. When the ultimate tensile strength is reached, the fibers break brittlely. The nature and zones of fibers breakage depend on the fiber physical and mechanical characteristics, on the tool cutting edge wear out or its radius, and on the fiber-matrix adhesive bonding. It should be noted that the brittle breakage or fracture of the fiber occurs at a different depth from the cutting surface, that is, from the tool entry surface. This leads to appearing of splintering, variable dimensional accuracy and surface roughness. Thus, the cutting speeds and the feeds of the cutting tools have a significant impact on the quality of the hole, as well as physical-mechanical and operational performances of parts made of composites.
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3 Purpose and Objectives of the Study The purpose of the study is to ensure the specified quality in terms of holes shrinkage and surface roughness in aerospace components made of composites. As a result it unsure high static and fatigue strength, service life, reliability, and efficiency of joining aerospace components made of composites. To accomplish this purpose, the following tasks have been stated: studying the influence of various cutting speeds, under conditions of the same automatic feed rate 0,1 mm/rev, onto values of holes shrinkage and surface roughness in structures made of composites of carbon fiber, fiberglass and organic fiber types. The object of study – holes in aerospace structural components made of composites.
4 Experimental Procedures The studies were carried out on the 4-mm-thick specimens of ELUR-P-1,0 carbon fiber, T-10–14 fiberglass, T42/1–76 organic fiber, and UOL-300-1A carbon fiber. The choice of these materials is due to the fact that their physical and mechanical performances overcome other fiber-reinforced composite materials, besides they are difficult-to-cut materials, but are widely used in structures of the aerospace and mechanical engineering components. Drilling was carried out with a spade drill bit made of VK-8 (HG30, DIN) alloy with a diameter of 6 mm. The choice of VK-8 (HG30, DIN) spade drill bit is due to its simplicity and low cost, as well as the existing positive experience of such drill bits using for processing the holes in fiberglass and carbon fiber [20]. For hole microprofile analysis, the specimens with drilled holes were cut in halfs in an axial (longitudinal) cross-section. The average shrinkage deviation from the nominal hole size was measured using a TMR 120 profilometer (Chine) with a digital readout. The microstructure of plastic was studied using an EVO-18 scanning electron microscope (Carl Zeiss Microscopy Ltd., Great Britain). Figure 3 schematically shows the study of composite drilling process, where: specimen (1), surface roughness to be measured (2), guides for measuring (3), t – specimen thickness, ddrill – drill diameter that is equal to hole entry diameter, dhole – specified diameter of hole in composite equal to drill bit diameters ddrill , dreal – real diameter of hole drilled in composite, dexit – diameter of exit hole after drilling. The difference between the diameters appears due to fibers fibrillation, shrinkaging and cracking of the hole material, that is, it is the result of drilling the composite and forces acting in the material. To determine the impact of drilling parameters on the holes quality indicators, the following specimens were studied: • made of carbon fiber ELUR-P-1,0, fiberglass T-10–14, organic fiber T42/1–76 – when studying the shrinkage of composites (the speed of the spindle rotation was changed from 450 to 4500 rpm) and when studying the structure parameters of the hole surface (cutting speed was changed from 10 to 90 m/min). The rate of the automatic feed during drilling was 0.1 mm/rev. • made of carbon fiber tape UOL-300-1A, when studying the hole shrinkage and surface structure; the cutting speed was 50 m/min, the automatic feed rate was 0.1 mm/rev.
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Fig. 3. Schemes illustrating the composites drilling process along the drill bit movement (a) and specimen that was cut for surface roughness measurement using profilometer (b).
5 Results of Studying the Holes Shrinkage and Surface Roughness in Composite Laminates 5.1 Results of Holes Experimental Studying in Specimens Made of ELUR-P-1,0, T-10–14, and T42/1–76 Composite specimens with holes of 6 mm diameter that drilled at a drilling speed of 10 m/min and 90 m/min (carbon fiber ELUR-P-1,0 specimens) and at a drilling speed of 30 m/min and 90 m/min (fiberglass T-10–14) were made. The results of study showed almost the same surface roughness at the holes exit for ELUR-P-1,0 carbon fiber specimens drilled at both speed of 10 m/min and speed of 90 m/min. Speed of 90 m/min. at the drilled hole entry resulted in a hole shrinkage of 0.025 μm, which is due to the thermal destruction of the material. Speed of 10 m/min. Resulted in the lowest surface roughness (Ra 3.2) at the drilled hole entry and gradually increasing along the hole depth to the maximum surface roughness (Ra 6.3) at the hole exit. This is explained by the amout of fibers that were uncut during the drilling process and by the material thermal destruction. The main reasons for defects occurrence are the high strength of the reinforcing fibers and the wear out of the tool cutting edges. For fiberglass T-10–14 at a both drilling speed of 30 m/min and at a drilling speed of 90 m/min the surface roughness is the same (Ra 3.2) on the all hole surfaces: at the entry, along all the depth and at the exit. Shrinkage was observed in the speecimens at a drilling speed of 90 m/min and it was 0.001 μm. For all the speecimens made of composites ELUR-P-1.0, T-10–14 and T42/1–76, the minimum shrinkage was observed in the range of the cutting wedge angle ϕ from 0° to 90° and from 180° to 270°, and the maximum - with angle ϕ from 90° to 180° and ϕ from 270° to 360°. This is due to the cutting forces reducing within the operating zones of the drill with angle ϕ from 0° to 90° and from 180° to 270°, which improves shrinkage indicators.
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In the zones with angle ϕ from 90° to 180° and from 270° to 360° – on the contrary, the cutting forces increase, which explains the low quality of the surface in these ranges. Table 1 presents the results of the study of the cutting speed effect on the holes shrinkage when drilling at spindle rotating rate from 450 to 4500 rpm for tested specimens. Table 1. Average values of hole diameter deviations, mm Composite type
Spindle Rotation Rate, rpm 450
1400
2500
4500
Carbon fiber ELUR-P-1,0
−0,009
−0,011
−0,015
−0,021
Fiberglass T-10–14
−0,004
−0,007
−0,009
−0,012
Organic fiber T42/1–76
−0,012
−0,018
−0,024
−0,031
As can be seen from the Table 1 data, when the cutting speed increases, the average value of shrinkage comparatively with the hole specified diameter increases as well for all the types of specimens made of composites. The minimum average value of shrinkage at all cutting speeds was observed when drilling fiberglass, the maximum organic fibers. The maximum effect on the holes shrinkage due to an increase in the cutting speed was found when drilling fiberglass - almost three times, and the minimum effect on shrinkage it’s when drilling carbon fiber - almost two times. Figure 4 shows the surface roughness microrelief of the holes drilled in the mode at a cutting speed of 50 m/min and automatic feed rates of 0.1 mm/rev in experimental specimens made of ELUR-P-1.0, T-10–14 and T42/1–76 composites. For all the composite specimens made of carbon fiber ELUR-P-1.0, fiberglass T-10– 14 and organic fiber T42/1–76 it was observed the following: • shrinkage at the drilled holes entries from 0.001 to 0.003 mm in all zones with angle ϕ from 0° to 360°; • appearing of delamination at the drilled holes entries and cracks along the hole depth; • tear-outs and uncut fibers along the hole depth with maximum fiber sizes up to 0.055 mm. The obtained results showed that holes accomplishing in the composites of carbon fiber ELUR-P-1.0, fiberglass T-10–14 and organoplastic T42/1–76 types at the mode with a cutting speed of 50 m/min and automatic feed rates of 0.1 mm/rev requires using the additional machining or another surface fininshing methods. 5.2 Results of the Carbon Fiber Specimens Experimental Studies Made of UOL-300-1A Tape In Fig. 5 it is shown a photo of a scaled area with sizes of 600 × 700 μm that demonstrates the hole surface structure after drilling the UOL-300-1A carbon fiber composite. Drilling was carried out at a cutting speed of 50 m/min and automatic feed rate of 0.1 mm/rev in the range of a cutting wedge angle ϕ from 90º to 180º and from 270° to
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Fig. 4. Microrelief of the hole surface roughness obtained in the drilling mode at a cutting speed of 50 m/min and an automatic feed rate of 0.1 mm/rev.
Fig. 5. Photo of hole surface in specimen made of the UOL-300-1A carbon fiber.
360°. In the range of the cutting wedge angle ϕ from 90º to 180º, the splitering height is up to 0.05 mm, which indicates uncut fibers and their “thickness” on the surface. A similar cutting process takes place in the hole areas with the cutting wedge contact angles ϕ from 270° to 360°. Such splintering negatively affects the processed surface accuracy and surface roughness. In the hole area with the cutting wedge contact angle ϕ from 0° to 180° it occurred cracking and delamination of fibers in the upper layers of the deformation zone of the composite. Measurements showed 1.0 to 1.5 mm spalling and up to 1.5 mm delamination. This is explained by the high mechanical properties of carbon fiber and while in the range
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of ϕ from 0° to 180° there is a sharp change in the external forces acting on the composite fibers, so at these points there is a sharp jump from tension to compression. The same edge effect with surface spalling and delamination of the composite was observed in the areas with angles ϕ = 90° and ϕ = 270°. In this case, the reason may be • lower strength of the adhesive bond of the upper ply fibers with the inner plies of the composite; • an excessive amount (more than 60% of total one) of fibers in composite; • one of the cutting force components, which apperas in the cutting process, is greater than lateral force acting along the cutting edge of the drill bit. Shrinkage was observed at the drilled hole entry and was 0.001 mm.
6 Conclusion The influence of the holes drilling mode on holes shrinkage and surface roughness in carbon fiber reinforced composite ELUR-P-1,0, in fiberglass T-10–14, in organic fiber T42/1–76 and carbon fiber UOL-300-1A was studied. It was found out that the surface roughness of the ELUR-P-1.0 carbon fiber specimens at a drilling speed of 10 m/min and 90 m/min and on T-10–14 fiberglass specimens at a drilling speed of 30 m/min and 90 m/min was provided at rate Ra 3.2…Ra 6.3, which is sufficient, in general, for holes drilled in composites. The opposite effect was obtained for the same specimens in the drilling mode at a cutting speed of 50 m/min and automatic feed rate of 0.1 mm/rev. It is recommended to carry out additional mechanical processing of hole or to change the type of drill bit. For all composites specimens of ELUR-P-1,0, T-10–14 and T42/1–76, the minimum shrinkage was observed in the range of the cutting wedge ϕ from 0° to 90° and ϕ from 180° to 270°, and the maximum shrinkage - with angle ϕ from 90° to 180° and ϕ from 270° to 360°, which provide approbation and confirmation of the work [20] statements. The results of the study of the cutting speed effect on the holes shrinkage in specimens made of carbon fiber ELUR-P-1,0, fiberglass T-10–14 and organic fiber T42/1–76 when they are drilled at spindle rotating rate from 450 to 4500 rpm showed that with an increase in the cutting speed, the average value of shrinkage increases for all types of composites specimens. As a result of the studying the hole surface structure after drilling the UOL-300-1A carbon fiber composite material at a cutting speed of 50 m/min and automatic feed rate of 0.1 mm/rev the following was found out: • in the zone of the cutting wedge ϕ from 90° to 180° and from 270° to 360°, the height of the composite splintering is up to 0.05 mm; • in the zone of the cutting wedge ϕ from 0° and 180° and ϕ = 90° and ϕ = 270°, chipping from 1.0 to 1.5 mm and delamination up to 1.5 mm were formed; • shrinkage was 0.001 mm at the drilled hole entry. Increasing the cutting speed up to 50 m/min showed admissible shrinkage of the holes in all the composites under study, but at the same time there is a risk of the hole surface quality reducing in composite specimens (increased surface roughness), which
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indicates the need for further research with drills of other types and finding out a range of increased cutting speed with minimal risks of the surface quality reducing for holes drilled in composite materials.
References 1. Kesarwani, S.: Polymer composites in aviation sector. Int. J. Eng. Res. Technol. (IJERT) 6, 518–525 (2017) 2. Mohammad, R.M.J., Mohammad, S.A.M., Khasri, A.: Sustainable Composites for Aerospace Applications. Woodhead Publishing Series in Composites Science and Engineering, pp. 155– 170 (2018) 3. Kiran, R., Vasudevan, A., Pugazhendhi, L.: A review on different hybrid composites for aircraft structures. Materials Today: Proceedings (2021). https://doi.org/10.1016/j.matpr.2020. 10.774 4. Vorobiov, I., Maiorova, K., Voronko, I., Boiko, M., Komisarov, O.: Creation and Improvement Principles of the Pneumatic Manual Impulse Devices. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) Integrated Computer Technologies in Mechanical Engineering – 2021. ICTM 2021. Lecture Notes in Networks and Systems, vol. 367, pp. 178–191. Springer, Cham (2022) 5. Campbell, F.C.: Manufacturing Technology for Aerospace Structural Materials/F.C. Campbell, p. 616. Elsevier (2006) 6. Le Cann, R.P., Guerin, S.: Portable automation for feeding and installing fasteners. Aerospace Manufacturing and Automated Fastening (AMAF). Conference and Exhibition, pp. 12–14. Toulouse, Frace (September 2006) 7. Liu, D., Tang, Y., Cong, W.: A review of mechanical drilling for composite laminates. Compos. Struct. 94(4), 1265–1279 (2012) 8. Advances in Machining of Composite Materials. In: Islam Shyha, Dehong Huo. Springer Nature Switzerland AG 2021, p. 552 (2021) 9. Liu, K., Ye, J., Tang, Z., Liu, W., Zhang, B.: Simulation and verification of machining deformation for composite materials. Journal of Wuhan University of Technology-Mater. Sci. Ed. 29(5), 917–922 (2014). https://doi.org/10.1007/s11595-014-1020-6 10. Kumara, D., Gururajab, S.: Machining damage and surface integrity evaluation during milling of UD-CFRP laminates: Dry vs. cryogenic Composite Structures 247(1), 112504 (2020). https://doi.org/10.1016/j.compstruct.2020.112504 11. Merino-Pérez, J.L., Royer, R., Ayvar-Soberanis, S., Merson, E., Hodzic, A.: Induced thermomechanical damage in the drilling of thermoplastic-toughened CFRP composites. In: Proceedings of 20th International Conference on Composite Materials, 19–24th July 2015. Copenhagen, Denmark (2015). https://doi.org/10.13140/RG.2.1.4499.5682 12. Agrawal, C., et al.: Experimental investigation on the effect of dry and multi-jet cryogenic cooling on the machinability and hole accuracy of CFRP composites. J. Mat. Res. Technol. 18, 1772–1783 (2022). https://doi.org/10.1016/j.jmrt.2022.03.096 13. Wang, H., Sun, J., Li, J., Lu, L., Li, N.: Evaluation of cutting force and cutting temperature in milling carbon fiber-reinforced polymer composites. Int. J. Adv. Manuf. Technol. 82(9–12), 1517–1525 (2015). https://doi.org/10.1007/s00170-015-7479-2 14. Ghafarizadeh, S., Lebrun, G., Chatelain, J.-F.: Experimental investigation of the cutting temperature and surface quality during milling of unidirectional carbon fiber reinforced plastic. J. Compos. Mater. 50(8), 1059–1071 (2016) 15. Merino-Pérez, J.L., Royer, R., Ayvar-Soberanis, S., Merson, E., Hodzic, A.: On the temperatures developed in CFRP drilling using uncoated WC-Co tools Part I: Workpiece constituents, cutting speed and heat dissipation. Compos. Struct. 123, 161–168 (2015)
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Influence of Physico-Mechanical Factors on Residual Technological Stresses and Strains of Composite Panels Andrii Kondratiev1(B)
, Viktor Kovalenko2 and Igor Taranenko3
, Tetyana Nabokina3
,
1 O.M. Beketov National University of Urban Economy in Kharkiv, 17 Marshal Bazhanov
Street, Kharkiv 61002, Ukraine [email protected] 2 Yuzhnoye Design Office, 3 Krivorozhskaya Street, Dnipro 49008, Ukraine 3 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
Abstract. Modern technology development is characterized by growing use of the laminated elements made of polymer composite materials in the panel structures for various applications. A significant part of these structures is subjected to residual technological stresses and process-induced deformations (warpage) because of different factors. This can lead to distortion of the geometry of the resulting structure and decrease in its bearing capacity under the effects of assembly stresses arising from the restraint of process-induced deformations of the panel during its setting in the frame. The objective of this work is the development of a method for assessment of the effect of inner structure of the polymer composite material on residual stresses and deformations, providing of recommendations on its optimal design. We consider polymerized composite panel, which is installed into the rigid frame of the prefabricated structure after its release from the moulding tool. It is assumed that the panel is moulded of six monolayers at the temperature of 100ºC. Based on our studies, we obtained the dependence, which allowed us to predict the index of maximum stress at the critical point of the composite structure depending on residual thermal stresses after manufacturing of the product and measurement of the maximum deflection. Initial structures of the composite were studied for various ranges of deviations of the reinforcing angles corresponding to the automated winding (layup), as well as manual layup. Implementation of the obtained results will significantly reduce the additional process operations and the number of rejects, and also improve the quality of the manufactured panel composite products. Keywords: Setting in the frame · Thermal warpage · Nonequilibrium structure
1 Introduction Panel structures made of polymer composite materials (PCM) are widely used in the modern technology [1]. Their use allows obtaining a strong and rigid structure of much lower weight compared to the metal analog [2, 3]. Moreover, the technology for creating the complex products of PCM is often less expensive and more practical owing © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 99–110, 2023. https://doi.org/10.1007/978-3-031-36201-9_9
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to elimination of a number of intermediate operations, thus reducing the labor intensity and production costs significantly [4, 5]. However, along with the obvious advantages of PCM use there are a number of issues limiting their wide acceptance [6, 7]. One of the most important problems in the manufacturing of PCM products is the occurrence of residual stress-strain behavior [8]. It results from the moulding of composite products at elevated temperatures, heterogeneity of the PCM structure and faults in the technological processes [9, 10]. Therefore, it can be difficult to install the deformed parts during assembly of the composite structures. It ultimately leads to the emergence of additional assembly stresses and decrease in the bearing properties of the structure, or to the need for additional machining and complication of the technological process. In addition, certain types of composite structures require high manufacturing accuracy. For example, deviations from the required accuracy may result in the distortion of the received/transmitted signal for the composite reflector antennas and elements of loadbearing structures of spacecraft optical telescopes [11], or deterioration of aerodynamic properties of the aircraft skin panels [12]. Residual stress-strain behavior is also observed in cases when nonsymmetric PCM structure is used. In real practice, there is a tendency to use the PCM packages with symmetric layup. However, in some cases it is difficult to achieve the package symmetry because of a specific feature of the product manufacturing technology. In this case, it is necessary to use a set of measures aimed at reducing the residual stress-strain behavior of PCM products. The objective of this work is the development of a method for assessment of the effect of inner structure of the PCM on residual stresses and deformations, issue of recommendations on its optimal design, and analysis of the various technological factors affecting the residual stress-strain behavior.
2 Literature Review As a rule, the laminated panels made of fiber PCM have a constant thickness and are divided into panels of symmetric or nonsymmetric structure in thickness [13]. Composite laminated panels are typically manufactured so that each layer with certain material properties, thickness and fiber orientation, located on one side relative to the middle plane of the panel, corresponds to the equidistant layer on the other side thereof with the same properties, thickness and fiber orientation [14, 15]. Laminated panels, which are symmetric in thickness, do not show warpage during their manufacture, and the appearance of cracks in such products is actually the only sign of the presence of residual stresses in them caused by the membrane interaction of layers with each other [16]. The cause of warpage of flat symmetric PCM panels can be inhomogeneity of the temperature conversion field at the stage of binder cure [17]. If such symmetry is broken for at least one pair of layers, for example, because of error in the technological process [18], the laminated composite panel is thermally unbalanced. In this case, membranebending interaction of the layers will be observed as a result of application of thermal loads during the panel manufacture. Therefore, warpage will appear in the form of a displacement normal to the middle surface, i.e. deflection. This property and behavioral features of nonsymmetric laminated composites remain little studied so far, although they are expectable [19]. Parametric study of the factors described above and their influence on
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the residual stress-strain behavior is given in a number of papers [9, 20]. Mathematical models for the determination of stress-strain behavior of anisotropic panels made of PCM are described in [21, 22]. The mathematical model of the reinforcing element is refined under conditions of one-sided contact with the moulding tool. Based on the Lagrange variational principle, eighth-order resolving equations for the natural boundary conditions are obtained. The papers [8, 23] deal with the change in the distribution of the residual stress field in PCM specimens with the layup sequence [0°/90°]n . Stresses on the free edge after cutting of specimen along the reinforcement direction have been analyzed. The calculated residual stresses are further used to analyze the development of the composite failure under uniaxial tension. As a result of modeling of the composite loading to failure taking residual stresses into account, it is found that the tension leads to longitudinal cracking of the matrix in layers where the orientation of reinforcement corresponds to the direction of the load application. At the same time, when the residual stresses are neglected, occurrence of such defects is not observed. The influence of the automated fiber layup process on the level of residual stresses is studied in [24]. This paper describes the optimization based on the response surface method to minimize residual stresses while maintaining the maximum performance. Mathematical modeling of the relationship between the technological process parameters and residual stresses is performed. Now a significant number of studies deal with the development of the most complete models of phenomena occurring in the process of manufacturing of complexshaped PCM products and their numerical implementation using the modern numerical methods, in particular, the finite element method [25, 26]. Experimental determination of the values of residual stresses is reduced to testing of elementary samples for tension, compression, etc. and comparison of the findings with the results obtained for specimens not subjected to internal stresses [27, 28]. Nevertheless, there are other methods for the determination of residual stresses, based on layer-by-layer removal of material from one of the specimen sides and measurement of its deflection, described in [29]. When we analyze the studies mentioned above, it should be noted that most of them are limited to the study of process-induced deformations of panels in their free state only and actually no studies deal with their behavior after setting and fixation in the structure. The models and algorithms which allow evaluating the behavior of PCM panels during their free warpage and setting in the rigid frame are developed in [30]. However, the authors of the paper consider only the flat rectangular and orthotropic panels with symmetric supporting conditions. These boundary conditions do not correspond to those arising during setting of the real structures in the frame. The papers [31, 32] describe the analytical studies of assembly stresses of flat plates and single-curvature panels of PCM on prepreg base made of carbon fibers of various structures featuring different thermal nonequilibrium degrees under various panel supporting conditions. However, these papers use the criterion of maximum normal stresses to assess the level of assembly stresses. It is not quite correct for the laminated orthotropic composite under study, which is subjected to the complex stress, and significant errors may occur in the assessment of the panel bearing capacity. Considering the above, it seems relevant to study the effect of spacing of thermal nonequilibrium layers relative to the middle plane of the composite panel on its residual stress-strain behavior.
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3 Research Methodology Now we shall consider polymerized composite panel installed into the rigid frame of the prefabricated structure after its release from the moulding tool. It is assumed that the panel is moulded of six monolayers of thickness δ at the temperature of ΔT = 100ºC. The temperature increment is taken uniform over the panel thickness. We proceed from the fact that structure of the panel (Fig. 1) is predetermined by some reasonable arguments, due to which it is preset structure [33]. One symmetric thermal equilibrium structure (Fig. 1, a) and three nonsymmetric structures with different degrees of thermal nonequilibrium (Fig. 1, b – d) are considered.
Fig. 1. Panel structures under study with different degrees of conditional thermal nonequilibrium: a – thermal nonequilibrium (ξ = 0); b – minimal (ξ = 0.167); c – medium (ξ = 0.334); d – high (ξ = 0.5)
We introduce the concept of the conditional degree of thermal nonequilibrium of the panel, assuming the panel generally comprising m layers. Then the conditional degree of the panel thermal nonequilibrium ξ is the total distance of pairs of thermal nonequilibrium layers from the geometric middle plane of the panel, referred to its thickness m −1 k nj δj · ni δi , (1) ξ= j=1
i=1
where nj , δ j – number and thickness of the j-th thermal nonequilibrium monolayer; k – number of pairs of thermal nonequilibrium monolayers in the panel (pairs of thermal nonequilibrium layers are counted from the middle plane); m – total number of monolayers in the panel; δ i – thickness of i-th monolayer. Then for the PCM panel structures shown in Fig. 1: a) ξ == 0/6 = 0; b) ξ == 1/6 = 0.167; c) ξ == 2/6 = 0.334; d) ξ == 3/6 = 0.5. We consider fiber reinforced plastic (GFRP), carbon fiber reinforced plastic (CFRP) and organic plastic as possible fillers for the panel. PCM physico-mechanical characteristics (PhMC) adopted for calculations are given in Table 1 [26, 32]. We assume that in the process of the panel setting, i.e. its forcible return to the restricting rigid frame, panel displacements completely return to the stressed state. Then we may consider mutually reversible problems of warpage of a fixed panel in the rigid frame and warpage of free panel identical to it according to the stress state occurring in the first case before the product is removed from the moulding tool and in the second case during mounting fixation of the previously free panel. Here determination of assembly
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Table 1. PhMC of PCM monolayers adopted for calculations PCM type/PhMC of the monolayer
Unidirectional PCM CFRP
GFRP
Organic plastic
No.1
No.2
No.3
No.4
No.5
No.6
Modulus of elasticity along the fibers, GPa
100
142.8
114.2
45
80
81.8
Modulus of elasticity across the fibers, GPa
10
9.13
8.1
10
5.5
5.1
Shear modulus, GPa
6
5.49
3.94
5
2
1.82
Poisson’s ratio
0.35
0.32
0.34
0.3
0.31
0.31
Coefficient of linear thermal expansion along the fibers ·106 ,1/°C
0
0
−0.01
8
−4
−4
Coefficient of linear thermal expansion across the fibers ·106 , 1/°C
30
27.7
30.6
25
60
79
Ultimate tensile strength along the fibers, MPa
900
1400
1322
800
1600
1850
Ultimate compressive strength across the fibers, MPa
700
1108
719
1000
300
235
Tensile strength across the fibers, MPa
50
34.5
35.3
50
16
7.9
Compressive strength across the fibers, MPa
120
186
133
40
300
53
Shear strength, MPa
75
74
112
60
30
24.4
Monolayer thickness, mm
0.12
0.12
0.12
0.15
0.12
0.12
stresses is an inverse problem with regard to the considered straight line – warpage of the polymerized panel after its release from the tool, when process-induced residual stresses are transformed into deflections (warpage) w. In order to assess the level (index) of stressψ, we use the energy failure criterion of Von Mises-Hill, which satisfactorily corresponds to the results of the laminated PCM stress tests [1, 15, 32] σ1i2 2 F1i
−
σ1i2 σ2i2 2 F2 F1i 2i
+
σ2i2 2 F2i
+
2 τ12i 2 F12i
= ψi ≤ 1,
(2)
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where σ 1i , σ 2i , τ 12i , F 1i , F 2i , F 12i – actual stresses in the i-th layer of PCM and its ultimate strength in the coordinate system 102, where direction 1 corresponds to the location of reinforcing fibers of the unidirectional monolayer. Stress index is understood as the numerical value of the right side of the adopted PCM strength criterion ψ i ≤ 1, which is equal to one in case of exhaustion of bearing capacity of the material at the considered point. The pattern of stress index values is different at each point of the panel and each monolayer of the package, so we will determine its maximum value ψ max for the panel as a whole, without specific localization of the monolayer position and the coordinates of the point. Software systems for the finite element analysis were used for the determination of the stress-strain behavior of the considered panel under the action of the temperature field. For the spatial analysis we used 4-node multilayer quadrangular shell element with flexural and membrane properties: ShellL type, six degrees of freedom (three translations and three rotations were considered per node). Figure 2 shows the dependences of the relative deflection and maximum value of stress index ψ max for the freely supported panel of 500 × 500 mm on the PCM structure conditional thermal nonequilibrium degree ξ. Analysis of the results, as expected, indicates the significant effect of spacing of the panel’s thermal nonequilibrium layers relative to its middle plane on the stress-strain behavior of the panel.
Fig. 2. Dependencies of relative deflection of the panel wmax (a) and maximum value of its stress index ψ max (b) on conditional degree of thermal nonequilibrium ξ of the PCM structure: No. 1, No. 2, No. 3 – CFRP; No. 4 – GFRP; No. 5, No. 6– organic plastic
For example, for unidirectional carbon fiber reinforced plastic at minimal value of spacing of thermal nonequilibrium layers the relative deflection is w max = 0.6, at medium value – wmax = 1.98, and at maximum value – 3.47. When the original PCM structure is transformed into the thermal equilibrium one, the deflection is equal to zero, i.e. no warpage is observed. There is a slight growth of maximum stress index ψ max with the increase in thermal nonequilibrium; however, the change is very small. For example, for CFRP the relative difference between ψ max for the minimum and maximum spacing does not exceed 6%, for GFRP – 11%, for organic plastic – 12%.
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4 Results Analysis of our findings allowed us to establish a relationship between the relative maximum deflection wmax and stress index of thermal nonequilibrium structures of various PCM (Fig. 3). The obtained dependencies allow us to establish maximum stress index of the local zone ψ max by measuring its maximum deflection wmax after moulding with the further reference to the total thickness of the panel δ Σ .
Fig. 3. Dependence of the index of maximum stress of freely supported panel of the various PCM on maximum relative deflection (R2 – coefficient of determination)
Initial structures were studied for the cases of different deviations of the monolayer reinforcing angles within –3º ≤ Δϕ i ≤ 3º and –5º ≤ Δϕ i ≤ 5º. Here we proceed from the assumption that the range of deviations ±Δϕ i = 3º corresponds to the mechanized or automated winding (layup) of the PCM structure, while ±Δϕ i = 5º corresponds to the manual layup [32, 34]. The study found that at the same deviations of angles in all monolayers in the positive or negative direction +Δϕ i = 3º, –Δϕ i = –3º and +Δϕ i =
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5º, – Δϕ i = –5º for all considered materials of the PCM filler the deviations wmax and ψ max from the ideal structure (±Δϕ i = 0º) did not exceed 1%. More significant influence is exerted by nonsymmetric distribution of the deviations of the reinforcing angles. So, for example, for the freely supported PCM panel based on carbon fiber: – at Δϕ i = +3º up from the middle surface and Δϕ i = –3º down from it wmax ≈ –11.7%; – at Δϕ i = +5º up from the middle surface and Δϕ i = –5º down from it wmax ≈ –18.6%. In both cases, the level of change in the maximum stress ψmax did not exceed 5%. On the assumption that the range of deviations of the reinforcing angles Δϕ i covers the largest deviations wmax and their respective ψmax , we can make a conclusion that changes in the maximum deflections in this range are either negligible or tending to decrease, and the changes ψ max lie within the accuracy of the calculations. Initial structures were studied for the case of changing of the dimensions of freely supported panel: we considered the panel dimensions of 250 × 500 mm and 500 × 250 mm (Table 2). As an example, Fig. 4 shows the patterns of the panel deflection distribution for PCM based on CFRP No. 1 with the minimal degree of thermal nonequilibrium at the various panel dimensions. Table 2. Results of calculations for the freely supported panels of the various conditional thermal nonequilibrium structure PCM material
Degree of conditional thermal nonequilibrium ξ =0 wmax , mm
ξ = 0.167
ξ = 0.334
ξ = 0.5
ψ max
wmax , mm
ψ max
wmax , mm
ψ max
wmax , mm
ψ max
500 × 250 mm panel CFRP No. 1
0
0.047
-1.2
0.049
-4.33
0.052
-8.51
0.054
GFRP No. 4
0
0.089
-0.86
0.11
-2.84
0.12
-5.16
0.13
organic plastic No. 5
0
0.023
-2.05
0.013
-8.14
0.014
-17.36
0.015
250 × 500 mm panel CFRP No. 1
0
0.047
1.2
0.049
4.33
0.052
8.51
0.054
organic plastic No. 5
0
0.089
0.86
0.11
2.84
0.12
5.16
0.13
organic plastic No. 6
0
0.023
2.05
0.013
8.14
0.014
17.36
0.015
We received quite predictable result on the significant change in the value wmax compared to the initial panel of size 500 × 500 mm, whereas the level of maximum stress
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Fig. 4. Example of the obtained patterns of deflection distribution for PCM based on carbon fiber with the minimal degree of thermal nonequilibrium ξ = 0.167 at the various panel dimensions
ψ max practically did not change. This conclusion indicates a significant dependence of the maximum stress index and maximum deflection of the panel on its overall dimensions.
5 Conclusions Based on the studies of the thermal stress-strain behavior of the composite panels at the various degree of thermal nonequilibrium of their structure, we have found the relationship between the maximum stress index and the relative maximum deflection, which was approximated for the various structures of different PCM classes by polynomials with the varying coefficients. It allows us to predict the index of maximum stress at the critical point of the structure depending on residual thermal stresses after manufacturing of the product and measurement of the maximum deflection. It is a conditional relationship, since coordinates of the points defining ψ max and wmax do not match. In addition, even for the same PCM structure formed of monolayers of various classes or featuring different properties the nature of change in this relationship can be different. The established relationship can be effective in the future for the analysis of combined effect of residual thermal assembly impacts or operational impacts and power loads on the product, taking into account the synergistic effect. The foregoing indicates the relevance of further research in terms of consideration of the level of thermal nonequilibrium of various PCM classes, including hybrid structures, in designing and manufacturing of the critical panel structures for the various applications.
