Special Topics in Structural Dynamics & Experimental Techniques, Volume 5: Proceedings of the 39th IMAC, A Conference and Exposition on Structural ... Society for Experimental Mechanics Series) 303075913X, 9783030759131

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Conference Proceedings of the Society for Experimental Mechanics Series

David S. Epp   Editor

Special Topics in Structural Dynamics & Experimental Techniques, Volume 5 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research

More information about this series at http://www.springer.com/series/8922

David S. Epp Editor

Special Topics in Structural Dynamics & Experimental Techniques, Volume 5 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021

Editor David S. Epp Sandia National Laboratories Albuquerque, NM, USA

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-75913-1 ISBN 978-3-030-75914-8 (eBook) https://doi.org/10.1007/978-3-030-75914-8 © The Society for Experimental Mechanics, Inc. 2022 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

Preface

Special Topics in Structural Dynamics & Experimental Techniques represents one of nine volumes of technical papers presented at the thirty-ninth IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held between February 8 and 11, 2021. The full proceedings also include volumes on nonlinear structures and systems; dynamics of civil structures; model validation and uncertainty quantification; dynamic substructures; rotating machinery, optical methods, and scanning ldv methods; sensors and instrumentation, aircraft/aerospace, energy harvesting, and dynamic environments testing; topics in modal analysis and parameter identification; and data science in engineering. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Special Topics in Structural Dynamics & Experimental Techniques represents papers highlighting new advances and enabling technologies for experimental techniques, finite element techniques, system identification, and additive manufacturing. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Albuquerque, NM, USA

David S. Epp

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Contents

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A Comparative Study of Joint Modeling Methods and Analysis of Fasteners . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricardo Garcia, Michael Ross, Benjamin Pacini, and Daniel Roettgen

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Historical Perspective of the Development of Digital Twins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Matthew S. Bonney and David Wagg

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Distributed Home Labs at the Time of the Covid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A. Cigada and S. Manzoni

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Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Hector Laos

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Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases . . . . . . . . . . . . . . . . . . 39 Hector Laos

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Vibration Control in Meta-Structures Using Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 D. Mehta and Vijaya V. N. Sriram Malladi

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Using Steady-State Ultrasonic Direct-Part Measurements for Defect Detection in Additively Manufactured Metal Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Erica M. Jacobson, Ian T. Cummings, Peter H. Fickenwirth, Eric B. Flynn, and Adam J. Wachtor

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Toward Developing Arrays of Active Artificial Hair Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Sheyda Davaria and Pablo A. Tarazaga

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Challenges Associated with In Situ Calibration of Load Cells in Force-Limited Vibration Testing . . . . 81 Kenneth J. Pederson, Vicente J. Suarez, Emma L. Pierson, Kim D. Otten, James C. Akers, and James P. Winkel

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Chapter 1

A Comparative Study of Joint Modeling Methods and Analysis of Fasteners Ricardo Garcia, Michael Ross, Benjamin Pacini, and Daniel Roettgen

Abstract One of the more crucial aspects of any mechanical design is the joining methodology of parts. During structural dynamic environments, the ability to analyze the joint and fasteners in a system for structural integrity is fundamental, especially early in a system design during design trade studies. Different modeling representations of fasteners include spring, beam, and solid elements. In this work, we compare the various methods for a linear system to help the analyst decide which method is appropriate for a design study. Ultimately, if stresses of the parts being connected are of interest, then we recommend the use of the Ring Method for modeling the joint. If the structural integrity of the fastener is of interest, then we recommend the Spring Method. Keywords Finite element modeling · Joint modeling · Fasteners · Structural dynamics

Nomenclature FEM Finite element method

1.1 Introduction One of the key components of any system in a structural dynamics analysis is the joints of the system. This work explores different techniques for modeling the joint system that use fasteners for linear models. There are several studies regarding the best methods to model these types of joints. Most of the research work attempts to address the nonlinearities in the system [1] caused by the joint. To be accurately predictive, it is important to eventually capture the nonlinearities. However, this work attempts to explore typical methods of modeling the joint in large structural dynamic system models, where one cannot computationally afford a high-fidelity model at the joint. These linear models are well suited for design studies and development of component specifications early on in a system’s initial design developments. During design studies, it is imperative that the analyst can generate several quick models of design parameters that can be assessed for structural dynamical performances. These structural performances can range from an assessment of the structural integrity of the parts and the fasteners to the motion of parts of the system to avoid impacts. During the design studies, the analyst is often faced with various methods of modeling the fastener. The analyst is often not afforded the time for deep study on the various methods of joining the materials. Consequently, this work explores various methods for one particular lap joint with fasteners. There are two typical concerns during design studies for the fastener. The first is the structural integrity of the parts being joined, and the second is the structural integrity of the fasteners and nut or insert. Stresses in the parts of interest are required to assess the structural integrity. However, some typical methods used for modeling the joints introduce stress singularities in the parts due to the use of rigid elements. This can lead to reporting incorrect stresses, especially as one appropriately refines the mesh. Hence, this work ultimately recommends using a method that assures proper reporting of the stress and can report the structural integrity of the fastener.

R. Garcia · M. Ross () · B. Pacini · D. Roettgen Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_1

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1.2 Modeling Methods for Fastener Joints There are several methods for modeling fastener joints, see Fig. 1.1. Here we compare some common methods with the proposed method (Ring Method). The methods we explore are the following: 1. 2. 3. 4. 5. 6.

Tied surfaces at joint (Tied Contact Method) Spring model of fastener (Spring Method) Beam model of fastener (Beam Method) Solid model of fastener (Plug Method) Cylinder of solid elements (Ring Method) Including preload and fastener properties (Ring-Beam Method)

1.2.1 Tied Surfaces at Joint: Tied Contact Method A straightforward method is to ignore the fastener and simply constrain the surfaces at the joint interface to move together. This removes the ability to do any post processing of the fastener itself. However, it is a typical method when the joint is not in an area of concern. In finite element terminology, this is typically referred to as “tied surfaces,” “glued surfaces,” or “tied contact.” Tying the surfaces at the joint together in this study is referred to as the Tied Contact Method.

1.2.2 Spring Model of Fastener: Spring Method In this technique, a spring element is used to connect the mating surfaces at the fastener shaft area and is the representation of the fastener. Rigid elements are used to connect one end of the spring element to the fastener location in one of the joining materials. A similar procedure is used for the other joining material. This is all depicted in Fig. 1.2. The fastener is not modeled with solid elements but represented with the spring element. If weight of the fastener is of a concern, concentrated masses can be added to the nodes of the spring element to account for the mass. This method is referred to as the Spring Method in this study.

Fig. 1.1 Bolt modeling representations used in study

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Fig. 1.2 Spring Method for representing fastener

Fig. 1.3 Beam Method for representing fastener

1.2.3 Beam Model of Fastener: Beam Method Another common method for modeling the fastener joint for structural dynamics applications is the use of beam elements in place of the fastener. Typically, the beam is discretized to at least four to five elements to allow for preload application into one of the elements. Enough elements are also needed to capture the bending stiffness. A contact zone or connected surfaces can be represented at the interface of the joint from Shigley’s contact pressure frustrum formula [2]. This is depicted in Fig. 1.3. The fastener is not modeled with solid elements but represented with the beam elements. In this method, preload can or does not have to be considered. In this study, we compare a beam with preload referred to as the Beam Method. Preload is found as [3]: Fi =

T , Kt dbolt

(1.1)

where T is the torque, Kt is the torque coefficient, and dbolt is the bolt diameter. The torque coefficient, Kt , also known as the nut factor, is a factor applied to account for the effects of friction. Typically, the torque coefficient for UNS Standard threads with coefficients of friction at 0.15 is 0.22 [4]. Calculating the forces to apply to the middle beam, see Fig. 1.3, is an iterative process to assure the correct force in the beams.

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Fig. 1.4 Plug Method for representing fastener

1.2.4 Solid Model of Fastener: Plug Method It is also common to see the fastener in a finite element model (FEM) to be represented with solid elements. In full system FEM that can have millions of elements, the threads are typically not modeled and are defeatured. The nut and head of the fastener are also defeatured and represented with cylinders. Including the washer is generally dictated by the analysis being conducted and the level of concern for the stress/strain near the fastener. Determining what is in contact or connected surfaces can also vary among analysts. We recommended using the contact zones shown in Fig. 1.4. In this study, we refer to this method as the Plug Method, since the solid fastener resembles a plug in the finite element model. Preload can be applied to a portion of the fastener shank. This is commonly introduced with a thermal strain on a portion of the solid elements representing the shank of the fastener, see Fig. 1.4. By knowing the desired preload force from Eq. (1.1) and the area of the shank, the desired stress in the shank can be determined and used for the iteration to find the appropriate thermal strain to get the correct preload.

1.2.5 Cylinder of Solid Elements: Ring Method The first two methods discussed, using spring or beam and rigid bar elements, Sects. 1.2.2 and 1.2.3, can potentially lead to erroneous stress predictions due to stress singularities. Though there are common methods to avoid reporting incorrect stresses in these cases, it is rather time consuming and difficult to automate. A simple method around this is to generate a cylinder of solid elements. Shigley’s formula for calculating the frustum can be used for determining the radius of the cylinder. In this study, the Ring Method is explored (Fig. 1.5).

1.2.6 Including Preload and Fastener Properties: Ring-Beam Method It is possible to include this technique with the previous mentioned methods. If it is desired to include a preload or obtain fastener forces, one can use this method in conjunction with the Beam Method, Sect. 1.2.3. This allows for obtaining the fastener loads for analysis. When reporting the stress in the joining parts, the analyst can easily remove the ring part that would have the stress singularities due to the rigid elements for the beam. This method is referred to as the Ring-Beam Method (Fig. 1.6).

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Fig. 1.5 Ring Method for representing fastener

Fig. 1.6 Ring Method with beam for preload or obtaining fastener forces

Fig. 1.7 Shigley’s frustum calculation [2]

1.2.7 Frustrum Calculation The method recommended for finding the geometry of the frustrum is that by Shigley [2]. In this method, the stiffness in a layer is obtained by assuming the stress field looks like a frustum of a hollow cone, see Fig. 1.7. Shigley recommends an ◦ angle, α of 30 , where the angle is typically between 25 and 33 degrees.

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1.3 Example Problem An example problem is used in this study to demonstrate the applicability of the different methods. It is represented as a cylinder with a plate at one end and a beam on top of the plate, see Fig. 1.8. There are eight ¼-20 fasteners used to connect the plate to the cylinder. The fastener properties used are those of steel. For this study, the fastener models were made to be very stiff in the attempt of obtaining stiffness values on the order of 1.0 × 107 lb/in. The plate, cylinder, and rectangular beam are aluminum. The photos shown in Fig. 1.9 are representative of the test model.

Fig. 1.8 Cylinder-plate-beam example with fasteners connecting plate to cylinder

Fig. 1.9 Test model photos

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Modal Shaker Input

Fig. 1.10 Modal shaker input

Accelerometers

Fig. 1.11 Accelerometer locations used in study

There were two experiments conducted. The first was a free boundary condition modal experiment. The second experiment performed was a burst random excitation at the base of the cylinder with the force applied in the Y-direction, as shown in Fig. 1.10. The burst random excitation signal was signal processed and removed the off times to provide an ergodic, stationary random signal. There are 28 attached triaxial accelerometers used in the experiment. However, this study focused on five triaxial accelerometers that were surrounding one of the fasteners as shown in Fig. 1.11. Figure 1.12 depicts the specific accelerometer gage number noted in this study. There are three gages noted as aft that are on the excitation side of the joint. There are two gages on the forward end of the joint near the beam. The forward/aft designation is from missile terminology, where the shaker and beam represent the rocket motors and the missile payload, respectively.

1.4 Results 1.4.1 Modal Comparisons The first comparison of the methods is a modal analysis. The modal analysis for a free boundary condition is shown in Table 1.1 with the first six modes being rigid body. In the modal analyses, the initial model (Tied Contact Method) was correlated to the test data by a previous analyst. Recall that the model for the Tied Contact Method was developed with surfaces at the joints constrained by multipoint constraints typically referred to as tied surfaces or glued surfaces. Then, the other models were developed with no tuning of the fastener method. The idea was to generate the methods as if no test data was available and see which performed the most accurate. As indicated in Table 1.1, the mode frequencies were insensitive to the modeling method used. The error percentage difference of each method compared to the test is shown Fig. 1.13. Generally, the largest error was noted at modes 11 and 15,

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3180

6585

3498

(A) Aft Left Gage

(B) Aft Center Gage

(C) Aft Right Gage

47594

47564

(D) Forward Left Gage

(E) Forward Center Gage

Fig. 1.12 Specific accelerometer gages Table 1.1 Modal frequency (Hz) comparison between test and simulation Mode 7 8 9 10 11 12 13 14 15 16

Test data 139 182 385 390 590 945 951 1039 1221 1288

Tied contact 141 182 398 398 543 948 948 1045 1323 1323

Plug 141 182 398 398 543 948 948 1045 1323 1323

Ring 141 183 398 398 551 948 948 1046 1322 1322

Ring-beam 139 180 393 398 544 949 952 1030 1293 1320

Spring 142 185 399 399 568 949 949 1047 1325 1325

Beam 139 180 393 398 551 949 952 1031 1293 1320

which are an axial and ovaling mode of the system, respectively. This may be a question of material properties as opposed to fastener issues.

1.4.2 Random Vibration Comparisons A typical study will require a random vibration analysis of the system for structural dynamics performance. In this regard, we compare accelerometer responses on two sides of the joint and see if there is any clear indication of a preferred modeling method for the fastener. The acceleration responses are noted in the auto-spectral densities shown in Figs. 1.14, 1.15, 1.16, and 1.17, where the X-direction is transverse and the Y-direction is axial. The responses were grouped into similar behaving responses with the Beam, the Plug, and the Tied Contact methods grouped together and shown in the right column of Figs. 1.14, 1.15, 1.16, and 1.17. This leaves the Spring, the Ring, and the Ring-Beam methods in the other group and shown in the left column of Figs. 1.14, 1.15, 1.16, and 1.17. In this particular experiment, the Spring and the Ring methods have responses that are similar and appear to be more accurate than the Beam, Plug, and Tied Contact methods. The Spring and Ring methods also lend themselves to easier implementation over the previous methods. Any of the methods, however, appear to be adequate and could benefit from furthering calibration.

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Fig. 1.13 Percent difference between test data and various methods for modal analysis

None of the methods do a particular good job of capturing the response at the forward center location, Fig. 1.17. This is an area of future study and probably due to nonlinear issues.

1.5 Conclusion This paper explores various methods of modeling fasteners. The hope is that it can provide the analyst with a method that is consistent with reporting stresses in the parts being joined and providing the fastener loads. Under this specific lap joint, it is recommended to use the Ring Method. It is suggested, however, to use the Spring Method if the structural integrity of the fastener is of interest. The Ring-Beam Method would be suitable if preload and stress near the fastener is of concern. It is beneficial that the Ring, the Ring-Beam, and the Spring methods are easily implemented. Any of the methods, however, appear adequate and could profit from furthering calibration. Future work should explore additional joints as well as the response at the forward center fastener gage location where none of the methods did particularly well at matching that response.

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Fig. 1.14 X-direction responses aft of bolted joint compared to test

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Fig. 1.15 X-direction responses forward of bolted joint compared to test

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Fig. 1.16 Y-direction responses aft of bolted joint compared to test

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Fig. 1.17 Y-direction responses forward of bolted joint compared to test

Acknowledgments Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy National Nuclear Security Administration under contract DE-NA0003525. SAND2020-13563C.

References 1. Brake, M.R. (ed.): The Mechanics of Jointed Structures: Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics. Springer, Cham (2018) 2. Budynas, R.G., Nisbett, J.K.: Shigley’s: Mechanical Engineering Design, 9th edn. McGraw-Hill Publishing Co., New York (2012) 3. Chambers, J.: Preloaded Joint Analysis Methodology for Space Flight Systems, Technical Report 106943, NASA (1995) 4. Norton, R.: Machine Design: an Integrated Approach. Pearson Prentice Hall, New Jersey (1998)

Chapter 2

Historical Perspective of the Development of Digital Twins Matthew S. Bonney and David Wagg

Abstract With modern advances in high-performance computing, design engineers have put a large focus on digital testing and simulations to inform new systems. In addition, recent market tendencies show a desire to reduce waste and for longer designed life. One major strategy used to meet these trends is the utilization of a digital twin. Digital twins are numerical analogues to physical systems such as aircraft, auto-mobiles, and power generation systems. With the wide applicability of the digital twin, an understanding of their development can give insight into the impact and direction of recent research. Understanding these advancements can also give confidence in both the technique of using a digital twin and the simulated predictions to various loading conditions. This chapter focuses on detailing the historical development of digital twins to the state-of-the-art research being done and specifically how it is relevant to the structural dynamics community.

2.1 Background Using computational models to simulate physical phenomena is by no means novel. Recently, however, there has been an increase in desire to integrate the computational models into the design process. Additionally, there is an increased need to expand the models to be accurate for the life cycle of a system. This is where the concept of twinning, especially digital twins, originates and the main motivation of recent research. The discussion of integrating a twin into designs dates back to NASA’s Apollo mission [1]. The twin for the NASA Apollo mission incorporated a physical cockpit to use during training and diagnostic testing. The nomenclature of digital twin is based on the work in product life-cycle management [2]. This was first published in the ASME Standard for Verification and Validation (V&V) in Computational Solid Mechanics (ASME V&V 10) [3]. In the ASME standard, a digital twin is defined as “Digital Twin is an integrated multiphysics, multiscale simulation of a vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its corresponding flying twin.” A common generalization of this definition is the use of the term physical twin instead of flying twin to incorporate non-aerospace systems. From this definition, there are a few main aspects, first being the multiphysics and multiscale simulations. The majority of the digital twins incorporate multiphysics, such as structural and fluid dynamics (such as aerodynamic pressures on an aircraft during flight). Using multiscale simulations, however, seems to be more flexible depending on the physical twin. For industrial systems, the multiscale aspects are typically included; however, academic systems do not tend to have multiscale since they can be designed to only contain a single length scale of importance. A second aspect of this definition is the incorporation of “physical models, sensor updates and fleet history.” This included model updating, grey-box modeling (combination of computational models and sensor data), and pure sensor-based models. The last aspect is the ability to “mirror the life of the flying twin.” Around the same time as the ASME standard was published, several other terms have been used to describe identical or similar ideas. For example, Digital Counterpart [4], Virtual Engine [5], Intelligent Prognostic Tools [6], and Mirrors [7] to mention a few. Many of these alternative names tend to focus on either a specific type of physics (electrical or hydraulic, for example), scale, or specifying computational/sensor-based models only. The term digital twin captures the underlying meaning for these systems and allows for a better understanding for a general audience. This chapter discusses some usages of the digital twins and describes different types of digital twins. Section 2.2.1 discusses how digital twins are used in the design of a new system, and Sect. 2.2.2 discusses the utilization during the

M. S. Bonney () · D. Wagg Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_2

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manufacturing process. In Sect. 2.3, discussion is held on how the digital twin is utilized to determine the life cycle and how it can be used for Verification, Validation, and Uncertainty Quantification (VVUQ). Finally, Sect. 2.4 discusses the different types of digital twins via a hierarchy with some concluding remarks in Sect. 2.5.

2.2 Pre-delivery Usage Currently, one of the major uses of a digital twin is based on the design/prototyping stages of a system. This is primarily due to the required time to see a system from design to decommission. Systems such as aircraft can have life spans that last for nearly 30 years. This creates a large time lag between modern digital twins and the complete validation over the entire life cycle.

2.2.1 Design Usage One of the major uses of a digital twin is to aid in the design and prototyping of a new system. This is particularly true for high-consequence systems that are sparsely manufactured such as aerospace or nuclear systems. One of the primary steps in the design process is the use of Computer-Aided Design (CAD) to create and test candidate designs. Particularly, a digital twin can help for design modifications [8] or long-term failure criteria [9]. In addition to CAD, digital twins also help aid in the prototype stages of the design. However, utilizing the data from prototype tests leads to an issue with how to incorporate this large amount of information, aka big data, into a digital twin and how an analyst visualizes the results. A large amount of research is being done in the field of big data [10, 11] and utilizing the Internet of Things [12, 13]. Another interesting aspect of using digital twins is the ability to incorporate data and models from various sources. One example of this is in the use of electronics that use a digital twin to incorporate a block chain methodology [14].

2.2.2 Manufacturing Usage An initial explanation of digital twins might give the impression that it is only useful for design and long-term monitoring; this however is not true. Digital twins can aid in the manufacturing process and enable modern research techniques. One use of digital twins is based on the ability to utilize sensor data and incorporate live data. Modern manufacturing utilizes a large amount of sensors for both the machinist and the manufacturer for quality control. The utilization of these sensors during manufacturing is called real-time manufacturing [15]. One main aspect that a digital twin can be used is in the use of CNC automated manufacturing. The digital twin allows for the transfer of information between design, purchasing, and manufacturing that can minimize the time between ordering and receiving a product [1]. Additionally, utilizing digital twins also increases the traceability of any issues and can account for any maintenance or other issues [16]. In addition to CNC, digital twins can also be greatly helpful for Additive Manufacturing (AM). In order to manufacture via AM, a specific path must be simulation and verified that the structure meets design requirements. A digital twin can utilize these path simulations both for verification and for design modifications [17, 18].

2.3 Asset Management Usage One major use of the digital twin involves the incorporation of full life-cycle monitoring and modeling. The advancements discussed in Sect. 2.2 are expansions of other current research by the utilization and centralization of sensors, simulations, and historical data. However, one of the major utilizations of the digital twin is the mirroring of scenarios to the physical twin. This section discusses how the digital twin can be used in the delivery and maintenance for the life span of the product.

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2.3.1 Life-Cycle Determination One of the advantages for using a digital twin is the ability to take long-term measurements and perform simulations to predict how the system changes during the time in operation. This ability to perform during the operation allows for both monitoring and predicting damage/failure. One aspect of this is the maintenance and intelligent prognosis [6]. The concept of intelligent prognosis is an interesting combination of monitoring data and future predictions. In [6], the authors utilize the Watchdog AgentT M to take in monitoring data and make updated predictions on the system. Another aspect of the digital twin is the one-to-one relationship between the digital and physical twins. This allows for models to use an accurate model for the specific system, including the manufacturing defects. The term as-manufactured geometry is commonly used to describe the physical system as opposed to the nominal geometry that does not account for manufacturing tolerances that can create issues particularly for nonlinear systems [19]. This is also important for the study of how manufacturing and assembly uncertainty affect the life span of a system and help identify required maintenance for the physical twin. In addition to predictions based on normal operation, the digital twin can also aid in the identification of damage through structural health monitoring (SHM). This is especially important for systems with long expected life, such as space vehicles [20], civil structures, and aircraft. While the work in [20] performs damage detection with no modeling, this type of work can be expanded to utilize the damage information into the digital twin and possibly make future predictions, such as in [21].

2.3.2 VVUQ The study of VVUQ with digital twins participates in two main aspects: model updating and uncertainty quantification. Model updating is not novel to digital twins as it has been a major part of structural dynamic modeling [22]. However, the advancements in model updating, particularly in nonlinear dynamics [23, 24], can easily be modified to utilize the wealth of information contained within a digital twin. One particularly interesting aspect of model updating is utilizing the associated sensors available to the digital twin. This includes the ability to perform tests periodically during the system’s operation. Two methods to incorporate these sensors for model updating are Bayesian networks [25] and Bayesian operational modal analysis [26]. In addition to model updating, the quantification of uncertainty, both physical and numerical, is also an important aspect of the digital twin [27, 28]. For digital twins, there are several sources of uncertainty including: • Sensor Uncertainty: All the sensors are calibrated to a specified tolerance based on calibration standards. Additionally, the location of sensor has physical dimensions that are not typically explicitly modeled. • Manufacturing/Assembly Uncertainty: Manufacturing parts is not perfectly exact but is specified within a certain range (typically very small). These ranges can build up and potentially cause discrepancy between the model and the physical twin. For digital twins that correspond to a singular physical system, this uncertainty is nearly zero, but can be larger is a digital twin is used to express more than one physical twin (such as specific aircraft model). • Averaging Error: Since the digital twin is used for the entire life expectancy of the system, parameters (such as stiffness) can change over time due to damage or degradation. Snapshot-based calculations can ignore the history of the parameters, but aspects such as predictions need to incorporate all the historical data. This error is associated with any averaging that is used incorporating data collected from multiple sources or instances. • Model-Form Uncertainty: It is very unlikely that the entirety of the experienced physics is modeled. However, very accurate approximations are used (such as finite elements). This uncertainty accounts for the physical to numerical errors and other assumptions made (linear vibrations, rigid connections, etc.) in the computational models.

2.4 Digital Twin Hierarchy As one can surmise from the history and discussion of digital twins so far in this chapter, digital twins are complicated systems. To better categorize the aspects of digital twins, the authors in [29] introduce a hierarchical view of the technology that is utilized in digital twins. This hierarchy is presented in Fig. 2.1 with increasing complexity and technology evolution closer to the tip of the pyramid.

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Fig. 2.1 Hierarchy organization of digital twins (taken from [29])

It is important to note that aspects of the digital twin are evolutions of conditional monitoring of plants such as those used for power generation. As a basic need, the ability to supervise is the first desirable aim. The ability to monitor the operations of systems is vital. A second basic need is the ability to manipulate operations based on the data recorded (such as environmental temperature and space availability). These aspects make up the first two levels of a digital twin with level one being Supervisory, the ability to monitor the system, and level two being Operational, the ability to make operational decisions based on the data recorded from the monitor data. The first two levels are very well established. Some authors have considered this a digital twin; however, modern interpretations classify these two levels as predigital twins, meaning that they have the capacity of becoming a digital twin but do not contain all the necessary abilities needed for a digital twin. The next level of sophistication is the incorporation of simulations to make predictions on the system. At this level, the authors in [27] utilize the name simulation digital twin. This builds on the supervisory and operational capacity of the predigital twin and incorporates numerical simulations to compute data that is not directly observed on the system. The inclusion of numerical simulations also gives the user a graphical interface to utilize. Simulation digital twins can both give predictions (such as maximum stress during normal operation) and a quantitative measure of the trust for predictions through the use of uncertainty quantification. This level is the current level of most digital twins. With the current evolution of digital twins residing primarily in level three, there are two more levels of advancements that are aspirations for digital twin sophistication. The first level of advancement is the introduction of intelligence through learning. This level introduces the concept of learning from data (via machine learning) and adds levels of scenario planning and decision support. The final level allows the digital twin to perform autonomously. Adding this level allows for the digital twin to perform routine decision-making, within specified ranges, without requiring a user decision. One simple example of this level would be to have an automatic climate control within a structure (such as a space system) that takes in temperature/humidity measurements, simulate what vents would provide the most efficient climate control, and to automatically turn on the heater/cooler and open the required vents. This hierarchy view of digital twins gives an understanding on what baseline information/developments are needed for the development of digital twins. The aspiration of intelligent/autonomous digital twins is currently a large focus of current research. Additionally, advancement in model simulations can also greatly advance aspects of digital twins. These are generally viewed as making the digital twin more accurate, perform simulations with less computational burden, or better incorporate the operational data into the simulations and decision-making.

2.5 Conclusions Despite being a modern buzz word, digital twins have a rich historical aspect to them. They date back to NASA’s Apollo mission as a method to monitor and simulate product life-cycle management and aid in personnel training. The evolution

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of digital twins is comprised of multiple layers involving experimental testing, numerical modeling, product life-cycle monitoring, and decision-making. There are two main time spans of interest for digital twins, pre-delivery, and asset management. The time span of predelivery is focused from the initial design phase until the product is delivered to the customer. For an initial design, digital twins aid in the general design by aiding the ability to make design modifications informed by simulating the failure criteria. Additionally, a digital twin can be very useful during the prototyping stage with data management, grey-box modeling, and model updating. After the delivery of the product to the customer, the digital twin is also very useful in the long-term monitoring and modeling of the system. To ensure that a digital twin corresponds nearly perfectly to the physical twin, a large amount of model verification, validation, and uncertainty quantification is performed. This can also be used to identify long-term aspects such as maintenance and monitoring using the data from structural health monitoring. Digital twins are still an evolving and expanding area of research. To better characterize the development process and the wide possibility of digital twins, a five-level hierarchy from various sources is examined to better understand the state-ofthe-art development and the possibilities available for future research. The development of digital twins is ongoing but still has an interesting history.

References 1. Rosen, R., von Wichert, G., Lo, G., Bettenhausen, K.D.: About the importance of autonomy and digital twins for the future of manufacturing. IFAC-PapersOnLine 48(3), 567–572 (2015). 15th IFAC Symposium on Information Control Problems in Manufacturing 2. Grieves, M., Vickers, J.: Digital Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems, pp. 85–113. Springer International Publishing, Cham (2017) 3. Schwer, L.E.: An overview of the ASME v&v-10 guide for verification and validation in computational solid mechanics. In: 20th International Conference on Structural Mechanics in Reactor Technology, pp. 1–10 (2009) 4. Nicolai, T., Resatsch, F., Michelis, D.; The web of augmented physical objects. In: International Conference on Mobile Business (ICMB’05), pp. 340–346 (2005) 5. Morel, T., Keribar, R., Leonard, A.: “Virtual engine/powertrain/vehicle” simulation tool solves complex interacting system issues. In: SAE Technical Paper. SAE International, 03 (2003) 6. Lee, J., Ni, J., Djurdjanovic, D., Qiu, H., Liao, H.: Intelligent prognostics tools and e-maintenance. Comput. Ind. 57(6), 476–489 (2006). E-maintenance Special Issue 7. Worden, K., Cross, E.J., Barthorpe, R.J., Wagg, D.J., Gardner, P.: On digital twins, mirrors, and virtualizations: frameworks for model verification and validation. ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg 6(3), 05, 030902 (2020) 8. Schleich, B., Anwer, N., Mathieu, L., Wartzack, S.: Shaping the digital twin for design and production engineering. CIRP Ann. 66(1), 141–144 (2017) 9. Boschert, S., Rosen, R.: Digital Twin—The Simulation Aspect, pp. 59–74. Springer International Publishing, Cham (2016) 10. Tao, F., Cheng, J., Qi, Q., Zhang, M., Zhang, H., Sui, F.: Digital twin-driven product design, manufacturing and service with big data. Int. J. Adv. Manufact. Technol. 94(9–12), 3563–3576 (2018) 11. Tao, F., Sui, F., Liu, A., Qi, Q., Zhang, M., Song, B., Guo, Z., Lu, S.C.-Y., Nee, A.Y.C.: Digital twin-driven product design framework. Int. J. Prod. Res. 57(12), 3935–3953 (2019) 12. Lee, J., Lapira, E., Bagheri, B., Kao, H.: Recent advances and trends in predictive manufacturing systems in big data environment. Manufact. Lett. 1(1), 38–41 (2013) 13. Schroeder, G.N., Steinmetz, C., Pereira, C.E., Espindola, D.B.: Digital twin data modeling with automationML and a communication methodology for data exchange. IFAC-PapersOnLine 49(30), 12–17 (2016). 4th IFAC Symposium on Telematics Applications TA 2016 14. Heber, D., Groll, M., et al.: Towards a digital twin: How the blockchain can foster e/e-traceability in consideration of model-based systems engineering. In: DS 87-3 Proceedings of the 21st International Conference on Engineering Design (ICED 17) Vol 3: Product, Services and Systems Design, Vancouver, 21–25.08. 2017, pp. 321–330 (2017) 15. Uhlemann, T.H.-J., Schock, C., Lehmann, C., Freiberger, S., Steinhilper, R.: The digital twin: demonstrating the potential of real time data acquisition in production systems. Procedia Manufact. 9, 113–120 (2017). 7th Conference on Learning Factories, CLF 2017 16. Ríos, J., Hernandez, J.C., Oliva, M., Mas, F.: Product avatar as digital counterpart of a physical individual product: Literature review and implications in an aircraft. In: ISPE CE, pp. 657–666 (2015) 17. Knapp, G.L., Mukherjee, T., Zuback, J.S., Wei, H.L., Palmer, T.A., De, A., DebRoy, T.: Building blocks for a digital twin of additive manufacturing. Acta Mat. 135, 390–399 (2017) 18. DebRoy, T., Zhang, W., Turner, J., Babu, S.S.: Building digital twins of 3d printing machines. Scripta Mat. 135, 119–124 (2017) 19. Cerrone, A., Hochhalter, J., Heber, G., Ingraffea, A.: On the effects of modeling as-manufactured geometry: toward digital twin. Int. J. Aerosp. Eng. 2014, 439278 (2014) 20. Zagrai, A., Doyle, D., Gigineishvili, V., Brown, J., Gardenier, H., Arritt, B.: Piezoelectric wafer active sensor structural health monitoring of space structures. J. Intell. Mat. Syst. Struct. 21(9), 921–940 (2010) 21. Seshadri, B.R., Krishnamurthy, T.: Structural health management of damaged aircraft structures using digital twin concept. In: 25th AIAA/AHS Adaptive Structures Conference, pp. 1675 (2017) 22. Friswell, M., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics, vol. 38. Springer Science & Business Media, Berlin (2013)

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23. Worden, K., Wong, C.X., Parlitz, U., Hornstein, A., Engster, D., Tjahjowidodo, T., Al-Bender, F., Rizos, D.D., Fassois, S.D.: Identification of pre-sliding and sliding friction dynamics: grey box and black-box models. Mech. Syst. Signal Process. 21(1), 514–534 (2007) 24. Worden, K., Barthorpe, R.J., Cross, E.J., Dervilis, N., Holmes, G.R., Manson, G., Rogers, T.J.: On evolutionary system identification with applications to nonlinear benchmarks. Mech. Syst. Signal Process. 112, 194–232 (2018) 25. Li, C., Mahadevan, S., Ling, Y., Choze, S., Wang, L.: Dynamic Bayesian network for aircraft wing health monitoring digital twin. AIAA J. 55(3), 930–941 (2017) 26. Au, S.K., Zhang, F.L., Ni, Y.C.: Bayesian operational modal analysis: theory, computation, practice. Comput. Struct. 126, 3–14 (2013). Uncertainty Quantification in structural analysis and design: To commemorate Professor Gerhart I. Schueller for his life-time contribution in the area of computational stochastic mechanics 27. Tuegel, E.J., Ingraffea, A.R., Eason, T.G., Spottswood, S.M.: Reengineering aircraft structural life prediction using a digital twin. Int. J. Aerosp. Eng. 2011, 154798 (2011) 28. Karve, P.M., Guo, Y., Kapusuzoglu, B., Mahadevan, S., Haile, M.A.: Digital twin approach for damage-tolerant mission planning under uncertainty. Eng. Fract. Mech. 225, 106766 (2020) 29. Wagg, D.J., Worden, K., Barthorpe, R.J., Gardner, P.: Digital twins: state-of-the-art and future directions for modeling and simulation in engineering dynamics applications. ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg 6(3), 05, 030901 (2020)

Chapter 3

Distributed Home Labs at the Time of the Covid A. Cigada and S. Manzoni

Abstract The well-known difficulties with the recent pandemic have forced people to find new communication means, especially concerning educational methods. University courses have been dramatically changed in their structure, just in a few weeks. While traditional lectures have been more easily switched to “on-line” methods and tools, classes mainly based on experimental lab activities have suffered more from the new forced approaches: a new and stronger effort had to be produced to guarantee the proper knowledge transfer. Many ways have been tried to export experimental labs into students’ houses, preserving and stimulating their curiosity. However, there was a risk to foster a more passive role; students watching a movie or listening to a faraway teacher could not have direct interaction with the instrumentation locked in not accessible labs, nor had the important chance to develop “hands-on” sessions. This paper deals with the ideas and attempts to preserve the value of the experimental activities during the COVID period, in which experimentation has also meant experimenting a new way of teaching; early attempts will be described up to a final proposal, which has been successfully tested with students of both the bachelor and the master of science. Keywords Smartphone · Educational laboratories

3.1 Introduction During the recent pandemic, it seemed that a real chance to perform experimental classes was completely lost. Students could not have access to the University classrooms, most of them had returned to their native countries: even if on-line teaching seemed to make the world smaller, some distances proved to be incredibly wider, as the personal contact and the possibility to interact with real instrumentation seemed unrecoverable. A proper problem introduction requires a short history related to the environment in which the project to get over the mentioned difficulties was born: this history relates to the situation of University studies, mainly in Mechanical Engineering, in Italy. Italy has a specific situation as, in University courses, Measurements are considered a discipline on its own. The advance of metrology issues and the complexity linked to a proper management of measurement systems and networks, together with the needed data management strategies, have led to consider a specific skill: the measurement specialist. This is the reason, especially in Mechanical Engineering, a long tradition exists in creating and managing experimental labs, to properly train students, since the bachelor courses. From the 1990s on, in many Italian universities an effort has been made not just in setting up experimental laboratories, but also to make them an effective educational tool, overcoming a series of barriers which leads to consider experimentation less important than modeling. Literature on this topic tends to focus on single experiments or tests, rather than working on a general method: one of the few examples in this direction is given by the works of the Portuguese group of Maria Teresa Restivo [1, 2].

A. Cigada () · S. Manzoni Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_3

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3.2 Experimental Laboratories as a Tool for Practitioner Engineering The first attempts to set up educational labs consisted in big rooms having desks equipped with the basic instrumentation: power supply, multimeter, frequency generator, data acquisition boards, and some sensors (Fig. 3.1). It was immediately recognized that the main effort was not just in setting up the lab, but in maintaining it. Maintenance was not just related to the relevant use of these facilities, creating wear and damage, but also to instrumentation aging and the need to substitute it. Safety issues had to be properly transferred to students working in the labs, instructors had to be trained: in the end all these aspects often made the choice of having experimental labs very expensive, sometimes not sustainable. Other issues were related to the need to provide quick instructions for the students, to help them learning how to use the instrumentation. This aspect required times not always available in courses, which need to run fast, usually around 3 months. The proposed experiments spanned across the whole world of measurements, every test having an introduction, some guided tests, and in the end some hints to develop some work on own, in which the instructor had the main role of a living manual. Particular care was devoted to the dynamic performances in measurements, being this the most tricky part in mechanical measurements: most examples, coming from the area of sound and vibrations, were modeled on the remarkable activity of Anders Brandt [3–7]. In the end, due to the usually high number of students attending courses, it is hardly possible to allow each individual to practice “hands-on sessions” as students usually work in small groups. The dynamics of social relations inside groups has to be carefully studied, as some students tend to hide themselves, not being very active, while others do most of the job, although in the end the single student activity has to be evaluated. In case we really wanted each single student to practice in person on the instrumentation, time and resources were not enough. This is the reason research on the best way to perform experimental activities in university classes never stopped. A first and almost immediate solution was trying to lower the high pressure on experimental labs, by transferring some activities into common classrooms. This was possible thanks to a joint initiative with National Instruments and PCB. The first company offered help in the use of a product, my-DAQ [8], which can be adapted to multiple uses, like a multimeter, a function generator, a spectrum analyzer, a data acquisition system, and many others, relying on virtual instrumentation developed on a computer with Labview [9]. PCB [10] offered some instrumentation out of production, still perfectly working, which could be used to work on simple experiments: they were microphones and accelerometers, giving many chances to develop small classroom projects, assuming that a table can pretend to be a bridge, fan coils are narrow band noise generator, a set of microphones can be used to get the speed of sound, and so on. In some specific cases, students could borrow the sensors for their tests, to be carried out on own: in one case, an archery champion asked to measure the bow vibrations while shooting an arrow, also getting interesting results to improve his performances. In addition, students had the chance to obtain a free of charge student edition of Labview; this solution was chosen because the Express VI family allows one to use the basic tools of data acquisition and spectral analysis even without being an expert in programming and in a very short time. The idea of having a laboratory easily transferrable everywhere was really challenging and effective in getting the goals of making lab activity easier and accessible to everyone: for this reason, this project was given a name: “Flying Lab” meaning the possibility of easily reach every place. But many problems were still not solved and a new revolution was starting.

Fig. 3.1 The first educational laboratories in Politecnico di Milano (left); the “Flying Lab”: every classroom can become a lab

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3.3 The New Revolution Offered by Microelectronics Although the “Flying lab” was an important step forward, still it did not solve some main issues, among the others the availability of sensors for all students, to get directly in touch with measurement problems. Moreover, Academia should train students to use what they will more easily use in their everyday life: the huge spread of the new low-cost MEMS sensors created a revolution not just in industry, as a reflection also on the new educational labs. At the same time, as sensors can be used by everybody, an enormous effort was made to increase awareness about metrology and its rules, to get good measurements: the new sensor performances had to be fully known to allow for the right choice in every application. New problems were behind the corner, as the reduced cost of each sensor modifies the general strategy in measurements; the new sensors have lower performances but allow denser networks; information redundancy becomes a main requirement, adding new issues related to networks of sensors, a topic seldom faced in measurement courses. The new systems come out equipped with microcontrollers: although their use is not complex, all the same the very short time allowed inside courses, especially in mechanical engineering, does not allow to learn even the basic and simple rules needed to start with their use. The first attempt was therefore to use analog output sensors, coupled to the already available boards, providing the power supply from the USB computer ports, to further push on the development of the “Flying Lab,” therefore making it richer and more autonomous. The set of available MEMS sensors is quite rich, as pressure, temperature, acceleration, sound, rotation, rotation speed, and many others can be measured, the main drawback consisting in the rather brittle connection between the sensors, often hosted on naked boards, and the data acquisition unit. The two most commonly adopted solutions have been Arduino [11] and STM32 [12] by STMicroelectronics, both having wide suites of already developed basic programs to perform the easiest tasks [13]. Students have also been invited to develop projects on own, and this served to develop awareness about the most common problems in experimental activities. A continuous interaction with developers has pointed out how the main problem for non-electronic engineers, using such devices, is the lack of a user friendly interface. Hence, a huge effort was made in helping to get over the barrier constituted by the needed software and hardware skills; this was recognized as a problem not only for education, but also for industrial applications. This is the reason why some products came out recently, with specific aids aimed at allowing everybody to use them. A line of products developed by STMicroelectronics for IoT has helped a lot also in education: the SensorTile.box [14] first, then the brand new STWIN [15] have been considered quite interesting. In the case of STWIN, a single board, created with the aim to prototype new measurement systems, without designing ad hoc boards, hosts many different sensors, a microcontroller, a wireless connection, a USB port, a Bluetooth antenna, and a slot for a micro-SD board; power to the system is provided by a small lithium battery, and some AI tools are already available on board. The burdensome task of microcontroller programming can be jumped over, as a simple smartphone app allows one to select the sensors to be used, the data rate, the storage output destination. Once the acquisition is over, data can be read back, thanks to a library developed in the most common programming languages for further evaluation. Though very powerful, some issues are still related to costs, not yet allowing to provide a board to each student, especially for crowded courses, unless a specific budget is provided. All the same this approach is still considered really powerful, as new MEMS sensors are available almost every day, making the available database richer and richer. This was the teaching standard we were working on, in our group, until February 2020.