References 1. Smetankina, N., Merkulova, A., Merkulov, D., Postnyi, O.: Dynamic Response of Laminate Composite Shells with Complex Shape Under Low-Velocity Impact. In: Nechyporuk, M.,
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Control of Stress Conditions in Growth of Copper Oxide Nanostructures Andrii Breus(B)
, Sergey Abashin , and Oleksii Serdiuk
National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine [email protected]
Abstract. Two methods of synthesis of copper oxide nanostructures are discussed with respect to the effect caused by the stress developed during the nucleation and growth of the nanostructures, namely, conventional heating in the oven, and plasma-enhanced heating obtained when using a cathode of a plasma-generating circuit as a holder for a copper sample. It is found that the rate of the temperature increase of about 5 °C/min utilized in the thermal process allows nucleation of 0D nanodots from grains of CuO oxide, which proceed their growth as separate 1D nanowires. However, the implementation of the plasma-enhanced heating results in drastic increase of the temperature increase rate, up to 150 °C/min, which is accompanied with mutual attachment of the nanodots and formation of roots of 2D nanosheets of copper oxide; moreover, the active boundaries of copper oxide grains serve as the nucleation centres. At that, the developed stress deforms the net od the nanosheets thus forming a stacked 3D nanostructure with a huge surfaceto-volume ratio which, in turn, makes the structure suitable for possible energy storage applications. Keywords: copper oxide · nanostructures · thermal growth · plasma-enhanced heating
1 Introduction Nanostructured oxides of various metals are considered as perspective materials for industrial applications [1]. Yuan et al. reported a successful development of copper oxide two-dimensional nanostructures decorated with carbon nanotubes, which allowed improving the energy capacity of lithium-ion batteries due to the highly-expanded surface-to-volume ratio of the composite material [2]. Similar nanostructures were grown in glow discharge plasma yet in that case carbon nanostructures were decorated with copper nanoparticles [3, 4]. Dense arrays of copper oxide nanowires were synthesized between two electrodes by use of simple thermal method to obtain the material with promising photocurrent switching under the action of violet light [5]. Schottky diode was developed for white light detection on a base of single nanowire with silver nanoparticles printed on it [6]. The importance of geometric shape of copper oxide nanostructures on electric properties was confirmed by Kimura et al. [7]. However, the growth mechanisms are still controversial for copper oxide nanostructures. In particular, the reasons of growth and nucleation of 2D CuO nanomaterials © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 111–122, 2023. https://doi.org/10.1007/978-3-031-36201-9_10
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at the temperatures above 700 °C, where nanowires are not growing, is unknown [8]. Self-assembly of individual nanowires under the action of the increased diffusion facilitated at the elevated temperatures is considered among the possible mechanisms [9]. According to a number of reports, single and bi-crystalline nanowires are formed in a wide range from room temperatures to about 700 °C, with average length of 1 to 20 µm and diameter of 90 to 300 nm [10, 11]; at that, external electric field causes increase in length of the nanowires thus adding some effect to the action of surface stress. At the same time, Dorogov et al. detected poly-crystalline nanowires mixed with bi-crystalline nanowires in arrays of thermally synthesized nanostructures, as well as synergetic effect caused by internal stress and defects of structure of copper oxide layer [12]. According to the experiments reported by Tran et al. [13], length and diameter of nanowires depend on the annealing temperature, and a conclusion about the leading role of copper diffusion and oxidation was made and theoretically described [14]. According to the research conducted by Mema et al. [15] tensile stress is beneficial for the increase of the nanowire length because of the decreased sizes of grains of copper oxide. It is speculated that the large number of grain boundaries formed at the process promote the delivery of copper atoms from a substrate to the nanowire roots, as well as decrease of the losses of copper on side surfaces of the nanowires due to the decreased surface area associated with the smaller roots [16]. As for the origins of the stress, it is believed to be caused by excessive quantity of volume created at the oxidation, as one should note that a lattice parameter changes from about 0.36 nm for copper to about 0.43 nm for copper oxide. A comprehensive analysis of the factors and stress mechanisms in nanostructured material is performed by Wang et al. [17], and as static as dynamic solutions should be taken into account to make accurate predictions of the stress development. The combination of stress-induced grain-boundary diffusion of copper supplemented with gradient of oxygen concentration, is considered to be the main driving force of the nucleation and growth of copper oxide nanostructures [18]. Role of stress was studied by Altaweel et al. [19] in their research of the oxidation process under the influence of atmospheric afterglow plasma discharge to a surface of a copper sample. According to the measurements, compressive stress was observed in Cu2 O layer, while the tensile stress – in CuO layer. At the same time, the regions located close to higher temperatures and highest temperature gradients, exhibited larger grain sizes, which was accompanied by the larger diameters of nanowires – which, in turn, can be considered as an argument for the grains as the nucleation centers, opposite to the dislocations etc. In the experiment, simultaneous growth of different types of nanostructures was observed on the substrate, and their morphology was related to the proximity of the investigated area to the heat source. Thus, the coolest regions shown the presence of nanodots (0D) followed by nanowires (1D), nanowalls (2D), and complex cabbage-like nanostructures (3D) observed in the most heated area. From these data, an assumption about the nucleation of 0D structures, their growth and transformation into 1D nanowires, combination of 0D nuclei into the roots of 2D nanosheets, and association of 2D nanoleaf into 3D structures can be made. The results are consistent with theoretical studies on the stress development [20] and formation of clusters [21]. The processes on the grain boundary are suggested by Jagtap and Chason [22] to be responsible for the residual stress in thin films; sizes of the grains, thickness of the films, as well as deposition
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rates and growth temperatures affect the development of the internal stress. The authors proposed a kinetic model to describe the evolution of stress via a competitional process between tensile and compressive stresses generated in the films [23]. Thus, static and dynamic stress appears to be a powerful tool in control the nucleation and growth process. It should be mentioned that formation of thermal induced stress gradients can be realized in a number of ways, such as control of distribution of a treatment plasma flow [24, 25], utilization of explosion-like processes [26], under the action of mechanical impact [27, 28] or friction forces [29]. At the same time, a lot of applications, green energy e.g. [30], are still waiting for the successful implementation of nanotechnologies, which is hindered by the unsolved problems, and many of them are conditioned by the lack of experimental data. Unfortunately, a lot of researches are focused only of the effects that are strongly dependent on a method applied to obtain a nanostructure; the effects that occurs at a transition from one method to another are neglected. In this paper the results of studies by use of scanning electron microscopy (SEM) of copper oxide nanostructures obtained by different treatment methods are considered with respect to the possible effect caused by the variation in stress conditions due to the change of the surface treatment technique.
2 Materials and Methods Two surface treatment techniques were addressed in the research due to their advantages and widest industrial dissemination, namely, conventional thermal synthesis, and plasma-enhanced thermal synthesis [8, 14]. For the thermal oxidation process, MTI Corporation Dual Zone Split Tube furnace OTF-1200X was engaged as a processing chamber, the schematic and the details of the process are reported [31]. Briefly, the copper samples were arranged in the reactor and exposed at atmospheric air conditions at the temperatures of 600 ºC, 700 ºC, 800 ºC and 1000 ºC for 5 h; at that, the stationary modes of the heating were reached after one hour of heating. After the treatment, the samples were cooled for 24 h in the furnace. As for the plasma-enhanced process, the reactor layout is described elsewhere [4, 16, 31]. The samples were put on a cathode in a current conducting circuit, which included plasma-generating power supply. At that, two modes of exposure to plasma fluxes were engaged: direct, i.e. without any cover, and capped, when a cap with orifices was put above the samples to provide oxygen delivery but prevent direct action of charged particles from plasma. The rate of heating was much faster for the plasma-assisted treatment, and the temperatures reached their stationary values for a time period of two minutes. It was assumed that the difference in the heating condition can be the factor that changes the modes of nucleation and growth for the obtained copper oxide nanostructures. After the treatment, all the samples were studied by means of a field emission scanning electron microscope (SEM).
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3 Experimental Results and Discussion The results of the conventional thermal growth technique applied to the copper samples, are shown in Fig. 1 and Fig. 2. In Fig. 1 the attention is paid to the pattern formed on a surfaces of copper oxide (Fig. 1, a-d) of copper sample after termination of the growth process. General view of copper oxide layer shown in Fig. 1, a, exhibits black area where copper nanowires are found (it should be noted that such area are usually described as “wet ash”, and are characterized by much lesser reflectivity). Figure 1, b, indicates an example of how the process of stress relieving affects the nucleation of nanowires, since the flaked area shows much denser array of the nanowires as compared to the rest of the surface. In addition, the relief of the stress results in the change of the growth dynamics since the nanowires are longer on the bubble, as it is shown in Fig. 1, c. At the same time, a patterned view of the array of nanowires grown on a surface of CuO oxide (Fig. 1, d), can indicate the dependence of the growth direction on the initial orientation of Cu2 O oxide where the oxidation and formation of CuO layer takes place. The patterned structure of the growth process can also be seen even after the peeling of the oxide layer, when a surface of the copper samples is uncovered (here the elevated temperature of 1000 °C was applied to enhance the effect). Figure 1, e, reveals the paths of oxidation, which are possibly associated with morphology of copper surface, as well as with stress developed in the surface layers because of heating of the sample. A magnified view of the pattern shows the sub-micrometer grains of copper oxide grown on the copper surface and confirms the anisotropy of copper diffusion and oxidation. Figure 2 shows the tilted view of the nanostructures to demonstrate the effects. Typical ratio of about 1:10 of thicknesses of CuO layer (8 µm) and Cu2 O layer (90 µm), and the difference in their morphology, where CuO layer is composed of grains with much lesser sizes, is shown in Fig. 2, a. It should be noted that the lowest part of Cu2 O layer in the figure is covered by CuO layer formed after the peeling off Cu2 O layer from Cu substrate. Interesting feature of the growth process is revealed in Fig. 2, b, where a flaked part of copper oxide is shown. It can be seen that hilltop is not covered with nanowires opposite to the hillslope covered with them, which indicates that the growth was terminated possibly because of formation of the crack. Figure 2, c exhibits a patterned structure of an array of nanowires, which is likely associated with the initial orientation of Cu2 O grain where oxidation takes place; this pattern is similar to that shown in Fig. 1, d. A magnified view (Fig. 2, d) of a tip of a nanowire shows a terrace-like structure on the tip, which speaks in favor of the growth mechanism of the nanowire, where copper atoms are delivered to the tip through the diffusion from the nanowire root and then attached to the places with higher energy of adsorption. At that, the almost constant diameter of the nanowires along their length allows concluding about the large difference in energies of adsorption of oxygen on the side surface of the nanowires, and on their tips to sustain the anisotropic growth of the nanowires. Figure 2, e, and Fig. 2, f, show the possible ways of tailoring the arrays of nanowires into a more complex structures: thus, soldering of two nanowires by use of the electron flux of SEM facility is demonstrated in Fig. 2, e, while the possibility of change of diameter of a nanowire in a specified point is shown in Fig. 2, f. As was pointed in Introduction, plasma-enhanced process of oxidation results in much faster growth of the sample temperature to reach the temperature saturation on time. It is assumed that the effects caused by the stress gradients, are much more pronounced
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Fig. 1. Copper oxide nanostructures synthesized under ambient air conditions at the temperatures of 700 °C (a-d) and 1000 °C (e, f): a – general view, where the arrays covered by the nanowires are distinguished due to the change of reflectivity (the area with the nanowires are much less reflective); b – flaked area with nanowires grown in a much denser array than in the area without flaking, thus indicating the relieved stress as a driving force of the process; c – a magnified view with the nanowires of increased length, which are grown on a bubble, as compared with the nanowires on the rest of the surface; d – SEM image that shows a patterned structure of nanowire arrays with respect to the tilt to the surface, which is possibly associated with the morphology of CuO later that depends on crystal orientation of Cu2 O grain; e – a patterned structure on a surface of copper sample, where copper oxide layer was peeled off; f – magnified view that shows the sub-micrometer grains of copper oxide that create the pattern on the copper surface and confirm the anisotropy of copper diffusion from Cu sample to the surface of copper oxide layer.
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Fig. 2. Copper oxide nanostructures synthesized under ambient air conditions at the different temperatures: a – 800 °C; b – 600 °C; c-f – 700 °C: a – typical ratio of thicknesses of CuO layer (8 µm) and Cu2 O layer (90 µm), and the difference in their morphology, where CuO layer is composed of grains with much lesser sizes (the lowest part of Cu2 O layer is also covered by CuO layer formed after the peeling off Cu2 O layer from Cu substrate); b – a flaked part of copper oxide where hilltop is not covered with nanowires opposite to the hillslope, which indicates the growth termination process possibly due to the crack formation; c – a patterned structure of a nanowire array, which is likely associated with the initial orientation of Cu2 O grain where oxidation takes place (similar pattern is also shown in Fig. 1, d); d – terrace-like structure on a tip of CuO nanowire, which indicates the growth mechanism of the nanowire where copper atoms are delivered to the tip through the diffusion from the nanowire root and then attached to the places with higher energy of adsorption; e, f – soldering of two nanowires, and diameter increase, respectively, by use of electron flux from SEM facility.
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in that case. Figure 3 shows SEM images of a sample treated for 20 min in oxygen plasma on a cathode that was partially covered by a cap to protect the sample from direct action of plasma ions and leave just the thermal effect of heating caused by plasma. Indeed, the fast development of the stress gradient changes the nucleation and growth of the nanostructures a lot, as it can be seen in Fig. 3, a. In the figure, arrays of 2D copper oxide nanostructures (nanosheets) grown on the sample surface covered by the cap and associated with the peeled off parts of copper oxide layer are shown. A magnified view of the nanosheets shown in Fig. 3, b, reveals the structures with a height of 3 to 4 µm, width of 1 to 2 µm, and thickness of 500 nm. As for the origin of the nanostructures, the globular nanostructures (Fig. 3, c) found on the perimeter of CuO grains in the region of the sample that was not covered by the cap, can mark the active grain boundaries as the nucleation area where the 0D nuclei (nanodots, similar to those shown in Fig. 1, f) coalescence or attach to each other to form the roots of the nanosheets. At that, the role of stress is demonstrated in Fig. 3, d, where a patterned structure of the nanosheets grown outside of the capped region of the sample is supposedly associated with a distribution of the internal stresse in CuO layer, and generation of active boundaries between the grains. SEM image showing the nature of the microstructures composed of stacked 2D nanostructures grown on the sample surface outside of the cap, is shown in Fig. 3, e. Again, the internal stress is obviously the main factor of such anisotropic pattern. It should be noted that the presence of ion bombardment does not affect significantly to the nanostructure growth since the arrays of nanosheets with a height of 3 to 5 µm and thickness of 500 nm are also found on the sample surface that was not covered by the cap, as it is shown in Fig. 3, f. The fact of low effect of the ion bombardment (but not treatment!) is confirmed also by the results shown in Fig. 4, where the ion energy was decreased (from 740 eV to 620 eV) while the ion flux was increased (the current to the same surface area was changed from 0.23 A to 0.2 A). Some peculiarities of the oxidation process were revealed at that. In particular, grains of copper oxide transformed under the action of oxygen in a region of the sample that was covered by the cap, are shown in Fig. 4, a. A magnified view, which reveals the primordial segmentation of the grains in the interior of the grains, is shown in Fig. 4, b. 2D nanostructures grown on the sample outside of the capped region yet in the close proximity to it (Fig. 4, c), confirm the absence of severe ion bombardment, in spite of the ion energy that exceeds the threshold where the sputter yield is unity. Again, surface nanostructures grown on the sample at the far distance of the capped region (Fig. 4, d), exhibit the same shape, yet the density of the nanostructures is a bit less, which can be explained by the higher stress in the central area. At the same time, the nanocolumns composed of stacked 2D nanostructures as shown in Fig. 4, e, confirm the high level of the stress, which is responsible for the deformation and stacking of 2D nanostructures to form the pattern shown in Fig. 4, f.
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Fig. 3. SEM images of a sample treated on a cathode and partially covered by a cap for 20 min in oxygen plasma (360 Pa, 740 V and 0.23 A): a – the arrays of the distorted nanosheets are grown on the sample surface covered by the cap and associated with the peeled off parts of copper oxide layer; b – magnified view of the nanosheets with a height of 3 to 4 µm, width of 1 to 2 µm, and thickness of 500 nm; c – globular nanostructures grown on the perimeter of CuO grains in the region of the sample that was not covered by the cap; d – patterned structure of the nanosheets grown outside of the capped region of the sample is supposedly associated with a distribution of the internal stresses in CuO layer; e – SEM image showing the morphology of the microstructures composed of stacked 2D nanostructures grown on the sample surface outside of the cap; f – an array of nanosheets with a height of 3 to 5 µm and thickness of 500 nm grown on the sample surface that was not covered by the cap.
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Fig. 4. SEM images of a sample treated on a cathode and partially covered by a cap for 30 min in oxygen plasma (360 Pa, 620 V and 0.2 A): a – grains of copper oxide transformed under the action of oxygen in a region of the sample that was covered by the cap; b – magnified view that reveals the peculiarities of the oxidation process, namely, primordial segmentation of the grains in the interior of the grains; c – surface nanostructures grown on the sample outside of the capped region yet in the close proximity to it; d – surface nanostructures grown on the sample at the far distance of the capped region; e – nanocolumns composed of stacked 2D nanostructures; f – magnified view of the stacks showing the sizes of 2D nanostructures of about 3 to 5 µm in diameter and 100–200 nm in thickness.
4 Conclusions The analysis of the experimental data allows concluding abouts the important points that should be considered at the development of theoretical models to explain the mechanisms of nucleation and growth of copper oxide nanostructures of various dimensionality, as
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well as predict the control factors of the transition between 0D, 1D, and 2D nanostructures. Both the stress and rate of stress development affect the process of synthesis yet in the different ways. Presence of stress can result in nucleation, if the value of the stress is rather high, and the conditions that accompany the oxidation of Cu2 O layer and its transition to CuO oxide satisfy the requirement. The rate of the temperature increase that is usually of about 5 °C/min in conventional thermal process allows nucleation of 0D nanodots from grains of CuO oxide, and these nuclei grow separately, thus forming an array of 1D nanowires. Opposite to that, the rate of the temperature growth of about 150 °C/min obtained for the plasma-enhanced method of heating results in mutual attachment of the nanodots generated on the active boundaries of oxide grains, which leads to the formation of the roots of 2D nanostructures (nanosheets). Moreover, the stress developed by use of the plasma heating, deforms the nanosheets and urges them to stack, thus forming the complex 3D nanostructures observed in the experiment. It should be noted that the latter structures can be very beneficial with respect to the energy storage applications due to their huge surface-to-volume ratio. Acknowledgements. The authors acknowledge the support from the project funded by National Research Foundation of Ukraine, under grant agreement No. 2020.02/0119. A. Breus acknowledges the support from the project sponsored by the NATO Science for Peace and Security Programme under grant id. G5814 project NOOSE.
References 1. Guo, B., et al.: Single-Crystalline Metal Oxide Nanostructures Synthesized by PlasmaEnhanced Thermal Oxidation. Nanomaterials 9(10), 1405 (2019) 2. Yuan, W., Qiu, Z., Chen, Y., Zhao, B., Liu, M., Tang, Y.: A binder-free composite anode composed of CuO nanosheets and multi-wall carbon nanotubes for high-performance lithiumion batteries. Electrochim. Acta 267, 150–160 (2018) 3. Breus, A., Abashin, S., Serdiuk, O.: Carbon nanostructure growth: new application of magnetron discharge. Journal of Achievements in Materials and Manufacturing Engineering 109(1), 17–25 (2021) 4. Breus, A., Abashin, S., Lukashov, I., Serdiuk, O., Baranov, O.: Catalytic Growth of Carbon Nanostructures in Glow Discharge. In: Ivanov, V., Trojanowska, J., Pavlenko, I., Rauch, E., Perakovi´c, D. (eds) Advances in Design, Simulation and Manufacturing V. DSMIE 2022. Lecture Notes in Mechanical Engineering, pp 375–383, Springer, Cham. (2022) 5. Ko, Y.H., Nagaraju, G., Lee, S.H., Yu, J.S.: Facile preparation and optoelectronic properties of CuO nanowires for violet light sensing. Mater. Lett. 117, 217–220 (2014) 6. Raveesh, S., Yadav, V.K.S., Paily, R.: CuO Single-Nanowire-Based White-Light Photodetector. IEEE Electron Device Lett. 42(7), 1021–1024 (2021) 7. Kimura, Y., Tohmyoh, H.: Effect of the geometric shape of nanostructures formed on Cu oxide thin films by the stress-induced migration on electrical characteristics. Scripta Mater. 210, 114469 (2022) 8. Baranov, O., Košiˇcek, M., Filipiˇc, G., Cvelbar, U.: A deterministic approach to the thermal synthesis and growth of 1D metal oxide nanostructures. Appl. Surf. Sci. 566, 150619 (2021) 9. Cao, F., et al.: Thermal-induced formation of domain structures in CuO nanomaterials. Physical Review Materials 1(5), 053401 (2017)
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10. Shyrokyi, Y., Kostyuk, G.: Investigation of the Influence of Crystallization Energy on the Size of Nanostructures During Copper Ion-Plasma Treatment. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds) Integrated Computer Technologies in Mechanical Engineering - 2021. ICTM 2021. Lecture Notes in Networks and Systems, vol. 367, pp. 57–66, Springer, Cham. (2022) 11. Emanuela, F., Marco, T., Tiziana, S., Daniela, C., Cosimo, M., Rachele, G.M.: Room temperature facile synthesis of CuO nanostructures by resistive heating. Physica E 60, 59–64 (2014) 12. Dorogov, M., et al.: CuO nanowhiskers: preparation, structure features, properties, and applications. Mater. Sci. Technol. 34(17), 1–10 (2018) 13. Tran, T.H., et al.: Effect of annealing temperature on morphology and structure of CuO nanowires grown by thermal oxidation method. J. Cryst. Growth 505, 33–37 (2019) 14. Baranov, O., Filipiˇc, G., Cvelbar, U.: Towards a highly-controllable synthesis of copper oxide nanowires in radio-frequency reactive plasma: fast saturation at the targeted size. Plasma Sources Sci. Technol. 28, 084002 (2019) 15. Mema, R., Yuan, L., Du, Q., Wang, Y., Zhou, G.: Effect of surface stresses on CuO nanowire growth in the thermal oxidation of copper. Chem. Phys. Lett. 512, 87–91 (2011) 16. Breus, A., Abashin, S., Lukashov, I., Serdiuk, O.: Anodic growth of copper oxide nanostructures in glow discharge. Archives of Materials Science and Engineering 114(1), 24–33 (2022) 17. Wang, J., et al.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24(1), 52–82 (2011). https://doi.org/10.1016/S0894-9166(11)60009-8 18. Xiang, L., Guo, J., Wu, C., Cai, M., Zhou, X., Zhang, N.: A brief review on the growth mechanism of CuO nanowires via thermal oxidation. J. Mater. Res. 33(16), 2264–2280 (2018) 19. Altaweel, A., Gries, T., Migot, S., Boulet, P., Mézin, A., Belmonte, T.: Localised growth of CuO nanowires by micro-afterglow oxidation at atmospheric pressure: Investigation of the role of stress. Surf. Coat. Technol. 305, 254–263 (2016) 20. Shypul, O., Myntiuk, V.: Transient thermoelastic analysis of a cylinder having a varied coefficient of thermal expansion Periodica Polytechnica. Mech. Eng. 64(4), 273–278 (2020) 21. Shyrokyi, Y., Kostyuk, G.: Erosion Processes on Copper Electrodes Applied to Growth of Nanostructures in Plasma. In: Ivanov, V., Trojanowska, J., Pavlenko, I., Rauch, E., Perakovi´c, D. (eds) Advances in Design, Simulation and Manufacturing V. DSMIE 2022. Lecture Notes in Mechanical Engineering. pp. 494–503, Springer, Cham. (2022) 22. Jagtap, P., Chason, E.: A unified kinetic model for stress relaxation and recovery during and after growth interruptions in polycrystalline thin films. Acta Mater. 193, 202–209 (2020) 23. Chason, E., Engwall, A.M., Rao, Z., Nishimura, T.: Kinetic model for thin film stress including the effect of grain growth. J. Appl. Phys. 123, 185305 (2018) 24. Baranov, O., Fang, J., Rider, A., Kumar, S., Ostrikov, K.: Effect of ion current density on the properties of vacuum arc-deposited TiN coatings. IEEE Trans. Plasma Sci. 41(12), 3640–3644 (2013) 25. Baranov, O., Romanov, M., Fang, J., Cvelbar, U., Ostrikov, K.: Control of ion density distribution by magnetic traps for plasma electrons. J. Appl. Phys. 112(7), 073302 (2012) 26. Korytchenko, K.V., et al.: Numerical simulation of gap length influence on energy deposition in spark discharge. Electrical Engineering and Electromechanics 1, 35–43 (2021) 27. Ugrimov, S., Smetankina, N., Kravchenko, O., Yareshchenko, V.: Analysis of Laminated Composites Subjected to Impact. In: Nechyporuk, M., Pavlikov, V., Kritskiy, D. (eds.) ICTM 2020. LNNS, vol. 188, pp. 234–246. Springer, Cham (2021). https://doi.org/10.1007/978-3030-66717-7_19 28. Smetankina, N., Merkulova, A., Merkulov, D., Postnyi, O.: Dynamic Response of Laminate Composite Shells with Complex Shape Under Low-Velocity Impact. In: Nechyporuk, M.,
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Required Geometric Parameters of Inflatable Air Shock Absorbers of Parachute Platforms Bench Dynamically Similar Models Olexander V. Betin , Igor V. Kaluzhynov , Denys O. Betin(B) and Olena Y. Betina
,
National Aerospace University «KhAI», Chkalova 17, Kharkiv 61070, Ukraine [email protected]
Abstract. Research has been carried out aimed at developing theoretical bases for determining the required geometric parameters of inflatable air shock absorbers of parachute platforms of bench dynamically similar models (BDSM). It was established that the most common in landing equipment were spherical, cylindrical with a bottom and torus shells or geometrically close to them. Operating conditions and loads acting on these shells in the inflatable air shock absorbers of parachute platforms are determined. Features of BDSM as an object of research of equipment landing on inflatable air shock absorbers of parachute platforms are revealed. The substantiation of the application of the main provisions of the similarity theory in the form of criteria and scales in the task of creating a BDSM and conducting experimental researches of the landing of a natural object (NO) on it is given. It is established that the overall dimensions, areas of external surfaces and volumes of the shells of air shock absorbers NO and BDSM must be geometrically similar. It is proposed to use the theory of calculation of thin-walled rotation shells for determination of wall thicknesses of spherical, cylindrical with the bottom and torus shells of air shock absorbers of BDSM. Formulas for determination of stresses acting in spherical, cylindrical with the bottom and torus shells of BDSM are obtained. Formulas for determining wall thicknesses of such shells have also been obtained, both for structurally similar shells made of identical materials, and for shells made of various materials. Keywords: Required Geometric Parameters · Inflatable Air Shock Absorbers · Parachute Platforms · Bench Dynamically Similar Models
1 Introduction In recent years, the landing systems used in the landing of unmanned aerial vehicles for various tasks and dropped from aircraft cargo and objects (especially equipment with crew inside) have undergone significant changes. Among the adjustments is the refusal to use exclusively soft-landing jet engines due to insufficient reliability. They can fail and lead to the death of the crew or the failure of the landing equipment. Although the loss of equipment usually doesn’t exceed the established 2%, there is always temptation to reduce them [1–6]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 123–133, 2023. https://doi.org/10.1007/978-3-031-36201-9_11
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Soft-landing jet engines have been replaced by parachute-shock absorbing systems that use soft tissue shells filled through inlets with atmospheric or gas cylinder-pumped air. The action of these devices based on the compression of air in the shell and its flow through outlet holes (valves). At the same time, the air inflatable shells absorb the kinetic energy of the impact, controlling outflow of air through outlet holes. The tissue shells themselves, due to the peculiarities of their material and the transience of the landing processes, are (or can be considered) weakly permeable (Fig. 1).
Fig. 1. Appearance of inflatable air shock absorbers with valves [4]
The most widespread in landing equipment were spherical shells, cylindrical shells with a bottom (more often of the “vertical cylinder” type) torus shells or geometrically close to them (Fig. 2). Although shells in the form of rectangular parallelepipeds or other more complex shapes may take place. All these shells have fairly large overall dimensions, but are rarely equipped with devices that increase the stability of the object on the platform when landing with wind demolition [1–4].
Fig. 2. Examples of the use of spherical (a), cylindrical with bottom (b) and torus (c) shells in the landing equipment [4–6]
The creation or modernization of the landing equipment and equipment used involves a complex of theoretical, development, research calculation and experimental work. The
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most effective method of research the behavior of landing equipment is physical modelling. A free flying dynamically similar model (FDSM) or the bench test dynamically similar model (BDSM) of natural object (NO) are used as object of researches [7–9]. Physical modeling of drop by parachute and landing the parachute platforms onto pneumatic shock absorbers (and, most often, onto inflatable airbags) on the specific nature of external and internal factors of functioning of all system should be separated into range of tasks [5]: 1. Modeling of process of separation of the landing NO (having certain lift-to-drag ratio) from host aircraft or any other type of delivery vehicle. 2. Modeling of process of unfolding of parachute and shock absorber systems of NO, and also their setting into pre-landing state (braking of NO of landing by parachute system to the optimum speed of steady descent, inflation of shock-absorbers and placement of the bottom part of the platform into necessary position). 3. Modeling of process of contact the platform of the NO with the ground surface under different conditions of wind drift and “suppression” of parachute system. 4. Modeling of process of landing the parachute platforms onto inflatable air shock absorbers of NO (movement of NO together with parachute and shock-absorber systems without achievement of extreme values of vertical overload for the equipment when landing). The first three tasks related to flight dynamics and modeling the airflow around NO surfaces and its parachute and shock-absorbing systems. The solutions of abovementioned modeling tasks are quite well considered, and there are no problems related to the implementation of fundamentally new theoretical studies, and it is most rational to use FDSM as a research object [5, 7–9]. When modeling the landing process of the parachute platforms of a NO on inflatable air shock absorbers, it is possible to use FDSM, but rationally – BDSM, since this modelling task, unlike the previous ones, is not related to the flight dynamics and modeling the of NO surfaces and its parachute and shock absorption systems by airflow. For theoretical foundation of dynamic similarity during landing, in addition to the basic provisions of the theory of similarity to motion modeling, it is necessary to use the laws of gas dynamics in relation to air shock absorbers. This is due to the fact that the central (or main) element of considered shock absorber systems is an air shock absorber, which can be considered as a kind of shell with an expanding (or shrinking) control volume. When ensuring its appropriate behavior under such a load, modeling of the landing process of a NO on a BDSM is achieved [10]. This task is relevant and important, especially in modern conditions of limited resources for the creation of new models of aviation equipment, and to solve it, appropriate design methods should be available and used. The purpose of this work is to theoretically substantiate the determination of the required values of geometric parameters of inflatable air shock absorbers of parachute platforms of BDSM.
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2 Literature Review on Operating Conditions and Loads Acting on Inflatable Air Shock Absorbers of Parachute Platforms and Modeling Features on DSM The shock absorber system includes a rigid cargo platform and inflatable air shock absorbers, which are located between upper and lower parts of the cargo platform. If the landing object has a wheeled or tracked chassis, then the cargo platform is placed between the chassis. The upper part of the cargo platform fits tightly to the body of the landing object and rigidly attached to it. Prior to the deployment parachute and shock absorber system, the air shock absorbers are placed in the assembled state between the upper and lower parts of the cargo platform and are attached to them [1–5]. After deployment pilot, brake and main parachutes (or multi-dome parachute system) the vertical velocity of the steady descent is V y = 25–30 km/h (approximately 7–8 m/s). The horizontal rate of descent V x is limited to 5 km/h (approximately 2 m/s) to prevent the landing objects from capsizing or overturning. All this allows receiving vertical overload for the equipment which not exceeds nlim y = 15 when landing [3, 4, 6]. The shock absorber system is unlocked. The lower part of the cargo platform moves downward under gravity and atmospheric air is filled through the inlet openings (located at the bottom of the platform and air shock absorbers). When touching the ground surface, the pressure inside the shock absorbers pin , is usually equal to the atmospheric pressure patm at the height on the landing site [1–3, 5]. In rare cases, landing takes place on sites located at altitudes markedly different from sea level. But if this is supposed to be done, then in calculations and experiments they take the value patm , corresponding to the landing site, and not patm (H = 0 m) = 1 atm = 101, 3 kPa according to the Standard atmosphere [1]. When the lower part of the platform touches the ground surface the inlet holes are closed. The compression of air in the tissue shells of the shock absorbers begins, as the speed Vy decreases, brake acceleration ay , overload ny and pressure pin in shock absorbers increase. The outlet holes (valves) are located at the upper part of the side walls of the shock absorber shells, but their opening and air outflow begins only when the difference the internal pin and external atmospheric patm pressures, acting on the shock absorber shells doesn’t exceed the p value. After some time, the ny overload reaches its maximum value, and the vent opens and activates the valves. For calculations of the wall thicknesses it is usually, that p = p = pin − patm = (0, 4...0, 5)1 atm = (0, 4...0, 5)101, 3 kPa [1–3, 5]. Further the air pressure releasing in shock-absorbers takes place up to the pressure pin = patm (H = 0 m) = 101, 3 kPa [1–3, 5]. Forces are transmitted between the inflatable air shock absorbers, cargo platform and landing object through contact surface of Sc area. As a rule, the landing time tlt (from contact of platform with ground surface up to full stopping of the movement of object of landing) does not exceed 0, 2 − 0, 3 s [1–5]. As for DSM, they are not exportable or saleable objects. Therefore, information about them (patents for original design solutions of units and model systems, methods of their design, manufacture and testing) is practically absent in the open foreign literature, and the appearing publications mainly relate to the history of the issue and analysis of the
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results of research on DSM. At the same time, the modeling task under consideration is relevant and important, especially in modern conditions, limited resources for the creation of new aircraft models, and appropriate design methods should be available and used to solve it.
3 Criteria and Scale of Similarity in the Task a Creating BDSM and Execution Experimental Researches of NO Landing on It Different types of modeling (mathematical, physical, analogue, etc.) are used in solving a number of scientific and engineering problems. As mentioned above, for the study the landing process of a NO on a bench DSM, it is rational to use physical modeling in which an experiment on a DSM in natural conditions is carried out with the subsequent transfer of the results of physical modeling to the NO [10]. To adequately simulate the landing on the BDSM and to transfer the results of model studies to the NO it necessary to fulfill the requirements of similarity theory (geometrical, kinematic and dynamic) both in terms of experimental conditions and parameters and characteristics of the NO and its BDSM (shape, mass-inertial parameters and elastic geometric characteristics, deformations under load, etc.). The use of similarity theory helps to formalize the conditions that must be met, both at all stages of creating a DSM and during experimental studies. Transferring the results of model researches of a BDSM to a NO to obtain reliable data about the most complex landing modes is possible only when velocities, accelerations, overloads, air pressure in shock absorbers and the required environmental parameters meet the accepted similarity conditions [10]. Regarding questions of similarity theory, we use its general provisions and results of theoretical and experimental studies of modeling the process of landing parachute platforms on air shock absorbers and accept [1–3, 10–12]. a) after contact of platform with the ground surface, the parachute system is “suppressed” and has no significant influence onto the process of landing; b) aerodynamic drag of air shock-absorbers and the platform is negligible in the process of landing; c) shells of air shock absorbers are impenetrable, i.e. air at landing flows out only through vent holes; d) pressure in air shock absorbers is distributed evenly on all surface of shells; e) the ideal gas law and adiabatic process of changing air condition in the shells of air shock absorbers are valid during landing due to short duration of the impact and, thus, exchange with structural elements of the platform and with the surrounding atmosphere is limited (and, most likely, is absent at all). Having made preliminary conclusions, let’s get to the essence. In order to ensure geometric similarity, the following ratios shall be observed (hereinafter index “n” determines the relation to the parameters or characteristics of the NO; and the index “m" to the corresponding parameters or characteristics of the BDSM.) n = k = const1 ; m
(1)
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φn = φm or kφ =
φn = 1; φm
(2)
2 Sn = 2n = k2 = ks ; Sm m
(3)
3 Wn = 3n = k3 = kw , Wm m
(4)
where n , m – randomly chosen analogous sizes (for example, characteristic sizes) of the comparable bodies (natural object and model); k – scale of the linear sizes; φn , φm – angles between corresponding lines of geometrically similar bodies; kφ – scale of angles; Sn , Sm , Wn , Wm – areas and volumes of geometrically similar bodies; ks – scale of areas; kw – scale of volumes. Two movements are geometrically and kinematically similar if: tn = kt = const2 ; tm
(5)
Vn n tm k = = = kv = const3 ; Vm m tn kt
(6)
n tm2 k kv an = = 2 = = ka = const4 , 2 am m tn kt kt
(7)
where tn , tm – proportional time intervals in which geometrically similar bodies pass similar paths in proportional time intervals; kt – scale of times; Vn , Vm – linear velocities of geometrically similar bodies; kv – scale of linear velocities; an , am – linear accelerations of the movement of geometrically similar bodies ka – scale of linear accelerations. Dynamic similarity of movements is observed if the conditions of compliance of all elements creating dynamic picture are met. In this case, the polygons of forces acting on geometrically similar bodies will be similar (and the overloads, of course, are equal). This means that forces of the same nature must be equally directed and proportional: Rn = kR = const5 , Rm
(8)
where Rn , Rm – forces of the same nature; kR – scale of forces. The theoretical researches carried out in [6, 10, 11] concerning the modeling of the landing process of parachute platforms on inflatable air shock absorbers made it possible to obtain a number of formulas for determining both these and other required scales of similarity. This work shows that if the scale of linear sizes k taken as the primary scale, the other scales that determine the design, parameters and movement characteristics at landing are derived scales that depend on the scale k and indirectly on the landing heights of the NO and BDSM. In particular, if the landing NO and BDSM takes place with the same heights above sea level (i.e. Hn = Hm = H ), then the considered scales kt = kv = k , (9)
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ka = 1,
(10)
kR = k2 .
(11)
In addition, the scale of the acting vertical overloads when modeling the landing of NO on the BDSM on inflatable air shock absorbers kol =
ny.n (tn ) = 1. ny.m (tm )
(12)
4 Results of Using the Similarity Theory and Theory of Calculation of Thin-Walled Shells of Rotation to Determine the Wall Thicknesses of Air Shock Absorbers of BDSM Modeling of process of landing the parachute platforms onto inflatable air shock absorbers by means of BDSM eliminates a range of traditional tasks of designing: choice of form of object of landing, mutual position of their aggregates, parachute platforms and inflatable air shock absorbers, determination of center of gravity, etc. At the same time, similarity conditions significantly complicate development of BDSM. In particular, to ensure dynamic similarity, air shock absorber shells with required valves are necessary and the walls of the shells are made of certain materials with certain thicknesses. Taking into account that valves of the same shape with NO were used for the BDSM, in the work [10] it was obtained that the ratio of the areas of outlet valves Sval.n (tn ) = k2,5 = ks.val , Sval.m (tm )
(13)
where ks.val – scale of the areas of valves of the air shock absorbers of the NO and the BDSM. As for the areas of external surfaces and volumes of geometrically similar air shock absorbers of a NO and a BDSM, they should be in constant ratios (3) and (4), respectively. Finally, at any intermediate landing point of a NO and its BDSM, the condition [16] must be fulfilled pin.n (tn ) = 1 = kp.in , pin.m (tm )
(14)
where kp.in – scale of air pressure inside the air shock absorbers of a NO and BDSM. In this work, we focus on determination the thickness h of the walls of the air shock absorbers of the BDSM in the form of spherical, cylindrical with the bottom and torus shells under uniform pressure p = p = pin − patm (Figs. 3, 4 and 5). Moreover, consider the most common case of landing a NO and a BDSM on sites with the same height above sea level (i.e. Hn = Hm = H ) with the same external atmospheric pressure patm .
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Fig. 3. Design scheme spherical shell
Fig. 4. Design scheme cylindrical shell with a bottom
Fig. 5. Design scheme of torus shell
For this case, according to condition (14), geometrically similar air shock absorbers of NO and BDSM in similar time intervals tn and tm must be under the same uniform pressure, i.e. p = pn (tn ) = pm (tm ). All three types of shells can be classified as momentless thin-walled shells of rotation and apply the same theory of strength calculation of elements of thin-walled structures under uniform pressure and away from the seal or edges [12].
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The equations for determining the meridional (longitudinal) σm and circumferential (ring) σθ normal stresses, as well as the movement ω in direction of normal to surface (radial movement) of each of the shells are of the following form [13, 14]: a) spherical σm = σθ =
pR2 pR ; ω= (1 − μ), 2h 2Eh
(15)
b) cylindrical σm =
pR2 pR pR ; σθ = ; ω= (2 − μ), 2h h 2Eh
(16)
c) torus σm =
pR 2k + sin φ pR pR2 · ; σθ = ; ω= [k(1 − 2μ) + (1 − μ) cos φ], 2h k + sin φ 2h 2Eh (17)
where R – radius of median surface in the cross section of the shell; E, μ – respectively, the modulus of elasticity of the first kind and Poisson’s ratio shell material; r – distance from the center of generating circle to the axis of rotation Z; φ – angular coordinate of a point on the generating circle; k = Rr = const – constant ratio for the torus surface (values σm and σθ are sufficiently accurate for k ≥ 2...3). Based on the concept of geometric similarity, it can be concluded that when modeling the landing process a NO on a BDSM, moving ω surfaces of geometrically similar air shock absorbers into similar time intervals tn and tm should also be geometrically similar, i.e. ωωmn (t(tnm)) = k . Using this conclusion in conjunction with the conclusion p = pn (tn ) = pm (tm ) in relation to dependencies (15)–(17), for the shells of a NO and a BDSM, we obtain following relations: a) spherical σm.n σθ.n En 1 − μm = = · ; σm.m σθ.m Em 1 − μn
Em 1 − μn hn = k · , hm En 1 − μm
(18)
σm.n σθ.n En 2 − μm = = · ; σm.m σθ.m Em 2 − μn
Em 2 − μn hn = k · , hm En 2 − μm
(19)
b) cylindrical
c) torus σm·n σθ·n En k(1 − 2μm ) + (1 − μm ) cos φ ; = = · σm·M σθ·m Em k(1 − 2μn ) + (1 − μn ) cos φ
k2
Em k(1 − 2μn ) + (1 − μn ) cos φ hn . = k · hm En k(1 − 2μm ) + (1 − μm ) cos φ
(20)
However, the direct use of formula (3) to determine the scale of areas ks = SSmn = geometrically similar bodies and formula (11) for determining scale of forces
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kR = k2 when modeling the landing of a NO on a BDSM in case of Hn = Hm , allows for structurally similar elements to obtain a scale of stresses σn kσ = = σm
Rn Sn Rm Sm
=
k2 Rn Sm kR · = = 2 = 1, Rm Sn ks k
(21)
i.e. equal stresses in such elements. Ratios (18)–(20) for structurally similar shells and made of the same materials do not contradict this conclusion, and the wall thicknesses are geometrically similar. So for all the shells in question in this case hn hn = k or hm = . (22) hm k If the shells of the air shock absorbers of the NO and the BDSM are made of different materials, then there will be no stress equality, although geometric similarity of deformations can be achieved with wall thicknesses that satisfy the ratios (18)–(20) for each of the shells in the form: a) spherical hm =
hn En 1 − μm · · , k Em 1 − μn
(23)
hm =
hn En 2 − μm · · , k Em 2 − μn
(24)
b) cylindrical
c) torus hn k(1 − 2μn ) + (1 − μn ) cos φ . (25) · k k(1 − 2μm ) + (1 − μm ) cos φ It should be noted that achieving full similarity (geometric, kinematic and dynamic) sometimes makes solving the task much more difficult. Most often, experimental researches have to be carried out in partial (incomplete) similarity, observing similarity only in relation to some of the parameters defining the phenomenon under study. Such a partial similarity is possible when simulating the landing of a NO with inflatable air shock absorbers of parachute platforms on a BDSM. hm =
5 Conclusions The purpose of the work has been achieved – theoretically justified the determination of the required geometric parameters of inflatable air shock absorbers for parachute platforms of BDSM. For this purpose, similarity theory and the theory of calculation of thin-walled torqueless shells of rotation are used. It is established that the overall dimensions, external surface areas and volumes of the air shock absorbers shells of a NO and a BDSM must be geometrically similar. Formulas for determining the stresses acting in spherical, cylindrical with a bottom and torus shells of the BDSM are obtained. Formulas for determining the wall thicknesses of such shells, both for shells made of same materials structurally similar shells, and for shells made of various materials, have also been obtained.
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References 1. John, D.A., Jr.: Fundamentals of Aerodynamics, 6th edn. McGraw-Hill Education, New York, NY, p. 1131 (2017) 2. Gardinier, D.J., Taylor, A.P.: Design and testing of the K − 1 reusable launch vehicle landing system airbags. AIAA 99–1757, 418–427 (1999) 3. Yongxiang, L.E.: Numerical Simulation and Optimal Design of the Process of Airbag Landing. Hunan University, Changsha (2010). (in Chinese) 4. Li, X.: Robust and High Order Computational Method for Parachute and Air Delivery and MAV System. Technical Report. Defense Technical Information Center. AD1050759 (2017) 5. Betin, A.V., Betin, D.A., Kaluzhynov, I.V.: Features of the application of physical modelling methods for the analysis of landing processes of parachute platforms on inflatable air shock absorbers. In: Open Information and Computer Integrated Technologies: Collection of Scientific Articles. , vol. 93, pp. 112–124. National Aerospace University “KhAI”, Kharkiv (2021). (in Russian) 6. Wahab, M.A., Jared, S.D., James, J.H., Douglas, A.L., Michael, T.V.: Design of a parachute canopy instrumentation platform. In: AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, pp. 1–9. Daytona Beach, FL, USA (2015) 7. Betin, A., Ryzhenko, A., Sadovnychiy, S., Ricardo, P.-F.: Simulation of aircraft flight dynamics by means of dynamically similar models. In: Modeling and simulation technologies, pp. 64–69. American Institute of Aeronautics and Astronautics, Boston, USA (1998) 8. Betin A., Sadovnychiy S., Ryshenko A.: Simulation of aircraft emergencies by means of dynamically similar models. In: Modeling and simulation technologies, pp. 328–334. American Institute of Aeronautics and Astronautics, Portland, USA (1999) 9. Sadovnychiy, S., Ryzhenko, A., Betin, A.: Flight control system damage simulation using freely flying models. The Aeronaut. J. 109(1091), 45–50 (2016) 10. Betin, O.V., Betin, D.O., Kaluzhynov, I.V., Betina, O.Y.: Criteria and similarity scales in the task of modelling the operation of inflatable air shock absorbers when landing. In: Integrated Computer Technologies in Mechanical Engineering, AISC, Springer Nature Switzerland AG 2021, pp. 344–352 (2021) 11. Wang, H., Hao, G.: Research on matching and evaluating for recovery system of airborne vehicle based on FEM. In: The 7th China CAE Annual Conference, Kunming, China, pp. 120– 127 (2011). (in Chinese) 12. Du, Z., Shao, P.: Dynamic finite element simulation of the aluminum alloy hull at landing. Acta Armamentar II 30(1), 1–4 (2009). (in Chinese) 13. Ye, L., Junying, M., Yongii, L.: A thin-walled structure with tailored properties for axial crushing. Int. J. Mech. Sci. 157–158, 119–135 (2019) 14. Rajabiehfard, R., Darvizeh, A., Darvizeh, M., Ansari, R., Alitavoli, M., Sadeghi, H.: Theoretical and experimental analysis of elastic–plastic cylindrical shells under two types of axial impacts. Thin-Walled Struct. 107, 315–326 (2016)
Shape of Part Edges with the Various Surface Finish of the Adjacent Surfaces Valeriy Sikulskyi1 , Kateryna Maiorova1(B) , Ihor Bychkov1 Svitlana Myronova1,2 , and Stanislav Sikulskyi2
,
1 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected] 2 Embry-Riddle Aeronautical University, Daytona Beach, FL, USA
Abstract. The subject of the study is the shape of the edges of parts with different surface texture of the surfaces, adjacent to the edge. It is shown that the edge is a transitional surface tangent to the edges of two intersecting surfaces. The types of intersecting adjacent surfaces combinations for forming the edges shape are determined: symmetrically orthogonal, symmetrically parallel, parallel-to-orthogonal, orthogonal-to angular, symmetrically angular, asymmetrically angular. The dependencies of the parts edges characteristics on the roughness and topography of the adjacent surfaces forming the edge are shown. The minimum values of edge roughness are achieved by reducing the roughness of the surface with the orthogonal, with respect to the edge, orientation of the surface lay. The values of the minimum radiuses of the edges rounding are fond out according to the criterion of the possibility of forming the edge of linear, i.e. simplified, shape. The edge shape depending on the combination of intersecting adjacent surfaces lay orientation is studied. Keywords: Roughness · Edge · Minimum Radius of Edge Rounding · Surface Lay · Adjacent Surfaces
1 Introduction The need in part edge representation as a mathematical model arises when simulating the technological process of parts processing. The edge shape, accuracy of manufacturing and the material physical characteristics determine the functional efficiency of the parts, assembly components and product as a whole during their functioning. The complication of the edge model creating is in the fact that edge is usually considered as the result of intersection of two theoretical surfaces (“nominal surfaces”) and described as a line, consequently. In current standards [1, 2], an edge is defined as intersection of two adjacent integral surfaces. Actually, when crossing the real surfaces, such an edge is not a line, but a transitional surface of a complex geometric shape. The edge dimensions and microrelief depend on the methods of part surfaces formation and on the equipment capabilities, as well as on the technological operations sequence (manufacturing route). Moreover, defects of adjacent intersecting surfaces are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 134–145, 2023. https://doi.org/10.1007/978-3-031-36201-9_12
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accumulated up on the edges due to various reasons during machining (as an example, a BUE – buit-up edge – effect). Thus, an edge should be considered as a transition surface that is tangent to the edges of two intersecting adjacent surfaces, which, in addition, has such peculiarities: – physical and chemical characteristics of it may differ significantly from the properties of other fragments of the part; – the values of tolerances for parameters of macrogeometry (size, form, orientation, location) and of microgeometry (surface texture, such as surface roughness and waviness) of it may be significantly less than the tolerances of the adjacent surfaces; – such a surface is important in ensuring the due product service life. Therefore, the design-and-technological viewpoint is that part may be represented as a set of surfaces and edges, which are determined by special technological processes in their formation.
2 The State-of-the-Art and the Study Purpose The edges of parts are a subject of special requirements those are divided into aesthetic, ergonomic and functional groups. The quality and dimensional parameters in the first two groups are assessed “subjectively”, thus they are assigned quite arbitrarily. The last group of requirements effects the physical processes while products functioning. The functional edges of the parts must be of a strictly specified geometry, microrelief and technical cleanliness of the surfaces. For example, hydraulic losses in spool valves and plunger pairs with sharp edges are an order of magnitude greater than in ones with edges are rounded. This property is widely used in mechanical engineering to reduce losses through the gaps of the pumping elements, to protect the hydraulic aggregates capillary slots from solid particles and suchlike. It should be noted that in the technical documentation it is usually paid insufficient attention to the edges description. The description of the nominal (“ideal” [2]) geometry of the parts shapes and surfaces is presented, as a rule, without considering the possibility of any deviations as well as the state of the edges. However, the state of the edges, that is their geometric shape and surface quality, significantly affect – and in some cases even determine – the service life of the product as a whole [3, 4]. Thus, the edges quality must be taken into account already at the product design stage, namely when creating a digital mockups (DMU) of parts and products [5]. This is especially important in the manufacturing of the microtechnology components and the parts of hydraulic control systems – in which the edge sizes, shape and roughness need to be indicated in the technical documentation [6, 7]. The edge of part has its own roughness in the horizontal (along the edge, longitudinal one) and vertical (transverse) directions [8]. The less the adjacent surfaces roughness, the less the longitudinal roughness of the edge. After conventional machining the edges roughness increases that is resulted in significant reducing the edges strength. Due to the low mechanical strength of the fine asperities on the edge these are destroyed in the first seconds of the part work in the product, meanwhile the edge microroughnesses valleys work as the microcracks initiators and resulted in appearing the chips.
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Conventional machining methods can provide, for example, the required roughness values the adjacent surfaces, but the edge roughness parameters will be three to four times higher than these values. The same tendencies are also observed for nonmetallic and composite materials [9–11]. Currently, more than a hundred methods and an order of magnitude more models of equipment are used for finishing and deburring the edges, but the process of their implementation with correct results of these operations is far from its successful solution [12, 13]. It should be noted that in recent years there has been a trend towards the unified complex system creation, which merge several methods for finishing and deburring the edges, but for the implementation of it the accurate data of the edges geometry and properties are required [14]. The latter ones immediately depends on the machining modes and tools geometry. As a result, the issue of edges rounding with exact dimensions providing is still a complex multi-technological problem to be solved and remains relevant in the present-day. Thus, the purpose of this study is to ensure the specified quality, geometry and shape of the parts edges by finding out the dependencies between the parts edges characteristics and the roughness and microrelief parameters of the adjacent surfaces that forming the edge.
3 Study of Relationship Between the Adjacent Surfaces Microrelief Orientation (Lay) and the Edge Shape There are distinguished four types of deviations of the nominal geometric surface shape (“the intended surface boundary”): macrodeviations (such as errors of form and flaws), waviness, roughness, and subroughness [16, 17]. Roughness is considered as a set of the finer spaced irregularities with relatively small steps, which form a surface microrelief. The step of such irregularities varies from 2 to 800 µm, and their height varies from 0.01 to 400 µm, l/Rmax < 50 [17]. The general requirements for the surface roughness of the base metal after machining, which are specified in the drawings, usually are as follows: sharp corners and edges of parts, except for technically reasonable cases, must be rounded with a radius of at least 0.3 mm; the radius value for edges roundings of parts prepared for solid and electrical insulating oxide protective coatings is at least 0.5 mm [18]. It should be noted that on the asperities, which actually form the roughness, there are even smaller irregularities – subroughness. Subroughness is largely determined by the structure of the processed material and by residual stresses are induced during the machining. Electron microscopy studies of surfaces show that subroughness is formed by irregularities having a height of 2…20 µm. Let’s consider the main parameters of microgeometry in detailed way, in accordance with the current standards [1, 2]. In Ukraine, the tolerancing of microgeometry and quantitative assessment of surface roughness in mechanical engineering is carried out using three height parameters – Ra , Rz and Rmax , two spacing parameters – Sm and S, and parameter tp ,which is a profile bearing length ratio [15]. The Ra parameter is considered preferable, since the assessing the Rz parameter (the average maximum height of profile) needs measurements of no less than five peaks and valleys of the profile along the evaluation length, which is not always feasible in practice.
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In addition, Ra more accurately represent the roughness, since it is integral, therefore, the surface roughness is most often estimated by the parameter Ra . All existing national standards are based on two-dimensional (profile) assessment of surface roughness, but the EU countries and the USA have recently adopted a number of recommendations on terminology and surface roughness assessment based on threedimensional (area) analysis. The advantages of two-dimensional assessing are due to the short time for measurements and the lower price of the instruments [6, 7]. The edge as a surface is formed by the intersection of adjacent surfaces that characterized with their own roughness and subroughness. Considering this, the following possible variants of edge surface formation depending on the microrelief lay (direction of dominant pattern) on intersecting adjacent surfaces are possible: symmetrically orthogonal, symmetrically parallel, parallel-to-orthogonal, orthogonal-to-angular, symmetrically angular, asymmetrically angular (Fig. 1).
Fig. 1. Edge formation variants with different directions of microrelief dominant pattern (lay) of intersecting symmetrically orthogonal surfaces (a), symmetrically parallel surfaces (b), parallelto-orthogonal surfaces (c), orthogonal-to-angular surfaces (d), symmetrically angular (e) and asymmetrically angular surfaces (f).
Depending on the lay direction of the intersecting adjacent surfaces, the line of their intersection, which is considered as an edge, will have different spatial shapes and dissimilarity of characteristics. So it would be quite advisable to introduce the concept of “edge roughness”. This term means the deviation of the edge along the line of intersection of the adjacent surfaces, which depends on the magnitude of the roughness and the lay direction of those intersecting surfaces. The content of this concept can be illustrated with the example of adjacent surfaces with a symmetrically orthogonal microrelief dominant pattern (lay). Figures 1 and 2 shows the surface lay of two adjacent intersecting planes, which have roughness with values Ra1 and Ra2 correspondingly.
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Fig. 2. Arrangement of the adjacent intersecting surfaces lays and the edge roughness for the case with the symmetrically orthogonal combination of adjacent surfaces lays.
The value of edge roughness Ra1 k is minimal when Ra2 is a zero value. And accordingly, the value of the edge roughness Ra2 k is minimal when parameter Ra1 is zero value. The maximum value of the edge roughness Ra k occurs when both surfaces roughnesses Ra1 and Ra2 have the maximum values. The maximum value of edge roughness can be defined as follows 2 (1) Rka = R2a1 + R2a2 . As an example let’s consider the value of the edge roughness Ra k for the case of a symmetrically parallel combination of both adjacent surfaces lay (Fig. 3).
Fig. 3. The edge roughness value Ra2 k for the case with the symmetrically parallel combination of adjacent surfaces lays.
In such a case, the roughness direction is parallel to the edge and the effect of the roughness values summation is eliminated. Therefore, here the value of the edge roughness depends only on the value of the surface roughness with the worst roughness, i.e. Rka = Ra2 = Ramax .
(2)
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Thus, the direction of the surface lay of the adjacent surfaces immediately affects the magnitude of the edge roughness. So as not to bore readers with detailed explanations for each case of adjacent surfaces lay combination, the authors presented the results of studies of the edge roughness value for the remaining possible variants of combinations of the surface lay in Tables 1 and 2.
4 Study of Edge Shape for the Different Combinations of Adjacent Surfaces Lay The importancy of the edges shape and its significancy for achieving the proper radius of the edge rounding in parts of hydraulic and fuel equipment is confirmed by studies in the works [19–21]. By results of analysis of the adjacent surfaces roughness and lay, the curve of possible edge was constructed for the cases of symmetrically orthogonal (Fig. 1, a) and symmetrically parallel (Fig. 1, b) combinations of the intersecting adjacent surfaces lay. The shape of the edge, resulted from the intersection of two adjacent surfaces with symmetrically orthogonal lay, constructed by means of CAD system, is shown in Fig. 4, a.
Fig. 4. The edge shape for the cases of symmetrically orthogonal (a) and symmetrically parallel (b) adjacent surfaces lay combinations.
The shape of such an edge in this case is a complex curvilinear spatial object, which depends on the magnitudes of spacing and height parameters of the roughness on the both adjacent surfaces, forming the edge. Reducing the edge curvature and simplifying of its shape is possible due to reducing of the roughness magnitude on the one of the adjacent surfaces or on the both of them. The edge shape in the case of symmetrically parallel adjacent surfaces lay combination (Fig. 4, b) has much more simple spatial shape, rather close to a straight one, and does not depend on the magnitude of roughness as well as the roughness spacing parameters of the both adjacent surfaces. Hence, such a variant of surface lay combination is an excellent opportunity to obtain the rounded edges with the minimum radius values in spite of rather high values of adjacent surfaces roughness.
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The shape of the edge, resulting from the intersection of surfaces with parallel-toorthogonal (c) and orthogonal-to-angular (d) combinations of surface lay, is shown in Fig. 5. The shape of the edge with parallel-to-orthogonal (Fig. 5, c) combination of surface lay is curvilinear in the plane that normal to the surface with orthogonal lay. The edge roughness in this case is much higher than the roughness of the adjacent surfaces. Obtaining the edge with the minimum roughness magnitude is possible due to reducing the roughness parameters of the surface with the orthogonal direction of lay (Fig. 5, d).
Fig. 5. The edge shape for the cases of parallel-to-orthogonal (c) and orthogonal-to-angular (d) adjacent surfaces lay combinations
The shapes of the edges with a symmetrically angular (Fig. 1, e) and asymmetrically angular (Fig. 1, f) direction of the surface lay combinations does not significantly differ from the previous case and are complex spatially curvilinear. A distinctive feature of them is the large edge roughness due to the inclination of the surface lay with respect to the resulting edge. The edge roughness in this case may be reduced only by reducing the roughness of the adjacent surfaces.
5 Study of the Edge Rounding Minimum Possible Radius for the Different Combinations of the Adjacent Surfaces Lay The study of the edge rounding minimum possible radius was carried out according to the accepted criterion – the possibility of forming an edge with a linear shape, as close to the straight one as possible. The values of both adjacent surfaces roughness were considered as dispersion field of random variables of each surface deviations. The minimum possible radius of edge, under this condition, was determined as the radius that able to form edge of surfaces with various deviations within the limits of their roughness. Based on the previously described conditions, the values of the minimum radii for various types of edges (shown in Fig. 1) were built, and the results are presented in Fig. 6. Dependencies for determining the minimum edge rounding radius for the case of symmetrically parallel adjacent surfaces lay (Fig. 1, b) are presented in Table 1. In Table
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Fig. 6. Schemes for determining the minimum possible radius of edge rounding for six different combinations of adjacent surface lay.
1: Ra k is the edge roughness; Rmin k is the minimum edge rounding radius. The shape of the edge, depending on the scheme, is as follows: a – spatial curvilinear spiral-shaped one, b – close to straight line, located in one plane; c – is curvilinear one, is in the
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plane normal to the surface with orthogonal lay; d – spatial curvilinear spiral one; e – curvilinear spiral one with a large roughness; f – curvilinear spiral one with a large roughness. Table 1. Main characteristics of edges for the different variants of adjacent surfaces lay. Rka R2a1 + R2a2
Rkmin
Symmetrically parallel
Ramax
1.5Ramax
c
Parallel-to-orthogonal
2.0Ramax
d
Orthogonal-to-angular
e
Symmetrically angular
f
Asymmetrically angular
Ramax R2a1 + R2a2 R2a1 + R2a2 R2a1 + R2a2
Scheme
Lay direction of adjacent surfaces
ha
Symmetrically orthogonal
b
3.5Ramax
3.5Ramax 2.0Ramax 2.0Ramax
The shapes of the edges are described in the previous section of this article according to the results of the study. The formulas take into account that a surface with a higher roughness value has a great influence on the edge roughness and as well as on the value of the minimum possible radius of edge rounding. For each case of edge formation by adjacent surfaces with different direction of surface lay, it is possible to determine the values of the minimum possible radius depending on the roughness of the adjacent surfaces. The results of calculating the minimum radius of edge rounding for symmetrically parallel adjacent surfaces are shown in Table 2. Using the data of this table, it is possible to determine the value of the minimum edge rounding radius depending on the roughness of the adjacent surfaces – surface 1 and surface 2. It should be noted that the radius of the edges rounding is the same and it is: – 0.15 mm when roughness of the surface 2 is Ra 0,1 and roughness of the surface 1 is in a range Ra 0.05…0.1; – 0.225 mm when roughness of the surface 2 is Ra 0,15 and roughness of the surface 1 is in a range Ra 0.05…0.15; – 0.375 mm when roughness of the surface 2 is Ra 0,25 and roughness of the surface 1 is in a range Ra 0.05…0.25; – 0.3 mm when roughness of the surface 2 is Ra 0,2 and roughness of the surface 1 is in a range Ra 0.05…0.2; – 0.45 mm when roughness of the surface 2 is Ra 0,3 and roughness of the surface 1 is in a range Ra 0.05…0.30; – 0.525 mm when roughness of the surface 2 is Ra 0,35 and roughness of the surface 1 is in a range Ra 0.05…0.35; – 0.6 mm when roughness of the surface 2 is Ra 0,4 and roughness of the surface 1 is in a range Ra 0.05…0.4;
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– 0.657 mm when roughness of the surface 2 is Ra 0,45 and roughness of the surface 1 is in a range Ra 0.05…0.675; – 0.75 mm when roughness of the surface 2 is Ra 0,5 and roughness of the surface 1 is in a range Ra 0.05…0.5.
Table 2. Minimum edge rounding radius for the case of symmetrically parallel lay orientation of adjacent surfaces, 10–6 m
Surface 2
Ra
Surface 1 0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.675 0.5
0.05
0.075 0.15
0.225 0.3
0.375 0.45
0.525 0.4
0.675 0.75
0.1
0.15
0.225 0.3
0.375 0.45
0.525 0.4
0.675 0.75
0.15
0.225 0.225 0.225 0.3
0.375 0.45
0.525 0.4
0.675 0.75
0.2
0.30
0.375 0.45
0.525 0.4
0.675 0.75
0.25
0.375 0.375 0.375 0.375 0.375 0.45
0.525 0.4
0.675 0.75
0.3
0.45
0.525 0.4
0.675 0.75
0.35
0.525 0.525 0.525 0.525 0.525 0.525 0.525 0.40
0.675 0.75
0.4
0.6
0.675 0.75
0.45
0.657 0.657 0.657 0.657 0.657 0.657 0.657 0.657 0.675 0.75
0.50
0.75
0.15
0.30
0.45
0.6
0.75
0.30
0.45
0.6
0.75
0.3
0.45
0.6
0.75
0.45
0.6
0.75
0.45
0.6
0.75
0.6
0.75
0.6
0.75
0.75
0.75
Analysis of the Table 2 data shows that that the radius of the edge rounding increases with the increase in the height parameters of adjacent surfaces roughness (Fig. 7).
Fig. 7. Micro burr on the edge of a fitting after boring operation Ti-6Al-4V 8.4 µm (a), 13.9 µm (b) for the cases of parallel-to-orthogonal adjacent surfaces lay combinations [22].
Edge roughness and the minimum radius of edge rounding to a greater extent is determined by the adjacent surface with the worse roughness.
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6 Conclusion It is proven that the edge of the part is a complex-shaped surface, which characteristics significantly depend on the roughness of adjacent surfaces and lay direction of the surfaces that adjacent to the edge. Cases of the possible edge forming depending on the combination of surface lay orientation on intersecting adjacent surfaces are considered: symmetrically orthogonal, symmetrically parallel, parallel-to-orthogonal, orthogonalto-angular, symmetrically angular, asymmetrically angular. It has been found out that the best edge performances are realized in case of symmetrically parallel orientation of the adjacent surfaces lay, slightly worse performances – with a parallel-to-orthogonal adjacent surfaces lay orientation. The importancy of the edge shape and its influence for obtaining the minimum radius of the edge rounding is confirmed. Simple dependencies are given for determining the edge roughness and the minimum radius of the edge rounding, which allow to determine the roughness height parameters and the edge shape. It has been found out also that with an increase in the adjacent surfaces roughness height parameters the edge rounding radius increases. Thus, to achieve an edge with high quality, its microrelief should be taken into account while planning the part machining technology (manufacturing route). Experimental verification of the stated provisions was carried out on samples of full-size cross-section panels and large panels (Fig. 6). With the introduction of the technological process, the following results were achieved: the forming time of double curvature panels with form control using a 3D scanner amounts to 5–6 h; panel finishing time amounts to 1–2 h. During forming, deviations of 0.7…0.9 mm from a given surface form, and 0.3…0.5 mm after finishing the form are achieved.