3.4 The Smartphone: A Complete Measurement Lab Parallel to the described progress, the same MEMS sensors gained a lot of attention, being essential tools in every smartphone: smartphones are complete and rich labs, equipped with a number of sensors measuring a lot of different quantities. During February 2020, all educational activities had a sudden stop due to the pandemic. At a first glance, the impact over laboratory activities was feared to be a disaster, as both students and instructors were working at home. Many solutions have been tried to overcome these difficulties [16, 17]. A first attempt to heal this trouble consisted in trying to move at least some lab activities at home. Paired to this, there was a long-lasting experience in managing the students’ psychological approach to experimental activities: since the start of the experimental projects in the early 1990s, a clear need was recognized to force each individual to directly interact with hardware and instrumentation: some students, less familiar with practical issues, tend to refuse this activity; written reports

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Fig. 3.2 Images from the home labs: uncertainty (left), calibration of a load cell (middle), step response of a thermometer (right)

were also considered an unavoidable completion to the lab activity, forcing students to pay attention to their work as real professionals, gaining awareness and critical sense. Three labs have been carefully selected, with short movies recorded at the trainers’ houses, not as a substitution, rather as an aid to the on-line explanation provided by the instructors. Measurements have not been collected in tables or files: the students have been invited to work on clips or photos sent to them, to get data on own starting from the readings: to check their attention wrong measurements were inserted too. In the end, the topics have been chosen in such a way that everybody could eventually and easily replicate the same measurements at home. The three chosen topics have been (Fig. 3.2): • Uncertainty: dimensional measurements with a caliper and a micrometer (available in many houses) • Calibration: a cantilever loaded with different weights and a camera as the displacement sensor (a webcam is on every notebook) • Step response of first-order systems, with a thermometer inserted in a hot water pot Anyway even if the effort to produce material for these experiments has been huge, a physical separation remained between the hardware and the students, creating a gap impossible to be filled. At this point, a clear idea came out: almost every student has a smartphone and a smartphone is already a complete lab in our hands [18–21]. It is equipped with many sensors, and it already has some data acquisition capability, calculation, and storage. These are the same MEMS sensors already introduced.. Up to here, there was not so much novelty: many apps already allowed one to directly connect to the smartphone sensors, though most of them were a sort of game, demonstrators rather than real data acquisition systems. Just to mention an example, most of them do not have any control on the sampling frequency, continuously varying, according to the load given by the other apps running on the smartphone; then no care is devoted to those metrological issues, changing a simple sensor into a measurement device. From this point of view, Phyphox, an app developed by a group of physicists from RWTH Aachen [18, 19, 21] was something completely different: the smartphone was converted into a real small laboratory, offering a suite of tools to directly connect to the available sensors, with specific attention to metrological issues. A series of already developed tests is also included in the app, then a calibration tool is provided; data files can be easily transferred after the tests through the e-mail, and there is also the possibility to transfer the system control to a computer, creating a local network. Every study course and every trainer can decide which is the best usage level: concerning engineering studies, the choice has been again to leave students as free as possible. Smartphones are only used to get data and export them for further work in the best preferred environment. The already developed experiments have just been used as a tool to inspire new ideas. In our case, the smartphone just offered the sensors and a data acquisition board to be carefully managed, as the data acquisition system is not a professional one. But this is considered quite useful for educational purposes. Concerning our activity in educational labs, after a short preliminary training and explanation of the basic functions of Phyphox, students have been invited to plan tests on their own, in their houses, after some discussion with the instructor. There was a clear awareness about the risk of such an option: a student left alone without any strong guidance could have had opposite reactions; an unconditional surrender or the opposite, as students, alone in their houses, worried or bored for the unreal general situation, could have had the time to think about their project and come out with good ideas, also thanks to the link to the instructor. It was believed that this approach could also serve to better fix the theoretical explanations, through their real use. Trying to push students along this second path, has forced all the instructors to spend much time providing assistance and discussing the problems each student met, but this stage was already part of the final exam. Luckily most reactions have been toward a good planning of nice projects.

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3.5 Some Preliminary Attempts with Small Groups on Research Projects Some preliminary approaches were already attempted during the last year to verify the feasibility of the proposed approach, with smaller groups, attending their labs in presence. During a short course held at the University of Miami, in the early 2020, on experimental methods for SHM, students had to develop a project on a bridge in their campus, caring about many different aspects, both theoretical and experimental; hence, they were also asked to get at least a rough description of the bridge dynamic behavior, to validate numerical models. The easiest way was to use Phyphox, leaving some smartphones laid on the sidewalk, to measure vibrations for some time (Fig. 3.3); luckily some heavy trucks crossed the bridge, which was equipped with speed bumps at both ends, a good help in providing a high excitation. Lack of synchronization, different smartphone sampling rates, and different positions along the bridge have not considered problems, rather hints which helped in deepening many aspects on the quality of measurements, much better than a perfect multi-channel data acquisition system, which in the end was used, as a verification of the ideas previously discussed. We also took the occasion to have a benchmark, by performing the same measurements with the SensorTile.box by STMicroelectronics, getting similar outputs and managing similar problems: the main differences are that this latter device is not part of the smartphone, though being controlled by this device and the presence of an on-board micro-SD card allows for longer measurements. A second interesting series of tests was about human structure interaction: this research poses many problems, as the effect of humans moving over a structure can be easily measured in terms of vibrations, strains, displacements, etc., while the input is nearly impossible to be directly measured, consisting in the force produced by each single individual jumping, bobbing, or just walking. Since some time ago, the only possibility to measure the crowd motion was to use DIC approaches, relying on the use of cameras taking movies of the moving crowds. More recently a new approach has been tempted: groups of students having their smartphones equipped with Phyphox have been asked to jump on a stadium grandstand or to walk on a very flexible and wobbling bridge, measuring each individual body acceleration (Fig. 3.4). Measurements provided by the accelerometers inside smartphones have allowed to compare different walking habits, to recognize the different step phases, the level of synchronization while walking or jumping, and to check the motion frequency against the structural response. This case too has been an interesting test, which has probably helped in a better comprehension of measurement basics, by working on real problems and solving real needs.

3.6 Some of the Most Interesting Projects The above described testing provided good satisfaction. Due to this reason, when the pandemic forced to stop any classroom activities, the use of Phyphox was immediately considered a possible solution for the experimental activities. As already stated, there was the will to leave students being the main players in their projects, having the smartphones as their data acquisition systems and then analyzing results with the best preferred software. Unfortunately, there was no time to write and deploy simple and robust basic data analysis software: hence only some hints have been given about the possibility to work with specific VIs written in Labview, or with Matlab GUIs, or again pointing at specific Python libraries.

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Fig. 3.4 Tests relying on an extensive use of Phyphox: a group of students jumping on a stadium grandstand (left and middle); a group of students walking on a wobbling bridge (right)

Fig. 3.5 Smartphone used to detect heart beat and breathe (left and middle); washing machine rotating speed measurement (right)

Many projects have demonstrated a good students’ sensitivity in acting as an engineer; we will report some of the most interesting ideas. A student, son of a cardiologist, just laid his smartphone on his chest and through acceleration measurements (including g – Fig. 3.5), he could detect both his heartbeat and his breath frequency. Of course, this was not an ECG, but a comparison with these data has offered the chance for a discussion about many aspects of filtering, as a tool capable of separating a low frequency range, typical of breath, from a relatively higher frequency band, responding for the heartbeat. Another project was aimed at verifying that the rpm value during the spinning cycle of a washing machine was the same as declared in the instruction manual, again putting the smartphone on the top cover of the washing machine (Fig. 3.5). This test has offered another chance to work with spectral analysis under steady-state conditions and with varying working conditions. Some students, fond of music, have created tools to verify the sound quality of their playing instruments, by recording a single note with the associated harmonics; in one case, a program to tune a guitar has been produced by comparing the spectrum of the theoretical note of each string with that produced by the playing instrument and then comparing the spectra up to perfect superposition. Other cases have been about vibrations produced by different machines, a coffee machine water pump, then a drilling machine (Fig. 3.6), cars or motorcycles under steady-state conditions or under variable regime; in one case, a student wanted to check his treadmill performances by the use of the clinometer and the accelerometer inside the smartphone. Many students worked on the lift accelerations, to recognize the number of travelled floors and to detect the door openings and closure for any eventual anomalous behavior. A student having the restless leg syndrome has recorded his leg movements along several hours, to detect frequency, amplitude, and number of cycles, trying to relate this to his health state. Another interesting experiment was carried out by some students in a group: during the pandemic, they were forced to live at their parents’ houses at the seaside or on the mountains. They recorded the atmospheric pressure for some time, trying to correlate results and find its change with the height above the sea level, or with the weather, also trying to define a reasonable uncertainty. In the end, as the hardest lock-down was at least slightly released, some students have carried out tests on real structures getting again interesting data for discussion on the dynamics of structures (Fig. 3.7).

3 Distributed Home Labs at the Time of the Covid

27

Fig. 3.6 Some tests from the students’ projects: vibrations produced by a coffee machine (left), kinematics of a salad spinner (middle), drilling machine (right)

Fig. 3.7 Vibration measurements on a pedestrian bridge in Milano

A number of students have recorded the acceleration of the floor or the noise related to living in close proximity of trains, heavy traffic or subway lines, trying to point out if any prevailing frequency component was present. One student developed a simple anti-intrusion system based on the recognition of a change in the RMS acceleration value at the floor while someone walked in the room. In the end, a student decided to put her smartphone in a salad spinner (Fig. 3.6) and by measuring the accelerations produced by the three axis MEMS accelerometer has tried to reconstruct on her own one of the experiences which is part of the standard packet in Phyphox.

3.7 Final Remarks and Conclusions Helping students with their projects demands a remarkable effort by the instructors and by the students themselves, but the continuous discussions have helped in better fixing the course contents and have made the final exam easier and smoother, starting from the project presentation. Even if it is difficult to compare the final results with those of the preceding years, due to the strong differences in the exam organization, results over around 200 tests have been slightly better this year than in the past. Students demonstrated to be satisfied anyway, and in a form, they have been asked to fill after closing their exam, on one side they claimed real classroom lessons, as these were anyway preferred; on the other side, they wrote they did not really think that they learned less due to the lack of laboratory.

28

A. Cigada and S. Manzoni

Being alone has probably convinced students that they had no backup solutions or that they could not hide in a working group. On the instructors’ side, even if the adopted solution has allowed to fix a problem which apparently had no solution, that is performing experimental classes at home, much has still to be done. A subtle balance has to be found, stimulating students to do as much as possible on own, at the same time maintaining a guidance on their activity and not making their tasks impossible. As an example, one point deserving further attention is the software to manage data: this should be provided in an essential form, without the need to use programs occupying a wide space on disk, preventing from a passive approach, but at the same time also keeping the main attention on the problem to be solved and not on the programming task. Even if the authors wish that any improvements in the presented approach will not be strictly necessary in the future, hoping in a fast return to the real experimental labs, it is strongly believed that some parts from the gained experience will remain: the use of a smartphone or of the new evaluation boards for the new MEMS sensors will remain, as these offer to every student the chance to work with real sensors at an almost null cost. The possibilities are really wide and will presumably grow in a near future, also because these tools can really help the implementation of the real internet of things. Acknowledgments The authors gratefully acknowledge all the students who have worked with passion producing most of the material presented in this paper and all the instructors who have helped in the design and development of this project. A special thanks to Sebastian Staacks and Christoph Stampfer from RWTH Aachen, who created Phyphox and have had a key role. The project has been developed partly thanks to the support of the PRIN Project “Life-long optimized structural assessment and proactive maintenance with pervasive sensing techniques.”

References 1. Restivo, M.T., de Fátima Chouzal, M., Abreu, P., Zvacek, S.: The role of an experimental laboratory in engineering education. In: Auer, M., Tsiatsos, T. (eds.) The Challenges of the Digital Transformation in Education ICL 2018. Advances in Intelligent Systems and Computing, vol. 917. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11935-5_61 2. Restivo, M.T., de Almeida, F.G., de Fatima Chouzal, M., Medes, J.G., Lopes, A.M. (eds.): Handbook of laboratory measurements and instrumentation. In: International Frequency Sensor Association (IFSA) Publishing Search for Publisher Publications (2011) 3. Brandt, A.: Some educational vibration measurement exercises. Sound Vib. Open Access. 50(1), 12–14 (2016) 4. Brandt, A., Kjær, C.: Flipping the classroom for a class on experimental vibration analysis. In: Conference Proceedings of the Society for Experimental Mechanics Series, vol. 8, pp. 155–159 (2016). 34th IMAC, A Conference and Exposition on Structural Dynamics, 2016; Orlando; United States; 25 Jan 2016–28 Jan 2016 5. Brandt, A.: Some cornerstones of signal analysis history. In: Conference Proceedings of the Society for Experimental Mechanics Series, vol. 6, pp 33–36 (2014). 32nd IMAC Conference and Exposition on Structural Dynamics, 2014; Orlando; United States; 3 Feb 2014–6 Feb 2014 6. Brandt, A.: Lab exercises for a course on mechanical vibrations. In: Conference Proceedings of the Society for Experimental Mechanics Series, vol. 6, pp. 15–20 (2014). 32nd IMAC Conference and Exposition on Structural Dynamics, 2014; Orlando, FL; United States; 3 Feb 2014–6 Feb 2014 7. Brandt, A.: ABRAVIBE – a toolbox for teaching and learning vibration analysis. Sound Vib. Open Access. 47(11), 12–17 (2013) 8. Chesnutt, C., Baker, M.C.: Incorporation of NI Mydaq exercises in electric circuits. In: Proceedings of the 2011 ASEE Gulf-Southwest Annual Conference 9. https://www.ni.com 10. https://www.pcb.com/ 11. Cvjetkovi´c, V.M., Stankovi´c, U.: Arduino based physics and engineering remote laboratory. In: Auer, M., Guralnick, D., Uhomoibhi, J. (eds.) Interactive Collaborative Learning ICL 2016. Advances in Intelligent Systems and Computing, vol. 545. Springer, Cham (2017). https://doi.org/ 10.1007/978-3-319-50340-0_51 12. https://www.st.com/content/st_com/en/support/learning/stm32-education.html 13. https://os.mbed.com/ 14. https://www.st.com/en/evaluation-tools/steval-mksbox1v1.html 15. https://www.st.com/en/evaluation-tools/steval-stwinkt1.html 16. Klein, P., Ivanjek, L., Dahlkemper, M.N., Jeliˇci´c, K., Geyer, M.-A., Küchemann, S., Susac, A.: Studying physics during the covid-19 pandemic: student assessments of learning achievement, perceived effectiveness of online recitations, and online laboratories. October 2020 Physics. arXiv: Physics Education 17. Pols, F.: A physics lab course in times of COVID-19. Electron. J. Res. Sci. Math. Educ. 24(2), 172–178 (2020) 18. Stampfer, C., Heinke, H., Staacks, S.: A lab in the pocket. Nat. Rev. Mater. 5, 169 (2020) 19. Staacks, S., Hütz, S., Heinke, H., Stampfer, C.: Advanced tools for smartphone-based experiments: phyphox. Phys. Educ. 53(4), 045009. https:/ /doi.org/10.1088/1361-6552/aac05e 20. Countryman, C.L.: The educational impact of smartphone implementation in introductory mechanics laboratories. 2015 PERC Proceedings, American Association of Physics Teachers 21. https://phyphox.org/

Chapter 4

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed Hector Laos

Abstract The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f (u), through a novel process. The roots of f (u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown. Keywords Gyroscopes · Closed-form solutions · Equations of motion · Routhian · Cubic polynomial f (u)

4.1 Background In the mid-1700s, Euler made a great contribution to the dynamics of the rigid body with the first solution for the heavy symmetrical top with one point fixed [1]. In the following years, many authors continued using the Euler equations [2, 3] alongside Newtonian mechanics and created the basis for gyroscopes and their applications. Currently, the modern books of classical mechanics use the Lagrangian [4] method because it greatly simplifies the derivation of the equations of motion (EOM) [5, 6]. Even further simplification is obtained using the Routhian method [7], as shown in Udwadia and Han’s [8] application of the Routhian. For the topic of closed-form solutions for the heavy symmetrical top with one point fixed, the author has only two references available, MacMillan [9] and Fetter [10], and these sources show that this topic requires further research. The formulas derived in this paper will hopefully serve the purpose of checking the results from numerical calculations and can also be used as a component of the controls for a gyroscope system.

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan). H. Laos () Oak Ridge National Laboratory, Oak Ridge, TN, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_4

29

30

H. Laos

4.2 Derivation of the EOM The EOM will make use of the Euler angles that are shown in Fig. 4.1 [11]. Figure 4.1(a) [11] shows the Euler angles θ , φ, ψ that define the motion of the system. Figure 4.1(b) shows in red and blue lines the paths of the center of gravity (spin and precession). The angular movement of axis 3 relative to z ( , nutation) is also shown. The Lagrangian (L) for the system [6] is given by L=

 I  2 I1  2 3 ˙ θ˙ + φ˙ 2 sin2 θ + ψ + φ˙ cos θ − Mgl cos θ, 2 2

(4.1)

Where M is the mass of the gyroscope, I1 = I2 is the principal equatorial moment of inertia through the center of gravity (CG) of the gyroscope, and I3 is the polar moment of inertia about the symmetry axis. The dots in (4.1) refer to the differentiation with respect to time t.   Because the coordinates φ and ψ are cyclic, the angular momentum pψ = I3 ψ˙ + φ˙ cos θ and pφ = I1 φ˙ sin2 θ + pψ cos θ are conserved. The angular velocities φ˙ and ψ˙ can be eliminated by using the Routhian (R) [7]:       R θ, θ˙ , t = L − pφ φ˙ pφ , pψ , θ − pψ ψ˙ pφ , pψ , θ .

(4.2)

The EOM is derived from d dt



∂R ∂ θ˙

 −

∂R = 0, ∂θ

(4.3)

and then the EOM is reduced to

θ¨ =





1 I1

2 sin3

θ

  Mgl  pφ − pψ cos θ pφ cos θ − pψ + sin θ. I1

Fig. 4.1 (a) Euler angles. (b) Trajectories of the center of gravity. The author owns one of these toy gyroscopes [11]

(4.4)

4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

31

4.3 The Cubic Polynomial f (u) The classical EOM is of the form:

u u0

du 

 = t, where t is time (s) [6] and is typically shown on the literature as a derivation

f u

from the conservation of energy equation. The roots of this cubic polynomial f (u) furnish the angles at which θ˙ changes in sign; in other words, the extreme values of the nutation (θ ) trajectory. The next section shows the importance of selecting the most suitable value of the root u3 to facilitate the creation of a closed-form solution. The author of this paper used a novel approach to formulate f (u) as shown in the following equations. The  general expression shown in (4.4) could be further reduced by making the angular momentums equal (i.e., p = pφ = pψ and then substituting in (4):

θ˙

θ˙ d θ˙ =

θ 

θ˙0 =0

θ0

p2 (1 − cos θ ) (cos θ − 1)  + sin θ 1 − cos2 θ I1 2



Mgl I1





sin θ dθ

(4.5)

The parameter q is defined as follows: 

p I1

2

 =q

Mgl I1

 (4.6)

,

and (4.5) is solved as 1 2 θ˙ = 2

   

θ  Mgl 1 (cos θ − 1) Mgl q sin θ dθ + I1 sin θ (1 + cos θ ) I1

(4.7)

θ0

θ˙ 2 =

2Mgl I1

θ θ0

q (cos θ − 1) + sin θ sin θ (1 + cos θ )

 dθ

(4.8)

Resolving (4.8) in terms of a new variable u = cos θ is convenient, as shown on the next line: ˙2

θ

θ sin θ = u˙ = β sin θ 2

2

2

θ0

where β =

2Mgl I1

 q (cos θ − 1) + sin θ dθ, sin θ (1 + cos θ )

(4.9)

and the expressions (4.10), (4.11), and (4.12) are used:     sin2 θ = 1 − cos2 θ = 1 − u2

θ θ0

−q q (cos θ − 1) q = + sin θ (1 + cos θ ) (1 + cos θ ) (1 + cos θ0 )

(4.10)

(4.11)

θ sin θ dθ = − cos θ + cos θ0 θ0

(4.12)

32

H. Laos

The final form of (4.9) is reached using

 u˙ 2 = β (1 − u) (u − u0 )

  q − 1 − u = f (u), (1 + u0 )

(4.13)

Where f (u) is a third-order polynomial. Regarding selection of the roots for f (u), the f (u) cubic polynomial has the following roots:  u1 = 1, u2 = u0 & u3 =

 q −1 . (1 + u0 )

(4.14)

Further advancement for a closed-form solution of (4.13) is achieved by making the last root u3 = 1. Therefore, f (u) will have a double root at u1 = u3 = 1, and the final form for (4.13) will be u˙ 2 = β (1 − u)2 (u − u0 )

(4.15)

 √ u˙ = u˙ (+),(−) = ± β (1 − u) u − u0 .

(4.16)

Notice that the parameters q and u0 are related to each other in (4.14) because of the u3 = 1 condition: q = 2. (1 + u0 )

(4.17)

4.4 General Closed-Form Solution of the EOM The following equations explain the algorithm that is used for the general closed-form solution of the EOM. The process starts by defining u0 (the initial condition of the variable u) as a function of a new parameter h:  u0 = f (h) =

 h−1 . h

(4.18)

Recalling the positive form of (4.16),  √ u˙ = u˙ (+) = + β (1 − u) u − u0 .

(4.19)

The integral form of this expression is

u u0

du = √ (1 − u) u − u0

t  β dt.

(4.20)

0

From a table of integrals [12], the following formula is found:

 

−1 d sech x =

−1 dx, √ x 1 − x2

(4.21)

which will be used for the solution of (4.20). The following change of variable is applied in (4.21): 1 − x 2 = uh − (h − 1) or

x=



h (1 − u).

(4.22)

4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

33

The differential of [x2 = h (1 − u)], results in dx =

−h du. 2x

(4.23)

Replacing (4.22) and (4.23) in the right-hand side of (4.21) results in

−1

−h 1 . du √ 2x x uh − (h − 1)

(4.24)

Further manipulation produces 1 2



du 1  ,  √ (1 − u) h u − h−1 h

(4.25)

and using the definition (4.18), the right-hand side of (4.21) is converted to

2

1 √

h

du 1 . √ (1 − u) u − u0

(4.26)

Replacing the result of (4.20) and (4.26) into the full expression of (4.21):

1 √ 2 h



  d sech−1 x =

x

−1 dx = √ 1 − x2

du 1  1 = √ β t. √ (1 − u) u − u0 2 h

(4.27)

Replacing (4.22) in (4.27) is obtained by the following equation:     1 β −1 t. h (1 − u) − sech h (1 − u0 ) = sech 2 h √  Because sech−1 h (1 − u0 ) = 0, (4.28) is further reduced to −1

sech

−1





1 h (1 − u) = 2



β t. h

(4.28)

(4.29)

Solving for the inner term in sech−1 , the following is obtained:    1 β t . h (1 − u) = sech 2 h 2

The following trigonometric identities will help to reduce (4.30):

(4.30)

34

H. Laos

cosh

sech

x  2 x  2

 =

cosh(x) + 1 2

(4.31)

2 cosh(x) + 1

(4.32)

=

When (4.32) is replaced in (4.30) and with u = cos θ , the following result is obtained:   2 1 − cos θ = h

1   . β cosh t + 1 h

(4.33)

With (4.33), defining the nutation angle θ as a function of time (t) for any parameter h ≥ 1 is feasible. This paper presents the closed-form solutions of two cases, h = 1 and h = 2, because with these parameters, the expression (4.33) will produce simple closed-form solutions.