References 1. ISO 13715-2017 Technical product documentation – Edges of undefined shape – Indication and dimensioning (2017) 2. ASME B46.1-2019 Surface Texture (Surface Roughness, Waviness, and Lay), 144 p. (2019) 3. Gupta, K., Gupta, M.K.: Developments in nonconventional machining for sustainable production: a state-of-the-art review. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 233(12), 4213–4232 (2019). https://doi.org/10.1177/0954406218811982 4. Nee, Y.C.: Andrew: Handbook of Manufacturing Engineering and Technology. Springer, London (2015) 5. Sikulskiy, V., Maiorova, K., Vorobiov, I., Boiko, M., Komisarov, O.: Implementation of reengineering technology to reduce the terms of the technical preparation of manufacturing of aviation technology assemblies. Eastern-Eur. J. Enterp. Technol. 3(1 (117)), 25–32 (2022). https:// doi.org/10.15587/1729-4061.2022.258550 6. ASME Y14.36-2018 Surface Texture Symbols (2018) 7. ISO 25178-2:2021 Geometrical product specifications (GPS) – Surface texture: Areal – Part 2: Terms, definitions and surface texture parameters (2021) 8. Stout, K., Blunt, L.: Development of methods for characterization of roughness in three dimensions. Butterworth-Heinemann Elsevier Ltd. (2000) 9. Maiorova, K., Vorobiov, I., Andreev, O., Lupkin, B., Sikulskiy, V.: Forming the geometric accuracy and roughness of holes when drilling aircraft structures made from polymeric composite materials. Eastern-Eur. J. Enterp. Technol. 2(116), 71–80 (2022)
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10. Niknam, S.A., Davoodi, B., Davim, J.P., Songmene, V.: Mechanical deburring and edgefinishing processes for aluminum parts—a review. The Int. J. Adv. Manuf. Technol. 95(1–4), 1101–1125 (2017). https://doi.org/10.1007/s00170-017-1288-8 11. Plankovskyy, S., Shypul, O., Tsegelnyk, Y., Brega, D., Tryfonov, O., Malashenko, V.: Basic principles for thermoplastic parts finishing with impulse thermal energy method. In: Kumar, K., Babu, B., Davim, J. (eds.) Handbook of Research on Advancements in the Processing, Characterization, and Application of Lightweight Materials, pp. 49–87. IGI Global (2022). https://doi.org/10.4018/978-1-7998-7864-3.ch003 12. Struckmann, J., Kieser, A.: Thermal deburring. ATL Anlagentechnik Luhden GmbH Publ. (2020) 13. Kohli, R., Mittal, K.: Preface. In: Developments in Surface Contamination and Cleaning, vol. 9, pp. xiii–xv. Elsevier (2017). https://doi.org/10.1016/B978-0-323-43157-6.00010-0 14. Sikulskiy, V., Maiorova, K., Vorobiov, I, Fomichev, P., Myronova, S.: Convergence technology of vehicle parts surface finishing. In: XIV International Scientific Conference, Katowice – Silesia, 29 June – 1 July 2022, Transport problems 2022, pp. 637–646. Wydawnictwo ´ askiej, Gliwice, Poland (2022). In press Politechniki Sl˛ 15. ISO 1101:2017 Geometrical product specifications (GPS) – Geometrical tolerancing – tolerances of form, orientation, location and run-out (2017) 16. Gillespie, L.K.: Deburring and Edge Finishing Handbook. Society of Manufacturing Engineers, Dearborn (1999) 17. Grigoriev, A.Y.: Scale dependence of shape of engineering surface irregularities. J. Friction Wear 36(5), 363–367 (2015) 18. Jane Wang, Q., Chung, Y.-W. (eds.): Encyclopedia of Tribology. Springer US, Boston, MA (2013). https://doi.org/10.1007/978-0-387-92897-5 19. Smith, G.T.: Surface texture: two-dimensional. In: Industrial Metrology. Springer, London (2002) 20. Denkena, B., Kästner, J., Göttsching, T.: 4 Microstructuring by means of cutting processes. In: Denkena, B., et al. (eds.) Microstructuring of Thermo-Mechanically Highly Stressed Surfaces. LNPE, pp. 28–57. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09692-6_4 21. Yang, S., Li, W.: Surface Quality and Finishing Technology. In: Surface Finishing Theory and New Technology, pp. 1–64. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3662-54133-3_1 22. Plankovskyy, S., Popov, V., Shypul, O., Tsegelnyk, Y., Tryfonov, O., Brega, D.: Advanced thermal energy method for finishing precision parts. In: Gupta, K., Pramanik, A. (eds.) Handbook of Advanced Machining and Finishing, pp. 527–575. Elsevier (2021)
Study of a Geometry Accuracy of the Bracket-Type Parts Using Reverse Engineering and Additive Manufacturing Technologies Kateryna Maiorova1(B) , Valeriy Sikulskyi1 , Iurii Vorobiov1 Oleksandra Kapinus2 , and Anton Knyr3
,
1 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected]
2 The Ukrainian-American Joint Venture “KODA”, 22, Bahaliia St., Kharkiv, Ukraine 3 Progresstech Ukraine, BC “Cubic-Center”, 3 Sholudenka St., Kyiv, Ukraine
Abstract. The subject of study is the accuracy of shape of bracket-type parts produced using reverse engineering and additive manufacturing technologies. The goal is to increase the geometry accuracy of the parts obtained by these technologies. Tasks: creating a “digital portrait” of a typical aircraft transmission bracket and converting it into a digital mock-up with specified geometry (dimensions and shapes); creating a 3D model for 3D printing using the digital mock-up and printing the aircraft transmission bracket at technological modes that provide specified geometry. The following results were obtained: a “digital portrait” of a typical aircraft transmission bracket, which consist of two halves, was created; 3D models, based on the “digital portraits” were generated and used as the transmission bracket digital mock-up with the specified geometry; AM technologies of fused-disposition modeling (FDM) were implemented on a 3D printer with two independent extruders, using the selected technological modes, for the aircraft transmission bracket 3D printing. Inspection of the 3D printed transmission bracket geometry was carried out by comparing the “digital portraits” and a digital mock-up, its results showed deviations in the range from −0.37 mm to +0.14 mm, which did not exceed the tolerance of ±0.5 mm. This asserted the adequacy of 3D printing modes for ensuring the specified geometry. The proposed combination of reverse engineering and AM technologies, on the example of the aircraft transmission bracket, allows the shortening the aerospace components production lead time with insuring their specified geometry. Keywords: Reverse Engineering · 3D Printing · Additive Manufacturing Technology · Digital Mock-up · Aircraft Transmission Bracket · Geometry Accuracy
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 146–158, 2023. https://doi.org/10.1007/978-3-031-36201-9_13
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1 Introduction Aerospace manufacturing is characterized by a large number of parts with geometric accuracy within ±0.1…0.3 mm. Such aircraft indicators as reliability and efficiency, on a significant extent depend on the ensured geometric accuracy of its assembly components and the aircraft as a whole. That is why, the coordinate measuring machines (CMMs) are widely used in aerospace manufacturing. They are indispensable when carrying out the large number of machining operations, assembly, measurements, as well as reverse engineering (re-engineering) operations and technical inspection of parts and tooling elements, namely stretching punches, surface standards, etc. [1]. The technology of reverse engineering based on CMMs is widely used by now in the repair and restoration of the aircraft parts and the product as a whole. This especially important for cases of modernization of products that were developed before the digital era. For such products, in the case of their modernization, it is necessary to create tooling, while their digitization becomes relevant [2, 3]. The stability of the aircraft assembly components geometric accuracy in the aviation industry is most often ensured by wide using the computer numerical control CNC machines [4]. But the main problem manufacturers faced with is the high cost of the material itself and its processing at the CNC machines. The production of workpieces with a specified geometry and allowances for their further processing on CNC machines can be done in two ways – either by traditional technology, like milling of a bulk monolithic metal, or by additive manufacturing (AM) technologies, namely by 3D printing. In both approaches, a significant factor in economic efficiency providing is the reduction of machining allowances, time and cost of manufacturing. In some cases, when using AM, these indicators are significantly lower than when milling on CNC [5, 6]. This is because instead of removing the material, 3D printing technology creates parts adding material layer by layer, so facilitates both manufacturing the aircraft assembly components of complex geometric shapes and effective waste minimization [7–9]. Among the other advantages of AM using is the possibility of parts manufacturing without the further using the additional milling operations for the contour allowances removing, but in such cases, it depends on the demands to geometric accuracy, physical and chemical properties and reliability of the aircraft assembly components. In some cases, it is extremely important to produce a workpiece by moulding of it with minimal processing allowances immediately from the surface mock-up that produced on a 3D printer, in turn. Taking into account the widespread use of reverse engineering and AM technologies, the studies that aimed at the implementation of reverse engineering in the process of digital mock-ups creating and inspection of the aircraft assembly components, produced by AM, are relevant. The accuracy inspection within the accepted technology is carried out by comparing with the digital mock-up, which is the primary source of co-ordination, the part produced on a 3D printer.
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2 Literature Review and State-of-the-Art The reverse engineering and additive manufacturing technologies implementation in the aerospace manufacturing is complex of various work that require radical changes in technological processes for minimization the use of production resources, for reducing the production cycle duration and expenses, and for improving the aircraft quality [10]. The paper [11] shows the prospects of using the reverse engineering and 3D printing, and gives a forecast for the possibility of lean production of the any aircraft assembly components in the near future. Reverse engineering of the aerospace components is mostly used at the stages of inspection and technical measurements as an element of automated or robotic assembly systems of the aircraft [12, 13]. Reverse engineering shows adequate outcomes if there is exact information on the geometry, structure, design, and functioning of either the part to be studied or the aerospace components, i.e., if a Digital Mock-Up (DMU) is used [14]. It should be noted that as of today, the aircraft that were manufactured using loft-template approaches of their production tooling are still in operation. The implementation of reverse engineering for maintenance and repair of the such aircraft helps to solve the issue of the lost documentation reproduction and its digitization, as well as to accomplish the technological transition from templatebased technologies to information technology systems of manufacturing with using the CNC machines and CAD/CAM software. The proof of this is the work [15], which presents a complex scheme of co-ordination the aerospace technology produced using the template-based approach with a DMU, obtained due to reverse engineering, and gives calculations of the labor intensity when implementing the reverse engineering into the existing established manufacturing. Despite the practical significance of such results, it should be pointed out that the computer-aided design (CAD) and methods of finite elements (FEM) inclusion in the technological chain of the reverse engineering will significantly increase the efficiency of prototyping and creating the new aerospace technology assembly components. In works [16, 17], it is proposed as last stage of the reverse engineering the manufacturing of parts using modern AM technologies of 3D printing. The authors affirm – but don’t confirm it with calculations – that such approach allows not only to reduce the allowances for further machining, but also save the time and expenses for these parts manufacturing. An alternative solution is proposed in the work [18], which suggests the finite element modeling methods using for preliminary correction of the 3D printing technology, by taking into account the material properties and temperature gradients while printing, to ensure the exact specified geometry of the part. However, the approbation of the given approach is not shown in the work itself. Additive manufacturing of assembly components in the modern aerospace industry currently exists in the form of 3D printing both certified plastics and metals [19–21]. The analysis of the publications allows to give a forecast that the combination of reverse engineering and AM technologies allows to shorten the lead time of the aerospace assembly components manufacturing with the ensuring of specified geometry and properties. However, today there is neither entire systematic knowledge nor experiences in the such production of aerospace components. For example, it is a “painful” issue when the parts are worn out and need to be replaced, but their manufacturing using traditional template-based approach causes certain difficulties due to the lack of appropriate documentation and technological tooling. Thus, in each case, solution should be individual
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and requires it full approbation. Aforementioned facts give grounds to state that it is expedient to carry out research in this field for the typical aerospace part or unit.
3 Purpose and Objectives of the Study The purpose of this work is to study the geometric accuracy of the aircraft transmission bracket using reverse engineering and additive technologies. This will make it possible to increase the reliability and efficiency of typical parts of the aircraft as a whole. To achieve this purpose, the following tasks were solved: • a “digital portrait”, that is point cloud of a typical part – the real part of the aircraft transmission – was produced and converted into its digital mock-up (3D model) with a specified geometry and shape; • on the ground of the part DMU it was further created a 3D model for the aircraft transmission bracket 3D printing and the part printing was accomplished using technological modes that ensure specified geometry and shape; • an inspection of the 3D printed aircraft transmission bracket was carried out by comparing the 3D printed part point cloud (that is, its “digital portrait”) with it digital mock-up that created by results of the real part measurements.
4 Experimental Procedures As a typical part for the study was chosen transmission bracket assembly component that consist of two parts – top and bottom halves, which in assembly form inner cylindrical surface with high demands to it accuracy (Fig. 1).
Fig. 1. Aircraft transmission bracket (a) and component parts of the bracket (b).
This choice is made because these typical parts have high demands to manufacturing accuracy, they are critical parts (the failure of which could have a catastrophic effect upon
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the aircraft) with demands of minimum weight providing, as well as high the rigidity of their shape, even when they consist of several components. The scheme of the experimental study sequence is presented in Fig. 2.
Fig. 2. Scheme of the experimental study sequence of the aircraft transmission bracket.
Scanning (three-dimensional measurement of the surface) carried out with determining either the points coordinates or a system of surface points in the accepted coordinate system, followed by mathematical processing to determine linear and angular dimensions, deviations of form, orientation, location. For parts measurements was chosen a three-dimensional portable scanner Space Spider of Artec (USA). Its high accuracy of measurement, ability of high resolution capturing of minimum size objects to be scanned, portability and the ability to work with the main 3D models formats became the decisive factors in this choice of equipment for the aircraft transmission bracket scanning. The selected Artec Space Spider scanner works using the method of structured white light and provides high accuracy (up to 0.05 mm) of the surface of the scanned object – a point cloud (its “digital portrait”) [22]. The point cloud data usually does not have any topological information and is therefore converted to mesh files with triangular STL elements. The next stage is an exporting the “*.stl” file and creating a digital mock-up by
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3D modeling in any available way: a set of surfaces of heterogeneous rational NURBS splines or a solid model in CAD systems (SolidWorks, CATIA, Geomagic Design X, etc., USA) [23–25]. After creating the DMU it is possible to correct the geometry and shapes of the aerospace component under studying. Following the creating of the aircraft transmission bracket digital mock-up, it is possible to produce it using either conventional technology – milling on CNC machine, or using the technology of 3D printing. The second option requires additionally: the of printing modes selection, an inspection – by comparing of the printed bracket point of cloud (its “digital portrait”) to its digital mock-up, and a decision on whether it’s possible to start it serial production or there is a need in correcting the technological modes of printing. 4.1 Creating a “Digital Portrait” (a Point Cloud) of the Aircraft Transmission Bracket and Its Digital Mock-Up (3D Model) of the Specified Geometry In Fig. 3 shown the obtained “digital portraits” of the aircraft transmission bracket parts before and after 3D texturing.
Fig. 3. “Digital portraits” of the aircraft transmission bracket components without texturing for bottom part (a) and top part (b) and after 3D texturing for bottom part (c) and top part (d).
To create a 3D model from the “*.stl” file, it was chosen the Geomagic Design X software. After a series of transformations (placing all the surfaces, trimming them and
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creating a closed “whole” surface), it was created, using the sketches, a three-dimensional model, which was used as the digital mock-up (Fig. 4).
Fig. 4. Image of a digital mock-up of the aircraft transmission bracket components in assembly.
This DMU was hereafter considered as the primary source of information for further measurements of the bracket shape accuracy. 4.2 The Aircraft Transmission Bracket 3D Printing at Selected Process Modes To produce the aircraft transmission bracket, the one of layer-manufacturing technologies, Fused Deposition Modeling (FDM), was chosen, and a 3D printer Raise3D E2 (China) with two independent extruders [23]. After preliminary technical preparation of the printer (checking and cleaning the table and nozzle, balancing the table, downloading the plastic filament, etc.), the “*.stl” file, with the generated 3D model of the digital mock-up, was prepared. To implement the printing process on the Raise3D E2 printer, the “*.stl” model was converted into G-code with the involvement of the slicer program. Such a solution helps to optimize and automatize the programming of printing process. Each vector of movement path in the G-code is described in the Cartesian coordinate system. The use of the G-code compiled by the slicers allows additional editing: when precise editing the code, when special settings if necessary (changes of temperature, the extruder heating during printing, etc.).
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The selected printer set up for the necessary technological modes are shown in Table 1. Table 1. The technological modes and the Raise3D E2 3D printer settings selected for the aircraft transmission bracket printing. №
Parameter
Parameter definition
The value of parameter
1
Quality
Sets the layer thickness
High quality: 0.10 … 0.05 mm
2
Walls
Specifies the number of walls (contours) of the model
High strength: 6 walls
3
Top/Bottom
Number of bottom and top layers to be completely filled
Average strength: 6 layers
4
Infill
Parameter that shows the percentage of the model filling with plastic
High strength: 55%
5
Material
Parameter for the extruder temperature and the table temperature setting
The values of these parameters are individual for each plastic
6
Speed
Parameter that affects the Quality printing: printer speed during the printing 25–35 mm/s process
7
Travel
Retraction is the process of plastic removing from an extruder when the tool isn’t printing. Improves print quality
Standard: Enabled
8
Cooling
Parameter that describes a cooling the extruded plastic
Standard: 80–100%
9
Support
Parameter for generating the Enabled special structures – which serve as supports for printing part elements “hanging” in the air
10 Build Plate Adhesion Parameter for creating the structures for better part adhesion to the table
Raft
In Fig. 5 it is the printed aircraft transmission bracket shown. The next step in this printed transmission bracket production was its machining for removing the allowance that was formed on the outer surface of the top half of the transmission bracket, and removing the additional supportive structure on the bottom half of the bracket assembly.
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Fig. 5. The aircraft transmission bracket manufactured by additive manufacturing technologies.
It should be noted that the removed allowance does not included into the allowance for surfaces that require the highest both quality and geometric accuracy. Besides, in this study the bracket structural strength performances were not taken into account. The FDM printed aircraft transmission bracket was used for measurements regarding the accuracy of dimensions and shape; it also may come in handy for correcting part dimensions when creating a program for CNC machines. 4.3 Realization of the 3D Printed Aircraft Transmission Bracket Inspection by Comparing Its “Digital Portrait” and Digital Mock-Up The “GOM Inspect” program was used to check and analyze the 3D printed bracket geometry accuracy. The printed bracket was scanned, in turn, by a three-dimensional portable scanner Artec Space Spider (USA), with subsequent conversion of the scanned data into a “digital portrait” of the printed bracket in STL format. These “digital portraits” were loaded into the GOM Inspect software along with the digital mock-up of bracket, and were compared with each other by means of the “best-fit” function. The results of inspection in a form of surfaces deviations, at the control points, of the transmission bracket top half, which was produced by additive manufacturing technologies, with the surfaces of the digital mock-up are shown in Fig. 6. The results of inspection in a form of surfaces deviations, at the control points, of the transmission bracket bottom half, which was produced by additive manufacturing technologies, with the surfaces of the digital mock-up are shown in Fig. 7.
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Fig. 6. The deviations of the aircraft transmission bracket top half, which produced using additive manufacturing technologies, by results of the inspection.
The analysis of the obtained values of surfaces deviations of aircraft the transmission bracket parts shows that the maximum deviations on the outer surface of the transmission bracket top half are −0.17 and +0.07 mm, and on the bottom, half are −0.39 and + 0.14 mm. It should be noted that deviations of up to −0.37 mm were observed on the working surfaces (plane of parts junction, inner cylindrical surface). These deviations are explained as follows: incorrect detaching of the supportive structure from the inner cylindrical surface of the bottom part of the transmission bracket assembly, as well as insufficient integrity of the upper layer and excess of the extruded plastic that was due to the accepted printing modes, namely “Top/Bottom” and “Infill”. It can be recommended to reduce the model filling parameter of the plastic “Infill” to 45% and increase the number of layers “Top/Bottom” to 8. The general analysis of the obtained deviations showed that the most of the obtained values are negative. This indicates the shrinkage of the material, which should be regulated by the melting temperature and the speed of the plastic feed into the extruder. But, anyway, the obtained indicators are within the tolerance field, which for the aircraft transmission bracket is ±0.5 mm. Therefore, the selected 3D printing modes can be considered as satisfactory ones, because they provide the specified geometry of the studied components of the aircraft transmission bracket.
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Fig. 7. The deviations of the aircraft transmission bracket bottom half, which produced using additive manufacturing technologies, by results of the inspection.
5 Conclusion It was implemented the reverse engineering technology for a typical aircraft transmission bracket that consists of two parts. Based on the “digital portraits” (the point clouds) in the STL format, in the Geomagic Design X software the 3D models were built, which used as the basis for the digital mock-up of the aircraft transmission bracket with it specified geometry. It was implemented the additive manufacturing technology, namely FDM, as 3D printing of the aircraft transmission bracket on the Raise3D E2 printer with two independent extruders, in accordance to the selected technological modes. Inspection of the 3D-printed aircraft transmission bracket was carried out by comparing its component parts’ “digital portraits” and a digital mock-up. The inspection results showed that the deviations ranged from −0.37 mm to +0.14 mm, which do not exceed the limit deviations (tolerance) of ±0.5 mm. It gives reasons to assert about the adequacy of the selected technological modes of 3D printing for ensuring the specified geometry when creating programs for CNC machines.
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Design of High Fatigue Life Joints of Fuselage Structures Considering Fracture Mechanics Dmytro Zhyriakov1(B)
, Oleksandr Grebenikov1 and Dmytro Konyshev2
, Andrig Humennyi1
,
1 Department of Airplane and Helicopter Design, Faculty of Aircraft Building, National
Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine {d.zhyriakov,o.grebenikov,a.gumennyy}@khai.edu 2 State Company “Antonov”, Tupoleva Street, Kiev 04128, Ukraine
Abstract. The design of safe and competitive jet transport structures involves a host of significant considerations. There are manufacturing/assembly technology, the strength of joints, safety, profitability etc. This review is focused on one of the design philosophies of designing fuselage joint based on fracture mechanics. This concept is widely known and still being researched for new areas of application (like a composite structure). Before starting to design joints of the fuselage, the designer must review existing structures, existing approaches to design a joint. This paper also has a review of some joints of the tail section of the fuselage with damage tolerance aspects. Keywords: butt joint · fail-safe · damage tolerance (DT) · widespread fatigue damage (WFD) · limit of validity (LOV)
1 Introduction Metal fatigue and other types of damage in airplane structure have proved to be unavoidable and must be considered when an airplane is designed or repaired [1]. Aircraft of Boeing and other corporations can safely operate with a certain amount of un-detected damage. The damage can be found during scheduled maintenance. Before 1978, structural strength was maintained by designing airplanes according to fail-safe certification requirements [2]. Fail-safe design required that the airplane be able to operate safety at up to limit load condition after a single structural element failed. Therefore, the remaining adjacent structural members must be able to carry the extra load from the failed part (see Fig. 1). Experience has shown that fail-safe design produces a structure with a credible safety record. Numerous in-service incidents have demonstrated the ability of airplanes to fly and land safely with significant structural damage. Continued attention to fail-safe structural features is critical for maintaining and continuously improving the level of safety of airplanes. There are many examples of fail-safe design in traditional built-up structures. The fail-safe design provides multiple load paths and damage containment. Fail-safe design by itself does not always ensure that structural failures will be obvious © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 159–173, 2023. https://doi.org/10.1007/978-3-031-36201-9_14
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and safe. On May 14, 1977, a Boeing 707–300 fright operated by Dan-Air Service crashed while on approach to Lusaka International Airport [3]. The accident was the result of a horizontal stabilizer failure caused by fatigue [4]. B 707-300 was certificated on the fail-safe design basis that was not required specific inspection of the fatigue critical details. There was routine general inspection of stabilizer, but they did not detect the fatigue cracking. This accident forced the FAA to reconsider the design principle. Beginning in 1978, the aviation regularity required that damage tolerance be evaluated on all-new airplanes [2]. In a damage tolerance evaluation, it is assumed that multiple cracks may be present. Crack growth is analyzed on the cracked structure to predict how many flights cycles the structure can safely endure before the crack grows to a critical length. With the development of reliable fracture mechanics analysis techniques, it became possible to relate detectable damage, damage growth, and critical damage size to establish inspection methods and frequency required to maintain safe operation [5]. A structural inspection program is then developed to ensure that enough inspection is scheduled to find cracks in the structure before they grow to critical length.
Fig. 1. Fail-safe and Damage tolerance comparison.
The present article reviews modern principles of design fuselage with considering such design principles fracture mechanics.
2 Modern Principles of Designing Fuselage Joints Unfortunately, the principles of safe design for aircraft are developed using examples related to real disasters [6]. As a result, the engineers realized that it is impossible to operate the aircraft without damage (they will always be present in one form or another). Therefore, the last and most reliable design method is damage tolerance. It should be noted that the design principle like fail-safe is used together with damage tolerance, and this leads to the safest structure of airplanes design. 2.1 Residual Strength and Crack Growth Up-front use of these techniques in recent models of airplanes (fail-safe and damage tolerance) significantly influenced the design, including the structural arrangement, materials and stress levels, accessibility, inspectability, and repairability [7].
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The damage tolerance consists of three main elements: residual strength analysis, crack growth analysis, and inspection program. From Fig. 2, the damage tolerance conceptualization diagrams the point of starting the inspection program is determined by fatigue life and called as the inspection threshold.
Fig. 2. Damage tolerance conceptualization.
The basis of damage tolerance concept is fracture mechanics: crack extension will occur when the stresses and strains at the crack tip reach a critical value [8]. This means that fracture must be expected to occur when stress intensity factor reaches a critical value. The critical value of stress intensity factor is a parameter that characterizes a measure for the fracture resistance of a material.
(1)
where: K – stress intensity factor; σ – gross area stress; L – crack length;
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n – number of tip of crack; Y – geometry correction factor (takes into account of detail configuration [9]); C – load redistribution factor (considers of load redistribution in complex structure due to large cracks in one or more members). As has already been mentioned, failure occurs when the stress intensity factor reaches to critical stress intensity factor of the noted material. From the above, one can say that residual strength is based on this statement. So, residual strength is the maximum allowable damage that a structure can sustain at a critical fail-safe level is the key to the level of damage growth and inspections needed to ensure damage detection. Crack-length-cycles data is transformed to crack growth rate data by calculating the local slope of the crack length cycle at some length of the crack. The data is generally linear on a semi-log plot. However, there are boundary conditions to the stress intensity factor that will affect the character of the rate of crack growth. First, there is some lower limit of the stress intensity factor at which there is only infinitesimal crack growth under either a repeated load or a static load. On the other hand, there is some maximum stress intensity factor (increased fracture) dependent upon plane stress or plane strain conditions, or any combination of them. Between these limits of crack growth, there is a typical range (steady growth of crack) that can be described in a somewhat linear form for all practical purposes. The rate of fatigue crack propagation per cycle is determined by the stress intensity factor. For the determination of flights cycles, Paris’s equation shows dependence of crack length on fights cycles [10]:
(2)
where: K = Kmax − Kmin – stress intensity factor over a range; c and m – constants of noted material determined experimentally depends on stress ratio. With a known crack growth rate and an initial crack size, it is possible, to sum up crack growth from one length to another under a given loading spectrum. As in fatigue damage, it is possible to add the cyclic growth at several different stress levels to predict a time for developing a given crack length. As in fatigue, the computation of the crack growth for a random load will be dependent on the specific character of the loading, i.e., and the sequence of the applied external loads [11]. When the maximum allowable damage (residual strength) and damage growth (crack growth) are defined, the inspection program must be assigned.
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2.2 Inspection Program Inspection program is a sequence of inspections of a fleet of airplanes with methods and intervals selected to achieve timely detection of damage before it reaches the damage limit. Structural inspection programs are typical developed by using rating systems for each of the three major forms of damage: fatigue, environmental and accidental damages. Fatigue damage is characterized by cumulative progression relating to airplane usage measure in flights. For environmental and accidental damages, damage detection must be created (for example stress corrosion. This is a very dangerous phenomenon, mainly due to residual stresses in structure, that very difficult to detect). This damage detection is based on statistical data and is intellectually protected by each airplane company. Damage detection is a function of fleet size, the number of cracks and the number and type of inspections. Three independent probabilities determine the certainty of damage detection: – Probability of inspecting an airplane with damage; – Probability of inspection a detail containing a crack; – Probability of detection a crack in the detail. All these parameters are determined by statistics data provided by operators as by the airplane manufacturer. 2.3 Widespread Fatigue Damage One more item must be considered during design of airplane structure is Widespread fatigue damage (WFD). The problem of multi-site damage and multiple interacting cracks is one experienced by many aircraft manufacturers and operators [12, 13]. Independent damage in similar stresses has long been recognized as a potential continuing airworthiness problem [11]. Fuselage structure is typically more susceptible to widespread damage fatigue [14] because of numerous similar details subjected to pressure cycle loads with moderate flight-by-flight variations (see Fig. 3).
Fig. 3. Widespread fatigue damage of lap splice [11].
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On April 28, 1988, a Boeing 737–200, operated by Aloha Airline, has experienced an explosive decompression during climb out at cruise altitude whereby the upper portion of the forward fuselage separated from the airplane (see Fig. 4).
Fig. 4. Fuselage failure due to widespread fatigue damage of lap splice [4].
Future investigation showed that the main causes were [4, 15, 16]: 1. WFD initiated from the hole edges of the rivet joint rows, due to disbonding of the cold bond adhesive with subsequent corrosion and fatigue cracking in multiple adjacent rivet location; 2. Aloha Airlines has not properly followed related service bulletins (SB) issued by Boeing and Airworthiness Directive (AD) issued by FAA, to conduct required maintenances and inspections; 3. Flight cycles for the airplane has exceeded its design service objective (DSO), but there was not any service termination point for airplanes at that time. A combination of the design deficiencies and improper maintenance combined to cause the cracking. Boeing changed design of lap joint on airplane starting from line number 292 to remove unwanted effects and increase the fatigue life of the fuselage lap joint. The lap splice design was improved with the incorporation of a hot bonded doubler, which was expected to substantially delay the onset of WFD (see Fig. 5). Using final element modeling, investigate the effect of new design into stress level in upper skin at critical location as shown below.
Fig. 5. Design of lap splice before and after the accident (old and new design) [4].
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Fig. 6. Finite element model of a) old and b) new design
Using ANSYS Workbench, models were created (see Fig. 6). Actual operation internal pressure was calculated using formula below: σref . =
7.5 · 73.82 p·R = = 15, 4 ksi, t 0.036
(3)
where: p = 7.5 psi– operation internal pressure Boeing 737 classic; R = 73.82 inches – radius of fuselage in the analyzed area; t = 0.036 inches – skin thickness; σref . = 15,4 ksi – reference stress applied to upper skin. In the old design, the following contact were used in analysis [17]: between upper and lower skin – Frictionless; between lower skin and stringer – Frictionless; between rivets and other parts – Rough. In the new design, the following contact in analysis were used [17]: upper skin and doubler –Bonded; lower skin and stringer – Frictionless; lower skin and doubler – Frictionless; between rivets and other parts – Rough. The symmetry boundary condition was used.
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For investigation of critical location (critical fastener row), upper skin was chosen due to countersink hole. Critical fastener rows are rows (first fastener row in the upper sheet and the third fastener row in the lower sheet, see Fig. 7) where the fastener holes subject to the highest bypass load and the highest bearing load among all fastener holes [18]. As example of critical location see Fig. 8, normal stress in Z-direction of Old design.
Fig. 7. Loading components in a mechanically fastened join [19].
Fig. 8. Normal stress (orientation Z-axis) of old design, critical location
To consider the singularity of highly loaded units (countersink edge of skin), the stress was evaluated in the middle of skin sheet, see Fig. 9.
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Fig. 9. Normal stress in upper skin in the middle of sheet, old and new design.
As we can see from Fig. 9, the bonded doubler (New design of lap splice) reduced stress at a critical location more than twice. This led to increasing the fatigue life of lap splice. This accident showed when the operational bounds have not been established, WFD can be a significant risk to threaten the safety of an airplane that is only based on DT analysis design for a single crack [20]. In March 1998, the FAA revised design concept and added new requirements to inspect or modify structure susceptible to WFD and also added requirement for all new airplanes to undergo a minimum two-lifetime full scale fatigue test to demonstrate freedom from WFD [15]. WFD in a structure is characterized by the presence of multiple structural details with cracks of sufficient size and diversity that the structure will no longer meet its damage tolerance requirements (e.g., maintaining the required residual strength after partial failure) [21]. The concern for WFD thus exists in large regions with similar structural details and the same significantly high stress levels, where coalescence of multiple damage origins may be catastrophic, and there is a lack of reliable and economic damage detection techniques available to control such unsafe conditions before they develop. Therefore, investigation is now required to determine whether special inspections or actions are required in addition to existing maintenance to ensure safety. One way to protect against WFD is limitation the operation time of aircraft. This limitation called Limit of Validity (LOV). Each airplane has limitation of time (number of flight cycles) of operation, after which operation of airplane is forbidden. WFD is research topic in demand nowadays due to unpredictable destruction and needs attention from airplane manufacturing companies and designers.
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3 Fuselage Joints of Tail Section So, all designed structures of airplanes (especially highly loaded areas of the fuselage – splice joints) must be satisfied to fail safe and damage tolerance design concepts. The sequence in which parts are assembled affects their dimensional variation. Therefore, engineers must understand and participate in determining the build sequence and create a drawing tree that reflects the sequence. Unless the designer knows the order in which the detail parts are assembled and the tooling methods used to assemble them, he or she cannot choose appropriate datums for machining and drilling those parts and will not be able to predict the dimensional variation of the finished assembly. To reduce the assembly time of the fuselage (especially the fuselage of a larger aircraft), it is advisable to divide it into several compartments along the fuselage (Fig. 10). The assembly of each compartment is carried out in a separate slipway. The docking of the compartments is carried out in the stock of the general assembly of the fuselage.
Fig. 10. Assembly technology division of the fuselage of a passenger aircraft.
This method of assembly requires the creation of a rational design of technological connections. Technological connections of the fuselages are made along the frames. All elements of the longitudinal set are cut along the fuselage. The transfer of axial forces acting in the stringers is carried out with the help of special butt fittings. The cabin pressure develops a radial growth in the skin and the radial growth is resisted by the frames and stringers in the longitudinal and circumferential splice joints. The curved panels are subjected to biaxial tension due to cabin pressure. The airplane fuselage may suffer from the damage due to longitudinal cracks and circumferential cracks under hoop stress and bending of the fuselage. The repeated tension loading is a critical fatigue condition, therefore, it must be a fail-safe design where an individual part failure can be sustained or retained until it is found and repaired before it results in
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a catastrophic failure. The propagation of longitudinal and circumferential cracks and their arrest feature is one of the main aspects of the damage tolerance design of the pressurized fuselage cabin. A fuselage of a modern aircraft is a stiffened shell commonly referred to as semimonocoque construction. This type of construction, contains longitudinal elements (longerons and stringers) and transverse elements (frames and bulkheads) which are intended to carry the major portion of fuselage bending moment, maintain shape of a fuselage cross section and stabilize fuselage skin. The fuselage skin withstands the shear caused by applied external transverse and torsional forces, contributes to carrying of the bending moment and undergoes the action of excessive cabin pressure. The most efficient structure is the one with the least number of joints or splices; therefore, the skin panels are as large as possible, limited only by available milling sizes. But all parts of the fuselage structure have their own technology and manufacturing limitation. One of them is a butt splice, which connects two sections of the fuselage together [22]. In Fig. 11 shown elements of typical butt splice of fuselage.
Fig. 11. Fuselage butt splice of skin and stringer [22].
Stringer splice locations are established by another set of rules. Since the skin and stringers are working together, they should both be spliced at the same location. This maintains the relative stiffness of the skin/stringer combination, which is desirable from a fatigue standpoint. These joints are analyzed for the design conditions like pressure difference, the ultimate flight loads conditions and for the ground load conditions with no pressure effects. The butt splice of the fuselage is riveted or bolted. Typically butt splice has two fastener rows common to frame chord and additional fastener row common to skin and splice plate only. It contributes to better fatigue at critical row. Due to location of the frame at the butt joint, this joint can be considered as stabilized single shear (the joint that has only one shear surface per one fastener) joint that is better for fatigue as well.
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Each airplane has pressure bulkheads (at FWD and AFT sides of fuselage) to take a pressure load. The cylindrical shell of a pressure-cabin is closed at the rear by dome in preference to a flat bulkhead (as in FWD side of fuselage). From a structural point of view, a hemispherical shell provides an ideal rear dome because the membrane stresses for a given amount of material are the least. Based on this argument, the best shape for the rear dome of a pressure cabin is the hemisphere. Assuming this to be correct, the problem of choosing the most efficient method of joining the hemispherical dome to the forward cabin shell and to the rear fuselage is difficult. Like a lap splice, butt splice can be subject of WFD that required specific design and analysis for this type of fuselage joint, that must be considered during design [23, 24]. In considering the design of the rear dome of pressure cabin, the objective is to achieve a minimum weight for the dome itself and a minimum amount of interference (combination of two separate stresses) stresses at the junction of the dome and fuselage wall. The large transport and passenger airplane’s fuselage pressure dome is located at the rear section of the fuselage compartment and usually, the top of the dome rim is canted slightly forward in the vertical plane to accommodate the taper of the aft bottom fuselage. The typical joint of cylindrical and taper section of the fuselage [22] is shown in Fig. 12.
Fig. 12. Rear pressure bulkhead joint [22].
The outer support ring of the pressure dome attaches to end splice the fuselage skin. The dome is designed to react an 18.8 psi burst pressure. The pressure bulkhead joints to fuselage with tension type fittings splicing the stringer and fuselage skin butt type splice. Composite materials have recently been widely used to achieve high weight efficiency of an aircraft and there is more and more investigation about composite material in aircraft structure [25, 26]. Airplane parts made from composite material must be analyzed for fracture mechanics as metal parts [27]. Composite rear pressure bulkhead
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has already used in such airplane as Boeing 787 or Airbus A320, and there are many investigations about design of composite pressure bulkhead [28, 29].
4 Discussion When designing joints for aeronautical structures, special attention should be paid to the safety and reliability of the joints. In the article, the joints of the fuselage and the pressurized zone, which are subjected to some specific loading, were discussed. But the description of the design methods can also be applied not only to the fuselage structure and to the other main components of the airframe structure, in particular the wing, and empennage. It is necessary to introduce constructive methods of increasing reliability in the design multi-way load transfer (fail-safe). Also, more attention should be paid to such a design principle as damage tolerance. To achieve a safe design according to the principle of damage tolerance, it is necessary that the design has the following important parameters: inspectability, repairability, accessibility, maintainability, a good stress intensity factor (material), and has good resistance for fracture. Fail-safe and damage-tolerance approaches, the last one emerged as the main design philosophy for aerostructures, allowing weight savings and at the same time increasing reliability and structural integrity in the presence of damages which may have occurred during the manufacturing process or during service [30]. WFD was mentioned in the article as well. This phenomenon requires careful further study, especially in such highly loaded joints in which there are large critical areas that can turn into one whole crack. It can lead to disaster. Critical areas are subject to WFD like first fastener row of butt splice, lap splice etc. During designing of airplanes parts with principles described above and based on fracture mechanics, designer faces with the following problems: 4. Determination of the minimum detectable crack length; 5. Prediction of the residual strength of the structure in a cracked condition and determination if the critical crack length at the required fail-safe load; 6. Establishment of expected load history; 7. Determination of the crack propagation curve form the minimum detectable crack size to the critical crack length (see Fig. 2); 8. Reliable inspection, considering the accessibility of the parts or structure under consideration. Fracture safe design has long been a matter of qualitative engineering judgement, the time has now arrived that reasonable quantitative predictions can be made, especially with respect to residual strength [31].
References 1. Arep’ev, A., Gromov, M., Shapkin, V.: Voprosy jekspluatacionnoj zhivuchesti aviakonstrukcij; Vozdushnyj transport. Russian Federation, Moscow (2002) 2. Ferrarese, J.A.: Damage-Tolerance and Fatigue Evaluation of Structure. Advisory Circular (1978)
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3. Editor Dan Air Services – Boeing – B707-332 (G-BEBP). Aviation Accident Database (1977). https://www.aviation-accidents.net/dan-air-services-boeing-b707-332-g-bebp/ 4. Tiffany, C.F., Gallagher, J.P., Babish, C.A.: Threats to aircraft structural safety, including a compendium of selected structural accidents/incidents. 167 5. Toor, P.M.: A review of some damage tolerance design approaches for aircraft structures. Eng. Fract. Mech. 5, 837–880 (1973). https://doi.org/10.1016/0013-7944(73)90054-4 6. Safarian, D.P.: Fatigue and Damage Tolerance Requirements of Civil Aviation. Lecture at the University of Washington (2014) 7. Goranson, U.G.: Fatigue issues in aircraft maintenance and repairs. Int. J. Fatigue 20, 413–431 (1998). https://doi.org/10.1016/S0142-1123(97)00029-7 8. Benedictus, R., Alderliesten, R.C.: International conference on damage tolerance of aircraft structures. Int. J. Fatigue 31(6), 997–997 (2009). https://doi.org/10.1016/j.ijfatigue.2009. 02.004 9. Chen, D.L., Weiss, B., Stickler, R.: A new geometric correction factor for a finite width center cracked plate loaded by two pairs of splitting forces. Int. J. Fract. 61, R43–R50 (1993) 10. Paris, P., Erdogan, F.: A critical analysis of crack propagation laws. J. Basic Eng. 85, 528–533 (1963). https://doi.org/10.1115/1.3656900 11. Understanding the New Widespread Fatigue Damage Rule Available online: https://www.boe ing.com/commercial/aeromagazine/articles/2012_q4/2/. Accessed on 28 August 2022 12. Jones, R., Molent, L., Pitt, S.: Understanding crack growth in fuselage lap joints. Theoret. Appl. Fract. Mech. 49, 38–50 (2008). https://doi.org/10.1016/j.tafmec.2007.10.002 13. Bakuckas, J.: Full-Scale Testing and Analysis of Fuselage Structure Containing Multiple Cracks. 452 14. Piascik, R.S., Langley, N., Willard, S.A.: The growth of multi-site fatigue damage in fuselage lap joints. 11 15. Wojnar, R.T.: Damage Tolerance and Fatigue Evaluation of Structure. Advisory Circular (1998) 16. Jones, R., Molent, L., Pitt, S.: Study of multi-site damage of fuselage lap joints. Theoret. Appl. Fract. Mech. 32, 81–100 (1999). https://doi.org/10.1016/S0167-8442(99)00029-4 17. özgün Ansys Contact Types and Explanations. Mechead.com (2021). https://www.mechead. com/contact-types-and-behaviours-in-ansys/ 18. Theoretical Analysis of Fatigue Failure in Mechanically Fastened Fibre Metal Laminate Joints Containing Multiple Cracks | Elsevier Enhanced Reader Available online: https:// reader.elsevier.com/reader/sd/pii/S135063071830027X?token=12D1C961FECE328889909 CBCD8AFA4DFC0F8A3423B22354ADD05A25FECFD96201A4E4C9843DC3827FD99 9716E1447724&originRegion=eu-west-1&originCreation=20220828180247. Accessed on 28 Aug 2022 19. Bolted Joints in Glass Reinforced Aluminium (Glare) and Other Hybrid Fibre Metal Laminates (FML) | Elsevier Enhanced Reader Available online: https://reader.elsevier.com/rea der/sd/pii/B9781845699901500028?token=A26C5ED23619A64C577542B1F3673D7A9 05854C1CD6FEE984163D154AA93E6A9F0E3EB4B22C6C6FEA376D85F2EAC945D& originRegion=eu-west-1&originCreation=20220828181825. Accessed on 28 Aug 2022 20. Durability and Damage Tolerance Analysis Methods for Lightweight Aircraft Structures: Review and Prospects | Elsevier Enhanced Reader Available online: https://reader.elsevier. com/reader/sd/pii/S258884042200004X?token=9AAD7F711F4C1F4F8452E0E0EB4237 48FC1B4FB9E4BFE9302FA5A973ABDC9EC29E49655B2766A543D25349BB039E3 74F&originRegion=eu-west-1&originCreation=20221007192441. Accessed on 7 Oct 2022 21. Pantelakis, S.P., Papanikos, P.: Fatigue crack growth and residual strength in a simple MSD problem. In: Gdoutos, E.E., Rodopoulos, C.A., Yates, J.R. (eds.), Problems of Fracture Mechanics and Fatigue: A Solution Guide; Springer Netherlands: Dordrecht, pp. 611–614 (2003). ISBN 978-94-017-2774-7
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22. Niu, M.C.-Y.: Airframe Structural Design, 2nd edn. Hong Kong Conmilit Press Ltd. (1999) 23. Galatolo, R., Nilsson, K.-F.: An Experimental and numerical analysis of residual strength of butt-joints panels with multiple site damage. Eng. Fract. Mech. 68, 1437–1461 (2001). https:// doi.org/10.1016/S0013-7944(01)00027-3 24. Liu, J., Zhao, A., Ke, Z., Li, Z., Bi, Y.: Investigation on the residual stresses and fatigue performance of riveted single strap butt joints. Materials 13, 3436 (2020). https://doi.org/10. 3390/ma13153436 25. Setlak, L., Kowalik, R., Lusiak, T.: Practical use of composite materials used in military aircraft. Materials 14, 4812 (2021). https://doi.org/10.3390/ma14174812 26. Zimmermann, N., Wang, P.H.: A Review of failure modes and fracture analysis of aircraft composite materials. Eng. Fail. Anal. 115, 104692 (2020). https://doi.org/10.1016/j.engfai lanal.2020.104692 27. Rufin, A.C., Jochum, R.L.: Fatigue analysis and design of composite-metal joints in primary aircraft structures. Adv. Mater. Res. 891–892, 588–593 (2014). https://doi.org/10.4028/www. scientific.net/AMR.891-892.588 28. Avikal, S. Structural Analysis of Rear Pressure Bulkhead of Typical Transport Aircraft. International Journal of Engineering and Advanced Technology 2019, 8, doi:https://doi.org/10. 35940/ijeat.D1004.0484S19 29. Venkatesh, S., et al: Design, development and certification of composite rear pressure bulkhead for a light transport aircraft. In: Proceedings of the Proceedings of 18th International Conference on Composite Materials, London (2011) 30. Tavares, S.M.O., de Castro, P.M.S.T.: An overview of fatigue in aircraft structures. Fatigue Fract. Eng. Mater. Struct. 40, 1510–1529 (2017). https://doi.org/10.1111/ffe.12631 31. Broek, D.: Elementary Engineering Fracture Mechanics, 1st edn. Springer Dordrecht (1982). ISBN 978-94-009-4333-9
Analysis of Gluing Assembly Defects of Ultra-Thin Composite Face Sheets with Honeycomb Core Vitaliy Gaidachuk1 , Tetyana Nabokina1 , Igor Taranenko1 Oleksandr Gaidachuk2 , and Andrii Kondratiev3(B)
,
1 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine 2 Ningbo University of Technology, 201 Fenghua Rd, Jiangbei Qu, Ningbo Shi, Zhejiang
Sheng, China 3 O.M. Beketov National University of Urban Economy in Kharkiv, 17 Marshal Bazhanov St.,
Kharkiv 61002, Ukraine [email protected]
Abstract. Sandwich structures owing to their high stiffness and low density are widely used in various industries. High requirements for the weight reduction of such structures at the regulated stiffness predetermine more extensive use of ultrathin composite face sheets of 2…3 monolayers and honeycomb core of 0.01…0.03 mm thick aluminium foil in them. However, whereas the face sheets show good physico-mechanical characteristics and high quality of the structure, the further process stages related to their assembly in the sandwich structure system are associated with the appearance of dimples in them. The objective of this paper is the theoretical and experimental analysis of defects occurring in the process of the sandwich structure assembly by gluing of ultrathin face sheets and honeycomb core. The paper proposes the mathematical model of defects in the form of deep discrete dimples of the sheets, based on which the depth of a dimple is measured depending on the process parameters, structural dimensions of honeycombs and sheets, and their physico-mechanical characteristics. The results of experimental measurements of continual and discrete dimples allowed us to establish the statistical nature of their depth and therefore to assess the quality of the process of glued assembly of the structure and permissible level of defects according to operational requirements for the product. The results of the work provide a way to solve a number of new tasks in the technology of assembling the composite products for aerospace engineering. In addition, they can be used in the manufacturing of other precision products for conversion applications. Keywords: Sandwich structures · Continual and discrete dimples · Thermal nonequilibrium structure
1 Introduction We are currently observing a steady increase in the production of structures made of polymer composite materials (PCM) which are used in various fields of technology [1, 2]. The range of application of PCM and structures based on them is constantly expanding © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 174–185, 2023. https://doi.org/10.1007/978-3-031-36201-9_15
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[3, 4]. Developments in aerospace engineering are characterized by more extensive use of sandwich structures with ultrathin composite face sheets of 2…3 monolayers and honeycomb core of aluminium foil of 0.01…0.03 mm thick in the dimensionally stable products for various applications [5, 6]. This structural design allows achieving some of the highest indicators of specific strength and stiffness at minimum weight; it is a decisive criterion for the effectiveness of structures of this class [7]. As the examples, we can mention the structural elements of large-sized reflector antennas, load-bearing structures of solar panels and optical telescopes, etc. [8] High manufacturing accuracy is one of the basic requirements for such dimensionally stable structures operating in outer space [9]. This accuracy can be measured by the deviations from geometrically defined shapes in fractions of a millimeter [10]. It is quite common that manufactured sandwich structures change their original geometry in the production process [11]. As a result, these structures are not suitable for use in terms of meeting the accuracy requirements [12]. Despite the fact that the technology for manufacturing of sandwich structures in the aircraft industry has already been proven [13, 14], specific nature of moulding of ultrathin face sheets requires the solution of a number of new process tasks. They primarily include the tasks associated with the peculiar features of the use of ultrathin sheets of the thermal nonequilibrium structure because of specific loading [15]. In addition, new challenges also arise in the process of assembly of such ultrathin skins and honeycomb core by gluing. Thus, in the process of experimental studies [16, 17] a number of technological problems have been identified (Fig. 1): – appearance of evenly distributed (continual) dimples of the face sheets in the cell area (according to the results of measurements at the depth of 50…70 µm, sheet thickness of 0.02 mm, and honeycomb cell size of 6 mm); – presence of single rare (discrete) dimples under the honeycomb cell at the depth of 200…270 µm, sheet thickness of 0.02 mm, and cell size of 6 mm.
Fig. 1. Typical defects occurring during assembly of sandwich structures with ultrathin composite face sheets and honeycomb core: 1 – continual dimples; 2 – discrete dimples
Within this framework, the objective of this paper is the theoretical and experimental analysis of defects occurring in the process of assembly of a sandwich structure by gluing ultrathin face sheets and honeycomb core.
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2 Literature Review Currently, close attention is paid to the issues of creation of sandwich structures with ultrathin composite face sheets and honeycomb core of the minimum surface mass. For example, the impact of distribution of defects in the honeycomb structures on the mechanical characteristics of structures is studied in [18]. To reduce the defect impact, the paper proposes the method of machine vision to monitor the strain of face sheets within each cell of the honeycomb core. The paper [19] presents the results of studies aimed at ensuring the bearing capacity of sandwich structures in cases where their characteristics go beyond the permissible values because of process deviations. The obtained results allowed improving the standard processes for the manufacturing of sandwich structures. However, they do not provide a way to establish a specific mechanism of the impact of technological imperfections on the bearing capacity of structures. Mechanical behavior of sandwich structures with ultrathin face sheets is studied in [20], with the analysis of dependence of the stiffness of such structures on the parameters of the face sheets reinforcement and process deviations. To eliminate the defects occurring in the process of assembly of sandwich structures by gluing, simultaneous curing of the face sheets and honeycomb core is proposed in [21]. The paper shows that increase in the pressure inside the structure by creating excess super-atmospheric pressure in the vacuum bag is the most effective way to reduce the face sheet swelling above cells of the honeycomb core. The paper [22] suggests that in case of insignificant loads in the panel plane and relatively small size of the honeycomb cell the number of continual dimples is determined by the geometrical properties of the face sheet, adhesive layer and adhesive fillet. Experimental and finite element monitoring of continual dimples in the face sheets of sandwich structures is carried out in [16]. Here the analytical model is proposed to predict the dimple profile. However, as noted by the authors, it is necessary to develop more advanced method for estimating the dimple size based on strain measurement data. This line of research is developed in [23]. This paper deals with studying of the continual dimples of face sheets with the focus on their experimental monitoring and mathematical prediction. Distribution of strains along the optical fibers embedded in the face sheet was controlled in the process of the structure fabrication. The paper proposes the refined analytical model based on the beam theory to predict the dimple depth based on the data on the experimental level of deformations. It is found in [24] that formation of the continual dimples of face sheets can be caused by the loss of stability of the honeycomb cell walls. It results in compression of zones above the cells of sheets in their plane, in particular, under condition of their supercritical deformation at the pressure of the structure gluing. The paper proposes the mathematical model taking into account the thermal expansion of sheets and honeycomb core at the temperature of the structure gluing, followed by cooling and fixing of the glue in the heated state, and then cooling of the glued structure to the normal temperature. However, the authors do not take into consideration the thermal strain of the face sheet itself, arising in the process of its cooling because of PCM thermal nonequilibrium. Analytical studies of the stresses of face sheets with different degrees of thermal non-equilibrium under various panel supporting conditions were conducted in [25, 26]. Based on the findings of these papers, the authors of [17] developed the analytical dependence for the depth of maximum continual dimple. Besides, this paper, with the use of results of [24], states that continual
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dimples are relatively small, and they are inevitable when ultrathin composite face sheets of 2…3 monolayers and aluminum foil honeycomb core are used in sandwich structures. The papers [17, 24] also show that the use of adhesives polymerized at lower temperatures (and the binders for the manufacture of structural skins) can reduce the size of continual dimple. For example, when the temperature drops from 175 °C to 80 °C, the dimple can be reduced by 2.6 times. The use of thermal nonequilibrium patterns for the face sheets, which significantly reduce the dimpling, also seems to be effective [25, 27]. However, here it is necessary to strictly observe the thermal equilibrium of the sandwich structure as a whole in order to avoid the structure buckling [24, 26]. Analysis of test specimens of structures shows [19, 23, 24] that the lack of perforation on the honeycomb faces can cause the discrete dimples of single cells of the panel, 4…5 times exceeding the continual ones. It may result in the pressure difference inside the defective cell and surrounding perforated cells.
3 Research Methodology Occurrence of the pressure difference in the defective honeycomb cell with the discrete dimple of the face sheet follows from the physical picture of the phenomenon. Dimples of the single cells (3 cells on the panel of 20 × 35 cm) in the test specimen of the structure, with the absolute symmetry relative to the face sheets, featured maximum deflections, which were several times different in magnitude from the continual dimples of neighboring cells by 4…5 times [19, 23, 24]. In case of the excess pressure in the defective cell, the value of maximum deflection in the cell center would be significantly less than in defect-free ones. Face sheet deflections because of temperature strains, which arise after gluing and cooling of the structure, are theoretically the same for all cells. Additional deflection of the face sheets of a defective cell in case of excess pressure in it would reduce the total deflection up to a change in sign (there could be bulging, rather than dimpling, of the face sheets of the defective honeycomb cell). Apparently, negative pressure in the non-perforated cell is generated as follows. During vacuuming of the structure assembly, the rarefaction occurs in all cells, including non-perforated cells, because of gaps between the face sheets and ends of the honeycombs. These gaps are actually eliminated only at full working vacuum and moulding temperature T f , at which the glue applied to the honeycomb ends is liquefied and then cured [19, 24, 28]. When the vacuum is removed, the ambient pressure is almost instantly restored in the perforated cells, whereas the expansion pressure is maintained in the defective cells for some time. After cooling of the structure to the normal temperature, this pressure decreases significantly in accordance with the ideal gas equation pV = mRT ,
(1)
where p – pressure in gas volume V; R – universal gas constant; T – temperature. If we neglect the change in volume of the defective cell after its cooling from temperature T f to normal temperature T 0 , then the Charles’s law follows from (1) pf p0 = , Tf T0
(2)
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from which p0 = pf
T0 , Tf
(3)
In the process of moulding, compression pressure acts on the honeycomb ends [19, 24]: σc =
pf ac sin β(1 + cos β), 3δc
(4)
as a result of which therecan be a loss of stability of honeycomb faces at pf > pf . If the condition pf < pf is met in the gluing process, then at the normal temperature the pressure (pa − p0 ) will act on the defective cell, where pa – atmospheric pressure. Face sheets above the defective cell and each of the honeycomb faces will be subjected to the uniform transverse pressure. Maximum deflection of the face sheets (Fig. 1) can be determined by the approximation formula (one term of the series is retained) [19, 24]: wmax =
192 b4r π 6 δ3
4
Ea barr
(pa − p0 )(1 − μab μba ) , 2 br + 2 ar [Ea μba + 2(1 − μab μba )Gab ] + Eb
(5)
where E a , E b , Gab – moduli of elasticity in the directions a and b of the panel (Fig. 2) and shear modulus; μab , μba – Poisson’s ratios; ar , br – effective dimensions of a cell: ar = ac (1 + cos β), br = 2ac sin β.
(6)
Fig. 2. Typical sandwich structure design
After substituting π 6 in (5) with its numerical value and stating the cell dimensions ar and br through ac and the angle β according to formulas (6), we obtain: wmax =
3.2ac4 (pa − p0 )sin4 β(1 − μab μba ) , δ 3 16Ea tg 4 β2 + 8tg 2 β2 [Ea μba + 2(1 − μab μba )Gab ] + Eb
(7)
For a single cell face, formula (5) is also valid, where we assume E c = E a = E b , Gab = Ec /2(1 + μc ), μab = μba = μc = 0.3, substituting δ c instead of δ, ac instead of
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br and h instead of ar . So, after all transformations we get the deflection of a single face of honeycombs wmax c1 = 0.18
a4 (pa − p0 ) c ,
2 4 δc3 Ec ahc + 2 ahc + 1
(8)
For the deflection of the face of the honeycombs with the cylindrical stiff double δg 2 2δ 3 δ3 , rather than 12(1−μ , where δ g – thickness ness equal to 12(1−μc ) 1 + 3 1 + δc c) of adhesive film of the double face, we obtain instead of (8) wmax c2 = 0.09
δc3 1 + 3 1 +
ac4 (pa − p0 ) ,
ac 2 δg 2 ac 4 E + 2 + 1 c δc h h
(9)
At δ g = δ c the deflection of double face of the honeycomb wmax c2 is 26 times less than the deflection of a single face wmax c1 . Numerical analysis of the deflection (8) shows that for the actual values of design and technology parameters of honeycombs [29] it is more than an order of magnitude higher than possible values under elastic deformation. Therefore, at ac = 6 mm; pa − p0 = 0.08 MPa; δ c = 0.02 mm; E c = 72 GPa; h = 9 mm; wmax c1 = 16 mm. At the same time, wmax c2 = 0.6 mm. The above example characterizes the pattern below in qualitative terms. Single faces of a cell operate in the supercritical area, i.e. zone of plastic hinges 1 is formed in each of them (Fig. 3) [30]. However, double faces still seem to show the elastic deformation. After removal of the load and heating, elastic deformations can partially or completely restore the shape of the cell. This physical picture is supported by the fact that upon heating of the experimental structure in the defective cell area the dimpling of the latter disappeared completely, and when the face sheet was pierced with a thin needle, dimples of the defective cell did not differ from the continual ones.
Fig. 3. Diagram of discrete dimple formation
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4 Experimental Research To confirm the theoretical results, the dimple depth was determined experimentally on the test sandwich structures. Three fragments of the structure of 12.5 mm thick and 180 × 360 plan size were made, with PCM face sheets of 0.3 mm thick based on LU-P-0.1 and ELUR-P-0.1 carbon tapes of 0.1 mm and 0.11 mm thick on epoxy-triphenol binder of thermal nonequilibrium structure of 0° /90° . We used the honeycombs of aluminium alloy 556 with a hexagonal cell with 6 mm side, foil of 0.02 mm thick and 12 mm high. VK-36 glue in the amount of 50 g/m2 was used for the assembly by gluing. According to preliminary estimates, the depth of continual dimples is within 0.04…0.07 mm, and discrete dimples – 0.15…0.30 mm, therefore the choice of the measurement method, tools and equipment to obtain the required measurement accuracy was an important task. As a result, to control both discrete and continual dimples we developed a measuring device of strain gauge type based on the horizontal milling machine. The device consisted of the adjustable table and stand with an indicator fixed to it (Fig. 4).
Fig. 4. Diagram of the dimple measuring device: 1 – displacement indicator; 2 – indicator arrester; 3 – ruby tip of the indicator movable axis; 4 – upper plate; 5 – stand; 6 – lower plate; 7 – fixed pin; 8 – threaded bushing; 9 – adjusting pin; 10 – structural element under study
The above device allowed us to control and measure the dimples with the sufficient accuracy. Working surface of the table is a polished plate with the dimensions of 700 × 300 × 10 mm. Lower surface of the plate accommodates four rigidly mounted supports in the form of two pins joined by the threaded bushing. The lower pins rest freely on the base plate. By rotating the threaded bushings and lower pins, we can change the height of the upper plate, as well as its horizontality on the level, and carry out precise supporting on four points. The indicator stand is rigidly attached to the base plate; console with the displacement indicator fixed thereto (division value – 0.001 mm) is mounted so that the indicator tip is approximately in the center of the working table.
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The experimental unit is freely installed on the table, and the indicator height is adjusted so that the measuring rod is raised by ∼ =0.5 mm. Dimples are measured as follows. The tip of the measuring rod is brought to the trace of end of each measured cell face in turn, the indicator reading Δi is recorded, and the average value is determined: fmid =
1 6 i , i=1 6
(10)
Then the point of the maximum dimple depth Δmax is found; the panel is moved in circular motions, and position of the indicator arrow is fixed at the point where it changed the direction of reading. This point corresponds the maximum dimple depth Δmax . The reading corresponding to the point of maximum dimple depth is subtracted from the average value of the indicator readings at the ends of faces, and the true dimple depth f exp is determined: fexp = fmid − max .
(11)
The parameters of statistical processing of the results are obtained by the known formulas
n
n 2 μ 100σ 100σ i i=1 i=1 (μ − μ i ) μ= ,σ = ,V = ,p= √ , (12) n n−1 μ μ n where μ – expectation value, σ – variance, V – coefficient of variation, p – accuracy index.
5 Results Results of measurements of 70 continual dimples are given in Table 1. Parameters of statistical processing of the results obtained by formulas (12) are as follows: μ = 56.7, σ = 7.27, V = 12.82%, p = 1.53%. The chart in Fig. 5 (a) shows the density of probability distribution for the continual dimples. Analysis of results of statistical processing of continual dimples indicates that the sample is sufficient for the reliability of the received data, since P < 5%. Results of measurements of 10 discrete dimples are given in Table 2. Parameters of statistical processing of the results obtained by formulas (12) are as follows: μ = 244, σ = 12.8, V = 5.24%, p = 1.66%. The chart in Fig. 5 (b) shows the density of probability distribution for the discrete dimples. Comparison of the experimental results of continual dimple depth with the numerical values obtained in [19, 24] indicate the adequacy of the proposed mathematical models and satisfactory matching of the dimple values in quantitative terms. Sampling of discrete dimples given in Table 2 was performed on two test structures. In this case, the accepted mechanism of formation of discrete dimples and their average statistical value can be considered confirmed. High coefficient of variation indicates both the random nature of occurrence of these defects and the relative arbitrariness of
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Table 1. Results of measurements and statistical processing of continual dimples of the structure f c , µm
No
f c , µm
No
f c , µm
No
f c , µm
No
f c , µm
1
49
15
42
29
67
43
67
57
67
2
56
16
54
30
71
44
49
58
61
3
53
17
55
31
49
45
47
59
48
4
65
18
64
32
61
46
61
60
54
5
62
19
68
33
69
47
72
61
52
6
44
20
59
34
57
48
64
62
60
7
54
21
47
35
55
49
62
63
66
8
48
22
49
36
59
50
66
64
49
9
62
23
61
37
51
51
55
65
55
10
67
24
64
38
48
52
46
66
56
11
53
25
56
39
47
53
48
67
62
12
57
26
48
40
53
54
50
68
61
13
49
27
61
41
57
55
52
69
53
14
61
28
63
42
64
56
52
70
55
No.
Table 2. Results of measurements and statistical processing of discrete dimples of the structure No.
1
2
3
4
5
6
7
8
9
10
f d , µm
233
247
238
222
274
263
224
252
248
239
Fig. 5. Density of probability of the depth of continual (a) and discrete (b) dimples
the honeycomb core walls’ inelastic deformation within the defective cell. Probability of their appearance estimated as the ratio of the number of defective cells to the total cells in the test structures was 0.004 and 0.008, respectively. These values for the test
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structures can be considered acceptable, since the honeycomb core perforation process is to be more stable with the well-proven technology for the panel manufacturing. Based on the theoretical prerequisites for the occurrence of discrete dimples, simple method for their elimination has been proposed and tested. The method consists in piercing of the sheet above the defective cell, followed by local heating of the structure at the temperature close to the panel gluing temperature (160…170 °C). Taking into account the local character and insignificant relative number of defective cells, as well as enhanced control of the honeycomb foil perforation, we can consider acceptable the elimination of a cell defect by piercing the face sheet followed by local heating of the repaired zone. Thus, our studies generally confirmed the reliability of theoretical results obtained earlier in [19, 24]. Therefore, the developed complex of technological support for the manufacturing of PCM panels for space applications can be recommended for implementation.
6 Conclusions The paper proposes the mathematical model of defects in the form of deep discrete dimples of the sheets, based on which the depth of a dimple is measured depending on the process parameters, structural dimensions of honeycombs and sheets, their physicomechanical characteristics, as well as the physical nature of occurrence of a local defect. The results of numerous experimental measurements of continual and discrete dimples allow establishing the statistical nature of their depth, which provides a way to assess the quality of the process of glued assembly of the structure and permissible level of defects according to the operational requirements for the product. Therefore, findings of the research allow solving a number of new tasks in the technology of assembling PCM structures for aerospace engineering and can be used in the manufacturing of other precision products for conversion applications.