4.5 Closed-Form Solution for h = 1 Formula (4.33) has the following form for h = 1:

1 − cos θ =

2 √  , cosh β t + 1

(4.34)

which reduces to √ β √ cosh β cosh

 t −1  = cos θ. t +1

(4.35)

The following trigonometric property is applicable to (4.35): √ β √ cos θ = cosh β cosh

   t −1 1  βt . = tanh2 2 t +1

(4.36)

Therefore, the closed-form solution is  cos θ = tanh

2

 1 βt . 2

(4.37)

The following table provides an example for comparison. The input data are shown on Table 4.1, which gives all the inputs required for the numerical calculations (MATLAB in-house code) and for the closed-form solution from (4.37). Figure 4.2 shows the results for θ = f (t), which was calculated using a numerical procedure and the exact solution with the closed-form formula (4.37). Figure 4.3 shows the calculated percentual error between these two expressions on the range from 0 to 1 seconds.

4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed Table 4.1 Input data for the case h=1

35

Input parameter name Theta initial location Theta initial angular velocity Phi initial angular velocity Psi initial angular velocity Inertia in axes xx and yy Mass of the wheel Acceleration of gravity Location of the CG Inertia in axis zz

Nomenclature θ0 θ˙0 φ˙ ψ˙

Value

I1, 2 M g l I3

Units rad 0 rad/s 11.23873 rad/s 209.49 rad/s 2.33 E-03 kg m2 0.1 kg m 9.81 s2 0.15 m 1.25 E-4 kg m2

Angular momentum Parameter Initial cosθ 0 Parameter

p = pφ = pψ q u0 h

0.0262 2 0 1

π 2

kg m2 s2

– – –

Fig. 4.2 Results for h = 1 using the numerical procedure and the closed-form formula

4.6 Closed-Form Solution for h = 2 In a similar way, Eq. (4.33) has the following form for h = 2:

1 − cos θ = (1)

1   . β cosh 2 t +1

(4.38)

The closed-form solution is  sec θ = 1 + sech

 β t . 2

(4.39)

36

H. Laos

Fig. 4.3 Percentual error for the h = 1 case Table 4.2 Input data for the case h=2

Input parameter name Theta initial location Theta initial angular velocity Phi initial angular velocity Psi initial angular velocity Inertia in axes xx and yy Mass of the wheel Acceleration of gravity Location of the CG Inertia in axis zz

Nomenclature θ0 θ˙0 φ˙ ψ˙

Value

I1, 2 M g l I3

Units rad 0 rad/s 9.18 rad/s 251.98 rad/s 2.33 E-03 kg m2 0.1 kg m 9.81 s2 0.15 m 1.25 E-4 kg m2

Angular momentum Parameter Initial cosθ 0 Parameter

p = pφ = pψ q u0 h

0.0321 3 0.5 2

π 3

kg m2 s2

– – –

The input data for this example are shown in Table 4.2 and define all the inputs required for the numerical calculations (MATLAB in-house code) and closed-form solution (4.39). Figure 4.4 shows the results for θ = f (t), which was calculated using a numerical procedure and the exact solution with the closed-form formula (4.39). Figure 4.5 shows the calculated percentual error between these two expressions on the range from 0 to 1 seconds. At first glance, the results from the numerical procedure and the closed-form solution look the same. However, the amplitudes of the nutation angle θ are so small that perceiving the difference between them is difficult. Using the data from the percentual error (%) (h = 1, 2) is recommended because these charts clearly show that the error increases with time and, as shown in Fig. 4.3, the error can be as high as 45%.

4 Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

37

Fig. 4.4 Results for h = 2 using a numerical procedure and the closed-form formula

Fig. 4.5 Percentual error for the h = 2 case

4.7 Conclusion Most of the time, the EOMs for the heavy symmetrical top with one point fixed are solved numerically because only a few closed-form solutions exist. This paper has shown that obtaining closed-form solutions is possible for EOMs that are able to satisfy the following conditions:

38

H. Laos

– The angular momentum p = pφ = pψ is equal. – The last root of the cubic polynomial has a value of 1 (u3 = 1), and a link between the parameters q and u0 is created (4.14): u3 =

q −1=1 (1 + u0 )

– Therefore, all the parameters p, q, and u0 are interconnected with each other (4.6): 

p I1

2

 =q

Mgl I1

 .

– Closed-form solutions are defined for all the positive values of the parameter h, where h defines the value of the initial angle θ 0 for the calculations according to (4.18):  u0 = f (h) =

 h−1 , for h ≥ 1. h

The results shown in this paper are only defined for the positive value of u˙ (or u˙ (+) ). Therefore, the solutions only cover a region of the total trajectory of the nutation angle θ . The percentual error (%) is higher at low θ values. For practical purposes, this could be considered an ideal situation. What started as research for closed-form solutions to check the numerical results for the EOMs of the heavy symmetrical top with one point fixed has evolved into developing a full-procedure to produce closed-form solutions for a subset of EOMs that are able to fulfill a series of conditions. A practical application of these results may be to incorporate formulas like (4.37) and (4.39) into the controls of gyroscopic systems.

References 1. Euler, L.: Du mouvement de rotation des corps solides autour d’un axe variable. Histoire de l’Academie Royale des Sciences. 14, 154–193 (1765) 2. Routh, E.J.: The Advanced Part of A Treatise on the Dynamics of a System of Rigid Bodies. MacMillan, London (1884) 3. Klein, F., Sommerfeld, A.: The Theory of the Top, vol. II. Birkhauser, Boston (2010) 4. Lagrange, J.L.: Mécanique Analytique, vol. 2. Mme Vve Courcier, Paris (1815) 5. Marion, J.B., Thornton, S.T.: Classical Dynamics of Particles and Systems, 4th edn. Saunders College Publishing)., Chapter 11 (1995) 6. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Pearson, ISBN 9780201657029 (2001) 7. Cline, D.: Variational Principles in Classical Mechanics, vol. 206, 2nd edn. University of Rochester (2019) 8. Udwadia, F.E., Han, B.: Synchronization of multiple chaotic gyroscopes using the fundamental equation of mechanics. ASME J. Appl. Mech. 75, 1–10 (2008) 9. MacMillan, W.D.: Dynamics of Rigid Bodies, vol. 245. Dover Publications, New York (1960) 10. Fetter, A.L., Waleka, J.D.: Theoretical Mechanics of Particles and Continua, vol. 172. Dover Publications (2003) 11. More details of the toy gyroscope can be found at: www.gyroscopes.com 12. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 7th edn. Elsevier (2007)

Chapter 5

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases Hector Laos

Abstract Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotorbearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon. Keywords Rotordynamics · Rigid body · Nonlinear · Equations of motion · Generalized forces

5.1 Definition of the Equations of Motion The EOMs are defined using the Lagrange equations [1]: d dt



∂L ∂ q˙i

 −

∂L = Fqi , ∂qi

(5.1)

where the Lagrangian L (L = TD − UD )is defined as the difference between the kinetic energy (TD ) and the potential energy (UD ). Also, i (1 ≤ i ≤ N) is the number of degrees of freedom, the generalized independent coordinates are qi , and the generalized forces are Fqi . The generalized coordinates in the vertical rigid rotor are u, v, θ , ψ, as shown in  Fig. 5.1. The Newtonian approach  (not the Lagrangian) will be used for the external forces including springs and dampers Fqi = −Fu , −Fv , −Fθ , −Fψ and for the effect of the rotor weight. This will cancel out the potential energy (UD ) in the Lagrangian, and the expression in Eq. (5.1) is reduced to the following [2]:

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan). H. Laos () Oak Ridge National Laboratory, Oak Ridge, TN, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_5

39

40

H. Laos

Fig. 5.1 Rigid-body rotor

d dt



∂TD ∂ q˙i

 −

∂TD = Fqi ∂qi

(5.2)

For the sake of clarity, the definition of the terms in Eq. (5.2), TD and Fqi , will be explained in different sections.

5.2 Kinetic Energy of a Rigid-Body Rotor The kinetic energy of a rigid-body rotor is as follows [3]: TD =

  1   1 1 md u˙ 2 + v˙ 2 + w˙ 2 + Id ωx2 + ωy2 + Ip ωz2 . 2 2 2

(5.3)

Figure 5.1 shows the vertical rigid-body rotor with its system of coordinates, the location of the center of gravity (CG), and the parameters that will be used to define the displacements and rotations at the CG. In the present work, only the lateral displacements—u, v—and the rotations—θ , ψ, φ—were considered. All lateral displacement and rotations coincide with the CG. The axial movement w is assumed to be decoupled from the lateral displacements. Therefore, w˙ will not be considered any further in this chapter. The rotor spin angle φ will be kept because the transient unbalance is a function of φ and its derivatives. ˙ ψ, ˙ φ. ˙ The angular velocity vector {ω} will be defined as a function of the angular velocities of the Euler angles: θ,   {ω}T = ωx ωy ωz

(5.4)

The process consists of a series of rotations starting from an initial axis {Y} as shown in the following sequence [3–5]: ˙ {ω} = ψ˙ {Y } + θ˙ {x1 } + φ, and

(5.5)

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

⎧ ⎫ ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎡ ⎤⎡ ⎤⎧ ⎫ cos φ sin φ 0 ⎨ θ˙ ⎬ cos φ sin φ 0 1 0 0 ⎨ ωx ⎬ ⎨ 0 ⎬ ⎨0⎬ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = 0 + − sin φ cos φ 0 ψ˙ . 0 + − sin φ cos φ 0 0 cos θ sin θ ω ⎩ ⎭ ⎩ y⎭ ⎩ ˙⎭ ⎩ ⎭ φ 0 0 1 0 ωz 0 0 0 1 0 − sin θ cos θ

41

(5.6)

Solving (5.6), the angular velocity vector {ω} is obtained: ⎧ ⎫ ⎧ ⎫ ˙ sin φ cos θ ⎬ ⎨ ωx ⎬ ⎨ θ˙ cos φ + ψ = − θ˙ sin φ + ψ˙ cos φ cos θ . ω ⎩ y⎭ ⎩ ⎭ ˙ sin θ ωz φ˙ − ψ

(5.7)

Replacing ωx , ωy , and ωz in the kinetic energy TD in Eq. (5.3) produces: TD =

  1   1  2 1 md u˙ 2 + v˙ 2 + Id θ˙ 2 + ψ˙ 2 cos 2 θ + Ip φ˙ − ψ˙ sin θ . 2 2 2

(5.8)

The expression of TD in Eq. (5.8) is replaced in Eq. (5.2) to define the EOM on the generalized coordinates. (5.9)

−

∂TD = md v¨ = −Fv ∂v

(5.10)

d dt



∂ TD ∂ ψ˙

 −



∂ TD ∂ θ˙

 −

  ∂TD = Id θ¨ + Ip φ˙ ψ˙ cos θ + Id − Ip ψ˙ 2 sin θ cos θ = −Fθ ∂θ

(5.11)



∂ TD ∂ u˙



∂TD = md u¨ = −Fu ∂u

d dt d dt



−

d dt

∂ TD ∂ v˙



    ∂TD = Id cos2 θ + Ip sin2 θ ψ¨ − Ip φ¨ sin θ − Ip φ˙ θ˙ cos θ + 2 Ip − Id ψ˙ θ˙ sin θ cos θ = −Fψ ∂ψ (5.12)

The expressions (5.9), (5.10),  (5.11), and (5.12)  are the nonlinear form of the EOM. Cancelling out the higher-order terms and at steady-state conditions, φ¨ = 0, φ˙ = Ω , the linear form for small angles can be obtained, as shown in the following equations: d dt

d dt

d dt

d dt









∂ TD ∂ θ˙

∂ TD ∂ ψ˙

∂ TD ∂ u˙

 −

∂TD = md u¨ = −Fu ∂u

(5.13)

−

∂TD = md v¨ = −Fv ∂v

(5.14)

−

∂TD = Id θ¨ + Ip Ω ψ˙ = −Fθ ∂θ

(5.15)

−

∂TD ¨ = Id ψ− Ip Ω θ˙ = −Fψ ∂ψ

(5.16)

∂ TD ∂ v˙ 





  These expressions will be linear if the generalized forces Fqi are in linear form as well.

42

H. Laos

Table 5.1 Displacement and velocities at the supports

Planes Displacement—1 Displacement—2 Velocity—1 Velocity—2

Plane x-z u − a sin ψ u + b sin ψ u˙ − a ψ˙ cos ψ u˙ + bψ˙ cos ψ

Plane y-z v + a sin θ v − b sin θ v˙ + a θ˙ cos θ v˙ − bθ˙ cos θ

5.3 Nonlinear Generalized Forces The nonlinear generalized forces are calculated using a Newtonian approach. Table 5.1 defines the displacements and velocities at the end points of Fig. 5.1 (1 and 2) on the x-z and y-z planes. The stiffness (kAxis# ) and damping (cAxis# ) at the supports at each plane are known where the numbers are end points 1 or 2. The generalized forces will take the following forms:     Fu = kx1 (u − a sin ψ) + kx2 (u + b sin ψ) + cx1 u˙ − a ψ˙ cos ψ + cx2 u˙ + bψ˙ cos ψ

(5.17)

    Fv = ky1 (v + a sin θ ) + ky2 (v − b sin θ ) + cy1 v˙ + a θ˙ cos θ + cy2 v˙ − bθ˙ cos θ

(5.18)

    Fθ = aky1 (v + a sin θ ) − bky2 (v − b sin θ ) + acy1 v˙ + a θ˙ cos θ − bcy2 v˙ − bθ˙ cos θ

(5.19)

    Fψ = −akx1 (u − a sin ψ) − bkx2 (u + b sin ψ) − acx1 u˙ − a ψ˙ cos ψ + bcx2 u˙ + bψ˙ cos ψ

(5.20)

The following equations show how the grouping of the common terms is defined: Fu = kxT u + kxC sin ψ + cxT u˙ + cxC ψ˙ cos ψ

(5.21)

Fv = kyT v − kyC sin θ + cyT v˙ − cyC θ˙ cos θ

(5.22)

Fθ = −kyC v + kyR sin θ − cyC v˙ + cyR θ˙ cos θ

(5.23)

Fψ = kxC u + kxR sin ψ + cxC u˙ + cxR ψ˙ cos ψ

(5.24)

The equivalent stiffness and damping shown in the previous expressions are shown in Appendix A.

5.4 Linear Form of the EOM   For steady-state conditions φ˙ = Ω , the linearized form of the EOM used to solve the eigenvalues is as follows: ˙ + [K] {q} = {0} , ¨ + (Ω [G] + [C]) {q} [M] {q}

(5.25)

The following EOM is used to solve the unbalance response ˙ + [K] {q} = {U } . ¨ + (Ω [G] + [C]) {q} [M] {q}

(5.26)

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

43

where the matrices ([M], mass, [G], gyroscopic, [C], damping, and [K], stiffness) and vector ({q}, displacement) from Eqs. (5.25) and (5.26) are defined using Eqs. (5.13), (5.14), (5.15), and (5.16) and the linearized form of Eqs. (5.21), (5.22), (5.23), and (5.24): {q}T = [u v

θ

ψ]

(5.27)

⎡

⎤ 0 0 md 0 ⎢ 0 md 0 0 ⎥ ⎥ [M] = ⎢ ⎣ 0 0 Id 0 ⎦ 0 0 0 Id ⎡

0 ⎢0 [G] = ⎢ ⎣0 0 ⎡ ⎢ [C] = ⎢ ⎣

⎡ ⎢ [K] = ⎢ ⎣ kxC

0 0 0 0 0 0 0 − Ip

cxT 0 0 cyT 0 −cyC cxC 0

⎤ 0 0 ⎥ ⎥ Ip ⎦ 0

⎤ 0 cxC − cyC 0 ⎥ ⎥ cyR 0 ⎦ 0 cxR

⎤ kxT 0 0 kxC − kyC 0 ⎥ 0 kyT ⎥ kyR 0 ⎦ 0 −kyC − md g + md g 0 0 kxR

(5.28)

(5.29)

(5.30)

(5.31)

The effect of weight has also been included in the stiffness matrix, [K]. Finally, the unbalance vector {U} [6] is ⎧ ⎪ ⎪ ⎨

mε Ω 2 mε Ω 2   {U } = ⎪ − Id − Ip β Ω 2 ⎪   ⎩ Id − Ip β Ω 2

⎫ cos (Ωt + δ) ⎪ ⎪ ⎬ sin (Ωt + δ) . sin (Ωt + γ ) ⎪ ⎪ ⎭ cos (Ωt + γ )

(5.32)

The eccentricity (ε) of the rotor creates an unbalance (mε). If the axis of rotation has an angle β relative to its geometric axis, it creates a rotational unbalance of magnitude (Id − Ip ) β. No angular acceleration component exists at steady-state conditions, but this effect will be considered in the solution of the nonlinear cases. The phase angles δ and γ are used to provide correlation between the unbalance and rotational unbalance.

5.5 Results from the Linear Example Table 5.2 contains the rotor data that was used for the examples and came from Friswell et al. [7]. The state-space technique was used for the solution of the eigenvalues [8]. Figure 5.2 shows the Campbell diagram for the operational range from 0 to 3800 rpm. The intersection points correspond to the vertical rigid-body eigenvalues. The X value denotes the speed () at which the eigenvalue was calculated, and the Y value is the actual eigenvalue result.

44 Table 5.2 Data from the Dynamics of Rotating Machines [7]

H. Laos Input parameter Moment of inertia (xx and yy) Polar moment of inertia (zz) Rotor mass Acceleration of gravity Height above CG Height under CG Upper stiffness (H) Lower stiffness (H) Upper stiffness (V) Lower stiffness (V) Upper damping (H) Lower damping (H) Upper damping (V) Lower damping (V) Rotational unbalance Phase angle (in β) Lateral unbalance Phase angle (in mε)

Symbol Id Ip md g a b kx1 kx2 ky1 ky2 cx1 cx2 cy1 cy2 β γ mε δ

Value 2.8625 0.6134 122.68 9.81 0.250 0.250 1.0E+6 1.3E+6 1.5E+6 1.8E+6 20.0 26.0 30.0 36.0 0.0 or 1.0E−3 π /2 1.0E−2 or 0.0 0.0

Units kg m2 kg m2 kg m/s2 m m N/m N/m N/m N/m N. s/m N. s/m N. s/m N. s/m rad rad kg. m rad

Fig. 5.2 Campbell diagram

Calculations of the unbalance response were also made at steady-state conditions using the complex form of the EOM [6]. Figures 5.3 and 5.4 show the results for the displacements u and v for an unbalance mε = 1.0E−2 kg.m and β= 0.0 rad. Figures 5.5 and 5.6 show the results for the displacements u and v for an unbalance mε = 0.0 kg. m and β = 1.0E−3 rad.

5.6 Procedure for Solving of the Nonlinear EOM The results from the unbalance response calculations in the linearized EOM were obtained assuming steady-state conditions, but this assumption does not hold for the nonlinear EOM. The process requires a differential equation solver, and for these calculations, MATLAB’s@ subroutine ode45 was used. Ode45 uses the fourth-order Runge−Kutta method. Also, a time step and the value of the angular acceleration of the driver will be required. For this procedure, it will be assumed that the constant angular acceleration (α), the angular velocity, and angular displacement will be defined as a function of α and time (t).

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

45

Fig. 5.3 u Amplitude (mε= 1.0E−2 kg.m and β = 0.0 rad)

Fig. 5.4 v Amplitude (mε = 1.0E−2 kg.m and β = 0.0 rad)

φ¨ = α = constant

(5.33)

φ˙ = α t

(5.34)

φ=

1 α t2 2

(5.35)

The unbalance vector {U} will require an extra term (Eq. 5.32) because of the angular acceleration. The expression for {U} is ⎫ mε cos (φ + δ) ⎪ ⎪ ⎬ mε sin + δ) (φ 2   {U } = φ˙ + φ¨ ⎪ − Id − Ip β sin (φ + γ ) ⎪ ⎪ ⎪   ⎭ ⎩ Id − Ip β cos (φ + γ ) ⎧ ⎪ ⎪ ⎨

⎧ ⎪ ⎪ ⎨

⎫ mε sin (φ + δ) ⎪ ⎪ ⎬ − mε cos (φ + δ) . ⎪ I β cos (φ + γ ) ⎪ ⎪ ⎪ ⎩ d ⎭ Id β sin (φ + γ )

(5.36)

46

H. Laos

Fig. 5.5 u Amplitude (mε = 0.0 kg. m and β = 1.0E−3 rad)

Fig. 5.6 u Amplitude (mε = 0.0 kg. m and β = 1.0E−3 rad)

In summary, the EOMs for the nonlinear system are md u¨ + Fu = U1 ,

(5.37)

md v¨ + Fv = U2 ,

(5.38)

  Id θ¨ + Ip φ˙ ψ˙ cos θ + Id − Ip ψ˙ 2 sin θ cos θ + Fθ = U3 ,

(5.39)

and 

Id cos2 θ + Ip sin2 θ



  ψ¨ − Ip φ¨ sin θ − Ip φ˙ θ˙ cos θ + 2 Ip − Id ψ˙ θ˙ sin θ cos θ + Fψ = U4 .

¨ φ, ˙ φare defined by Eqs. (5.33), (5.34), and (5.35). The values for φ,

(5.40)

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

47

5.7 Results for the Nonlinear Example For the nonlinear example, an angular acceleration of α = 0.01 rad/s2 and a time step of δt = 0.001 s will be used along with the initial conditions for all the parameters. The displacements and their first derivatives are zero. For comparison purposes, the data for the steady-state calculations will be used for the calculations using the nonlinear EOMs. The results are shown in Figs. 5.7 and 5.8. The results from the nonlinear calculations compare well with the steady-state values. As shown in Fig. 5.7, the u amplitude of 0.02831 m at 1291 rpm is comparable to the u amplitude of 0.02901 m at 1289 rpm shown in Fig. 5.3. In a similar way, the v amplitude of 0.02442 m at 1556 rpm shown in Fig. 5.8 is comparable to the v amplitude of 0.02500 m at 1556 rpm shown in Fig. 5.4. The calculations made at steady-state conditions will always have a larger amplitude than the nonlinear results because in steady-state conditions, infinite time for the amplitude to grow at the critical speed exists. That is not true for the nonlinear case where the angular acceleration (α) must always be greater than zero.

Fig. 5.7 u Amplitude (mε= 1.0E−2 kg.m and β = 0.0 rad)

Fig. 5.8 v Amplitude (mε= 1.0E−2 kg.m and β = 0.0 rad)

48

H. Laos

As shown in Fig. 5.9, the u amplitude of 0.004941 m at 1290 rpm is comparable to the u amplitude of 0.005009 m at 1289 rpm shown in Fig. 5.5. Similarly, the v amplitude of 0.02524 m at 1556 rpm shown in Fig. 5.10 is comparable to the v amplitude of 0.02567 m at 1556 rpm shown in Fig. 5.6. All these results indicate that the unbalance force in the nonlinear cases is producing similar results as in the steady-state conditions for the set of imbalances that were chosen. The next section will show how a change in the β angle can bring the system into a chaotic mode.

Fig. 5.9 u Amplitude (mε = 0.0 kg. m and β= 1.0E−3 rad)

Fig. 5.10 v Amplitude (mε = 0.0 kg. m and β= 1.0E−3 rad)

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

49

5.8 Example That Exhibits Chaotic Behavior As shown in Eqs. (5.37), (5.38), (5.39), and (5.40), the EOMs for θ and φ are the ones that contain more higher-order terms, which may lead to sensitivity in nonlinear behavior. The author has corroborated this fact by doing multiple calculations using different values of mε and β and found that the system response is very sensitive to β. The same system parameters will be used for the next calculations, except that the value for β will be increased from 1.0E−3 rad to 1.0E−2 rad (one order of magnitude increment). The results are shown in Figs. 5.11 and 5.12. Notably, in Figs. 5.11 and 5.12, the first two critical speeds (1289 and 1556 rpm, which are associated with u, v,respectively) remain linear. The amplitude at 1289 rpm in Fig. 5.11 is one order of magnitude larger than the value in Fig. 5.9. The amplitude at 1556 rpm in Fig. 5.12 is one order of magnitude larger than the value in Fig. 5.10. However, once the speed reached the last two critical speeds (which are related to θ, ψ), the response was nonlinear. For the next critical speed, the maximum amplitude is located at 2059 rpm, which is slightly lower than the linear eigenvalue of 2068 rpm. Also, the curves (at u and v) no longer behave in a strictly vertical form, which is the so-called “softening spring” nonlinear behavior.

Fig. 5.11 u Amplitude (mε = 0.0 kg. m and β= 1.0E−2 rad)

Fig. 5.12 v Amplitude (mε = 0.0 kg. m and β= 1.0E−2 rad)

50

H. Laos

Fig. 5.13 Orbits on the x-y plane shown in time (vertical axis)

Fig. 5.14 Orbits on the x-y plane for the entire operational range

The next linear critical speed is predicted at 2737 rpm; starting at about that speed, a change in the behavior in the system can be observed. At 3393 rpm, a “jump” of very short duration but of high intensity occurs. This behavior is depicted in full in the orbits shown in Figs. 5.13 and 5.14. After the jump, the system reaches steady-state conditions and remains there for the duration of the calculations. Definitions like “softening spring,” “hardening spring,” and “jump” are common in nonlinear theory, and a short explanation of the Duffing equation in the next section will provide a better understanding of these topics.

5.9 Duffing Equation The Duffing equation [9] will be used to explain some of the nonlinear features that were observed in the vertical rigid-rotor example. The Duffing equation is as follows: x¨ + cx˙ + ωn2 x ± μx 3 = F cos (ωt + φ) .

(5.41)

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

51

The nonlinear stiffness μx3 can have a positive or negative sign and will define the system as a hardening or softening spring. The frequency response is calculated from the following expression:

   3 2 2 3 F = ωn − ω A + μA + [cωA]2 , 4 2

(5.42)

Where A is the amplitude of the solution for x. x = A cos (ωt)

(5.43)

Figure 5.15 shows the frequency response as a function of the excitation frequency (ω) with c = 0.1 , ωn2 = 1.0, F = 1.0 and values of μ = − 0.003 → 0.040. Brennan et al. [10] show results in similar fashion as in Fig. 5.15. The red line represents the frequency response in the absence of any nonlinearity. The magenta and green lines correspond to hardening spring cases for different degrees of μ. The cyan line is a case for softening spring example. The blue lines are “exclusion zones,” which means no vibrational activity will occur in this unstable area, and the curve will have to follow a different path on the run-up or coast-down. Figure 5.16 shows the paths used to avoid the unstable zone. On the run-up from the peak amplitude at point a, no other possibility exists but to go to b and then continue all the way up in speed to c. On the coast-down, point c will move to d, and the only way to be able to continue the descend in frequency is to move up to point e. The same parameters used to determine the frequency response will be used to calculate the transient for μ = 0.0400. This is shown in Fig. 5.17. The peak amplitude shown in Fig. 5.17 is very similar to the peak value shown in Fig. 5.16 that was calculated for steadystate conditions with the same parameters. Figure 5.17 clearly depicts the jump from the peak value in the manner predicted for a hardening-spring, nonlinear system (μ = 0.04). With a change in parameters, inducing a chaotic response is possible, as shown in Fig. 5.18. The set of adimensional parameters used in Fig. 5.18 were c = 0.3, ωn2 = −1.0, F = 0.37, and μ = 1.0. The negative sign in the ωn2 term only means that the linear stiffness is negative. These examples clearly illustrate the response of a nonlinear system and help put the results shown in Figs. 5.11, 5.12, 5.13, and 5.14 into perspective.

Fig. 5.15 Frequency response

52

H. Laos

Fig. 5.16 Frequency response (μ = 0.0400); jump phenomena—paths to follow

Fig. 5.17 Transient result for μ = 0.0400

5.10 Conclusions – The results from the nonlinear calculations compare well with the steady-state values if the unbalances are in the low- to moderate-value range. – The practice of accelerating the rotor during the run-up reduces the amplitudes of motion in the rotor-bearing system, which facilitates crossing through the critical speeds. The same angular acceleration value (but with the opposite sign) should be used during the coast-down to minimize the amplitudes of motion as well.

5 Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

53

Fig. 5.18 Chaotic response

– Large unbalances will trigger a nonlinear response, and the θ , ψ terms are the most sensitive to this effect, which explains the need to have good control of the rotational unbalance in the rotor-bearing system. – Notably, in Figs. 5.11 and 5.12, the first two critical speeds (1289 and 1556 rpm, which are associated with u and v,respectively) remain linear. – The nonlinear motions are triggered with large values of β, and the nonlinear effects showed up in the form of softening spring and chaotic motions for the last two critical speeds, respectively.

A.1 Appendix A kxT = kx1 + kx2 kxC = −a kx1 + b kx2 kxR = a 2 kx1 + b2 kx2

kyT = ky1 + ky2 kyC = −a ky1 + b ky2 kyR = a 2 ky1 + b2 ky2

cxT = cx1 + cx2 cxC = −a cx1 + b cx2

54

H. Laos

cxR = a 2 cx1 + b2 cx2

cyT = cy1 + cy2 cyC = −a cy1 + b cy2 cyR = a 2 cy1 + b2 cy2

References 1. Lalanne, M., Ferraris, G.: Rotordynamics Prediction in Engineering, vol. 1, 2nd edn. Wiley, New York (1999) 2. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines, vol. 158. Cambridge University Press, New York (2010) 3. Lalanne, M., Ferraris, G. Rotordynamics Prediction in Engineering, vol. 2, 2nd edn. Wiley, New York (1999) 4. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines, vol. 157. Cambridge University Press, New York (2010) 5. Vance, J.: Rotordynamics of Turbomachinery, vol. 121. Wiley, New York (1988) 6. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines, vol. 246. Cambridge University Press, New York (2010) 7. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines, vol. 247. Cambridge University Press, New York (2010) 8. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines, vol. 39. Cambridge University Press, New York (2010) 9. Thomsom, W.T., Dahleh, M.D.: Theory of Vibration with Applications, vol. 448, 5th edn. Pearson/Prentice Hall (1998) 10. Brennan, M.J., Kovacic, I., Carella, A., Waters, T.P.: On the jump-up and jump-down frequencies of the duffing oscillator. J. Sound Vib. 318, 1250–1261 (2008)

Chapter 6

Vibration Control in Meta-Structures Using Reinforcement Learning D. Mehta and Vijaya V. N. Sriram Malladi

Abstract This chapter considers using reinforcement learning (RL) to adaptively tune frequency response functions of meta-structures. RL algorithm tunes the stiffness of the spring of the lumped multi-DOF system, as the lumped mass is varied. As some of the lumped masses are modified by 10%, the spring’s stiffness is tuned to maintain the original bandgap. A Q-Learning algorithm is used for RL, wherein the Q-value is updated based on Bellman’s equation. The results compare the frequency response functions of the terminal masses of the baseline and varied mass structure. Keywords Reinforcement learning (RL) · Bandgap · Q-Learning · Reward function · Stiffness

6.1 Introduction Structures built using periodic lattices result in unique frequency response functions (FRFs) with bandgaps. Systems with such dynamics are called meta-structures. The Bragg scattering in phononic crystals motivates the development of these mechanical systems with periodic resonators [1]. The bandgaps in the FRFs are regions in elastic waves that do not propagate [2], thereby reducing vibrational energy. A meta-structure is built using periodic lattices made of lumped spring– mass unit cells. Figure 6.1 presents a schematic of the unit cell considered for this chapter. The spring–mass system model was chosen for the simplicity of modeling and detailed analysis. In the present case, the masses of stiffness of the unit cell are (m1 = 4/17) kg, (m2 = 5/17) kg, (k1 = 1000) N/m. The stiffness between m1 and m2 is varied through the spring k2 . The total masses of the 5 unit cells are maintained at 5 kg, and a feasible stiffness for spring is chosen for analyzing the output. (k2 ) is kept variable as it is the parameter that would be changed to obtain the desired bandgap and its location. All the forces and deformation occur in the longitudinal direction. The spring stiffness k1 is kept constant throughout the analysis. Each unit cell consists of (2n + 1) masses, where n is the number of unit cells. Therefore, five unit cells consist of 11 masses. Six masses weighing 4/17 kg and five masses weighing 2/17 kg are used in simulations. This chapter employs reinforcement learning (RL) technique in designing the lumped spring–mass model that would result in a meta-structure with a bandgap at a desired frequency range. A specific Q-Learning algorithm under the RL concept is applied to the stiffness of the springs (k2 ) to optimize and tune the location of the bandgap. In this chapter, to validate the use and robustness of the RL algorithm, the lumped spring–mass model is subject to varying parameters into three cases. In case 1, the spring stiffness of the meta-structure is designed such that the bandgap occurs at a particular frequency. In case 2, the spring stiffness is tuned to compensate the uncertainty present in lumped mass. Case 3 considers the case of unexpected change in one of the lumped masses. Like other machine learning algorithms, RL does not need historical data and a labeled data set. In RL, the system undergoes a trial and error method based on different inputs, and for every desired output, system gets a reward based on which it takes an action. The goal of the RL algorithm is to increase the rewards over time and find the optimal solution. In this chapter, RL is applied to estimate the spring stiffness (k2 ) for a five unit cell lattice structure. In total, there are 5 parameters estimated for each of the three cases discussed earlier.

D. Mehta () · V. V. N. S. Malladi Vibrations, Intelligent Testing and Active Learning of Structures, Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_6

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Fig. 6.1 2DOF unit cell of a periodic lattice

Fig. 6.2 RL framework

6.2 Q-Learning Algorithm Framework Under the RL concept, there are many algorithms like Q-Learning, Monte Carlo, and Deep Deterministic Policy Gradient. For finding the optimal stiffness combinations in this chapter, Q-Learning algorithm is used. Q-Learning algorithm is a basic formulation that directs the agent to take an action based on maximization of a reward. “Q” represents the quality of the action in a given state [3]. The framework is shown in Fig. 6.2. Environment is the plant, and agent is a device that performs an action (input) on the environment (plant). Based on the action, plant provides a desired output (state). The state drives the reward function. Rewards increase as the output gets closer to the optimal solution. For this experiment, environment is the meta-structure, agent is assumed as a device that changes stiffness, states are the bandgap and bandgap location, and action is changing the stiffness. The reward function specifies rewards for being in a desirable state [4]. Q-Learning algorithm uses the Bellman’s formula shown in Eq. 6.1: Qnew (st , at ) ← Q(st , at ) + α.(rt + γ .maxQ(st+1 , a) − Q(st , at )),

(6.1)

where st and at are the state and action, respectively, at time t, α is the learning rate, rt is reward at time t, and γ is the discount factor. Learning rate α determines how much does the newly acquired information override old information, and the discount factor γ determines the significance of rewards that can occur in the future. Both the factors range from 0 to 1. α 0 means that the agent learns nothing and exploits only past information. γ of 0 causes the agent to consider only current rewards. For the current Q-Learning algorithm, an epsilon greedy strategy is used, wherein epsilon signifies the exploration rate. As the agent learns through iterations, the initial epsilon value decays over time. The values considered for the results are shown in Eq. 6.2: α = 0.1; γ = 0.99; I nitialEpsilon = 1; MinimumEpsilon = 0.01.

(6.2)

The epsilon value decays over time by using Eq. 6.3:  =  min + ( max −  min ). exp−decay ∗i ,

(6.3)

where i represents the current iteration. The two additional factors that influence the convergence of the Q-Learning algorithm are episodes and steps. Each episode resets the structure to its initial state, and within each episode, algorithm performs steps in loops to find the optimal action. Using the algorithm and state-space model, results have been examined according to case 2 where an uncertainty is added to the baseline structure shown in Fig. 6.3.

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Fig. 6.3 (a) Is FRF of the baseline structure with both masses increased by 10%. (b) Is the average reward over 500 iterations for the corresponding masses

For case 2, the masses are varied by 10% to reflect the manufacturing errors of the structures. The variation in masses causes bandgap’s location to shift. The goal is to maintain the same bandgap location using the k2 values obtained from Case 1. The reward function for case 1 and case 2 is shown in Eq. 6.4: reward =

1 , abs(100 − bf req + 0.01)

(6.4)

where bf req is the bandgap location frequency. If the location frequency is close to 100 Hz, reward increases. As observed in Fig. 6.3a, the RL algorithm effectively maintains the bandgap location combination of masses by learning the optimal values over 500 iterations. The monotonic nature of the reward function validates the robustness of the algorithm. The stiffness values for Fig. 6.3a are shown in Eq. 6.5: k21 = 1500 N/m; k22 = 1400 N/m; k23 = 1400 N/m; k24 = 600 N/m; k25 = 200 N/m,

(6.5)

where for k2j , j represents the stiffness of the spring k2 in the j th unit cell.

6.3 Conclusion All the three cases represent one of the many possible variations in the environment conditions. For case 3, where an additional mass is added at a random location, a modification in the reward function can yield optimal parameters depending upon the desired output. The examination of the results shows that the RL algorithm is a promising approach toward solving uncertain inputs and can be applied to complex systems for reduced vibrations. Modifications in the type of algorithm and the parameters can affect the accuracy of the algorithm. Change in the number of episodes, number of steps, exploration and exploitation rates, reward function formulation, learning rate, and the discount factor can help in achieving a more optimal solution. The key advantage of RL is that it can be used for any modification in the environment. It can be implemented for any system given its generalized algorithm [5]. The model can be effectively used for structures subjected to random vibrations as well. Few disadvantages include: accurate design of the reward function, larger computation time, large amount of data needed, and applying constraints [5]. Although the current RL algorithms may have disadvantages, they will play a significant role in the future in many applications like vibration and self-driving cars where the environment is uncertain and a labeled data set is not available.

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References 1. Reichl, K., Inman, D.: Lumped mass model of a 1d metastructure for vibration suppression with no additional mass. J. Sound Vib. 403, 75–89 (2017) 2. C. Sugino, Ruzzene, M.: Merging mechanical and electromechanical bandgaps in locally resonant metamaterials and metastructures. J. Mech. Phys. Solids 116, 04 (2018) 3. Mattison, T.: Demystifying deep reinforcement learning. In: neuro.cs. ut. ee. Computational Neuroscience Lab. (2015) 4. Wiering, M., van Otterlo, M. (eds.): Reinforcement Learning. Springer, Berlin (2012) 5. Nian, R., Liu, J., Huang, B.: A review on reinforcement learning: introduction and applications in industrial process control. Comput. Chem. Eng. 139, 106886 04 (2020)

Chapter 7

Using Steady-State Ultrasonic Direct-Part Measurements for Defect Detection in Additively Manufactured Metal Parts Erica M. Jacobson, Ian T. Cummings, Peter H. Fickenwirth, Eric B. Flynn, and Adam J. Wachtor

Abstract With the increasing availability and implementation of additive manufacturing in a variety of safety-critical implementations, there is a demand for non-destructive in situ quality control processes that ensure part-to-part and build-tobuild repeatability. Traditional destructive evaluation methods are not financially sustainable for most additive manufacturing applications due to the long build times and the expense of manufacturing parts specifically for destruction. Acoustic wavenumber spectroscopy, a rapid steady-state non-destructive evaluation technique that has successfully identified defects such as delamination in carbon fiber reinforced panels and weld cracking, has been modified to collect in situ direct-part measurements in a laser powder bed fusion machine for 304L stainless steel builds. During each layer of the build, a laser Doppler vibrometer records the part surface response to an ultrasonic steady-state excitation. The result is a 3D inspection volume constructed alongside the part, offering detail of the potential faults and defects within. Various signal processing techniques were used to identify features sensitive to lack-of-fusion defects within the part. Statistical and machine learning methods were used to identify whether a direct-part response measurement is “nominal” or “abnormal.” X-ray computed tomography was used to verify the defect regions post-build and allow for accurate truth data generation. These validated features will enable simultaneous in situ data collection and processing. The operator may be alerted of potential defects and their locations as defect-indicative features form, allowing the operator to pause the build or adjust the print parameters depending on part specifications. Keywords Laser powder bed fusion · Acoustic wavenumber spectroscopy · Ultrasonic inspection · Lack of fusion · Quality control

7.1 Introduction Additive manufacturing (AM) of metal parts is increasingly the focus of interest in many applications because of the increased design space offered over traditional subtractive manufacturing methods. However, there is a lack of standardization that assures that AM parts have the same level of part-to-part and build-to-build repeatability as traditional manufacturing methods. Inherent variability of the AM process causes variations in mechanical and material properties between individual parts and build layers. These variations contribute to the formation of defects, which in turn affect the final specimen’s performance [1]. AM materials are unique in their anisotropy, material anomalies, location-specific properties, and residual stresses, all of which must be understood and related to their fatigue strength and other fracture properties before industry-wide standardization. E. M. Jacobson () · A. J. Wachtor Los Alamos National Laboratory, Engineering Institute, Los Alamos, NM, USA e-mail: [email protected] I. T. Cummings Los Alamos National Laboratory, Engineering Institute, Los Alamos, NM, USA Los Alamos National Laboratory, Space and Remote Sensing, Los Alamos, NM, USA P. H. Fickenwirth Los Alamos National Laboratory, Test Engineering, Los Alamos, NM, USA E. B. Flynn Los Alamos National Laboratory, Space and Remote Sensing, Los Alamos, NM, USA © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_7

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It is necessary to find and implement a quality control (QC) method that is fast, reliable, and can be implemented during the build process to inspect and validate build quality at each layer. This in situ inspection method must be fast enough to inspect large areas during the build process, must be able to withstand the harsh environment within the build chamber, and must provide enough detail to detect small defects, none of which the current post-build evaluation methods can provide [2]. The end result should be a defect-free, structurally sound part, with each layer inspected, qualified, and approved for use in safety-critical applications.

7.1.1 Defects in Additively Manufactured Parts Various defects can form in metal AM parts that are not found in traditionally manufactured parts because they are unique to the manufacturing process. Common defects in metal AM parts include porosity, high surface roughness, cracking, and delamination. In most instances, formation of these defects occurs when process parameters are improperly selected for either the material or the geometry of the part [3]. Defects most commonly found in laser powder bed fusion (L-PBF) parts, and their respective build parameter solution, are displayed in Table 7.1. This study focuses on bulk porosity due to lack-of-fusion (LOF) defects in L-PBF processes. Bulk porosity can occur when melt pools do not fuse with the previous layer’s solidified material. This means that the melt pool is not deep enough to reach the layer below or not large enough to reach the adjoining solidified tracks. LOF defects appear when the hatch spacing of the build is too large, scan speed is too fast, or laser power is too low. Melt tracks that would normally connect and solidify together are then left with large pores potentially filled with powder (Fig. 7.1). Foster et al. show that LOF defects are mainly caused by improper beam positioning (hatch spacing) and laser parameters (laser power, laser speed) [3]. L-PBF machine manufacturers often utilize proprietary technology, making process parameter standardization difficult between manufacturers and machine platforms. Recent developments in QC and in-process inspection for L-PBF include monitoring the build process for each layer and controlling the build process parameters to enforce part-to-part and layerto-layer repeatability. The AM community is still considering the merits of adaptively controlling process parameters during the build process in order to reduce the presence of defects.