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Information Technology for Determining the Flight Performance of a Paraglider Wing Dmytro Kritskiy(B)
, Oleksandr Karatanov , Serhii Koba , and Andrii Pohudin
National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine [email protected]
Abstract. The article describes the main features of the use of IT in the development of paragliders that allows, on the basis of a geometric model of a paraglider, to determine its main aerodynamic parameters, and flight simulation using certain aerodynamic parameters. The classification of paragliders is considered (basic types, types and classes). The necessity of automating the design of paragliders is substantiated. An overview of automated paraglider design systems is given, their main advantages and disadvantages are described. The main attention is focused on the problems of aerodynamic design and analysis. The analysis of the reliability of the calculated aerodynamic data is provided. Keywords: paraglider · aircraft · performance · design · analysis
1 Introduction Paragliding is a very young type of aircraft (LA). In literally 20 years, paragliders have undergone a rapid evolution from a slightly improved parachute to an aircraft that allows them to fly hundreds of kilometers. In the course of the development of paragliders, a wealth of experience in their creation was accumulated, the technical capabilities of the developers have significantly increased. Nevertheless, the task of developing a new aircraft did not become trivial [1]. The need for it appears, firstly, due to the obsolescence of existing models of paragliders, secondly, because of the need to improve safety, and thirdly, because of the emergence of new technical solutions that increase flight characteristics and, accordingly, toughen competition in the market, etc. Despite the fact that the paraglider is the simplest and cheapest of the existing aircraft heavier than air, it is a rather complex technical system. The most critical stage in the development of a paraglider, or any aircraft, is the general design, which includes a technical proposal and conceptual design [2]. To ensure the competitiveness of the new paraglider model in the context of fierce competition, a narrow market and limited material resources and development time, a higher degree of accuracy in predicting aircraft characteristics is needed already at the early design stages. In turn, the flight performance of an aircraft is largely determined by its aerodynamic model [3]. This has been proven many times over in the history of aviation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 186–199, 2023. https://doi.org/10.1007/978-3-031-36201-9_16
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This work is aimed at developing a technique for constructing a paraglider model, creating an information technology that allows, on the basis of a geometric model of a paraglider, to determine its main aerodynamic parameters, and flight simulation using certain aerodynamic parameters. A paraglider is a non-motorized aircraft, a glider with a soft two-shell wing, inflated through the air intakes by an incoming air stream. The lift force is created due to the counter-flow of air flowing around the wing profile. It is she who maintains a certain speed relative to the air (the vector of the lift force can be directed not only up, but also forward). This airspeed is limited only by the complex drag force (wing, lines, pilot), for the constant overcoming of which the stored altitude is consumed. Therefore, in order to fly, the paraglider continuously spends altitude (gliding) [4–7]. Paragliding design is almost always about finding compromises. Most often, the flight characteristics of the apparatus and flight safety are on opposite scales. Flight performance means, first of all, the aerodynamic quality of the wing - the ratio of its horizontal speed to the rate of descent. The higher the quality, the more efficiently the paraglider uses the energy of gravity, and the further the pilot can fly from a given altitude. In addition to quality, important characteristics also include the maximum and minimum flight speed, ease of launch, and maneuverability. Safety basically means the ability of the soft wing to maintain its shape or restore it if collapse did occur. Safety is assessed by a series of tests - their set is determined by independent organizations such as the German DHV or the European CEN. The test pilot provokes the paraglider to enter certain dangerous flight modes and looks at how long it will take to restore the wing and whether active human actions are needed for this. Unfortunately, flight performance and safety are inversely related. Aircraft dominated by LH are designed for experienced pilots who are able to remain calm in a critical situation, correctly classify a dangerous mode and take conscious, timely actions to get out of it. Apparatus for beginners, on the other hand, rarely fold and recover from additions in a couple of seconds without the participation of the pilot. However, the “safe wing” model will always be inferior in its qualities to the sports models.
2 Classification Let’s consider the classification, that is, the division into main types, types and classes of paragliders. Paragliders are conventionally divided into several large classes: – in terms of carrying capacity - single and double; – as applicable - for free and motorized flight; – by target audience - training, for weekend pilots, athletes, athletes of the international level; – for sport purposes - paragliders for aerial acrobatics, cross-country, record flights; – according to the design features - one- and two-layer, two- and three-four-row, according to the nature of the stiffnesses used in the toe and the tip of the ribs, etc.;
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– in the speed range - ordinary paragliders, speedflyers for flying in strong winds in dynamic flows, speedgliders for high-speed gliding from the mountains. Due to the fact that the paraglider, being a purely sports device, is becoming more and more popular, the main classification parameter, which determines certain design solutions, is safety. In order for the pilot to be able to unambiguously determine which aircraft is in front of him, as well as in order not to release an insufficiently strong and reliable aircraft to the market, there is a certification of paragliding equipment [2]. Today in the world there are two certification systems for paragliders - the German DHV and the pan-European EN. The German system is older and more subjective, that is, the decision of which class the paraglider belongs to is made directly by the test pilot. At the moment, the EN certification system is the most modern and the most objectively assessing the behavior of the paraglider. Paragliders according to the EN certification system are divided into four classes: – A - paragliders with maximum passive safety and high resistance to exit from normal flight. Designed for all pilots, including pilots at all stages of training. – B - paragliders with good passive safety and resisting exit from normal flight. Designed for all pilots, including pilots at all stages of training. – C - paragliders with moderate passive safety and potentially dynamic responses to turbulence and pilot error. Returning to normal flight may require precise pilot action. Designed for pilots who are proficient in wing deployment techniques, who fly actively and regularly and have a good understanding of the use of a reduced safety wing. – D - paragliders with potentially harsh reactions to turbulence and pilot error. Returning to normal flight may require precise action. Designed for pilots with extensive experience in wing deployment, very active flying, with significant experience in turbulence and well aware of the features of using such a wing. The flight characteristics of paragliders are inversely related to the level of safety: the more reliable the wing, the worse it flies. The elongation of modern serial paragliders is in the range of 4.5..8.5 (up to 10..13 on experimental wings). Paragliders of about 10 years ago had lower aspect ratios, which was due to the use of thin wing profiles (to reduce drag) and the inability to provide the necessary rigidity. However, the maximum elongation of a certified paraglider is 6.7, only sports prototypes have more. Elongation is interconnected with arch: as a rule, flatter wings behave worse in extreme conditions, and the designer is forced to reduce the aspect ratio of such a wing. So, modern wings have the following aspect ratios: – – – – –
category EN A - 4.5… 5.1; category EN B - 4.8… 5.8; category EN C - 5.6… 6.5; category EN D - 6.0… 6.7; COMPETITION (uncertified) - 6.5… 8.5.
The paraglider as a product is supplied in the following basic configuration: paraglider, harness, rescue system, packaging (backpack, bag).
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The main structural elements of the paraglider are shown schematically in the diagram (Fig. 1). Wing. The dome is a soft wing, consisting of an upper bearing surface (VP), a lower surface (LP) and a set of ribs.
Fig. 1. Paragliding structural elements
Loose Ends. The riser or V-line is a construction of belts and buckles designed to transfer the load from the pilot to the line system (Fig. 2). Sometimes rigid elements are also used in the design. To change the installation angles in the design of the ends, a trimmer and an accelerator are used. Suspension system (PS) necessary for secure and comfortable fixation of the pilot to the SS. It is assumed a sitting or lying position during the flight. It can house a speed system, a rescue parachute, and devices. Water ballast, etc. Rescue system. The rescue parachute should be able to rescue the pilot from a minimum height in the event of a paraglider failure (i.e., if it is impossible to land on it). The simplest parachute is a flat circle in cutting with 16, 18 or 24 lines (depending on the materials of the canopy and lines), reduced to one halyard. The parachute opens according to the percussion pattern of opening, when the canopy, which is partially filled after the throw and uncoupling, pulls out the lines from the honeycomb. The materials used in various elements of the paraglider and PS are shown in Table 1.
3 Paragliding Design Technique Designing paragliders has one significant difference from working on airplanes or gliders: the behavior of a soft wing is almost beyond computer simulation. A paraglider is an aeroelastic system: the incoming air flow not only flows around the wing, but also
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Fig. 2. An example of the design of risers: 1 - load-bearing slings; 2 - control line; 3 - carabiners for attaching slings; 4 articulated fastening of lines of row C; 5 - toggle; 6 - toggle fastening to row D; 7 - accelerator; 8 - trimmer; 9 - a carabiner for the suspension system; 10 - stirrup for accelerator control Table 1. Materials used in the manufacture of paragliders Paragliding element
Requirements
Material
Dome
Durability, airtightness, low elongation (poorly stretched), light weight
Bologna, varnish, NYLON ripstop (Carrington, Gelvenor)
Lanyard system a) slings b) free ends
Strength, low elongation, minimum diameter, wear resistance, strength, light weight
Aramid (kevlar), dyneema (long molecular polyethylene) Webbing HP 20… 25 LTKP 25-1000
Suspension system
Durability, convenience, light weight
Capron, avisent, LTKP 44-1600 Webbing HS 45… 50
influences its shape. The change in shape affects the nature of the flow, which again changes the shape - this is an endless chain of mutual influences. There is a fundamental work [2], covering all aspects of the design and manufacture of a paraglider, allowing its development without the use of computers. The technique proposed in this work includes the following main stages of creating a paraglider: Stage 1. Determination of the wing area and its shape in plan. Stage 1.1. The choice of the specific load on the wing. Stage 1.2. Calculation of the area depending on the weight of the pilot. Stage 1.3. Wing extension selection. Stage 1.4. Span calculation.
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Stage 1.5. Creation of the wing geometry in plan. Stage 2. Selection of the wing profile of the paraglider. Stage 2.1. Determination of the aerodynamic quality of the system. Stage 2.2. Profile selection. Stage 2.3. Determination of the parameters of the air intake. Stage 3. Determination of the number of wing sections. Stage 4. Construction of the geometry of the system in the plane of symmetry. Stage 5. Construction of the geometry of the system in frontal projection. Stage 6. Construction of the cutting shape of the wing panels. Stage 7. Design of a sling system. Stage 8. Making a paraglider. Stage 9. Test of the paraglider. However, the techniques described even in this book are so laborious that the need for design automation is obvious. In addition, a number of the most important design procedures are not sufficiently formalized, which results in a significant share of intuitive design decisions (the designer must “guess” the optimal value of the aerodynamic quality, “successfully” choose the wing layout and the number of its sections, the optimal air intake area, etc.) [8]. These features (apparently inevitable at this stage of development) lead to a large number of design iterations “synthesis – analysis”, associated with the refinement of the initially accepted initial data. There are several methods for designing paragliders that allow the development of a paraglider with the required characteristics [2, 5, 9]. However, these techniques imply an iterative approximation to the specified characteristics through the construction of several prototypes. The number of prototypes can go up to ten, which significantly increases the labor intensity and cost of developing a new paraglider model, which increases the cost of production models and slows down the process of updating the model range and introducing design improvements. The main source of information for developers is prototype testing, however, the use of programs for calculating the aerodynamics of rigid wings can make it possible to evaluate the airfoil flow in general terms and give quantitative estimates of the wing characteristics in non-extreme modes (without violating the wing geometry) in the early stages of design, which will simplify and speed up the process development. Due to the above-mentioned iteration of the design process, it was decided to focus on the problems of aerodynamic design of the wing and the analysis of design solutions as the most laborious. The design methodology does not require fundamental changes, since the tasks of developing other units and assemblies of the paraglider are well developed and do not cause significant difficulties. However, the modern capabilities of CAD and modeling systems allow, at relatively low cost, to supplement the existing methodology with the capabilities of virtual simulation [10]. This will allow us to predict some of the nuances of the wing’s behavior even before the prototype stage. The widely available programs for the design of paragliders and kites are mostly focused on building a three-dimensional model of the wing and line system and generating patterns taking into account the physical properties of fabric materials: the extensibility of the fabric, the “mattress” of the wing sections. At the same time, the flying
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qualities of the future paraglider will be entirely determined by the designer’s experience, and taking into account some characteristics (for example, the center of profile pressure) is possible only with the help of constructions and calculations by third-party methods (including graphic constructions on paper). A significant drawback of most of the existing available systems is the lack of an aerodynamic calculation module. It should be noted that a number of paragliding companies have their own design systems, including those with aerodynamic analysis capabilities. However, these systems are not available on the market and it is impossible to talk about their characteristics with the due degree of reliability. The most advanced of the free software for designing kites and paragliders is Surfplan/Gliderplan Hobby version (Fig. 3). The program allows user to create water kites with an inflatable balloon in the leading edge, as well as paragliders and kites inflated by a stream of air (parafoils), has a user-friendly interface, many settings, it is possible to use most modern features like oblique ribs.
Fig. 3. Surfplan interface
The program can print patterns for patterns on a regular A4 printer. User can also print to a virtual printer as PDF files, open them, for example, in CorelDraw, put them together and print to a large format plotter. Another free program for designing kites (which can be used for paragliders) is Foilmaker (Fig. 4). Several good flying kites have been developed with this program, but its capabilities are not sufficient for industrial applications.
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Fig. 4. FoilMaker interface
Both programs described above allow user to design a kite or paraglider of any shape, however, they do not take into account aerodynamics at all. Almost all available design systems do not have the capabilities of aerodynamic analysis of the constructed models. A rare example of an available design system with the ability to conduct aerodynamic calculations of the constructed model is fwDesign + XFLR5 (Fig. 5, 6).
Fig. 5. FwDesign interface
The free version of fwDesign allows user to design paragliders and parafoil kites and export their models to the XFLR5 aerodynamic analysis program. Export of the model and patterns in dxf format is possible in the commercial version of the program,
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Fig. 6. XFLR5 interface
or (in the case of non-commercial use) - with the help of the authors of the program. XFLR5 is an aerodynamic analysis tool that allows user to determine important profile characteristics, perform virtual blowing of models, and build polars. For these purposes, XFLR5 implements CFD methods (computational fluid dynamics methods). The advantages of the fwDesign system are such that this program (its commercial version) is widely used in Sky-Country, (Kharkiv), which produces world-class paragliders and kites. However, due to a number of shortcomings of the aerodynamic part of the program (XFLR5), the results of aerodynamic calculations differ significantly from the real characteristics of the paraglider, which makes it possible to use this part of the system only for comparative and qualitative analysis of profiles. The new PARATAILOR system has appeared on the market, which is relatively recently developed, but already has many advantages over its predecessors. In particular, the design process is in many respects more intuitive and well-visualized (Fig. 7), the authors have foreseen in advance various possibilities for implementing structural wing reinforcements and different options for their execution (oblique ribs, mylar stiffeners, fishing lines in the leading and trailing edges, semi-ribs). In addition, one of the newest developments in the field of paragliding aerodynamics, the “shark profile”, has been implemented. Although, due to the novelty of the system, some functions and interface elements still look unfinished, in general the system was highly appreciated by independent paraglider designers. The PARATAILOR system, like fwDesign, has an aerodynamic module, allowing to calculate the aerodynamic characteristics of the paraglider according to its geometric model (Fig. 8). The capabilities of this module and their comparison with similar capabilities of fwDesign and the real characteristics of a flying prototype are of interest for research.
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In addition, there are specialized software environments for aerodynamic and hydrodynamic analysis used in industrial applications. The most famous are Flow Vision and Flow Simulation for Solid Works. Thus, one of the most important directions for improving the design process of paragliders at the present time is to increase the objectivity of aerodynamic analysis at the early stages of development.
Fig. 7. PARATAILOR interface
Fig. 8. Aerodynamic module PARATAILOR
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4 Algorithm for Determining the Performance Characteristics of a Paraglider Wing As stated earlier, the general task of designing a paraglider is largely provided with ready-made software that implements the stages of determining the main structural and geometric characteristics. Therefore, in this work, the main attention is focused on the problems of aerodynamic design and analysis. There is a preliminary determination of the geometric and aerodynamic characteristics of the airfoil in the fwDesign system and a refined aerodynamic analysis in the PARATAILOR CFD Analysis system. The analysis of the reliability of the calculated aerodynamic data is provided based on the test results in the MicroFlight simulator environment and according to the LK8000 flight computer. At the system level, the task of obtaining output data (flight characteristics) based on the specified input (geometric parameters) should be solved. The result of processing the 3D model of the paraglider is to obtain its approximate flight performance (Fig. 9). It remains to determine exactly what tasks need to be performed at the processing stage.
Fig. 9. Detailing the system being developed
A separate task, not directly related to processing, but necessary for the functioning of the system, is the conversion of a 3D model from fwDesign format to PARATAILOR format.
5 Software Architecture The software includes the Windows operating system, the standard Microsoft Office package, the fwDesign system, the PARATAILOR system, the MicroFlight simulator with the supplied Spider paraglider model, and the PC version of the LK8000 flight computer. The structure of data flows and user interaction is shown in Fig. 10. The transfer of information between the system components is carried out manually by the user. The required profiles are exported from the fwDesign model, which are then converted to a PARATAILOR-compatible format using Excel. The system user builds a paraglider model in PARATAILOR using data from fwDesign. The model recreated
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in PARATAILOR is analyzed in PARATAILOR CFD Analysis. The user configures the calculation settings as needed and simulates the paraglider blowing. Virtual blowdown results are saved in Excel format.
Fig. 10. Scheme of information exchange between system components
If necessary, simulation is performed for several specified parameters (weight, paraglider area, drag coefficients). The resulting set of results is brought together and analyzed by the user in the Excel environment. Further, according to the data in Excel, the necessary coefficients for the configuration files MicroFlight and LK8000 are calculated. In the MicroFlight simulator, the weather parameters and the flight scenario are set, after which the flight is performed, during which the flight performance of the model is evaluated. The Leonardo database contains a suitable flight track for a Discovery 4 paraglider (if possible, the flight should have a long gliding interval before landing). This track is loaded into the LK8000 and played in repeat mode. While playing the track, the flight computer evaluates weather conditions using the track information and builds the configuration of the flight zone, taking into account the characteristics of the paraglider, the direction and speed of the wind, the terrain, and the flight height above the relief. In the course of playing, the correspondence of the polar configuration of the paraglider to the real characteristics of the paraglider is assessed.
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6 Conclusion The authors carried out a detailed classification of paragliders according to the main types, types and classes. An analysis of the work on the design of paragliders was carried out, the features of this process were explained, and the need for its automation was substantiated. An overview of automated paraglider design systems is given, their main advantages and disadvantages are described, while the main attention is focused on the problems of aerodynamic design and analysis. An analysis of the reliability of calculated aerodynamic data is given, and the main features of the use of IT in the development of paragliders are described, which allows, based on the geometric model of a paraglider, to determine its main aerodynamic parameters, as well as flight simulation using certain aerodynamic parameters.
References 1. Kritsky, D.N., Druzhinin, E.A., Pogudina, O.K., Kritskaya, O.S.: A method for assessing the impact of technical risks on the aerospace product development projects. In: Shakhovska, N., Medykovskyy, M. (eds.) CSIT 2018. AISC, vol. 871, pp. 504–521. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-010690_36. https://www.scopus.com/inward/record.uri?eid=2-s2.0-85057811342&doi=10.1007% 2f978-3-030-01069-0_36&partnerID=40&md5=beb8ccdc4b6461d10abc97d5b36ab800 2. Landell-Mills, N.: Paragliding explained by Newtonian physics, pp. 1–29 (2022). Independent Research 3. Vermeer, L.J., Sørensen, J.N., Crespo, A.: Wind turbine wake aerodynamics. Progr. Aerosp. Sci. 39(6), 467–510 (2003). https://doi.org/10.1016/S0376-0421(03)00078-2 4. Landell-Mills, N.: How airplanes generate lift is disputed (2019, pre-print). https://doi.org/ 10.13140/RG.2.2.34380.36487 5. Chen, Y., Lin, B., Lin, J., Wang, S.: Experimental study of wake structure behind a horizontal axis tidal stream turbine. Appl. Energy 196, 82–96 (2017). https://doi.org/10.1016/j.apenergy. 2017.03.126. ISSN 0306-2619 6. Tang, H., Lam, K.-M., Shum, K.-M., Li, Y.: Wake effect of a horizontal axis wind turbine on the performance of a downstream turbine. Energies 12, 2395 (2019). https://doi.org/10.3390/ en12122395 7. Nakamura, Y., Fukamachi, N.: Visualization of the flow past a frisbee. Fluid Dyn. Res. 7(1), 31–35 (1991). https://doi.org/10.1016/0169-5983(91)90004-3 8. Kritskiy, D.N., Druzhinin, E.A., Karatanov, A.V., Kritskaya, O.S.: Content management method of complex technical system development projects. In: Shakhovska, N., Medykovskyy, M.O. (eds.) CSIT 2019. AISC, vol. 1080, pp. 293–303 (2020). https://doi.org/ 10.1007/978-3-030-33695-0_21. https://www.scopus.com/inward/record.uri?eid=2-s2.085076981282&doi=10.1007%2f978-3-030-33695-0_21&partnerID=40&md5=a1a433082 d5210a2c2b59a8b1296ac76
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9. Maiorova, K., Vorobiov, I., Boiko, M., Suponina, V., Komisarov, O.: Implementation of reengineering technology to ensure the predefined geometric accuracy of a light aircraft keel. East.Eur. J. Enterp. Technol. 6(1), 114, 6–12 (2021). https://doi.org/10.15587/1729-4061.2021. 246414 10. Kritskiy, D., Yashin, S., Koba, S.: Unmanned aerial vehicle mass model peculiarities. In: Shkarlet, S., Morozov, A., Palagin, A. (eds.) MODS 2020. AISC, vol. 1265, pp. 299–308. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-581244_29. https://www.scopus.com/inward/record.uri?eid=2-s2.0-85091130983&doi=10.1007% 2f978-3-030-58124-4_29&partnerID=40&md5=96c9bc1235ee042b8dab641625a87998
Information Modeling
Research of Mathematical Models Based on Optimization Paraboloid and Alternative Method of Regression Valeriyi Kuzmin , Maksym Zaliskyi(B) , Yuliia Petrova , and Oleksii Holubnychyi National Aviation University, Huzar av. 1, Kyiv 03058, Ukraine [email protected]
Abstract. The regression models are widely used to analyze empirical data obtained from the various branches of human life. In this case, the classical approach using the ordinary least squares method is most often used. Higher requirements for the quality of prediction lead to the spread of more complex approximating functions and the simultaneous use of alternative methods of the regression analysis. This paper considers an example of mathematical models building for empirical dataset based on exponential and hyperbolic functions for the ordinary least squares method and the case of minimizing absolute deviations. Obtaining numerical values of the parameters for mathematical models in case of exponential and hyperbolic functions using became possible through the usage of an optimization paraboloid. To choose the best mathematical model, three criteria have been utilized: a) minimum of standard deviation, b) minimum of absolute deviations sum, c) minimum of maximal deviation. The resulting final version of the approximation can be used to solve the prediction problems. Keywords: Mathematical Models building · Ordinary Least Squares · Minimization of Absolute Deviations Sum · Optimization Paraboloid · Prediction · Checking Data for Linearity
1 Introduction In the process of empirical data approximation, a very important element is the choice of an appropriate mathematical model. Most often, a set of standard programs is used for these purposes, which include many approximating functions, but the formal approach often leads to unsatisfactory results for solving prediction problems [1, 2]. In some cases, in order to achieve good predictive properties of the approximating functions, it is reasonable to decrease accuracy. The standard deviation is usually taken as a measure of accuracy [3]. In the process of mathematical models building, many subtleties and nuances must be taken into account.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 203–214, 2023. https://doi.org/10.1007/978-3-031-36201-9_17
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Recently, the theory and methodology of mathematical models building has significant development. Alternative methods and techniques for these models are created that take into account heteroskedasticity, are more robust, and others [4]. The mathematical models building techniques use the apparatus of regression analysis, computational geometry, probability theory and mathematical statistics, cluster analysis and others [5, 6]. The classic and most common method of regression analysis is the ordinary least squares (OLS) method. Computational geometry allows to reliably identify convex, concave, rectilinear and asymptotic sections, as well as calculate extreme points, inflection points, asymptotes, connection points of segment sections and others [7].
2 Literature Analysis and Problem Statement Methods for mathematical models building based on regression analysis have obtained sufficient attention in modern scientific and technical literature [8–11]. Alternative methods of regression analysis began to develop intensively in the 80s of the last century [4, 12]. Today, the method of least absolute deviations (LAD) [13, 14] has become widely used among alternative methods. According to [15], the LAD method is more robust than the classical least squares method. The LAD method is used in various areas of human activity when building mathematical models and processing statistical data, such as econometrics [16], geography [17], biology [18], control systems [19] and so on. Among other alternative methods of regression analysis, M-regression, ridge regression, and the weighted median regression method are also being developed [4, 20]. It should also be noted that when building regression models, along with simple functions, more complex ones are used, such as exponential, hyperbolic, segmented. However, in this case, it is necessary to use more complex optimization methods [21]. The mathematical models building is performed in various applications: a) in radar [22, 23]: to increase the accuracy of data processing; b) in cybersecurity [24, 25]: to synthesize the algorithms for cyberattacks detection; c) in electrical engineering [26, 27]: to design data-driven decision-making systems; d) in aviation [28, 29]: to identify the flight situation; e) during equipment intended use optimization [30, 31]: to select the optimal quantity of equipment depending environmental conditions. All mentioned models in most cases are based on OLS techniques. This paper considers a comparative analysis of the OLS and LAD methods using several approximation functions for a concrete example of empirical data.
3 Methods and Materials 3.1 Initial Dataset Preprocessing Consider an example of statistical data of the dependence in form of two-dimensional array containing explanatory variable (EV) and response variable (RV) with sample size equal to 91 items [32].
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Table 1. Initial dataset. EV, x
RV, y
EV, x
RV, y
EV, x
RV, y
EV, x
RV, y
EV, x
RV, y
5
16.2
12
15.5
18
13.7
24
11.6
33
10.2
6
16.2
14
13.2
18
13.4
24
9.9
34
12.6
7
14.2
14
14.2
18
13
25
11.8
34
10.9
7
13.8
15
12.9
18
10.6
25
12.8
34
10.8
8
13.5
15
13.3
19
13.8
26
11
34
10.4
8
14.2
16
13.7
19
13.2
26
11
35
12.1
8
15.5
16
13
19
13.9
27
13.1
35
10.9
9
15.8
16
12.3
20
13
28
9.8
36
12.2
9
16.2
16
13.6
20
12.8
29
10.3
36
11
14
16
14.3
21
11
30
10.7
36
10.3
10
9
14.6
17
13.8
21
11.5
30
9.8
37
10.4
10
15.6
17
13
21
14.7
31
11.4
37
10.1
11
14.8
17
12.7
21
13
31
11.9
38
9.8
11
13.8
17
12.5
22
12.6
31
9.6
38
10.3
11
14.7
17
10.8
22
10.4
31
12.3
39
10.2
11
18.3
17
13.5
22
11.3
32
10.4
39
10.8
11
13.7
17
13.4
24
12.3
32
10.3
42
10.8
12
16.3
18
13
24
11.8
32
11.8
43
10.7
12
13.1
The mathematical model obtaining for empirical data must begin with testing data for linearity (Table 1). To test data for linearity (nonlinearity), we use the technique developed by the first author of the paper [33]. This technique assumes performing approximation of the data using a linear function and the OLS. As a result, equation was obtained and it has a form y(x) = 15.903 − 0.1518x.
(1)
In this case, the range of the cumulative curve of residuals is 19.479, and the standard deviation is 1.114, so the critical ratio is 17.485. Therefore, with a probability of 0.98, according to the table of critical values from [33], we can conclude that the data under study are nonlinear and should to be approximated by functions that are more complex. 3.2 Approximation Using a Parabola of the Second Degree Using the OLS method, we will perform approximation of the initial data using the parabola of the second degree. After computation, equation of such form was gotten: y(x) = 17.6985 − 0.3407x + 0.004058x2 .
(2)
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The approximation results are shown in Fig. 1. The standard deviation for the resulting parabola is 1.053. The radius of curvature of the parabola was also calculated, which for the first and last points of the initial data is 140.224 and 123.22, respectively. As can be seen from the results of calculations and visual analysis, the parabola of the second degree describes the data well over the interval of their observation. However, at abscissa 41.979, the parabola reaches its minimum value and then increases. 20
y
18
16
14
12
10
8
x 0
10
20
30
40
50
Fig. 1. Approximation by the second degree parabola.
Therefore, the approximation by a parabola is not suitable for solving prediction problems. 3.3 Approximation Using a Hyperbolic Curve To describe a hyperbolic curve, we can use equation of the form y(x) =
a . 1 + bx
where a and b are the unknown approximation coefficients to be determined. Consider an approach to determining these coefficients. To do this, the total set of points is divided into two approximately equal subsets. In this case, the subsets will contain 53 and 38 empirical points. This division was justified by the geometric structure of the initial data. Each of the subsets will be approximated by parabola of the second degree using the OLS. After computation, we obtain two equations of the form y(x) = 16.1622 − 0.09682x − 0.0044019x2 .
(3)
y(x) = 15.4299 − 0.2215x + 0.002542x2 .
(4)
The preliminary calculation of the coefficients a and b of the approximating hyperbola was carried out based on the method of selected points. Since there are two unknown
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coefficients, two points were chosen, the abscissas of which were determined by the average cluster value of the two subsets x1 and x2 , and the ordinates y1 and y2 by the equations of parabolas (3) and (4). As a result, a system of equations was obtained a − x1 y1 b = y1 , a − x2 y2 b = y2 . After solving the system of equations, the values of the coefficients in the first iteration of approximation were obtained, which are a = 17.667 and b = 0.01914. To solve the problem of optimizing the coefficients of the hyperbolic curve (in the sense of minimizing the standard deviation), the standard deviation for a set of five options for each coefficient for total of 25 possible combinations will be calculated. The calculation results are shown in Table 2. Table 2. Standard deviations. S(a, b)
b = 0.01514
b = 0.01714
b = 0.01914
b = 0.02114
b = 0.02314
a = 16.667
1.083
1.106
1.242
1.442
1.672
a = 17.167
1.177
1.073
1.105
1.24
1.434
a = 17.667
1.377
1.166
1.079
1.116
1.247
a = 18.167
1.645
1.36
1.172
1.099
1.139
a = 18.667
1.952
1.62
1.361
1.193
1.13
The data from Table 2 can be approximated by OLS using a three-dimensional optimization paraboloid of the following form: S(a,b) = c0 + c1 a + c2 a2 + c3 b + c4 b2 + c5 ab, where c0 , …, c5 are the unknown coefficients of the paraboloid to be calculated. Having performed the calculation, the following equation of the optimization paraboloid was obtained S(a,b) = 30.564 − 4.622a + 0.182a2 + 1135b + 12070b2 − 91.234ab. Let’s find the optimal values of the hyperbolic curve coefficients. To do this, partial derivatives are calculated and equated to zero. After solving the system of linear equations, we obtain aopt =
−c1 − c5 bopt 2c2 c3 − c5 c1 , bopt = 2 . 2c2 c5 − 4c2 c4
For the resulting optimization paraboloid the optimal values of the coefficients are found aopt = 16.796 and bopt = 0.01646. Therefore, the equation of the optimal hyperbolic curve was obtained 16.796 . 1 + 0.01646x The approximation results are shown in Fig. 2. y(x) =
(5)
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y Hyperbolic approximation for the first iteration
15
Optimal hyperbolic approximation
10
5
x 0
20
40
60
80
Fig. 2. Approximation using the hyperbolic curve.
3.4 Approximation Using an Exponential Function The mathematical model of the exponential function is described by equation of the form y(x) = ea+bx , where a and b are the unknown approximation coefficients to be determined. To calculate unknown coefficients, it is necessary to solve equations system of the form: ⎧ n n ⎪ ⎪ ln yi = na + b xi ; ⎨ i=1
i=1
n n n ⎪ ⎪ xi ln yi = a xi + b xi2 . ⎩ i=1
i=1
i=1
After calculation, the exponential function of the form was obtained: y(x) = e2.7846−0.01202x .
(6)
The calculated coefficients are not optimal, since the logarithm operation was performed, which transformed the initial dataset (after which the assumption about the normal distribution of deviations in the initial dataset was broken). Therefore, these coefficients will be refined according to the method described in the previous paragraph. The results of calculations of standard deviations for various options of coefficients are given in Table 3. Data from the Table 3 were approximated by the optimization paraboloid of the following form S(a,b) = 310.9 − 245.1a + 48.58a2 − 5218b + 27080b2 + 2111ab.
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Table 3. Standard deviations. S(a, b)
b = –0.01602
b = –0.01402
b = –0.01202
b = –0.01002
b = –0.00802
a = 2.7446
1.866
1.502
1.222
1.131
1.311
a = 2.7646
1.693
1.352
1.131
1.148
1.431
a = 2.7846
1.531
1.227
1.09
1.223
1.592
a = 2.8046
1.388
1.139
1.108
1.348
1.784
a = 2.8246
1.271
1.101
1.185
1.514
2.002
For the resulting optimization paraboloid the optimal values of the coefficients aopt = 2.799 and bopt = −0.01272 are found. Therefore, the equation of the optimal exponential curve was obtained y(x) = e2.799−0.01272x .
(7)
The approximation results are shown in Fig. 3. 20
y Exponential approximation for the first iteration
15
Optimal exponential approximation
10
5
x 0
20
40
60
80
Fig. 3. Approximation using the exponential curve.
3.5 Approximation Using the Method of Least Absolute Deviations Let’s repeat the performed calculations for the hyperbolic and exponential functions using the method of least absolute deviations. The results of calculating the sum of absolute deviations for various options of the coefficients of the hyperbolic curve are given in Table 4.
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(a, b)
b = 0.01514
b = 0.01714
b = 0.01914
b = 0.02114 106.02
a = 16.667
75.278
77.32
88.114
a = 17.167
82.301
74.091
77.763
89.144
a = 17.667
99.641
80.973
74.026
78.655
b = 0.02314 126.538 105.81 90.296
a = 18. 67
125.07
98.429
81.323
75.166
80.235
a = 18.667
154.313
122.329
98.475
82.672
77.842
Data from the Table 4 were approximated by the optimization paraboloid of the following form (a,b) = 2632 − 404.2a + 16.074a2 + 101600b + 1109000b2 − 8226ab. For the resulting optimization paraboloid the optimal values of the coefficients aopt = 16.875 and bopt = 0.01681 are found. Therefore, the equation of the optimal hyperbolic curve was obtained using the LAD method 16.875 y(x) = . (8) 1 + 0.01681x The results of calculating the sum of absolute deviations for various options of the coefficients of the exponential curve are given in Table 5. Table 5. Sums of absolute deviations. (a, b)
b = –0.01602
b = –0.01402
b = –0.01202
a = 2.7446
143.547
110.553
84.489
b = –0.01002 77.286
b = –0.00802 93.243
a = 2.7646
127.127
96.947
78.177
79.807
104.314
a = 2.7846
112.97
87.012
75.393
86.418
118.873
a = 2.8046
101.138
80.643
76.23
97.369
136.999
a = 2.8246
92.047
76.874
83.262
112.385
156.948
Data from the Table 5 were approximated by the optimization paraboloid of the following form (a,b) = 27631 − 21818a + 4328a2 − 463256b + 2407180b2 + 187397ab. For the resulting optimization paraboloid the optimal values of the coefficients aopt = 2.781 and bopt = −0.01204 are found. So the equation of the optimal exponential curve was obtained using the LAD method y(x) = e2.781−0.01204x .
(9)
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4 Comparative Analysis of Alternative Mathematical Models To select the best mathematical model among the models (6)–(9), a comparative analysis was conducted. At the same time, model (2) was discarded from consideration, since it is absolutely not suitable for forecasting purposes, although it has a minimum standard deviation (1.053). To implement the comparative analysis scheme, standard deviations, sums of absolute deviations and predictive values were calculated. The predicted values were chosen for explanatory variable equaled to 80. The calculation results are shown in Table 6. The graphic representation of the obtained models is shown in Fig. 4. Table 6. Comparative analysis of mathematical models. №
Model name
Formula
Standard deviation
Sum of absolute deviations
Yield forecast for x = 80
1
Hyperbolic OLS
y(x) =
1.073
74.561
7.251
Hyperbolic LAD
y(x) =
1.072
74.461
7.197
3
Exponential OLS
y(x) = e2.799−0.01272x
1.088
75.733
5.936
4
Exponential LAD
y(x) = e2.781−0.01204x
1.093
75.702
6.159
2
16.796 1+0.01646x 16.875 1+0.01681x
As can be seen from Table 6 and from the graphs in Fig. 4, the mathematical model (8) is the most preferable both in terms of the minimum standard deviation and the minimum sum of absolute deviations. In addition, this model is characterized by the minimum value of the maximum deviation. Despite the fact that the least squares method is classical and is preferred in most cases, in the considered example for the studied data, the method of least absolute deviations turned out to be the best for all criteria. Therefore, the model (8) is chosen as the final best approximation.
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y
Initial dataset Hyperbolic OLS approximation (5) Hyperbolic LAD approximation (8) Exponential OLS approximation (7)
15
Exponential LAD approximation (9)
10
x
5 0
20
40
60
80
Fig. 4. Alternative mathematical models.
5 Conclusions The paper is devoted to comparative analysis of several mathematical models in order to choose the best of them. To build mathematical models, the classical method of ordinary least squares was used, as well as the method of least absolute deviations, which has recently become widespread among alternative methods of regression analysis. Initially, the test was used to check the data for linearity, which confirmed the need for more complex models. Using the parabola of the second degree gives unsatisfactory forecasting results due to its inconsistency with the geometric structure of the data (since the parabola tends to increase for yield values greater than 41.979). The hyperbolic and exponential models follow the geometric structure of the data. Since both models have two unknown coefficients, a 3D optimization paraboloid was used to minimize the sum of absolute deviations and standard deviation. It gives possibility to obtain four mathematical models according to the proposed optimization scheme. The result of the comparative analysis showed that the hyperbolic model based on the LAD method has been chosen as the best model. This model is characterized by the minimum standard deviation, the minimum sum of absolute deviations and the minimum value of the maximum deviation. The results of research and the proposed methodology can be used during the mathematical models building in various branches of scientific activity.
References 1. Weisberg, S.: Applied Linear Regression. Wiley, New York (2014)
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A Model of a Transport Multi-section Conveyor Based on a Neural Network Oleh Pihnastyi1(B)
, Anna Burduk2
, and Olha Ivanovska3
1 National Technical University “KhPI”, Kharkiv, Ukraine
[email protected]
2 Wroclaw University of Science and Technology, Wroclaw, Poland
[email protected]
3 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected]
Abstract. The construction of a model for predicting flow parameters of a distributed transport system is considered. The rationale for using a neural network to model a transport system consisting of a large number of individual sections is given. It is shown that the model of a transport system, which is based on a neural network, can be successfully applied to predict the flow parameters of a transport system consisting of a very large number of sections. The architecture of the neural network and the technique of forming a data set for its training are proposed. To train the neural network, the back propagation method of error was used. The effect of the occurrence of peak values for the flow parameters of the transport system is studied. The application of the model for predicting the peak values of the flow parameters of the transport system is considered. Assessed the results of predicting the output flow of the transport system. Keywords: Control · PDE-model · Distributed System · Conveyor
1 Introduction One of the important problems of the leading enterprises of the mining industry is the reduction of energy costs for the extraction and transportation of minerals [1–3]. The conveyor type transport system is the main method of transporting material [4–6]. The average cost of transporting a unit mass of material is 20% of the total cost of coal mining [7]. Energy costs are especially noticeable for long multi-sectional transport systems and branched transport systems, the characteristics of some of them are given in Table 1. As the length of the transport system increases, transportation costs increase. Energy saving is achieved by increasing the load level with the material of the transport system [1, 14, 15]. To increase the load level of the transport system, control the speed of the conveyor belt or the value of the output flow from the accumulating bunker entering the input of a separate section is used [16–18]. The design of the control system is simplified if the transport conveyor is divided into sections [2, 5]. For a qualitative analysis of the transport systems parameters, equations of system dynamics [16] and aggregated © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 215–226, 2023. https://doi.org/10.1007/978-3-031-36201-9_18
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Conveyors name
Length, km
Sections
From the Bu Craa mine to the coast at El Aaiún, [5]
128.7
11
The Henderson Coarse Ore Conveying System [3, 4]
24.0
3
Worsley Alumina, Australia [8]
105
2
Shondoni Overland Conveyor [9]
20.5
Curragh Conveyor Australia [10, 13]
20
1
Power, kW
Speed, m/sec
Capacity (t/h) 2000
12700
4250
4.5
2270
6.5
2400
7.5
2500
The Dead Sea Works 18.11 Company [11] Flying Belt conveyor 7.2 Bra [12]
1500
Sasol’s Impumelelo project [5, 13]
28
2
8950
Çöllolar Lignite Open Pit Mine, [2]
17.4
26
46300
Cement plant in Bangladesh [5]
16.5
1
Neyveli Lignite Corp 14.0 [2]
8
2520
5.4
Open Cast Mine Reichwalde [2]
13.5
6
19350
5.5
Coarse ore conveyor system Minera Los Pelambres [2]
12.7
3
25000
Kaltim Prima Coal, Indonesia [9]
12.6
1
8.4
4500
Twistdraai Coal Mine 12.4 [9]
3
5
2400
Enlow Fork, USA [9] 10.4
6
Kaltim Prima Coal, Indonesia [9]
9.2
6.5
2400 9350
6.5
6000 8700
10000 7.3
4500 (continued)
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Table 1. (continued) Conveyors name
Length, km
Sections
Power, kW
Speed, m/sec
Capacity (t/h)
Tianjin China Port Authority [6]
8.98
1
6000
5.6
6000
Obajana Cement [9]
7.0
5.0
2400
2.54
200
3.5
1500
Baumgartner Tunnel, 6.18 [6] Barcelona Tunnel, Spain [6]
4.71
1
equations are used. Before the development of the analytical model of the conveyor, the finite element method and the finite difference method were used to model the transport system that operated in non-stationary modes.
2 Formal Problem Statement An increase in the throughput and length of transport systems leads to the tendency separating a long transport conveyor to separate sections, and to use transport systems with a branched internal route containing a large number of converging and diverging sections. As a rule, the finite element method and the finite difference method were used to calculate the flow parameters of the transport conveyor, consisting of one section [4, 19–25]. The analytical model of the transport conveyor [26] can be successfully applied to simulate a transport system consisting of a large number of sections. In [27], an analytical model is used to describe a three-section branched conveyor. An analytical model of the main multi-section conveyor is presented in [28]. If the conveyor belt speed for each section of the transport system, consisting of a very large number of sections, is constant, then the PiKh-model is an excellent tool for simulating such a transport system. If the speed of the conveyor belt changes over time, then the analytical model will require significant computational resources. Additional resources are needed to solve the transcendental equation that determines the transport delay. This circumstance imposes a limitation on the maximum number of sections of the transport system. It should be assumed that the analytical model can be success-fully applied to a transport system consisting of several dozen separate sections. If the number of sections reaches the specified limit, then a more advanced model is required to design an effective control system for the flow parameters of the transport conveyor. This paper proposes the use of a neural network for modelling a multi-section conveyor. It should be noted that the researchers of conveyor systems have already shown interest in this class of models. Regression models [29–31] and neural network models [32–35] were used to analyze the strength characteristics of the transport conveyor. The feasibility of using the regression model of the conveyor section for the design of the belt speed control system was studied in paper [30]. In [36–39] models based on a neural network are considered to control the belt speed. The advantage of these papers is that the
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regression model and the model based on a neural network were built in accordance with the actual experimental measurement. However, when designing multi-section transport conveyors, the ability to use an experimental data set is associated with great difficulties. This work proposes a solution to this problem through the use of a data set formed on the basis of an analytical model.
3 Analytical Model of the Conveyor Section A conveyor is a type of production line [40]. The rock material moves along the transportation route at the same speed equal to the speed of the belt. The system of equations for modelling the movement of material along the transport route has the form [18, 26]: ∂[χ ]0 (t, S) ∂[χ ]0 (t, S) + a(t) = δ(S)λ(t), [χ ]1 (t, S) = a(t)[χ ]0 (t, S) ∂t ∂S +∞ 0, S < 0; δ(S)ds = 1, H (S) = [χ ]0 (t0 , S) = H (S)Ψ (S), 1, S ≥ 0, −∞
(1) (2)
where the parameters [χ ]0 (t, S) (t/m), [χ ]1 (t, S) (t/h) is the linear density of the rock and the flow of the rock at time t(h) at the point of the route determined by the coordinate S(m), S ∈ [0, Sd ], Sd is the length of the conveyor line. a = a(t) (m/h) is the conveyor belt speed; λ(t) is the intensity of the arrival of the rock on the conveyor line at the point with the coordinate S = 0. Using dimensionless variables [26] Ψ (S) Td S t [χ ]0 (t, S) , ψ(ξ ) = , γ (τ ) = λ(t) , (3) , ξ= , θ0 (τ, ξ ) = Td Sd Θ Θ Sd Θ Td λ(t) Td g(τ ) = a(t) , ϑ(τ ) = σ (t) , Θ = max Ψ (S), , H (ξ Sd ) = H (S), (4) Sd Sd Θ a(t) τ=
the system of Eqs. (1, 2) takes the dimensionless form ∂θ0 (τ, ξ ) ∂θ0 (τ, ξ ) + g(τ ) = δ(ξ )γ (τ ), θ0 (0, ξ ) = H(ξ ) · ψ(ξ ), ∂τ ∂ξ
(5)
where Td is the time during which the rock passes the entire transportation route. The system of equations (3) corresponds to a system of characteristics: γ (τ ) dξ d θ0 (τ, ξ ) = g(τ ), ξ |τ =0 = β, = δ(ξ ) , θ0 (0, β) = H(β)ψ(β). dτ dξ g(τ )
(6)
Integration of the system of equations (6) allows to obtain a solution in the form: γ (τξ ) θ0 (τ, ξ ) = H (ξ ) − H (−G(τξ )) + H (−G(τξ ))ψ(−G(τξ )), g(τξ )
(7)
τξ = G −1 (G(τ ) − ξ ) = τ − τξ , G(τ ) = ∫τ0 g(ω)d ω, G −1 (G(τ )) = τ,
(8)
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where τξ is the value of the transport delay. For a section whose length is ξ = 1, the output parameters can be calculated using the following formulas: θ0 (τ, 1) = (1 − H (1 − G(τ )))
γ (τ − τξ ) + H (1 − G(τ ))ψ(1 − G(τ )). g(τ − τξ )
(9)
For the steady-state mode, expressions (9) and (10) have a simpler form: θ0 (τ, 1) = γ (τ − τξ )/g(τ − τξ ), θ1 (τ, 1) = θ0 (τ, 1)g(τ ).
(10)
In the initial period of movement of the conveyor belt (1 − G(τ )) > 0, the output parameters of the conveyor section are determined by condition (5): θ0 (τ, 1) = ψ(1 − G(τ )) · θ1 (τ, 1) = g(τ )ψ(1 − G(τ ))
(11)
The obtained expressions for calculating the output parameters θ 0 (τ , 1) and θ 1 (τ , 1) of the conveyor section will be used to form a data set for training the neural network of a multi-section conveyor.
4 Conveyor Section Model Using a Neural Network The route diagram of a branched conveyor system consisting of several dozen separate sections is given in [41]. The transport route diagram contains four input conveyors (section m = 1, 2, 4, 5) and two output conveyors (section m = 7, 8). Inside the transport route there are nodes where the material flows converge, and nodes where the material flows diverge. For simplicity, let assume that the input bunker is not controllable. The material coming from the previous section into the bunker instantly leaves it and enters the next section. The bunker only provides the required direction of movement of the material [13]. Regulation of the value of the material flow is carried out with a change in the belt speed (10), (11). The model under consideration can be extended to an arbitrary number of sections, which makes it possible to model transport routes of a rather complex configuration. To describe the functioning of a separate section of the conveyor dimensionless variables (3), (4) are used. An additional section parameter is the value of the transport delay (8). The measurement of the transport delay in production conditions is associated with a number of difficulties [42]. The calculation of the value of the transport delay is performed by measuring the value of the belt speed, which is equivalent to solving Eq. (13). But solving Eq. (13) leads to the fact that the advantage of using a model based on a neural network is lost in comparison with the analytical model. It is the cost of computing resources associated with solving Eq. (13) that makes it difficult to use an analytical model to describe a transport system consisting of a large number of sections and opens the prospects for using a model based on a neural network. Therefore, if in real production conditions it is not possible to directly measure the amount of transport delay, this parameter is not advisable to include in a set of parameters for an individual section. Similar considerations can be given for the parameters of the input material flow γm (τ ) = θ1m (τ, 0) and the material output flow θ1m (τ, 1) for the inner sections (11), (12), the values of which are determined through the value of
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the transport delay τξ m . In addition, it should be pointed out that for calculating the parameters γm (τ ) = θ1m (τ, 0), θ1m (τ, 1) of the inner sections, the information should values and values of the conveyor belt speed gm (τ ) for the time interval be stored on their τ − τξ m ; τ . Taking into account the above arguments, as the nodes of the input layer of the neural network, we choose the intensity of the input stream of the material and the speed of the conveyor belt of the input sections (m = 1, 2, 4, 5). The values of the nodes of the output layer of the neural network will correspond to the values of the output flow for the final sections (m = 7, 8). Also, for simplicity, let’s assume that for a node that contains an input material stream and two outgoing material flows, the ratio of the values of the output flows is constant γ2 (τ )/γ3 (τ ) = const, γ3 (τ ) = 0. For the model of the transport system, let’s use the input parameters γm (τ ), gm (τ ), ξ1 of the input sections m = 1, 2, 4, 5. We introduce the notation for the parameters of the input layer of the neural network x3m−2 = γm (τ ), x3m−1 = gm (τ ), x3m = ξm , m = 1..M.
(12)
y1 = θ17 (τ, ξ7 ), y2 = θ18 (τ, ξ8 ).
(13)
The output parameters y1 and y2 correspond to the output material flows for sections m = 7.8. Using a similar approach, let’s determine the input and output parameters for an aggregated model of a transport system with an arbitrary number of sections. The topology of the hidden layer of the neural network for models of the conveyor section using part of the parameters (12) was considered in [35]. For forecasting, one hidden layer with six nodes was used. As an activation function, the Logistic function was selected. Weights are initialized with random values. In [37], the topology of the neural network 4-20-1 was considered. Four input parameters are used, including speed gm (τ ). The output parameter of is the output flow of the material. The inner layer contains 20 nodes. In [38], the topology of a neural network of the form (m1 − m2 − 14) or (4 − 9 − 14) was considered. The number of nodes in the input layer is m1 = 4. The number of nodes in the hidden layer is m2 = 2m1 +1. When choosing the topology in our work, the topology of the neural network [35, 37, 38] is analyzed and taken into account the recommendations given in [43, 44]. Ultimately, the topology 9-3-2 are chosen. This architecture corresponds to the transport system model of 4 sections with parameters x3m−2 = γm (τ ), x3m−1 = gm (τ ) and one node whose value is one. The hidden layer contains 3 nodes. The output layer contains 2 nodes (24). The activation function has the form f (x) = 4(1 + exp(−x))−1 . The length ξm of the conveyor is different.
5 Preparation of a Data Set for Training a Neural Network Learning a neural network requires a data set that contains a wide range of values. Such a dataset is difficult to obtain on existing transport systems. This is due to the following reasons: 1) transport system is unique in design. Transport system sections have different lengths; 2) the functioning of the transport system is carried out in a narrow range of flow parameters, in which provides economic feasibility; 3) lack of measuring equipment in the required places of the transport route; 4) confidentiality of production information.
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Therefore, to form a data set for training a neural network, let’s use the analytical model (5). The values of the parameters γm (τ ), gm (τ ) are oscillatory in nature, limited by the minimum and maximum values. Given this fact, as well as the experience of using periodic functions to analyze the accuracy of approximation of predicted results using a neural network [45], flow parameters γm (τ ), gm (τ ) that determine the state of the transport system and initial condition ψm (t) were represented as: mπ
mπ
g0 m , gm (τ ) = g0 m + sin mπ τ + , γm (τ ) = γ0 m + γ0 m sin mπ τ − 4 2 3 (14)
mπ 3+m 3+m , g0 m = , γ0 m = ψ0m = . ψm (t) = ψ0 m + ψ0 m sin mπ ξ + 4 8 24 (15) The data set for training the neural network is generated by using the analytical model (5). For the time interval 0 ≤ τ ≤ 2, the material output flow θ1m (τ, ξm ) γ (τ −τ ) m ξm gm (τ ), ξm ≤ Gm (τ ), θ1m (τ, ξm ) = gm (τ −τξm ) (16) gm (τ )ψm (ξm − G(τ )), ξm > Gm (τ ) This behavior of the output flow of the material θ1m (τ, ξm ) is explained by the presence of a transition period during which the value of the output flow is determined by the distribution of the material with a linear initial density ψm (ξ ) along the m-th section. The profile of the linear density θ0m (τ, ξm ) is formed by two parameters, intensity incoming material flow γm (τ ) and the speed gm (τ ) separated section. The duration of the transition period τtrm is determined by the speed gm (τ ) of the conveyor belt and the length of the section ξm (8). For the transition period, the output flow of the material with the conveyor section θ1m (τ, ξm ) is not related to the input flow of the material γm (τ ) and the speed gm (τ ). The transition period of the m-th section is characterized by the average transport delay time τtrm ∼ τξm . The transition period for the considered transport system (Fig. 1) can be estimated by the value τtr ∼ max(max(τtr1 , τtr2 ) + τtr3 , τtr4 , τtr5 ) + τtr6 + max(τtr7 , τtr8 ).
(17)
Substituting the values τtr m ∼ τξ m allows us to obtain the transition period τtr ∼ τtr1 + τtr3 + τtr6 + τtr7 ≈ 2.0 + 0.9 + 0.9 + 1.2 = 5. The values of the parameters of the transport system of the time interval 0 ≤ τ < τtr that corresponds to the time of the transition period should be excluded from the data set intended for training the neural network. The reason is that during this period of time the output material flow θ1m (τ, ξm ) is determined by the initial distribution of the material ψm (ξ ) along the transport route, and not by the parameters γm (τ ) and gm (τ ), when the output flow of the transport system is independent of the initial distribution of the material (5). The transport system is a distributed system. Transport delay in a distributed system plays an important role in generating output values of flow parameters. An interesting fact is that with a variable speed of the conveyor belt g1 (τ ) for the first section, the value of the transportation delay τ1 (τ ) is constant. The constant value of the delay at a variable belt speed is determined by one of the special cases of solving Eq. (8). The integration of the equation allows
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us to make an important conclusion that an increase in the frequency of oscillation of the speed of the conveyor belt leads to a decrease in the amplitude of the oscillation of the transport delay. The value of transport delay for sections m = 5, 6, 7, 8 can be considered constant. Thus, the absence of transport delay in the data set for training the neural network should not lead to a significant error.
6 Prediction Model Analysis For training the neural network, a data set was used, which was formed in accordance with the provisions of the previous partition. This data set is pushed in [46]. To calculate the weight coefficients of the neural network, the back propagation method of error was used. The updated weight value for each era is calculated based on its old value and error determined by the parameters of the output layer Wj,k,n+1 = Wj,k,n − α∇Ej,k,n , where the learning rate is equal α = 10−5 . The error Ej,k,n was distributed between the nodes in proportion to the values of the weight coefficients. Weight coefficients were initialized with random values in the range [0.0; 1.0] with uniform distribution density. For some parameter options, the learning process reached 300,000 eras. As the input nodes of the neural network for modeling, the characteristics γm (τ ), gm (τ ) of the sections 1, 2, 4, 5 on the interval 0 ≤ τ ≤ Tk = 100 are used. The results obtained correspond to a neural network with an architecture of 3–10–1 (the input layer contains three nodes with values 1, gm (τ ), γm (τ ); the output layer contains one node θ1m (τ, ξm ); the hidden layer contains r 2 10 nodes). The prediction error is estimated by MSEm = N1r N r=1 (zm,r − ym,r ) , where Nr = 9000 is the amount of data for testing. The value of the indicator is MSE1 = 0, 001; MSE2 = 0, 0009; MSE4 = 0, 001; MSE5 = 0, 0173. MSE5 value is significantly higher than MSE1 ; MSE2 ; MSE4 . A high value MSE5 corresponds to the presence of peak values of the output flow θ15 . Figure 1 and Fig. 2 show the prediction of the output flow, m = 7.8.
Fig. 1. Prediction of the output flow of material θ17 (τ, 1)
The prediction error is MSE78 = 0, 22. A model provides a satisfactory prediction for peak values of the output flow of the material θ18 (τ, 1). For the output flow θ17 (τ, 1),
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Fig. 2. Prediction of the output flow of material θ18 (τ, 1)
Fig. 3. Prediction of the output flow of material θ13 (τ, 1)
Fig. 4. Prediction of the output flow of material θ16 (τ, 1)
the model averages the peak small values of the function, while trying to repeat the behavior of the function for peak maximum values. We explain the difference in the
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prediction for flow θ17 (τ, 1) and θ18 (τ, 1) by the fact that the output flow θ17 (τ, 1) has a significant spread between the height of the group of maximum peak values and the group of minimum peak values. The approximation of the output flows θ13 (τ, 1), θ16 (τ, 1) is fairly well presented, for the intermediate sections m = 3 and m = 6. The prediction results are given in Fig. 3 and Fig. 4. The prediction error MSE3 = 0, 022 is and MSE6 = 0, 25. For last sections, the prediction error remains at the same level as the prediction error of the previous section. We attribute this fact to the fact that the flows after the sixth section diverge, and the total prediction error also decreases.
7 Conclusion The results of the analysis the model using a neural network show that the neural network is a good enough tool for predicting the value of flow parameters of an industrial transport system, which consists of a very large number of divided sections. An important consequence of the analysis of the PiKh–model of the transport system (5) is that peak loads in the transport system also arise for the case of a smooth change in the magnitude of the incoming material flow γm (τ ) and the speed gm (τ ) of the conveyor belt. The peak value is many times bigger than the amplitude of the background wave. The simplest explanation of the peak value effect can be built on the analysis of the simple superposition of the waves different length. The occurrence of this effect is one of the causes of damage to transport systems. One of the problems in studying the influence of the appearance of peak values on the parameters of the transport system is the difficulty of obtaining them under industrial conditions due to the unpredictable nature of the occurrence. The prediction model allows you to identify these situations and ensure their elimination by controlling the flow parameters of the transport system. To reduce the prediction error in the formation of the data set for training the neural network, the data that corresponds to the transition mode should be excluded. In this paper, the technique is given for the estimate the value of the duration transitional mode for the many sections transport system. The analysis of the transport system model shows that the reduction of prediction errors can be achieved by including as an additional node into the input layer the flow parameter, which is the speed of the conveyor belt. An important result of the conducted research is the conclusion that for transport systems with a high frequency of oscillation of the conveyor belt speed, the oscillation amplitude of the transport delay value is significantly less than the average value of the transport delay. This allows us to consider the duration of the transport delay as a constant value and, accordingly, to conclude that this parameter has a negligible effect on the prediction results. Such an assumption provides a reason why transport delay is optional for inclusion in the set of parameters of the input layer of the neural network. The assumptions obtained in this paper determine the prospects for further research.
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Simulation of the Impact of Migration Caused by the Russian War in Ukraine on the Dynamics of COVID-19 in Poland Dmytro Chumachenko1(B)
, Adam Wojciechowski2
, and Sergiy Yakovlev1,2
1 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected] 2 Lodz University of Technology, Lodz, Poland
Abstract. The novel coronavirus pandemic, which began at the end of 2019, is still the cause of morbidity and mortality for a large number of people in the world to this day. Many factors, including population migration, influence the dynamics of the epidemic process. The escalation of the war that Russia started in Ukraine in February 2022 caused the forced displacement of the population of Ukraine both within the country and in European countries. Poland received the most significant number of refugees. Therefore, this study aims to test the hypothesis of the influence of migration processes caused by the escalation of the Russian war in Ukraine on the dynamics of the COVID-19 epidemic process in Poland. To do this, a simple machine learning model was built to predict new and fatal cases of COVID-19. The model showed high accuracy when checking its adequacy both on the sample from January 25, 2022, to February 23, 2022, and on the sample from February 24, 2022, to March 25, 2022. This proves that the migration flows caused by the escalation of the Russian war in Ukraine are not a critical factor in the growth of the incidence of COVID-19 in Poland. Keywords: Epidemic model · machine learning · polynomial regression · public health informatics · COVID-19 · war
1 Introduction COVID-19 is a severe acute respiratory infection caused by SARS-CoV-2 coronavirus. The disease can occur both in the form of an acute respiratory viral infection of a mild course and in a severe form [1]. The most common symptoms of the disease are fever, dry cough, fatigue, and loss of taste and smell. The most common complication of the disease is viral pneumonia. Complications can lead to acute respiratory failure requiring oxygen therapy and respiratory support [2]. First reported in December 2019 in China, the infection has spread worldwide. In March 2020, WHO declared it a global pandemic [3]. At the beginning of September 2022, more than 615 million cases were registered worldwide, more than 6.5 million of which were fatal [4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 227–237, 2023. https://doi.org/10.1007/978-3-031-36201-9_19
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The first case of infection with the new coronavirus in Poland was on March 4, 2020, in a person who arrived from Germany. From March 14 to March 20, 2020, an epidemic threat regime was in effect in Poland, and from March 15, a cordon sanitaire was introduced at the state’s borders, restricting movement across the border [5]. From March 20 to May 15, 2020, an epidemic regime was in effect in Poland, and from May 16, 2020, a state of epidemic danger was in effect. On December 15, 2020, the Council of Ministers approved the National Immunization Program against COVID-19 [6]. The national vaccination campaign included four phases of vaccination of the population, starting with health workers and ending with the general vaccination of the rest of the adult population. As of September 2022, 6.2 million cases were registered in Poland, of which more than 117 thousand were fatal [4]. At the beginning of autumn 2022, 60.33% of the population was vaccinated, 59.74% received the complete vaccination course, and 36.71% received a booster dose. On February 24, 2022, Russia launched an invasion of Ukraine, striking most of the centers of Ukraine. The escalation of the war brought not only large-scale destruction and the killing of thousands of people but also affected the health care system of Ukraine [7]. Also, hostilities stimulated the forced migration of the population both within the country and to the countries of the European Union. Poland received the most refugees from Ukraine [8]. At the beginning of September 2022, more than 6.1 million people crossed the border with Poland, and almost 1.4 million citizens of Ukraine received the status of temporary protection in the country [9]. The escalation of Russia’s war in Ukraine has also spurred the spread of COVID-19. People hiding from the bombings were in bomb shelters without keeping their distance or adhering to preventive measures. During the evacuation, people rode overcrowded trains and stood in long lines to cross the border. At the same time, when entering a new country, there was practically no control of documents, and it was impossible to trace the medical history. Therefore, this study aims to test whether the forced migration of the population of Ukraine, caused by the escalation of the Russian war, influenced the dynamics of the epidemic process in Poland. The pandemic has also spurred the development of data-driven medicine, activating many research groups worldwide. Thus, models, methods, and information technologies have been developed aimed at solving problems of medical diagnostics [10], analysis of medical data [11], processing of medical images [12], determining the information content of factors affecting the dynamics of the epidemic process [13], the study of molecular structures [14], the study of social factors influencing the development of a pandemic [15], the study and optimization of the transport system [16], etc. Therefore, to test the hypothesis formulated in this study, it is proposed to use the simplest machine learning method to model the dynamics of the spread of COVID-19 and the dynamics of mortality. Research is part of a complex, intelligent information system for epidemiological diagnostics, the concept of which is discussed in [17].
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2 Materials and Methods To test the hypothesis of the influence of migration processes caused by the escalation of the Russian war in Ukraine on the dynamics of the COVID-19 epidemic process in Poland, the following methodology is proposed: Develop a simulation model of the epidemic process of COVID-19 based on the method of polynomial regression to predict cumulative new and fatal cases of COVID-19. To test the adequacy of the model, calculate the forecast dynamics of cumulative new and fatal cases of COVID-19 in Poland 30 days before the start of the escalation of the Russian war in Ukraine. Assess the accuracy of the forecast. To calculate the forecast dynamics of cumulative new and fatal cases of COVID-19 in Poland for 30 days after the start of the escalation of the Russian war in Ukraine. Assess the accuracy of the forecast. To compare the accuracy of those calculated using forecasts for different periods. Assess the difference in accuracy. This study is the first stage in the study of the impact of the Russian war in Ukraine on the dynamics of infectious diseases. Therefore, to build a simulation model for COVID19, the simplest machine learning method, polynomial regression, was chosen. Regression is a statistical dependence of the mathematical expectation of a random variable on the values of another random variable or several random variables. Regression aims to find the relationship between the original variable and several regressors [18]. Therefore, regression models are convenient for finding and describing the trend in the time series analysis. At the same time, time is an independent argument. The simplest type of regression model is linear regression. The basis of the linear regression model is the hypothesis of the existence of a discrete external factor that impacts on the process under consideration, and the relationship between the factor and the process can be represented as a linear function [19]. The linear regression model has the following form: Y (t) = α0 + α1 ∗ X (t) + εt ,
(1)
where α 0 and α 1 are regression coefficients, εt is the model error. However, linear regression can only be applied when there is a linear relationship between both dependent and independent variables. In the case of non-linearity, a polynomial regression model should be applied [20]. The polynomial regression model has the following form: Y (t) = α0 + α1 ∗ x1 (t) + α2 ∗ x22 (t) + α3 ∗ x33 (t) + αn ∗ xnn (t),
(2)
where α 0 and α 1 are regression coefficients, y is the dependent variable, x is the independent variable. Thus, using a polynomial regression model, the non-linear relationships between variables can be simulated and also, they can be used to predict time series that have non-linearity. To assess the accuracy of the model, mean absolute percentage errors (MAPE) were calculated for various forecast durations. MAPE is a metric that tells how far apart
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our predicted values are from our observed values in a regression analysis, on average. MAPE looks like this: MAPE =
1 n |Oi − Pi | ∗100% , i=1 n Oi
(3)
where Pi is the predicted value for the ith observation, Oi is the observed value for the ith observation, n is the sample size
3 Results Data on the incidence and mortality of COVID-19 in Poland, presented in the World Health Organization COVID-19 Dashboard [21], were used for the experimental investigation. The software implementation of the model is made in the Python programming language. To verify the model, a forecast of the dynamics of the COVID-19 epidemic process was calculated 30 days before the start of the escalation of the Russian war in Ukraine. Figure 1 shows the forecast for cumulative new cases of COVID-19 in Poland from January 25, 2022, to February 23, 2022.