Table 7.1 Common defects in metallic AM components, their size, and potential solution [2–4] Defect Pores (gaseous) Pores (elongated) Balling Unfused powder Cracking and delamination Surface roughness

Size (μm) 5–20 50–500 Varies 100–150 Varies Varies

Potential solution Decrease laser speed, adjust laser power Increase laser power, decrease laser speed, decrease hatch spacing Decrease laser power, decrease laser speed Increase laser power, decrease laser speed Alter scan path strategy, heat build plate Increase laser power, perform post-processing

Fig. 7.1 Microscopic images of pores at two locations of 304L SS part built by L-PBF

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7.1.2 Quality Control and Non-destructive Evaluation AM parts are still tested via QC standards that were created for traditional manufacturing methods, including destructive evaluation. It is not financially feasible to continue this approach, because unlike subtractive manufacturing, metal AM does not guarantee that parts even from the same build have the same defect populations and resulting material properties, making destructive testing largely irrelevant. Non-destructive evaluation (NDE) is a method that allows the quality of a part to be determined without destroying the final product. Traditional NDE inspection methods, e.g. X-ray computed tomography (XCT), for conventionally manufactured parts are not always suitable for use on AM parts, because parts may be too complex to accurately image, surface roughness may distort the image, the surfaces of interest may not be accessible, or the part is too large to scan [1]. XCT is also time-consuming and expensive and can have difficulties detecting small out-of-plane defects, such as cracks [5]. NDE methods used for AM parts include high-speed cameras, temperature-monitoring systems, proximity sensors, and ultrasonic excitation. Most commonly, these tools are used to monitor the melt pool size/temperature or the surface vibration response. Most often, data is collected in situ, analyzed post-build, and compared with XCT data. Some existing in situ NDE methods that work well with the AM process include monitoring build temperature with thermocouples or a thermal-imaging camera or monitoring melt pool geometry with a high-speed camera [2]. Other existing NDE methods that use ultrasonic excitation via laser to measure changes in wavelength include interferometry, phased array, and spatially resolved acoustic spectroscopy (SRAS) techniques. Smith et al. used SRAS to measure surface acoustic wave velocity generated by a Q-switch laser to identify pores on the order of 100 μm in Ti-6Al-4V parts built on a Renishaw AM 250 [6, 7]. Cerniglia et al. applied interferometry as an ex situ inspection of Inconel 600 parts with blind holes from 150 to 500 μm [8]. Wang et al. and Li et al. applied phased array techniques to image flat-bottomed holes in titanium parts [9, 10]. Other ultrasonic techniques use a transducer as the source of excitation and receiver, including time of flight (ToF) and impedance measurements. Slotwinski et al. monitored ultrasonic wavespeed using a transducer as the source and receiver in cobalt-chrome parts and correlated density (porosity) to changes in measured ultrasonic wavespeed [11]. Sturm et al. utilized an ultrasonic transducer as the source and receiver for an impedance-based monitoring technique that monitored differences in the dynamic response of parts with a healthy control dataset [12]. Reider implemented time-of-flight measurements to characterize voids (2 mm in diameter) and porosity (up to 3%) in Inconel 718 parts built with an EOS M280 [13]. This work utilizes the benefits of steady-state ultrasonic excitation and rapid non-contact laser Doppler vibrometer (LDV) measurements during the build process. The response data for each layer penetrates below the surface, providing more information than images do. Unlike other ultrasonic monitoring methods, this data can quickly produce image-like plots with minimal processing for human and computer visual inspection. This flexible data can be analyzed with a variety of processing and detection algorithms.

7.1.3 Acoustic Wavenumber Spectroscopy Acoustic wavenumber spectroscopy (AWS) is an ultrasonic rapid inspection technique developed at Los Alamos National Laboratory [14]. AWS was originally created to identify defects on large thin-walled structures in a fraction of the time required by traditional inspection techniques. An ultrasonic transducer, physically attached to the structure of interest, emits a single-tone steady-state harmonic excitation. A scanning LDV measures the out-of-plane steady-state response in a raster scan across the structure’s surface. The raw time series data is divided into “blocks” that represent pixels in the final image. These blocks of data are transformed into the frequency domain via dot product with a complex exponential at the excitation frequency (7.1). V (x, y, z) =

1 #T v (x, y, z, t) e−j 2πf0 t t=0 T

(7.1)

Where v(x, y, z, t) is the LDV time response at pixel (x, y) on layer z, f0 is the ultrasonic excitation frequency, T is the timelength of the sliding window, and V(x, y, z) is the complex-value amplitude and phase response at the excitation frequency. The result is a complex-valued wavefield map of the structure’s steady-state response. Further processing is conducted to estimate the structure’s thickness. First, wavenumber is calculated from the complex response data. Thickness is then mapped to wavenumber through material-dependent dispersion curves. This NDE method has been applied to large thin-wall structures and successfully identified cracking, delamination, and corrosion. For a more detailed explanation of the wavenumber and thickness calculation process performed in AWS, see [15].

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This method utilizes steady-state measurements at a single frequency, allowing for data collection at a rate orders of magnitude faster than traditional inspection methods and rapid enough to collect measurements in between layers of AM parts. For the application discussed in this chapter, the AWS processing method has been modified to exclude wavenumber estimation and instead use the complex velocity response to identify features that indicate defects. AWS was first combined with additive manufacturing on a fused deposition modeling machine (TAZ Lulzbot 5; 3.0 mm ABS filament with 100% density; 0.35 mm extrusion nozzle diameter; excitation at 80.5 kHz) [16]. It was discovered that an increase in local wavenumber correlated with the presence foreign object defect, delamination, and local heating defect. This work builds upon previous AWS experiments, expanding the applications to an industrial L-PBF system and small defects unable to be detected by the wavenumber estimation approach. Recent work by Cummings et al. investigates applying AWS to geometrically complex metal structures [17]. In that work, a detection method that combines AWS data with the corresponding STL build file to identify common defects in lattice structures, such as cling-on particles and broke struts, was developed. This study builds upon recent work by Jacobson et al. [18].

7.2 Background 7.2.1 Materials An electro-optical systems (EOS) M290 L-PBF system was used with 304L stainless steel (SS) metal powder. The EOS M290 has a maximum build size of 250 mm × 250 mm × 325 mm. The sintering laser can scan up to 7000 mm/s at 400 W. The laser focus diameter is 100 μm [14]. The manufactured parts analyzed in this work consist of 16 cylinders (1 cm diameter; 3 cm height) constructed using a variety of process parameter settings. These cylinders were designed with a range of parameters intended to produce porous defects, such as LOF and keyhole pores. Simplified melt pool geometry calculations from previous melt track experiments (of varying laser power and speed) were used to define the build parameters of the 16-cylinder build (Fig. 7.2a). The cylinders in the four corners (numbers 1, 4, 13, and 16) were assigned nominally optimized parameters and are considered control (220 W; 1 m/s; 78 μm). The build parameter ranges of all cylinders are as follows: • Laser power: 180–290 W. • Laser speed: 0.9–1.1 m/s. • Hatch spacing: 65–90 μm. This study focuses on identifying the defects of part 3 (220 W; 1 m/s; 65 μm) (Fig. 7.2b).

Fig. 7.2 (a) Configuration and build parameters of the 304 L SS 16-cylinder build; (b) image of part 3 with defect region

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7.2.2 In Situ Data Acquisition Process An LDV system (Polytec OFV-505 & OFV-5000) was used to capture the steady-state part response from the ultrasonic excitation at a scanning speed of 32,000 mm/s. Note that the pixel resolution is limited, but not set, by the laser focus diameter. Two galvo-mirrors (Thorlabs GVS012) were used to steer the laser beam across the part surface. This configuration is compact enough to fit under the top cover of the M290 printer that houses the system’s optics. Data was recorded with an NI USB-6363 at a sample rate of 2 MS/s. Python code was used to control the transducer, LDV, DAQ, and mirrors. Acquisition was triggered by monitoring the interlayer movement of the systems stages and recoater arm. An ultrasonic transducer with dimensions of 2 mm thickness and 1.5 inch diameter was used to excite the build plate at a frequency of 176 kHz. The transducer was potted into the build plate and coated in epoxy, as to not interfere with the normal build process. Excitation and acquisition can occur immediately after sintering without affecting the build or solidification process (Fig. 7.3). The build and measurement process consists of four steps. Steps A, B, and D occur normally during a build. Step C is part of the AWS data acquisition process (Fig. 7.4a–d). (a) The recoater arm spreads a thin layer of powder across the build plate (or previous layer) surface (b) Loose powder is exposed to the laser, melting a thin 2D cross-section of the part according to the design data (c) After lasing and before the build plate is adjusted to the next layer position: • The ultrasonic transducer generates a single-frequency steady-state tone. • The LDV raster scans the exposed layer’s out-of-plane velocity response. (d) The build plate is set at the appropriate position and powder is spread across the surface to repeat the build cycle. The recoater arm return speed was reduced from 500 mm/s to 32 mm/s to allow approximately 15 seconds for the interlayer measurement. The total layer time is approximately 50 seconds, including sintering, ultrasonic response measurement, and powder recoating for the current part configuration (a 5 cm × 5 cm region with a pixel resolution of 100 μm).

Fig. 7.3 (a) Transducer potted in build plate; (b) sample 304L SS cylinders built on EOS M290

Fig. 7.4 L-PBF process – (a) powder layer; (b) laser melts powder according to build file; (c) AWS measurement begins with simultaneous ultrasonic excitation and LDV raster scan to measure the surface response of solidified part; (d) build plate moves to set position, recoater arm spreads loose powder over the build plate and previous layer surface, and process repeats

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This 16-cylinder build completed in 10.4 hours, with 3.1 hours used for the AWS system scanning the 5 cm × 5 cm region. The time for AWS to analyze a single cylinder (1 cm × 1 cm region) is approximately 0.124 hours with a 100 μm resolution. In comparison, an XCT scan of a single cylinder took 3.5 hours with a 35 μm resolution. There exists a trade-off between measurement scan speed and pixel resolution for both methods. Further development of this in-situ prototype will allow for this measurement time to be decreased in the future. After scanning each layer, the Fourier transform of the time response is calculated over a sliding window at the excitation frequency, resulting in a complex response at each pixel. Both the results of the Fourier transform and the time response data are saved for each layer. Upon build completion, the result is a 3D volume (X & Y pixels over Z layers) of complex values at each measurement location.

7.3 Analysis 7.3.1 Post-build Data Processing Presently, all data processing (beyond the Fourier transform) is performed in MATLAB post-build. Other information collected during the build is also used in data processing, including layer thickness, layer count, and XY grid spacing. It is possible to modify the post-build data processing to occur in situ with slight modifications, but that is not within the scope of this research. The loose powder has a low-amplitude near-random response that is eliminated from the complex part response through a denoising algorithm. Though it is beneficial to remove the powder response, filtering using denoising algorithms may eliminate or smear the signatures of small defects. A recent, nearly identical build with upgraded hardware shows signal improvement and the possibility of not requiring a denoising algorithm. The selected algorithm is called Block Matching and 4-Dimensional (BM4D) filtering, a volumetric extension of the original BM3D operation performed on images. This algorithm was selected since it is considered one of three state-of-theart hybrid denoising algorithms that does not rely on machine learning [19]. First, the algorithm performs a basic estimate of the noise-free image by grouping similar blocks, performing a hard threshold on each group, and creating the basic estimate by aggregating overlapping blocks. The algorithm performs a final estimate using the basic estimate. The process is the same as before, but a collaborative Weiner filter is implemented in place of the hard threshold [20]. This algorithm is applied to the complex scan data over the entire build volume (Fig. 7.5a–d). The BM4D MATLAB function used in this study was developed by Matteo Maggioni from Tampere University of Technology [21]. A logical mask is created across the entire measurement region to separate the parts from each other and the build plate to allow for analysis of individual parts. To separate part from powder, the magnitude of the scan region was integrated over all layers (Fig. 7.6a). Since the powder exhibits a random near-zero response, powder regions have a low integral value. In contrast, the parts have a high integral value. A magnitude threshold value is set to separate part (high magnitude response) from powder (low magnitude response), identifying each pixel as “part” (with a value of “1”) or “powder” (with a value of “0”) in a logical mask (Fig. 7.6b). Individual parts are identified as groups of connected pixels and saved into their own arrays (Fig. 7.6c). Each part’s complex response is saved with the mask applied, allowing each part to be analyzed individually (Fig. 7.6d).

Fig. 7.5 (a and b) Raw AWS measurements of the full build plate showing layer 100; (c and d) Denoised AWS measurements

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Fig. 7.6 Processing workflow of identifying and separating individual parts – (a) integrated volume over all layers; (b) logical mask of all parts; (c) isolated logical mask of a single part; (d) masked real AWS data of a single part

Fig. 7.7 Defect is present in these selected layers (a. 330; b. 333; c. 335), but is not visible in layer (b) 333 due to nodal lines obscuring the detrended response

As the geometry changes continuously during the build, it is possible for nodal lines to appear in the AWS measurement data. These regions have little to no structural response due to the excitation frequency, amplitude, and location. Unfortunately, these regions also obscure the part response, whether it is nominal or due to a defect. This affect can be mitigated in future builds by selecting a particular frequency in which the nodal lines do not exist within in the varying part geometry. Also, use of multiple excitation frequencies and/or multiple transducers can shift the nodal lines or eliminate them entirely with combined multi-frequency data. A complex 2D quadratic surface g is fitted to each part layer p using a least-squares estimate to demean the nominal response of the part and build plate. The separated real and imaginary fits are then combined into a single complex fit, subtracted from the masked denoised response Z(x, y, z) from V(x, y, z), then smoothed to the mean of each 6 x 6 pixel area. The mean of each part layer is shifted to zero making it easier to detect anomalies as they now stand out from the near-zero detrended surface Zˆ (x, y, z). The result is a value that represents absolute error of the LDV measurements predicted by a complex 2D quadratic fit for each layer (7.2). Although this does not eliminate the nodal lines, it normalizes the nominal part response, so any signature abnormalities are high in amplitude and everything else is near zero (Fig. 7.7). Zˆ (x, y, z) = Z (x, y, z) − g (Z (x, y, z))

(7.2)

7.3.2 Defect-Indicative Features After the build was completed, indications of defect were present in part 3 (both visually and quantitatively). The remainder of this analysis focuses on part 3. Three additional features indicative of defect were calculated from the denoised and ˆ detrended AWS data Z(x, y)p . These features are a subset of a larger feature space calculated over all layers of part 3. This subset was selected from visual representation of the defect region and statistical separation of nominal vs. defect signatures.

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The first is the absolute value of the denoised and detrended data (7.3): $ $ $ $ f1 (x, y, z) = $Zˆ (x, y, z)$ = |Z (x, y, z) − g (Z( x, y, z ))|

(7.3)

ˆ Where Z(x, y)p is the 2D quadratic g(Z) fit minus the LDV denoised response Z at pixel location (x,y) and layer z. The second feature is the Mahalanobis distance within 5 × 5 × 3 voxel in x, y, and z. The Mahalanobis distance computes the distance between the observation point and the distribution of observations. Within each voxel (xvoxel , yvoxel , zvoxel ), the center pixel (xc , yc , zc ) is considered the observation point and the other pixels within the voxel are considered as the observation distribution. The data is padded with zeros so that each voxel center pixel represents the physical bounds of the part. This feature is calculated using the real, imaginary, and magnitude components of the denoised and detrended data (7.4).   f2 (x, y, z) = mahal Zˆ (xc , yc , zc ) , Zˆ (xvoxel , yvoxel , zvoxel )

(7.4)

The third feature is the radial gradient of the relative velocity. The 2D XY gradient is taken from the absolute value of the data using the MATLAB function imgradient (7.5). The input to this function is the denoised and detrended magnitude response data, or feature one. The output are the gradients in X and Y across all pixels and layers. $ $ % & $ $ Gx (x, y, z) , Gy (x, y, z) = imgradient $Zˆ (x, y, z)$

(7.5)

For each layer, a reference pixel is determined (xi , yj ). The difference between that reference pixel and all other pixels in the current layer z is calculated, added together, and saved at the reference pixel coordinates. This is repeated for all pixels and layers. Eq. (7.6) is repeated for the gradient in the Y orientation (Gy ).       dGx xi , yj , z = Gx xi , yj , z − Gx xp , yq , z

(7.6)

From the XY gradients of relative velocity, the radial gradient is calculated by taking the square root of the sum of squares (7.7). f3 (x, y, z) =

 dGx (x, y, z)2 + dGy (x, y, z)2

(7.7)

The fourth feature is proportional to the normalized error in relative strain energy. The rate of change of strain energy (U˙ ) is proportional to the difference in relative velocity squared Z over the distance d between the two points of interest squared (7.8). Since the input to this calculation is denoised and detrended velocity, this feature does not calculate relative strain energy, but the difference in relative strain energy. 2

$ $2 $ˆ $ $Z2 − Zˆ 1 $

Z U˙ ∝ 2 = d |d2 − d1 |2

(7.8)

This feature is calculated between a reference pixel (xi , yj ) and a calculation pixel (xp , yq ) within the current layer z, similar to feature three (7.9).     Zˆ xi , yj , zk − Zˆ xp , yq , zk f4 (x, y, z)  2  2 xi − xp + yj − yq Features two and four are complex, while features one and three exist only as magnitudes (are not complex).

(7.9)

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7.3.3 X-Ray Computed Tomography Labeling High- and low-resolution XCT data was collected for part 3. The low-resolution measurement (35 μm) took 3.5 hours and scanned the entire cylinder. The high-resolution measurement (7 μm) scanned roughly one-third of the part and took 2.5 hours. The XCT data was aligned to the AWS measurements and interpolated onto the AWS coordinate system. The interpolated XCT data was used to create labels for the AWS data – whether that pixel of response was nominal (0) or indicative of a defect (7.8) (Fig. 7.8). Proper alignment requires visible features on both AWS and XCT datasets, preferably spanning the entire height of the part for X, Y, and rotation alignment. A small fiducial was added to the last few layers of the cylinders to aid with alignment, but should have extended through the height of the part. Fiducials on the part should also provide at least two reference surfaces for z-alignment. Since the completed parts are removed from the build plate by wire EDM or bandsaw, the only reliable surface is the top (the last build layer). After aligning and interpolating the XCT data, labels were created to assign to the AWS pixels. However, there exist discrepancies between the two datasets – the XCT data shows what is inside the part (e.g., single pores) at every resolution step, while the AWS data shows the cumulative structural response of the layers below the current one. For example, pores on layer 400 will show as individual pores in the XCT data, but will show as a region of low response in the raw AWS data because layer 400 also includes the response of layers below (Fig. 7.9). The number of layers having significant contributions to the AWS response is not known, but it is estimated to be a function of material and defect size. To estimate the region contributing to an AWS response, the XCT data was integrated 40 layers below the layer of interest. First, the XCT intensity data was normalized (Fig. 7.10a). An initial linear threshold at the 36th percentile was used to identify pores (under threshold) and part (above threshold) (Fig. 7.10b). The initial threshold results were integrated over the previous 40 layers to estimate the region of influence on the AWS responses (Fig. 7.10c). A threshold was applied at the 86th percentile to once again determine regions of part (under threshold) and pores (over threshold) (Fig. 7.10d). Modifications

Fig. 7.8 (a) Cross section of XCT showing bulk porosity from high-resolution measurement; (b and c) voids shown in a 3D render from highresolution measurement

Fig. 7.9 (a) Interpolated XCT intensity; (b) damage mask derived from interpolated XCT intensity; (c) real AWS part response of layer 400

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Fig. 7.10 (a) Normalized and interpolated XCT intensity; (b) binary threshold of XCT intensity; (c) binary threshold integrated over 40 layers below analysis layer; (d) binary threshold applied to integration results; (e) final modified defect mask

Fig. 7.11 Low-resolution XCT data of layer where pores are first identified (left: gray scale, right: jet). Yellow arrow indicates voids while black arrows indicate high Z particles (other material powder contamination with a higher intensity) Table 7.2 Measurement type and earliest sighting of pore formation

Measurement Low-resolution XCT (Fig. 7.11) Interpolated XCT F1 – Absolute error F2 – Mahalanobis distance F3 – Relative strain energy F4 – Radial gradient

Pores first visible Layer 212 Layer 211 Layer 214 Layer 212 Layer 214 Layer 212

were made to the mask perimeter to ensure the dimensions aligned to that of the AWS data. Structuring elements were used in MATLAB to “clean up” the mask and eliminate erroneous pixels (Fig. 7.10e). First, areas less than 4 pixels were eliminated using bwareaopen. Then, the mask was dilated using imdilate with a disk structuring element with a radius of 1 pixel. Last, isolated areas less than 30 pixels were eliminated by once again using bwareaopen.

7.3.4 Results Alignment and proper labeling are not guaranteed for this dataset. Future builds will incorporate more features to aid in alignment. The use of machine learning algorithms rely on accurate labels, so the results described below will include only visual comparisons. We can detect regions of porosity verified with XCT data from the denoised and detrended AWS data and the four features. Features one through three directly indicate regions of defect, while feature four bounds regions of defect. We can also detect the formation of LOF pores to within three layers of when they first appear in the XCT data. Table 7.2 shows at which layer pores are first detected for each feature. Presently, this detection is visual, but it can be implemented numerically to search layers for a certain threshold. The absolute largest margin of error is within three layers. However, this does not take into account the possible alignment discrepancies with this dataset. Features two (Mahalanobis distance) and four (radial gradient) detect pore formation sooner than features one and three. Features one through three show the presence of defect and the region of the defect. Feature four is capable of detecting change in defect and forms a perimeter around the region. Below is a compilation of the interpolated XCT, defect mask, and

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Fig. 7.12 Montage of interpolated XCT, defect mask, and four features from layers 210–215. Yellow arrows indicate pore presence

four features of layers 210–215, around the region of first pore formation (Fig. 7.12). Yellow arrows indicate present defect (lack of fusion pores). Similar plots highlighting layers 150-525 are located in Appendix A.

7.3.5 Future Work Through the work presented in this chapter, it has been confirmed that AWS can be used as an in situ measurement system to detect the presence of defects. However, this study has revealed the shortcomings of the geometry selected for this particular build which lead to ambiguity in aligning the AWS measurements to the XCT validation data.

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Future work will seek to increase the signal-to-noise ratio in the AWS measurements and to incorporate more fiducials into the part geometries to allow for proper alignment. This is crucial for obtaining accurate data labels to apply machine learning algorithms with this data. The STL file may be used in combination with the XCT data to enhance the accuracy of the data labels. Parts will also be built with intentional geometrical features to investigate the resolution limits and influence of unfused powder on the surface response measurements.

7.4 Conclusion Acoustic wavenumber spectroscopy was successfully integrated with an EOS M290 L-PBF system, measured the complex velocity response of each build layer, and was used to identify regions of bulk porosity post-build that were confirmed with XCT imaging. Limitations of this data set have been identified, mainly in relation to properly aligning the XCT measurements and creating ground truth labels. New experiments are planned to address these limitations and further study the relationship between measured AWS response and defect size. The eventual goal of this project is to automatically monitor and detect defect in situ, and avoid defect by either stopping the build or altering the build parameters. This study will be repeated with proper ground truth labels to automatically detect defects in situ. Concepts in this study may be altered to allow in situ processing on a layer-by-layer basis, including the denoising algorithm, and improving and automating the detection algorithm. Acknowledgments Research presented in this chapter was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project number 20190580ECR. The authors would like to thank George T (Rusty) Gray III for graciously providing the images used in Fig. 7.1 and Brian Patterson and Lindsey Kuettner of LANL MST-7 for providing XCT data.

A.1 Appendix A: Montage of Figures Below is a montage of six plots across layers 150–525 in increments of 25 layers. The figures show the interpolated XCT data, defect mask created from the XCT data, and the four features. The color scale remains the same as defect develops. Pores will show as empty regions (blue) in the XCT data but will show as full regions (red) in all other plots. Defect forms at layer 211. Defect is present in all figures representing layers 225–525. Nodal lines interfere with the responses of the following layers: 350, 450–500. The defect is still present in the XCT data, but not always present in the AWS data.

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References 1. Seifi, M., et al.: Progress towards metal additive manufacturing standardization to support qualification and certification. (in English). JOM. 69(3), 17 (2017). https://doi.org/10.1007/s11837-017-2265-2 2. DebRoy, T., et al.: Additive manufacturing of metallic components – process, structure and properties. (in English). Prog. Mater. Sci. 92(92), 113 (2018). https://doi.org/10.1016/j.pmatsci.2017.10.001 3. Foster, B.K., Reutzel, E.W., Nassar, A.R., Dickman, C.J., Hall, B.T.: A brief survey of sensing for metal-based powder bed fusion additive manufacturing. in SPIE, vol. 9489, no. 94890B, in Dimensional Optical Metrology and Inspection for Practical Applications IV (2015), p. 10, https://doi.org/10.1117/12.2180654 4. Everton, S.K., Hirsch, M., Stravroulakis, P., Leach, R.K., Clare, A.T.: Review of in-situ process monitoring and in-situ metrology for metal additive manufacturing. Mater. Des. 95(95), 15 (2016). https://doi.org/10.1016/j.matdes.2016.01.099 5. Waller, J., Parker, B., Hodges, K., Walker, J.: Nondestructive Evaluation of Additive Manufacturing. NASA, PowerPoint, Las Cruces (2014) 6. Smith, R.J., Hirsch, M., Patel, R., Li, W., Clare, A.T., Sharples, S.D.: Spatially resolved acoustic spectroscopy for selective laser melting. (in English). J. Mater. Process. Technol. 236(236), 10 (2016). https://doi.org/10.1016/j.jmatprotec.2016.05.005 7. Smith, R.J., Li, W., Coulson, J., Clark, M., Somekh, M.G., Sharples, S.D.: Spatially resolved acoustic spectroscopy for rapid imaging of material microstructure and grain orientation. (in English). Meas. Sci. Technol. 25(055902), 12 (2014). https://doi.org/10.1088/0957-0233/25/ 5/055902 8. Cerniglia, D.: Defect detection in additively manufactured components: laser ultrasound and laser theromgraphy comparison. Proc. Struct. Integr. 8, 154–162 (2018). https://doi.org/10.1016/j.prostr.2017.12.016 9. Wang, X., Li, W., Zhou, Z., Zhang, J., Zhu, J., Miao, Z.: Phased array ultrasonic testing of micro-flaws in additive manufactured titanium block. Mater. Res. Express. 7(10) (2020). https://doi.org/10.1088/2053-1591/ab6929 10. Li, W., Zhou, Z., Li, Y.: Application of ultrasonic array method for the inspection of tc18 additive manufacturing titanium alloy. Sensors. 19(20), 4371 (2019). https://doi.org/10.3390/s19204371 11. Slotwinski, J.A., Garboczi, E.J.: Porosity of additive manufacturing parts for process monitoring. AIP. 1581, 1197–1204 (2014). https://doi.org/ 10.1063/1.4864957

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12. Sturm, L., Albakri, M., Williams, D.C.B., Tarazaga, D.P.: In-situ detection of build defects in additive manufacturing via impedance-based monitoring. Solid Freeform Fabr. Symp. 27, 1458–1478 (2016) 13. Rieder, H., Spies, M., Bamberg, J., Henkel, B.: On- and offline ultrasonic characterization of components built by SLM additive manufacturing. in AIP Conference Proceedinga, vol. 1706, no. 130002 (AIP Publishing LLC, 2016), p. 7, https://doi.org/10.1063/1.4940605. [Online]. Available: https://aip.scitation.org/doi/pdf/10.1063/1.4940605 14. Flynn, E.B., Jarmer, G.S.: High-speed, non-contact, baseline-free imaging of hidden defects using scanning laser measurements of steady-state ultrasonic vibration, in presented at the 13th International Workshop on Structural Health Monitoring (2013) 15. Flynn, E.B., Chong, S.Y., Jarmer, G.J., Lee, J.-R.: Structural imaging through local wavenumber estimation of guided waves. (in English). NDT&E Int. 59(59), 10 (2013). https://doi.org/10.1016/j.ndteint.2013.04.003 16. Koskelo, E.C., Flynn, E.B.: Scanning laser ultrasound and wavenumber spectroscopy for in-process inspection of additively manufactured parts, in SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, Las Vegas, NV, vol. 9804, no. Nondestructive Characterization and Monitoring of Advanced Materials, Aerospace, and Civil Infrastructure (2016), p. 21, doi: https:// doi.org/10.1117/12.2222130. [Online]. Available: https://www-spiedigitallibrary-org.lanl.idm.oclc.org/conference-proceedings-of-spie/9804/ 1/Scanning-laser-ultrasound-and-wavenumber-spectroscopy-for-in-process-inspection/10.1117/12.2222130.full?SSO=1 17. Cummings, I.T., Jacobson, E.M., Fickenwirth, P.H., Flynn, E.B., Wachtor, A.J.: In-process defect detection for additively manufactured metal lattices, in presented at the ASME IMECE (2020), 24368 18. Jacobson, E.M., Cummings, I.T., Fickenwirth, P.H., Flynn, E.B., Wachtor, A.J.: Defect detection in additively manufactured metal parts using in-situ steady-state ultrasonic response data, in presented at the ASME IMECE (2020), 24336 19. Goyal, B., Dogra, A., Agrawal, S., Sohi, B.S., Sharma, A.: Image denoising review: from classical to state-of-the-art approaches. (in English). Inf. Fusion. 5(2020), 25 (2020). https://doi.org/10.1016/j.inffus.2019.09.003 20. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 16 (2007). https://doi.org/10.1109/tip.2007.901238 21. Maggioni, M., Katkovnik, V., Egiazarian, K., Foi, A.: A nonlocal transform-domain filter for volumetric data denoising and reconstruction. IEEE. 22(1), 15 (2012). https://doi.org/10.1109/TIP.2012.2210725

Chapter 8

Toward Developing Arrays of Active Artificial Hair Cells Sheyda Davaria and Pablo A. Tarazaga

Abstract The human cochlea perceives frequencies over a range of 20 Hz to 20 kHz, while a section of the organ of Corti in the cochlea transduces a particular frequency. The cochlear amplifier then amplifies or compresses the signal based on the stimulus level. An individual artificial hair cell (AHC) made of a piezoelectric beam with a feedback controller replicates the cochlear amplifier at a particular frequency. However, to capture a wider frequency range and mimic the tonotopic basilar membrane, an array of AHCs is required. Thus, numerical modeling of an array of active beams with different geometries is the focus of this chapter. With the dynamics of a single active artificial hair cell established, an array of AHCs with self-sensing characteristics is developed. A sample array is modeled using a few sensors to transduce a small set of frequencies in the human speech frequency range. The AHC array is simulated in Simulink and its response is controlled using a nonlinear cubic damping feedback control law that was presented in the authors’ previous work (Davaria, S., Malladi, V.V.S., Motaharibidgoli, S., Tarazaga, P.A.: Cochlear amplifier inspired two-channel active artificial hair cells. Mechanical Systems and Signal Processing, 129, 568–589 (2019)). The response of the active system to complex stimuli with multiple frequencies is analyzed. The design, modeling, and feedback control techniques developed in the current work will be applicable to future sensor arrays and cochlear implants with more beam elements. Because each sensor will utilize the active AHC technology, the total array will offer advantages over a passive series of cantilevers or traditional sensors. Keywords Artificial hair cell · Array · Self-sensing · Cochlea · Frequency selectivity

8.1 Introduction The cochlea serves several important roles in hearing [1] including decomposing the input acoustic signals into their constitutive frequency tones, increasing vibrations induced by small sound pressure levels, and suppressing high-level sounds [2]. The first role corresponds to the tonotopic mapping of the cochlea and its frequency selectivity. The frequency decomposition takes place in the basilar membrane that extends from the base to the apex of the cochlea. High-frequency inputs are encoded near the basilar membrane’s base, while low frequencies are transduced near its apex [3]. The decomposed signals are processed by the cochlea’s hair cells and are amplified or compressed based on their levels. This amplification or compression function is attributed to the cochlear amplifier and is nonlinear [4, 5]. According to studies on the mammalian cochlear amplifier, the amplitude compression is around 0.33 dB/dB [2, 6, 7]. This compressive rate expands the ability of the cochlea to transduce a broad range of stimuli and it can be replicated via sensors and cochlear implants to achieve better dynamic ranges with cochlea-like nonlinear properties. The work herein adopts resonance-based sensors and incorporates nonlinear controllers into these systems to introduce compressive nonlinearities into their dynamics. Each element of the sensor array is a piezoelectric cantilevered beam with a cubic damping feedback controller [8, 9] to improve the dynamic performance of the system. In the present study, each sensor corresponds to a single frequency of transduction. Therefore, variations in the geometries of beam sensors allow the entire array to transduce multiple frequencies in a manner similar to the basilar membrane’s tonotopic mapping. In the literature, several studies have focused on creating arrays of artificial hair cells (AHCs) with tonotopic properties [10– 13], while only a few of them have considered mimicking the cochlear amplifier [14–17]. The authors’ previous work on developing active AHCs focused on mimicking the amplification and compression functions of the biological cochlea via

S. Davaria () · P. A. Tarazaga Vibrations, Adaptive Structures, and Testing Lab, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_8

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piezoelectric cantilever beams operating near a Hopf bifurcation [18, 19]. In contrast to the previous study where a single AHC with two sensing channels was created, the present work concentrates on the array of AHCs and studies the system’s behavior as it is subjected to a complex base excitation rather than a stepped sine input. Furthermore, AHCs in the array are modeled as quadmorph self-sensing piezoelectric beams. Therefore, the piezoelectric layers are used for sensing the output of the AHC and supplying the control voltage to the system. This modeling approach is beneficial in future implementation of the array, as the control voltage for each beam is a function of its voltage output. This chapter consists of five sections. The next section presents the AHC array modeled in this work. The third section discusses the response of a single AHC to a complex input before analyzing the response of the array. Subsequently, the fourth section displays the simulation results for the active AHC array. Finally, the last section summarizes the outcomes of the work discussed in this chapter.

8.2 Array of Self-Sensing Artificial Hair Cells In this work, an eight-channel active AHC array is designed and its behavior is studied numerically. Each AHC in the array is modeled as a self-sensing single-channel AHC and its length is determined such that the fundamental frequency of the beam matches a particular frequency [20]. Array frequencies are selected in the human speech range [21] and are tuned to 200 Hz, 500 Hz, 800 Hz, 1 kHz, 3 kHz, 5 kHz, 8 kHz, and 10 kHz by adjusting the length of the AHC cantilevers. The active array is shown schematically in Fig. 8.1. In order to create an active array, a cubic damping controller is used for each AHC and its gains are tuned based on the dynamics of the AHC. The cubic damping controller removes the linear damping of the AHC by tuning it to the Hopf bifurcation and injects cubic damping into the system. The controllers adopted for self-sensing AHCs are designed based 3 on the cubic damping controllers introduced in the authors’ previous papers [18, 19], as Vc = α1 V˙sensed − α3 V˙sensed . The controllers have two different gains: (1) the linear damping gain (α 1 ) that is a fixed value for each controller and used to remove the linear damping and (2) the cubic damping gain (α 3 ) that can take different values based on the desired nonlinear behavior expected from the system. The gains used for the AHCs in this work are tabulated in Table 8.1. The next section investigated the response of a single AHC to complex stimuli before embedding the AHC in the array.

Fig. 8.1 Schematic of the AHC array with its fundamental frequencies

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Table 8.1 Schematic of the AHC array with its fundamental frequencies Gains α 1 (s) α 3 (s3 /V2 ) Gains α 1 (s) α 3 (s3 /V2 )

a

AHC1 3.59 × 10−4 1 × 10−11 AHC5 9.27 × 10−5 6 × 10−11

AHC2 2.27 × 10−4 5 × 10−11 AHC6 7.18 × 10−5 5 × 10−11

b

5 Uncontrolled Controlled

3 2 1 0 –1 –2 –3

AHC4 1.60 × 10−4 5 × 10−11 AHC8 5.08 × 10−5 5 × 10−9

4.5 Uncontrolled Controlled

4 Sensed Piezo Voltage - volt (V)

Sensed Piezo Voltage - volt (V)

4

3.5 3 2.5

volt = 0.584a1.0

2

volt = 0.815a0.38

1.5 1 0.5

–4 –5

AHC3 1.79 × 10−4 8 × 10−11 AHC7 5.68 × 10−5 1 × 10−9

0

2

4

6 Time (s)

8

10

12

0

0

1

2 3 4 5 Base Acceleration - a (m/s2)

6

7

Fig. 8.2 (a) Time-domain response of the active and passive AHC for a 7 m/s2 chirp input acceleration; (b) input-output plot

8.3 AHC Array Simulation Results This section evaluates the response of a single AHC to complex inputs as a starting point prior to studying the behavior of the array. The AHC used in this study is the second beam of the array shown in Fig. 8.1. Therefore, it is expected that this AHC affects the input signals that contain a frequency component near 500 Hz. Two types of signals are used in this study: a chirp signal and a periodic input. The chirp signal is applied to the AHC between 250 Hz and 2184.5 Hz and its amplitude is changed from 0.025 m/s2 to 7 m/s2 . The time response of the AHC in response to the highest examined input level is shown in Fig. 8.2a. As displayed in Fig. 8.2a, the maximum response of the uncontrolled AHC is compressed in the active AHC’s output. This maximum response corresponds to the input frequencies near the fundamental frequency of the AHC. The relationship between the input and output of the system is studied using the plot illustrated in Fig. 8.2b. The input-output plot shows nearly one-third power-law relationship between the base acceleration and the voltage. This shows that the AHC can mimic the mammalian cochlea’s behavior when excited by a complex stimulus. It is important to note that as the response spectra of the AHC are time-variant, the values for the sensed piezoelectric voltage are calculated using the short-time Fourier transform (STFT). In the second step, a multi-tone sine signal is used as the input and the response of the AHC was computed. The results for this part of the study are shown in Figs. 8.3b, 8.4b, and 8.5b in the next section. As the single AHC showed desired behavior in response to complex stimuli, a similar study is performed on an array of AHCs in the next section with the periodic input.

8.4 AHC Array Simulation Results Inthis section, the behavior of the AHC array for various input levels is studied. A signal in the form of, z¨ (t) = A sin (2π × 200t) + sin (2π × 500t) + sin (2π × 1000t) + sin (2π × 3000t) + sin (2π × 5000t) + sin (2π × 8000t) +

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Sensed Piezo Voltage - volt (V)

c

AHC2

0.25

d

AHC3

0.1

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Uncontrolled Controlled

1.2

AHC4

0.1

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0.2

0.08

0.08

0.15

0.06

0.06

0.1

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0.02

1 0.8 0.6 0.4 0.2 0

e

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10000

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h6

AHC7 Uncontrolled Controlled

0

2000

× 10–4

4000 6000 8000 10000 Frequency (Hz) AHC8

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5

0.008

0.015

0

2000 4000 6000 8000 10000 Frequency (Hz)

g 1.5 × 10–3

AHC6

0.01

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0

4

1

0.006 0.01

3 0.004 2

0.5 0.005

0

0.002

0

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2000 4000 6000 8000 10000 Frequency (Hz)

1

0

5000 Frequency (Hz)

10000

0

0

2000 4000 6000 8000 10000 Frequency (Hz)

0

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2000

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Fig. 8.3 Magnitude sensed voltage of the AHC array for 0.025 m/s2 input level

Sensed Piezo Voltage - volt (V)

a

b

AHC1 15

Uncontrolled Controlled

3

1.5

2

1

AHC4 1.4 Uncontrolled Controlled

1.2 1

10

0.8 0.6 5

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1

0.2 2000

4000 6000 8000 10000 Frequency (Hz)

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0.3

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g

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h

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0 0

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AHC3 2

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2000 4000 6000 8000 10000 Frequency (Hz)

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AHC8

0.035

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0.03 0.025

0.04

0.02

0.08 0.03

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0.015

0.06 0.01

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0.04

0.05 0 2000 4000 6000 8000 10000 Frequency (Hz)

0.005 0

0

0 0

0.01

0.01

0.02 0

5000 Frequency (Hz)

10000

0

5000 Frequency (Hz)

10000

0

5000 Frequency (Hz)

10000

Fig. 8.4 Magnitude sensed voltage of the AHC array for 5.5 m/s2 input level

sin (2π × 10000t), where z¨ (t) is the base acceleration and A is the excitation amplitude is applied to the array and the voltage response of each AHC is obtained. First, the signal’s amplitude is set to 0.025 m/s2 and the response of the AHCs to the stimulus is shown in Fig. 8.3. Figure 8.3 illustrates the output of the AHCs in two modes, i.e., passive and active or uncontrolled and controlled, against each other. As shown in this figure, the response of each AHC is amplified at its fundamental frequency. For each AHC, amplification occurs in two steps: (1) amplification by the passive system due to the resonance-based nature of the AHC and

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Sensed Piezo Voltage - volt (V)

a

Sensed Piezo Voltage - volt (V)

c

AHC2

4

Uncontrolled Controlled

volt = 2.662a1.0

5

0.8

volt = 0.333a1.0

1

volt = 4.321a0.33

Uncontrolled Controlled

1

volt = 0.673a1.0

2

AHC4

1.4 1.2

1.5

3 10

d

AHC3

2

Uncontrolled Controlled

Uncontrolled Controlled

volt = 0.238a10

0.6

volt = 0.745a0.33

volt = 0.295a0.23

volt = 0.340a0.33

1

0.5

0

0

0.4 0.2

0 0

e

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AHC1 15

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0.3

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0.1 0.08

0.2

0.04

0.05

0.02

0

2000 4000 6000 8000 10000 Base Acceleration - a (m/s2) AHC8

0.035

Uncontrolled Controlled

0.03 0.025

volt = 0.009a1.0

0.03 volt = 0.034a0.35

0.02

volt = 0.006a1.0

0.02

6

volt = 0.007a0.35

0.01

0.01

0 2 4 Base Acceleration - a (m/s2)

Uncontrolled Controlled

0

0.015

0.01

0

h

AHC7

0.06

0.04

0.06

volt = 0.064a0.24

2000 4000 6000 8000 10000 Base Acceleration - a (m/s2)

0.05

volt = 0.021a1.0

volt = 0.046a1.0

0.15

g

AHC6

0.12

0 0

0 0

2 4 Base Acceleration - a (m/s2)

6

volt = 0.003a0.25

0.005 0

2 4 Base Acceleration - a (m/s2)

6

0

0

2 4 Base Acceleration - a (m/s2)

6

Fig. 8.5 Input-output plots for the AHCs in the array

(2) nonlinear amplification by the feedback control law with respect to the passive system. As the AHC works in resonance region, the response is amplified when the frequency of the input matches the natural frequency of the beam. This provides an initial amplification for the signal and replicates the frequency selectivity of the cochlea. Subsequently, the controller injects nonlinearity to the AHC near the natural frequency such that the response is amplified or compressed with respect to the passive system. The amplification or compression provided by the controller depends on the input level and the cubic damping gain of the controller. To show the compression function of the active AHC array, the input level is increased to 5.5 m/s2 and the results are displayed in Fig. 8.4. As shown in Fig. 8.4, the response of each AHC is compressed relative to the passive system for 5.5 m/s2 input level. It is evident from Fig. 8.4 that even though the active system compresses the output for each AHC at its fundamental frequency, the response level at the natural frequency remains significantly higher than any other frequencies. Therefore, the active system can detect the output at fundamental frequencies of the AHCs. To investigate the compressive rate of the array, the input-output plot for each AHC at the fundamental frequency is plotted in Fig. 8.5. In these plots, the input level is varied from 0.025 m/s2 to 5.5 m/s2 . According to Fig. 8.5, the response of the AHCs is amplified or compressed by a one-third power function of the excitation level and the AHCs can mimic the biological cochlea’s compressive nonlinearity. As shown in Fig. 8.5, each AHC’s mode changes from amplification to compression at a particular input strength that depends on the dynamic of the AHC and the cubic damping gain used in the controller corresponding to that AHC. The input-output plots show the good performance of the array when the system is excited by a complex stimulus.