Fig. 1. Forecast of COVID-19 cumulative new cases in Poland, 25.01.2022 – 23.02.2022
Figure 2 shows the forecast for cumulative COVID-19 fatalities in Poland from January 25, 2022, to February 23, 2022. To assess the adequacy of the constructed model, forecast errors for 7, 10, 20, and 30 days were calculated for cumulative new and fatal cases of COVID-19 in Poland. Accuracy values are presented in Table 1.
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Fig. 2. Forecast of COVID-19 cumulative death cases in Poland, 25.01.2022 – 23.02.2022
Table 1. MAPE of forecast for the period 25.01.2022 – 23.02.2022 Duration of forecast
New cases
Death cases
7 days
0,7063%
0,5275%
10 days
0,8984%
0,5998%
20 days
1,5942%
0,7369%
30 days
1,4177%
0,7749%
To assess the impact of migration processes on the dynamics of COVID-19, a forecast of the epidemic process of COVID-19 in Poland was calculated for 30 days after the start of the escalation of the Russian war in Ukraine. Figure 3 shows the forecast for cumulative new cases of COVID-19 in Poland from February 24, 2022, to March 25, 2022.
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Figure 4 shows the forecast of cumulative COVID-19 fatalities in Poland from February 24, 2022, to March 25, 2022. Table 2 presents the accuracy scores for predicting cumulative new and fatal cases of COVID-19 in Poland from February 24, 2022, to March 25, 2022.
Fig. 3. Forecast of COVID-19 cumulative new cases in Poland, 24.02.2022 – 25.03.2022
Fig. 4. Forecast of COVID-19 cumulative death cases in Poland, 24.02.2022 – 25.03.2022
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Figure 5 shows the forecast of daily new cases of COVID-19 in Poland from February 24, 2022, to March 25, 2022. Figure 6 shows the forecast of daily COVID-19 deaths in Poland from February 24, 2022, to March 25, 2022. Table 2. MAPE of forecast for the period 24.02.2022 – 25.03.2022 Duration of forecast
New cases
Death cases
7 days
0,3458%
0,0926%
10 days
0,3865%
0,1187%
20 days
0,6761%
0,1094%
30 days
1,0101%
0,1315%
Fig. 5. Forecast of COVID-19 daily new cases in Poland, 24.02.2022 – 25.03.2022
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Fig. 6. Forecast of COVID-19 daily death cases in Poland, 24.02.2022 – 25.03.2022
To assess the absolute indicators, the deviations of the forecast dynamics of COVID19 from February 24, 2022 to March 25, 2022 from the actual incidence rates in Poland were calculated. Figure 7 shows the forecast deviation for cumulative new cases. Figure 8 shows the forecast deviation for cumulative fatalities.
Fig. 7. Deviation of the forecast of cumulative new cases of COVID-19 in Poland
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Fig. 8. Deviation of the forecast of cumulative death cases of COVID-19 in Poland
4 Conclusions The Russian invasion of Ukraine on February 24, 2022, started the most significant war in Europe since World War II, bringing with it problems. One of the consequences of a military invasion is stimulating the spread of infectious diseases. Therefore, this study tested the hypothesis of the influence of migration processes caused by the Russian war in Ukraine on the dynamics of the COVID-19 epidemic process in Poland, which received the most significant number of migrants from Ukraine. For this, a simulation model of the COVID-19 epidemic process was built based on the simplest statistical machine learning method - polynomial regression. The model was verified on data on the incidence and mortality of COVID-19 in Poland 30 days before the start of the escalation of the war. The model’s accuracy from January 25, 2022, to February 23, 2022, was 98.41% - 99.3% for cumulative new cases, and 99.48% - 99.23% for cumulative fatal cases. The model was applied to calculate the predictive dynamics of COVID-19 in Poland for 30 days after the start of the escalation of the war. The model’s accuracy for the period from February 24, 2022, to March 25, 2022, was 98.99% - 99.66% for cumulative new cases, and 99.87% - 99.91% for fatal cases. Thus, the simulation showed a slight difference in forecasting accuracy for the periods before and after the escalation of the Russian war in Ukraine. This suggests that migration flows from Ukraine to Poland are not a critical factor in the spread of COVID-19 incidence. This study is the initial stage of the investigation of the impact of the Russian war in Ukraine on the dynamics of infectious diseases. In the future, it is planned to develop
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models to study specific factors that may affect changes in the patterns of the spread of infectious diseases among the population. Acknowledgments. The study was funded by the National Research Foundation of Ukraine in the frame-work of the research project 2020.02/0404 on the topic “Development of intelligent technologies for assessing the epidemic situation to support decision-making within the population biosafety management”.
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14. Tkachenko, A.S., et al.: Semi-refined carrageenan promotes generation of reactive oxygen species in leukocytes of rats upon oral exposure but not in vitro. Wien. Med. Wochenschr. 171(3–4), 68–78 (2020). https://doi.org/10.1007/s10354-020-00786-7 15. Zakharchenko, O., et al.: Multifaceted nature of social media content propagating COVID-19 vaccine hesitancy: Ukrainian case. Proc. Comput. Sci. 198, 682–687 (2021) 16. Davidich, N., et al.: Advanced traveller information systems to optimizing freight driver route selection. In: Proceedings - International Conference on Developments in eSystems Engineering, DeSE, pp. 111–115 (2020). https://doi.org/10.1109/DeSE51703.2020.9450763 17. Bazilevych, K.O., et al.: Intelligent decision-support system for epidemiological diagnostics. I. A Concept of architecture design. Cybern. Syst. Anal. 58(3), 343–353 (2022). https://doi. org/10.1007/s10559-022-00466-x 18. Boyle, J., et al.: Regression forecasting of patient admission data. In: Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 3819–3822 (2008). https://doi.org/10.1109/IEMBS.2008.4650041 19. Olive, D.J.: Linear Regression, 1st edn. Springer Cham (2017). 494 p. https://doi.org/10.1007/ 978-3-319-55252-1 20. Ostertagova, E.: Modelling using polynomial regression. Proc. Eng. 48, 500–506 (2012). https://doi.org/10.1016/j.proeng.2012.09.545 21. WHO COVID-19 Dashboard. Geneva: World Health Organization (2020). https://covid19. who.int/. Accessed 01 Sept 2022
Limiting Properties of Boundary Value Problems Solutions for Some Higher-Order Equations Nina Padalko1
, Anatoliy Padalko2
, and Halyna Padalko3(B)
1 Lesya Ukrainka Volyn National University, Lutsk, Ukraine 2 Lutsk National Technical University, Lutsk, Ukraine 3 National Aerospace University – Kharkiv Aviation Institute, Kharkiv, Ukraine
[email protected]
Abstract. The work is devoted to studying the limiting properties of functions as solutions of some differential equations in plane domains using methods developed in approximation theory. We study solutions of one boundary value problem for some higher-order equations. The article aims to study the behavior of solutions in the half-plane of higher-order differential equations with different requirements of smoothness in coordinates, depending on the properties of the limiting data. We have found such a form of the inverse theorem, from which one could see the influence of various requirements of smoothness in coordinates on the differential properties of limit functions. Keywords: higher-order equations · boundary value problem · limiting property
1 Introduction The boundary value problem is a differential equation or a system of differential equations with given linear relationships between the values of the desired functions at the beginning and end of the integration interval [1]. The solution of the boundary value problem is sought in the form of a linear combination of solutions of homogeneous Cauchy problems corresponding to a given equation for linearly independent vectors of initial conditions and a solution of an inhomogeneous Cauchy problem with arbitrary initial conditions [2]. The problem of the existence and uniqueness of solutions to boundary value problems is relevant in the theory of ordinary differential equations [3]. For equations of higher orders and in the linear non-autonomous case, finding adequate conditions for the existence of a solution to a boundary value problem is a significant problem. It requires information about the fundamental system of solutions of a homogeneous equation. Mathematical models of various phenomena using differential equations with deviating arguments have been known for a long time and are still used [4–6]. At the beginning of the study of such an instrument, the deviation of the argument was constant. However,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 238–248, 2023. https://doi.org/10.1007/978-3-031-36201-9_20
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arbitrary continuous or measurable deviations, as well as functional differential equations, began to be used [7]. In such equations, the values of the highest derivative may depend on the behavior of the process under study over the entire given time interval. Therefore, this study is devoted to studying the limiting properties of functions as solutions of certain differential equations in plane domains using methods developed in approximation theory. We study solutions of one boundary value problem for some higher-order equations. Purpose of the work: to study the behavior of solutions in the half-plane of differential equations of higher orders with different requirements of smoothness in coordinates, depending on the properties of the boundary data.
2 Problem Statement Let’s consider the boundary value problem: to find a solution U (x, y) to the equation (−1)l+1
2s ∂ 2l U s+1 ∂ U + = 0, (−1) ∂x2l ∂y2s
where l, s ∈ N, y > 0, −∞ < x < ∞, which satisfies the conditions ∂ j U = ϕj (x)..(j = 0, 1, . . . , s − 1), ∂yj y =0
(1)
(2)
limited in G: = {y > 0, −∞ < x < ∞} and such as lim U(x, y) = 0.
y→∞
(3)
For l = s = 1 problem (1)–(2), there is a classical Dirichlet problem for the Laplace equation in the upper half-plane. If l = s, then, as can be seen from (1), the requirements for smoothness of the desired function U (x, y) are different for different coordinates. In this regard, the question arises of how this fact affects the boundary properties of the solution to the boundary value problem (1)–(2). Previous studies have shown that when l, s ∈ N solving problem (1)–(2) is given in the form s−1 U(x, y) = Uj (x, y), (4) j=0
where U j (x, y) is the functions that are built after the corresponding ϕ j and have studied the limiting properties of functions U j (x, y) depending on the properties of the functions ϕ j , −∞ < x < ∞. It was established in previous studies that in the half-plane y > 1 the decoupling U (x, y) is an infinitely differentiating function, and its properties do not depend on the properties of the limit functions ϕ j (x). Therefore, the decisive factor is the behavior of the solution in the horizontal strip 0 < y < 1, −∞ < x < ∞, in which the differential-difference properties of the solution U (x, y) are completely determined by the corresponding properties of the limit functions ϕ j (x), −∞ < x < ∞.
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Let us find a form of the inverse deviation theorem from which we can see the influence of various smoothness requirements in coordinates on the differential properties of limit functions. This theorem has the inverse theorem of polynomial approximation of functions. Denote by F the operator F: {ϕ} → {U ϕ }, which associates a function ϕ (x, 0) defined on y0 = 0, with a function U ϕ : = U ϕ (x, y), defined on G: = {y > 0, −∞ < x < ∞}. Since there is a bijective correspondence between the lines y0 = 0, y1 = const > 0, x ∈ (-∞; ∞), the function U ϕ (x, y1 ) can be considered as given on y0 = 0. We define a function U ϕ (x, y1 ) on the real axis OX and again consider the operator of continuation F of the function U ϕ (x, y1 ) into the region G, then we denote the value of the operator UUϕ(x, y ) (x, y), (x, y) ∈ G. 1
(5)
Let the value of the operator F have the properties: Uϕ+g (x, y) = Uϕ (x, y) + Ug (x, y),
(6)
where ϕ = ϕ (x), g = g(x) ∈ L p , 1 ≤ p ≤ ∞, −∞ < x < ∞. For arbitrary points there is a valid equality UUϕ(x, y ) x, y2 = UUϕ(x, y ) x, y1 .
(7)
1
2
For a fixed n ∈ N and given function ϕ, the value U ϕ of the operator F exist in the range G n times almost everywhere function differentiable with respect to x in case 1 ≤ p < ∞ and a continuously differentiable function for p = +∞, and the inequality holds for the total derivative: n d Uϕ (x, y(.)) ≤ M ϕL , (8) n p dx yσ Lp
where 0 < σ ≤ n, M = const > 0 and ϕLp =
∞ −∞
1 |ϕ| dx p
p
.
The properties (6), (7) imply the truth of the equality Uϕ−Uϕ (x, y1 ) x, y2 − Uϕ−Uϕ (x, y2 ) x, y1 = Uϕ x, y2 − Uϕ x, y1 .
(9)
(10)
Relation (10) is essentially used in the proof of the following theorem.
3 Results 3.1 Theorem Let G: = {y > 0, -∞ < x < ∞}, ϕ (x) ∈ L p (−∞, ∞), 1 ≤ p ≤ ∞ be the value U ϕ of the operator F: {ϕ} → {U ϕ } has the properties (6), (7), (8) and for fixed integers k ≥ 0 the condition is performed Uϕ (·, y) − ϕ(·) ≤ Ayk ω(y), (11) L p
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y > 0, A = const > 0, where ω (t), t > 0 is type function order n ∈ N, which under the k ≥ 1 satisfying condition ∞ ω(t) dt < ∞, (12) t 0 and the condition: ψ(t, σ) :=
ω(t) tσ+1
(13)
is monotonically descending for t > 0. Then the function ϕ almost everywhere at (−∞, ∞), coincide with a function that has a completely continuous derivative ϕ (k – 1) (x) and the derivative ϕ (k) (x) ∈ L p (−∞, ∞), 1 ≤ p ≤ ∞ (continuous for p = + ∞), and its ωn (ϕ (k) ; y) order n satisfies the relation
1 C1 yn y ω(u)du 1 σ+1 , k = 0, (k) y ω(u)duu 1 ω(u)du ωn (ϕ ; y) ≤ 0 0 is fixed. Let us divide the interval (0; d/2] by a system of points d/2, d/22 , …, d/2N and choose a natural N > 1 so that the relation is performed d d < h ≤ N −1 . N 2 2 Given the identity d ϕ(x) = Uϕ x, y0 (x) + 2
(16)
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d d Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j 2 2 j=1 d + ϕ(x) − Uϕ x, y0 (x) + N , 2
+
N−1
compose for the function ϕ (x) the difference of the order n with a step h: n n r n h (ϕ) := ϕ(x + rh) := S1 (x, h) + S2 (x, h) + S3 (x, h), (−1) r=0 r
(17)
(18)
where n d d n = , Uϕ x + rh, y0 (x + rh) + S1 (x, h) := nh Uϕ x, y0 (x) + (−1)r r=0 r 2 2 (19) N−1 d d (20) S2 (x, h) := nh Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j , j=1 2 2 d S3 (x, h) := nh ϕ(x) − Uϕ x, y0 (x) + N . (21) 2
Let us estimate L p the norm of each of the quantities S i (x,h), i = 1, 2, 3. Since dk Uϕ (x, y(x)) dxk
(22)
is a derivative of the function U ϕ (x,y(x)), −∞ < x < ∞, with respect to the independent variable x, then denoting d , (23) ω(x) = Uϕ x, y0 (x) + 2 and also giving the difference of the n-th order by the n-fold integral and using (8), we obtain the estimate (24) where K 1 = const > 0 which does not depend on h. From the condition of theorem (6) with k = 0 monotonicity of ω (t) and choice of N, we obtain n d r n S3 (·, h)Lp = (−1) ϕ(x + rh) − Uϕ (x + rh, y0 (x + rh) + N ) r 2 r=0 n d d n ≤ (25) ω N = K2 ω N ≤ K2 ω(h), r=0 r 2 2
Limiting Properties of Boundary Value Problems Solutions
where K 2 > 0 does not depends on h. Let us note d d vj (x) := Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j , 2 2
243
(26)
than S 2 (x,h) can be presented as S2 (x, h) =
N−1 j=1
N−1 nh vj (x) = j=1
h
h
(n)
0
0
∂n vj (x + t1 + . . . + tn )dt1 . . . dtn . ∂xn (27)
From identity (3), property (8) and the condition of theorem (11) for k = 0 we obtain
(28)
Let us estimate the total majorant, taking into account that ψ(t, σ) =
ω(t) σ+1
(29)
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is a monotonically descending function for t > 0,
(30)
The lower Darboux sum for the Riemann integral 1 ω(u)du 1 uσ+1 N+1
(31)
2
is written in parentheses for the partition of the interval under consideration. Than 1 ω(u)du n S2 (·, h)Lp ≤ K3 h , uσ+1 h where K 3 = const > 0, does not depend on h. Thus, for k = 0 we get 1 1 ω(u)du ω(u)du n ≤ K h , ωn (ϕ, h) ≤ K1 hn + K2 ω(h) + K3 hn 4 σ+1 u uσ+1 h h
(32)
(33)
where K 4 > 0, does not depend on h. In case k = 0 the theorem is proved. Case k ≥ 1. First, let us prove that under equality (15) by passing to the limit in N → ∞ we obtain the representation of the function ϕ (x): ∞ d d d Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j . + ϕ(x) = Uϕ x, y0 (x) + j=1 2 2 2 (34) Valid for the private sum S n (x) of this series N −1 d d d Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j , + SN (x) = Uϕ x, y0 (x) + j=1 2 2 2 (35) using (15) and (11), we obtain (36) This implies that the series (32) coincides on average (in the metric L p ) with the function ϕ (x), −∞ < x < ∞.
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For p = + ∞, series (32) coincides uniformly to a continuous function ϕ (x), −∞ < x < ∞. In the case 1 ≤ p ≤ ∞, according to F. Riesz’s theorem, we find that the series coincides almost everywhere on −∞ < x < ∞ till ϕ (x). For arbitrary q ∈ N, q ≤ k consider the series ∞ d d d (q) (q) (q) Uϕ x, y0 (x) + j+1 − Uϕ x, y0 (x) + j + Uϕ x, y0 (x) + j=1 2 2 2 (37) is composed of total order derivatives with respect q to x terms of series (32). Using property (8), differentiated by x, identity (3), condition of theorem (11), we consequently obtain
(q) (q) d − Uϕ ·, y0 (x) + 2dj Uϕ ·, y0 (x) + 2j+1 Lp
(q) (q) d d = x, y0 (x) + j Uϕ(·)−Uϕ ·,y (x)+ d x, y0 (x) + 2j+1 − Uϕ(·)−U 2 d 0 ϕ ·,y0 (x)+ j+1 2j 2 Lp
(q) (q) d d
x, y (x) + j+1
x, y (x) + j ≤ U −U 0 0 d 2 2 ϕ(·)−Uϕ ·,y0 (x)+ 2dj Lp ϕ(·)−Uϕ ·,y0 (x)+ j+1 2 Lp ⎡
⎤ ≤ M⎣
ϕ(·)−Uϕ ·,y0 (x)+ dj 2
σ
≤ MA
d 2j+1
d 2 j
k ω dj
2σ d 2j+1
+
Lp
d 2j+1
+
d ϕ(·)−Uϕ ·,y0 (x)+ j+1 2 σ
k
d ω j+1 σ 2 d 2j
d 2j
< M(k, d)ω
Lp
⎦
1 , 2j
(38) where M (k,d) = const > 0 does not depend on j. Taking into account the fact that for all j = 1, 2, … the estimate fair
2−(j−1) 2−j
we obtain
1 1 ω(u) du ≥ ω j , u 2 2
d d (q) ·, y0 (x) + j+1 − Uϕ ·, y0 (x) + j 2 2 Lp 2−(j−1) ω(u) du. ≤ 2M(k, d) u 2−j
(39)
(q) Uϕ
(40)
Since, taking into account that d/2N < h ≤ d/2N−1 the series ∞
2−(j−1)
j=1 2−j
ω(u) du = u
0
1
ω(u) du, u
(41)
coincides, it follows from here that the series (35) coincides by −∞ < x < ∞ on average in the understanding of the L p metric (if convergence p = + ∞ is uniform). According
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to the Riesz theorem, there is a sequence S q nm (x) of partial sums of the series (35), which, for all q = 1, 2, …, k, coincides almost everywhere up to ϕ q (x) ∈ L p (−∞; ∞). Assuming that x 0 is one of the points of convergence for all q = 1, 2, …, k let’s consider the relation
(42)
From this relation we get that for m → ∞ almost for all x ∈ (−∞, ∞) for q = 1, 2, …, k we have x
ϕq−1 (x) − ϕq−1 (x0 ) = ∫ ϕq (t)dt.
(43)
x0
Since almost everywhere ϕ 0 (x) = ϕ(x), than ϕ(x) almost everywhere on (−∞, ∞) coincides with a function that has an absolute continuous derivative of the (q−1) order and derivative of the q: ϕ q (x) = ϕ q (x) ∈ L p order, which is the average boundary of the sequence (S (k) n (x)). This proves that almost everywhere for all x ∈ (−∞, ∞) the equality is true ∞ d d d (k) (k) + − U . x, y U x, y x, y ϕ(k) (x) = U(k) + + + (x) (x) (x) 0 0 0 ϕ ϕ ϕ j=1 2 2 2 (44) Let us estimate the integral modulus of the n-th order of derivative ϕ (k) (x). To do this, we note that, similarly to the previous equality, we have an equality that is true for almost all x ∈ (−∞, ∞). d ϕ(k) (x) = Uϕ(k) x, y0 (x) + j 2 ∞ d d Uϕ(k) x, y0 (x) + m − Uϕ(k) x, y0 (x) + m−1 . + m=j+1 2 2 From this equality, using (38), we obtain (k) ϕ (·) − U(k) ·, y (x) + d 0 ϕ 2j −Uϕ(k)
d ·, y0 (x) + m−1 2
≤ 2 M(k, d)
∞
Lp
∞ (k) d ≤ Uϕ ·, y0 (x) + 2m m=j+1
Lp 2−(m−2)
m=j+1 2−(m−1)
ω(u) du u
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2−(j−1)
= 2M(k, d) 0
ω(u) du ≤ 2n+1 M(k, d) u
2−j 0
ω(u) du. u
247
(45)
Let us introduce the notation (t) := 0
t
ω(u)du . u
(46)
A function Ω(t) has the properties of the modulus of smoothness of an n order function. In this case, the inequality (k) ϕ (·) − U(k) ·, y (x) + d ≤ B d , (47) 0 ϕ 2j Lp 2j with B = const > 0, allows us to assume that we have L p , an estimate of the deviation of the function ϕ (k) (x) from d (k) Uϕ x, y0 (x) + j (48) 2 type of estimate in the condition of the theorem (for k = 0). Therefore, based on the case k = 0 considered earlier, we obtain the statement 1
(u)du n (k) , C1 = const > 0. (49) ωn ϕ ; y ≤ C1 y uσ+1 y The transformation of the right side of the last inequality leads to estimate (11) for k ≥ 1. Indeed, for integration by parts we set u1 : = Ω(u), dv1 = du/uσ +1 , then du1 = (ω(u)du)/u, v1 = –1/σuσ , we obtain y 1 1 (u)du ω(u)du n n (u) C1 y = C1 y + σ+1 uσ+1 uσ+1 1 y y σu y 1 ω(u)du ω(u)du + yn . (50) ≤ C2 yn−σ u uσ+1 0 y The theorem is proved.
4 Conclusions Consequently, the solutions of one boundary value problem for some higher-order equations and the relationship between the increase in smoothness in coordinates and the limiting properties of the solutions are investigated. We have found such a form of the inverse theorem, from which one could see the influence of various requirements of smoothness in coordinates on the differential properties of limit functions.
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References 1. Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22(1), 64–69 (2009). https://doi.org/10.1016/j.aml.2008.03.001 2. Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic Boundary Value Problems. LNM, vol. 1991, p. 423. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12245-3 3. Sevindir, H.K., Demir, A.: The solution of initial boundary value problem with time and spacefractional diffusion equation via a novel inner product. Adv. Math. Phys. 2018, 1389314 (2018). https://doi.org/10.1155/2018/1389314 4. Chumachenko, D., et al.: Investigation of statistical machine learning models for COVID19 epidemic process simulation: random forest, K-nearest neighbors, gradient boosting. Computation 10(6), 86 (2022). https://doi.org/10.3390/computation10060086 5. Chumachenko, D.: On intelligent multiagent approach to viral Hepatitis B epidemic processes simulation. In: Proceedings of the 2018 IEEE 2nd International Conference on Data Stream Mining and Processing, DSMP 2018, pp. 415–419 (2018). https://doi.org/10.1109/ DSMP.2018.8478602 6. Chumachenko, D., Sokolov, O., Yakovlev, S.: Fuzzy recurrent mapping in multiagent simulation of population dynamics systems. Int. J. Comput. 19(2), 290–297 (2020) 7. Shi, Y.G., Gong, X.B.: Linear functional equations involving Babbage’s equation. Elemente der Matematik 69, 195–204 (2014). https://doi.org/10.4171/EM/263
Method for Correcting the Mathematical Model in Case of Empirical Data Asymmetry Valeriyi Kuzmin , Yuliia Petrova , Maksym Zaliskyi(B) and Oleksandr Lavrynenko
,
National Aviation University, Huzar Av. 1, Kyiv 03058, Ukraine [email protected]
Abstract. To obtain a reliable and accurate mathematical model describing the results of an observation, it is necessary to repeat a large number of experiments. In the same time, such experiments may be very expensive and not always possible. However, the usage of special processing techniques makes it possible to increase accuracy for various estimates while processing datasets with small sample size. The main purpose of this paper is to show the necessity to correct the mathematical model in the case of asymmetry in the range of the observed data. The paper proposes the method for detecting this asymmetry and calculating an appropriate correction for the mathematical model. The asymmetry correction is carried out using basis function that takes into account the statistical characteristics of the observed data and quantiles of normal distribution. The results of research can be used during data processing in case of small sample size for various data-driven branches. Keywords: Mathematical Model Building · Data Processing · Data Correction · Asymmetry of Probability Distribution · Outliers
1 Introduction Numerical methods are often used during the mathematical models building in various fields of activity, and the problem of statistical processing of experimental data is very hot, especially in the case of insufficient data. This task is very important for applications where high accuracy is required or the variance of individual measurements is very high [1, 2]. As a rule, to obtain a reliable and most accurate model of the results of an experiment, it is necessary to conduct a large number of experiments, which may be expensive and not always possible [3, 4]. However, the usage of special mathematical processing (the application of new approaches and methods to data processing) makes it possible to obtain more accurate estimates while processing experimental data, especially in the case of datasets with a small sample size. It should be noted that the most common method for approximation of experimental data is the least squares method (ordinary and weighted) [5], which gives good and efficient results if the following postulates are respected, namely, the symmetry of the experimental data variance and the normal law of probability distribution. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Nechyporuk et al. (Eds.): ICTM 2022, LNNS 657, pp. 249–260, 2023. https://doi.org/10.1007/978-3-031-36201-9_21
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2 Literature Analysis and Problem Statement The process of mathematical model building or finding an approximating function must always take into account the features of the probability distribution for the studying data range [6, 7]. The usage of new approaches, technologies and even new methods is essential in the construction of mathematical models [8]. One of the main tasks of mathematical statistics is a comprehensive study and analysis of data [9–12]. At the same time, it is known that complete characteristic of any sample of experimental data is the law of probability distribution within the data range [13]. The question of the distribution law identification using data sample has been studied for a long time [14]. At the same time, it should be noted that, despite the huge number of methods, researchers rarely manage to obtain a consistent assessment of the data in terms of their belonging to any particular distribution law [1]. The analysis of literature [15–20] for this problem shows that in practice the procedure for distinguishing between symmetric and asymmetric distribution laws is quite well developed. It should be noted that the mentioned tasks are solved in various branches of industry as shown in [21– 26]. The accurate and reliable models building allows cleaning the observed dataset and reducing the forecasting error [27, 28]. The development of new statistical methods and approaches focused on the processing a small number of observations was caused by the impossibility of usage traditional methods of mathematical statistics, which are not suitable for small samples processing. Nowadays scientists raise the problem of the urgent need to apply new approaches and methods to the processing of small samples [29]. Among the new approaches is the method of rectangular contributions using to construct an estimates of the distribution function. Research on the possibilities of this method led to the development of a series of other methods based on the usage of contribution functions. But for the case when there are several cross sections and each section has a different number of repeated observations, it is interesting to reduce all cross sections to one basic section and further consider all experimental points as one cross section. After that, for the obtained base section, it is necessary to determine the probability distribution law with greater reliability and obtain its analytical description. This paper considers new approaches and criteria that allow taking into account the influence of asymmetry of probability distribution and the insufficiency of empirical data. The aim of this paper is to introduce a correction, which is determined by optimizing several variants of approximation by equidistant functions and finding the final one, which satisfies the equality principle of the number of points above and below the approximating function within the data range.
3 Methods and Materials Let build a rational mathematical model using the example of observed data given in [1]. These data are presented in Table 1. The data of this example are remarkable in the sense that six experimental values are given in each section. In most cases, while mathematical models building, the classical method of least squares is used, which assumes that the law of distribution of empirical points obeys the normal law of probability distribution.
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The method of least squares is given preference even when these assumptions are not met [9]. Table 1. Initial dataset. Section number
Explanatory variable, x
Response, y
Average yield
Standard deviation
Median
1
0
2,35 2,75 2,89 3,85 5,52 5,94
3,88
1.518679
3,37
2
12
4,31 4,78 4,84 5,83 6,51 7,52
5,63
1.222741
5,335
3
18
5,69 6,46 7,02 8,02
6,80
0.98056
6,74
4
24
6,00 6,89 7,96 8,32 8,38 9,96
7,92
1.362503
8,14
5
30
7,53 7,97 8,32 9,43 9,54 11,06
8,98
1.29619
8,875
6
36
7,58 8,22 8,63 9,33 9,38 12,48
9,27
1.71394
8,98
7
38
8,05 8,45 8,63 8,83 9,52 10,62
9,02
0.923464
8,73
8
60
5,55 7,25 10,17 10,70
8,42
2.44040
8,71
However, the subsequent statistical analysis of empirical data in our example shows that data under study have the property of positive asymmetry. Since only six empirical values are given in each section, it is not possible to reliably determine in each section to which probabilistic law the points belong. Therefore, a new approach is proposed that will allow reducing the empirical points of all sections to one generalized (reduced) section. Table 1 also shows the calculated average yields for each section, their standard deviations and medians. The considered example is characterized by the fact that in each section several repeated values of empirical points are given. Although, in general, in most experiments, only one experimental point is taken in each section. Therefore, the results of this experiment allow us to find a more correct approximating function. As an approximating function, a parabola of the second degree was taken y(x) = a + bx + cx2 . as it was proposed in [1]. The specific numerical equation of the parabola, obtained by the ordinary least squares method (LSM), has the form: y(x) = 3, 539 + 0, 2527x − 0.002827x2 Obtained parabola is LSM parabola (Fig. 1).
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x
Fig. 1. Data approximation using LSM parabola.
When obtaining this equation, it was assumed that the distribution of experimental data is symmetrical and corresponds to a normal distribution. However, the studying data have the property of positive asymmetry. In general, least squares method assumes the symmetry of the empirical data in the scatter range, and in the presence of significant data asymmetry, a correction must be introduced. Let show that the empirical dataset contains not symmetrical data. To do this, it is necessary to find the probability distribution law for empirical data within the scatter range. Since we have no more than six points in each section, this is very small for determining the distribution law in one section. A new method for reducing the data for each section into one common sample is proposed, but with some transformations. This will make it possible to more reliably determine the law of the probability distribution. In order to obtain the distribution law inside the probabilistic scatter band, the following actions were carried out: 1. The sixth section was chosen as the base one, because where the maximum yield was observed. 2. All points belonging to Sects. 1– 5, 7 and 8 were projected onto the base section using equidistant parabolas. The number of these parabolas is 38, and they are all equidistant to the LSM parabola. 3. As a result, we obtained a reduced sample of 44 points, which was ranked in ascending order (Table 2) for subsequent statistical analysis. The mean (8.9708), standard deviation (1.3728) and median (8.681) were found for obtained data. The coefficient of variation is 0.153028. For the initial visual statistical analysis, a histogram of the reduced sample was constructed (Fig. 2). Visual analysis of the histogram shows the presence of asymmetry. The skewness for the presented data and its error xi − x3 = 0, 437546. Sk = Nσ3
Method for Correcting the Mathematical Model in Case
σSk =
253
6 = 0, 357351. N +3
Student’s coefficient, which is the ratio of Sk to σSk does not provide clear evidence of the asymmetry of the studied experimental data. Therefore, we will use a different approach to prove the presence of significant positive asymmetry in the base section of the data under study. Table 2. Reduced ranked sample. Ur
n
Ur
n
Ur
n
Ur
1
5,998
12
7,926
23
8,716
34
2
6,996
13
8,183
24
8,82
35
9,936
3
7,118
14
8,22
25
8,956
36
10,328
4
7,49
15
8,26
26
9,283
37
10,436
5
7,58
16
8,266
27
9,316
38
10,618
6
7,588
17
8,323
28
9,318
39
10,953
7
7,648
18
8,366
29
9,33
40
10,956
8
7,698
19
8,446
30
9,336
41
11,148
9
7,783
20
8,63
31
9,376
42
11,373
10
7,866
21
8,638
32
9,38
43
11,456
11
7,886
22
8,646
33
9,82
44
12,48
9,826
Observed frequency
n
Sample values
Fig. 2. Histogram of the reduced ranked sample.
We assume that the range of all experimental data is homogeneous. Therefore, the law of probability distribution is constant throughout the band. Then the established
254
V. Kuzmin et al.
probability distribution law can be applied to the entire range of data under study. To confirm the significance of positive asymmetry, we present a table of absolute values of the left and right half-ranges of data relative to the median in the given section (generalized sample). A comparative analysis of peer-to-peer values was performed. Statistical analysis of these half-ranges will more reliably establish the presence of asymmetry. Note that the symmetric distribution of the probability is characterized by equal number of sign alternations in such inequalities (Table 3). Table 3. Comparative analysis of the absolute values of the lower (left) and upper (right) halfranges of data in the initial sample. n
Left half-range
Half-range comparison
Right half-range
n
Left half-range
Half-range comparison
1
0,035
2
0,043
3
Right half-range
=
0,035
12
0,795