8.5 Conclusion In this chapter, an array of active artificial hair cells was modeled and studied numerically. The array consisted of eight quadmorph cantilevers with fundamental frequencies between 200 Hz and 10,000 Hz in the speech frequency range. The AHCs lengths were chosen based on the target frequencies considered for the array. A cubic damping controller was used for each AHC and the system was tuned to the Hopf bifurcation. As an AHC array is intended to mimic the tonotopic basilar membrane and the cochlear amplifier, a complex input was used in this work. First, the response of a single AHC to a chirp input and a periodic multi-tone sine signal was obtained. Results showed that the AHC was able to detect the

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signal’s frequency component that matched the AHC’s fundamental frequency. Furthermore, it could amplify or compress the detected signal component at a compressive rate of about one-third. In the next step, the periodic signal was applied to the array and its amplitude was varied. The input-output curves showed that the targeted compressive rate (about one-third) was achieved and the output of the system was amplified or compressed depending on the input level. Therefore, the array’s AHCs showed desired behavior near the fundamental frequencies of the beams. The results also displayed that the AHC could mimic the frequency selectivity of the cochlea. Furthermore, it was observed that even for the highest input amplitude studied in this work where all the AHCs in the system compressed the response, the output at each fundamental frequency was detectable. In summary, this study showed the compatibility of the control law developed for single AHCs with AHC arrays that are excited by complex stimuli. Therefore, the active AHCs can be used as suitable candidates in an array format for sensor development or cochlear implants, as they can mimic the frequency selectivity and compressive nonlinearity of the cochlea. Lastly, the numerical study presented in this chapter provided a framework for future experimental analysis of AHC arrays. Acknowledgments The authors would like to acknowledge the generous support from the National Science Foundation (NSF) (Grant No.1604360) that provided the funding for this project. Dr. Pablo A. Tarazaga would also like to acknowledge the John R. Jones III Faculty Fellowship. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References 1. Dallos, P.: The active cochlea. J. Neurosci. 12, 4575–4585 (1992) 2. Hudspeth, A.: Making an effort to listen: mechanical amplification in the ear. Neuron. 59, 530–545 (2008) 3. Hudspeth, A.J.: How the ear’s works work. Nature. 341, 397 (1989) 4. Davis, H.: An active process in cochlear mechanics. Hear. Res. 9, 79–90 (1983) 5. Ashmore, J., Avan, P., Brownell, W., Dallos, P., Dierkes, K., Fettiplace, R., Grosh, K., Hackney, C., Hudspeth, A., Jülicher, F.: The remarkable cochlear amplifier. Hear. Res. 266, 1–17 (2010) 6. Robles, L., Ruggero, M.A.: Mechanics of the mammalian cochlea. Physiol. Rev. 81, 1305–1352 (2001) 7. Ruggero, M.A., Rich, N.C., Recio, A., Narayan, S.S., Robles, L.: Basilar-membrane responses to tones at the base of the chinchilla cochlea. J. Acoust. Soc. Am. 101, 2151–2163 (1997) 8. Joyce, B.S., Tarazaga, P.A.: Developing an active artificial hair cell using nonlinear feedback control. Smart Mater. Struct. 24, 094004 (2015) 9. Joyce, B.S., Tarazaga, P.A.: Mimicking the cochlear amplifier in a cantilever beam using nonlinear velocity feedback control. Smart Mater. Struct. 23, 075019 (2014) 10. Zhao, C., Knisely, K.E., Colesa, D.J., Pfingst, B.E., Raphael, Y., Grosh, K.: Voltage readout from a piezoelectric intracochlear acoustic transducer implanted in a living Guinea pig. Sci. Rep. 9, 1–11 (2019) 11. Song, W.J., Jang, J., Kim, S., Choi, H.: Influence of mechanical coupling by SiO 2 membrane on the frequency selectivity of microfabricated beam arrays for artificial basilar membranes. J. Mech. Sci. Technol. 29, 963–971 (2015) 12. Jung, Y., Kwak, J.-H., Lee, Y.H., Kim, W.D., Hur, S.: Development of a multi-channel piezoelectric acoustic sensor based on an artificial basilar membrane. Sensors. 14, 117–128 (2013) 13. Harada, M., Ikeuchi, N., Fukui, S., Toshiyoshi, H., Fujita, H., Ando, S.: Micro mechanical acoustic sensor toward artificial basilar membrane modeling. IEEJ Trans. Sensors Micromach. 119, 125–130 (1999) 14. JIAN-SI Y 2005 Study of stereocilia mechanics with applications to biomimetic sensor design 15. Lim, K., Park, S.: A mechanical model of the gating spring mechanism of stereocilia. J. Biomech. 42, 2158–2164 (2009) 16. Ammari, H., Davies, B.: Mimicking the active cochlea with a fluid-coupled array of subwavelength Hopf resonators. Proc. R. Soc. A. 476, 20190870 (2020) 17. Rupin, M., Lerosey, G., de Rosny, J., Lemoult, F.: Mimicking the cochlea with an active acoustic metamaterial. New J. Phys. 21, 093012 (2019) 18. Davaria, S., Malladi, V.S., Avilovas, L., Dobson, P., Cammarano, A., Tarazaga, P.A.: Special Topics in Structural Dynamics & Experimental Techniques, vol. 5, pp. 95–99. Springer (2020) 19. Davaria, S., Sriram Malladi, V.V.N., Tarazaga, P.A.: Bio-inspired Nonlinear Control of Artificial Hair Cells, pp. 179–184. Springer International Publishing, Cham (2019) 20. Davaria, S., Tarazaga, P.A.: MEMS scale artificial hair cell sensors inspired by the cochlear amplifier effect. In: Bioinspiration, Biomimetics, and Bioreplication 2017, p. 101620G. International Society for Optics and Photonics (2017) 21. Baskent, D., Shannon, R.V.: Speech recognition under conditions of frequency-place compression and expansion. J. Acoust. Soc. Am. 113, 2064–2076 (2003)

Chapter 9

Challenges Associated with In Situ Calibration of Load Cells in Force-Limited Vibration Testing Kenneth J. Pederson, Vicente J. Suarez, Emma L. Pierson, Kim D. Otten, James C. Akers, and James P. Winkel

Abstract The difference in mounting configuration between flight and test can significantly impact the effectiveness of the test in environmental vibration testing. Many tests are performed with large electrodynamic shakers, which utilize interfaces that seek to replicate a fixed base, such as slip tables and head expanders. This fixed base configuration is rarely seen in flight configurations; rather a more realistic configuration would include a flexible mounting structure with its own compliance and dynamics. This causes significant over- and under-tests in various frequency bands depending on the differences between the test article and fixture dynamics. The traditional way of avoiding these high loads is to limit the acceleration responses at multiple locations on the test article. However, the effectiveness of this approach is highly dependent upon the validity of the test article’s analytical in order to derive accurate acceleration response limit specifications. Also, this technique requires limiting the acceleration responses at many locations throughout the test article, which may not be implementable due to such things as access issues and cleanliness issues. An improved environmental vibration testing technique known as force limiting incorporates measurements of the forces between the test article and shaker system interface and limiting them to a specification that more accurately replicates the interface impedance of the structure the test article will be mounted to in flight. In effect, this transforms the high mechanical impedance at the test article to shaker interface to more closely match the mechanical impedance of the flight interface, which avoids producing the unrealistically high interface loads. Typically, force gauges or load cells are used to measure these interface forces. However, utilizing load cells can present a multitude of challenges depending upon such things as their installation method, geometric layout, and test fixture setup. Regardless, it is important to perform an in situ calibration of the load cells prior to vibration testing at any significant levels. This chapter will discuss the challenges associated with utilizing load cells during the NASA Evolutionary Xenon Thruster – Commercial (NEXT-C) gridded ion thruster proto-flight vibration test performed at the NASA Glenn Research Center’s Structural Dynamics Laboratory. Keywords Environmental · Fixed base · Force gauges · Force limiting · Glenn Research Center · Gridded ion thruster · In situ calibration · Load cells · Mechanical impedance · NASA · NEX-C · Proto-flight · Response limiting · Structural Dynamics Laboratory · Vibration testing

9.1 Introduction Force limiting has existed in some form since the late 1980s when analytical studies on controlling two degree of freedom (DOF) systems were first undertaken [1]. Over the next decade, force limiting proved to be an excellent means to account for the interface impedance difference between test and flight. These methods gained significant popularity due to the advent of commercially available and economical three-axis load cells. Also, multi-input controllers that could sum voltage signals streamlines test setups. By the late 1990s, force limiting was implemented on some of the highest profile tests, such as Cassini’s Spacecraft system level test [2, 3]. These load cells, while innovative, had limitations that still exist in modern state-of-the-art load cells. They are not as “plug and play” as accelerometers and for this reason are subject to misuse by

K. J. Pederson () · V. J. Suarez · E. L. Pierson · K. D. Otten · J. C. Akers NASA Glenn Research Center, Cleveland, OH, USA e-mail: [email protected] J. P. Winkel NASA Langley Research Center, Hampton, VA, USA © The Society for Experimental Mechanics, Inc. 2022 D. S. Epp (ed.), Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75914-8_9

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Fig. 9.1 Z-axis test setup

test engineers. Force-limiting techniques have continued to improve and a comprehensive guide can be found in the NASA Technical Handbook “Force Limited Vibration Testing” [4]. Additionally, force limits can be enacted in vibration analysis in situations where negative margins are unrealistic due to a fixed base constraint. A method and case study of this can be found in reference [5], which discusses how force limiting was used on the CoNNeCT SCAN Testbed. This chapter discusses the common pitfalls of improperly using load cells during force-limited vibration testing, including the effect of mounting configuration on load cell sensitivity, and the in situ calibration performed to remedy this. At NASA Glenn Research Center’s Structural Dynamics Lab (SDL), a recent test on the NEXT-C thruster exhibited some of the potential consequences of these pitfalls. This test is referenced throughout this chapter to provide a real-world example of these known challenges. The NEXT-C ion thruster is a commercialized version of NASA’s Evolutionary Xenon Thruster (NEXT-C) built by Aerojet Rocketdyne. This thruster will provide in-space propulsion technology demonstration of the Double Asteroid Redirection mission (DART). The DART mission is part of the Planetary Defense Program and aims to alter the trajectory of a candidate asteroid, proving that this technology is capable of redirecting a future asteroid off a potentially hazardous earth bound trajectory. The thruster went through environmental vibration testing at SDL in November 2019 and, as of the writing of this paper, is tabbed to launch in July 2021. The thruster is mounted to the spacecraft through three radially oriented joints. Figure 9.1 depicts this test setup. In situ calibration is required to account for the parallel load path introduced by the pre-load bolt, shown in yellow in Fig. 9.2. There are two common methods to perform in situ calibration of load cells, and both include imparting a known load into the test setup and making sure the load cells accurately measure that load. One model uses a modal hammer to impact the test article with a known force and the other utilizes a base-driven excitation. For both the modal hammer and base-driven excitation, frequency response functions (FRF) are analyzed. For the modal hammer technique, the FRF between the load cell force and hammer force should equal one at low frequency signifying that the measured force into the structure is being accurately captured by the load cells. For the base-driven technique, the FRF magnitude between the base acceleration and load cell force, which is 1 divided by the apparent mass, should be equal to 1 (total suspended mass) at low frequency. Depending on how far off the FRF is from one, the load cell’s sensitivity would then be adjusted. The NEXT-C testing was performed utilizing the in situ calibration because there was no identifiable hard point to impact the ion thruster. Also, for structural health monitoring purposes, the test plan included a low-level random test that would adequately excite the rigid body motion. The two main force-limited testing topics discussed in this chapter are (1) the unknowns associated with which mounting hardware should be included in the measured mass of the test article and (2) the handling of load cells that are not aligned with the test axis and updates to the sensitivity of summed channels.

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Fig. 9.2 Generic force-limited test setup

9.2 Test Setup The NEXT-C thruster utilized three mounting locations to interface with the spacecraft. Each lie on the perimeter of the thruster 120◦ apart from each other. Each mounting location has a triaxial load cell mounted between the thruster and the fixture. Additional figures showing closeups and schematics of this test setup are given throughout this chapter. Figure 9.1 labels the three load cells as A, B, and C as well as the global test axes as Y and Z. Local coordinate systems of relevant load cells are shown in Fig. 9.6.

9.3 Apparent Mass Calculation An equation describing the apparent mass of single degree of freedom oscillator’s response to a base-driven excitation is derived in reference number [6]. Equation (9.1), shown below, displays the equation with a slight modification that removes the damping term. ⎡ 

F (ω) = m⎣ A (ω)

ωn2 − ω2 −ω2

2 ⎤ ⎦

(9.1)

The excitation frequency analyzed must be well below the test article or fixtures dynamics to avoid any amplification due to resonances. A good “rule of thumb” is the frequency band analyzed is 5× or 10× below the test article’s firstmode frequency. This apparent mass is then compared to the actual measured mass to determine the appropriate load cell sensitivities. The actual measured mass is a debated topic for in situ load cell calibration calculations, and this thruster test was no exception. At first glance, the actual measured mass of a test article may seem trivial – simply measure the test article’s mass using a scale. Upon further evaluation, there was concern about including various pieces of mounting hardware in the actual mass calculation. Any hardware on the “free” end of the load cells should certainly be included, but it is less clear if the measure mass should incorporate the pre-load bolt, washers, and actual load cell itself. These seemingly small mass contributions become more significant as they become a larger percentage of the total mass. The implications of including or excluding these controversial mass contributions is discussed in relation to the NEXT-C testing. When a base-driven acceleration excitation is applied to the fixture and the force is measured at the load cell, the apparent mass should be equal to the mass of the test article. Determining the measured mass is less straight-forward, especially when considering which mounting hardware to include in the calculations. A generic example of a force-limited test setup is shown in Figs. 9.2 and 9.3. It is important to note the test setup below is purely notional to display the individual components of a typical joint with an integrated load cell. This test setup would have serious flaws if it were to be used in practice. In the event a load in an axis other than normal to the load cell is applied to this structure, the load cell will be carry a moment as well as a force. Typical three-axis load cells, including those used on the NEXT-C testing, are not capable of measuring moments. This causes the load cell to not tell the whole story of the load path through the structure. In a real-world force-limited test, force plates are common which include a rectangular plate with force links or load cells in each corner. This way, any moments can be

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Fig. 9.3 Exploded force-limited test setup

Fig. 9.4 “Free end” of load cell Table 9.1 NEXT-C test part masses

Part Ion thruster Bolt Washers Load cell

Mass per part (lbs) 36.2 0.14 0.01 0.6

Quantity 1 3 9 3

Total mass 36.2 0.42 0.09 1.8

reacted at the corners with linear forces. Similarly, in the NEXT-C test setup, moments could be reacted by linear forces at the three interface locations. In the exploded figure, the hardware referred to as being on the “free end” of the load cell (as defined in Fig. 9.4), can be seen as the purple “Test Article”, the furthest right red washer, and the part of the yellow bolt that extends beyond the load cell. The NEXT-C thruster and some auxiliary mounting hardware were weighed on the scale. The weights are tabulated in Table 9.1. The weight of the test article alone was 36.2 lbs. The total mass on the “free end” of the load cell includes the test article, three washers, and a small portion of the three preload bolts, totaling ~36.3 lbs. The “free end” is depicted in Fig. 9.4. Unfortunately, it is not immediately evident whether and what other mounting hardware should be included in the measured mass. The total mass of the test article and auxiliary hardware includes nine washers, the bolt, and three load cells, totaling 38.61 lbs. Depending on whether the measured mass includes free-end hardware only, all mounting hardware, or somewhere in between, the mass varies from 36.3 to 38.61 lbs, a 6% difference. Ultimately, any deviation from the correct measured mass will directly contribute to a shift in the acceleration spectrum achieved by test article due to the force limits that engage in certain frequency bands. It is hard to say with certainty, but a 6% difference in force measurements could have caused the NEXT-C test to not meet various test requirements such as input acceleration levels staying within tolerance bands. This crux of the problem is identifying which auxiliary components should be included in the measured mass. At first glance, the measured mass would equal that of the test article; however, it is unclear if the load cell, washers, and preload bolt’s mass should be considered in their entirety or at least in some portion. It is still unclear which components and proportions should be included in the measured mass. This chapter does not provide an answer to this question; rather it brings the issue to light and suggests how important it is to properly document and account for the uncertainty. Perhaps, the best way to approach this situation is to perform the calculations with only the test article’s mass and again with all of the

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Fig. 9.5 Off axis load cell generic setup

mounting hardware included. These two calculations would at least bound the problem and provide a worst- and best-case scenario. This issue likely contributes a slight, if any, error to individual load cell measurements.

9.4 Load Cell Summing Implications on In Situ Calibration In the single degree of freedom oscillator discussed previously, a single load cell sensitivity would be updated according to the in situ calibration results. Two complications to this simple example occur when (1) the measurement of the total force in the test axis requires summing multiple load cells and (2) the load cell axes are not aligned with test axes. Both of these situations are common in the aerospace industry and were exhibited in the NEXT-C vibration test, which is discussed below (Fig. 9.5). In the figure above, a biaxial load cell, each with its own sensitivity, is used to limit the force at joint A. The fixturing requires the load cell to be installed at some angle (θ ) off the test axes. The joint is to be limited in the axis of test to a certain force spectrum. This requires the components of each load cell in the axis of test to be calculated and summed together. The total force in the test axis is calculated using the sum of these load cells and some trigonometry, notionally shown below. −Fx cos (θ ) + Fy cos (θ ) = Ftot

(9.2)

Where Fx and Fy are equal to the force in each axis of the biaxial load cell. These forces are actually the result of multiplying the voltage coming out of the sensor’s charge amplifier by their sensitivity factor (volts/force). This equation could be broken down to the equation below where α and β are the individual load cell’s manufacturer-supplied sensitivities and Vx,y is the voltage coming out of each load cell’s charge amplifier. −αVx cos (θ ) + βVy cos (θ ) = Ftot

(9.3)

To perform the in situ calibration, a low-level excitation is applied to the base of the test article and the apparent mass is calculated from the force over acceleration transmissibility function. The next question is how to update the sensitivity of the load cells. Likely, each individual load cell’s sensitivity is not equally affected by the specific mounting configuration, rather each sensitivity is affected independently. Unfortunately, the apparent mass calculation provides a mass for the entire test article rather than the mass applied to each load cell. This prohibits tuning each load cell’s sensitivity individually. Instead, a blanket adjustment factor is applied to each load cell’s sensitivity, shown below as γ . % & γ −αVx cos (θ ) + βVy cos (θ ) = Ftot

(9.4)

γ is the apparent mass divided by the measured mass. This is problematic because the recovered force in each individual direction likely has an inaccurate calibration factor applied to it. After performing the in situ calibration, the following equations may be inaccurate because γ is not specific to either. Fx = γ αVx cos (θ ) Fy = γβVy cos (θ )

(9.5) and (9.6)

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Fig. 9.6 NEXT-C load cell schematic

In other words, the total force in the direction of testing may be accurate; however, the force contribution from each individual load cell likely is not. If Fx or Fy must be used for summing force in another axis, their contribution is likely inaccurate and will make limiting on this other axis problematic. Also, the individual forces cannot be compared to analytical model data accurately. This issue occurred during the NEXT-C testing in the lateral axes where the load cells used for limiting were not aligned with the axis of excitation. For example, the Z axis setup schematic is shown below. As you can see in Fig. 9.6, load cells A and B are not aligned with the axis of test, so each component of the load cells are used along with some trigonometry to calculate their force in the axis of excitation. The C load cell is aligned with the axis of test and no trigonometry is required. In this scenario, the total force in the Z axis can be calculated using the following equation, which assumes that the load cell labels from Fig. 9.6 are equal to the manufacturer-supplied sensitivity multiplied by the voltage coming from the load cell’s charge amplifier (i.e., Ay = *volts). Ftot,Z = Ay cos (ϕ) − Az sin (ϕ) − By cos (ϕ) − Bz sin (ϕ) + Cz

(9.7)

After performing in situ calibration and applying the sensitivity adjustment factor, this equation can be written as follows: & % Ftot,Z = γ Ay cos (ϕ) − Az sin (ϕ) − By cos (ϕ) − Bz sin (ϕ) + Cz

(9.8)

Now the total force in the Z axis can be limited to this summed force level. A challenge arose because the NEXT-C test plan also wished to force limit in the Y and X axes. The force sum equation in the Y axis is shown below. Ftot,Y = −Ay sin (ϕ) − Az cos (ϕ) − By sin (ϕ) + Bz cos (ϕ) + Cy

(9.9)

The force terms in the Y-axis force sum that appear in the Z axis force sum will be inaccurate. For example, the force measured from the Ay load cell will be equal to γ Ay . This γ adjustment factor was found during in situ calibration in the Zaxis and is likely not accurate for the individual Ay load cell. Therefore, the force sum in the Y-axis will carry this inaccuracy. When the ion thruster is configured for Z-axis testing, there is no way to excite it in the Y axis at low frequencies to perform in situ calibration and check the accuracy.

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At certain frequencies, the load cells included in these force sums would go in and out of phase. This led to the force sums equations to equal close to zero total force, while the individual load cells measured high force levels. To overcome this, the NEXT-C project requested the force limit equations be applied to each individual load cell as well as the sum. These individual load cell force limits may have been engaged slightly inaccurately due to the applied blanket sensitivity updates. During the NEXT-C testing, another situation arose due to limitations of the VR9500 controller, where all signals being used as part of an equation must go into the same controller box. The SDL at GRC has 4-input VR9500 modules. To calculate total force in a direction, the output of five load cells had to be reduced to four signals, which required a charge summer and caused a similar issue to that described in the preceding paragraph. If force signals are summed before the charge amplifier, identifying the force in individual load cells is challenging. Tuning the sensitivity of individual load cells after their charge is summed is not possible. Ultimately, tuning individual load cell sensitivities ends up being a lot of guess work and foregone in favor of a applying a blanket adjustment factor to all load cells. Each force is then scaled by this adjustment factor regardless of how far off the actual sensitivity is. The NEXT-C test setup posed additional challenges in properly using the load cells. When using any load cell, there is a balancing act when setting the joint pre-load. It is imperative that the preload is set such that the load cell does not crush when under compressive loads or slip when under shear loading. The load cell could crush if the static preload force plus compressive dynamic forces and moments exceeds the load cell’s compressive strength. The load cell would slip if the shear force exceeds the normal force times to coefficient of static friction. It is a good idea to avoid using torque values to predict pre-load on a load cell joint. There is significant variance on pre-load for a given a torque value due to factors that are hard to identify and quantify such as flange and bolt material and surface finish, torque application device tolerances, and normal joint relaxation. It is highly recommended to use long-time constant charge amplifiers when pre-loading load cell joints. These long-time constant charge amplifiers allow the static pre-load to be measured for up to 24 hours after being applied. These same long-time constant charge amplifiers cannot be used for the vibration test as their response has a significant time lag. A future study could be performed to quantify the effects of an in situ calibration when it is performed by exciting an axis other than the test axis. Ideally, a biaxial load cell would be preloaded in a custom mounting configuration and then excited at its base in a few various axes. First, it should be excited in an off-axis direction, as described above. Then the apparent mass and accompanying sensitivity update factor would be calculated. Next, the test article should be excited in-line with either load cell and then the apparent mass and accompanying sensitivity update factor should be calculated. The difference in these sensitivity update factors would be an indicator of the inaccuracy when updating load cells, as described above in the NEXT-C test. Performing this study with a wide array of load cells, mounting configurations, and preloads would provide insight into how robust the sensitivity update factor is to varying conditions. Ideally, this study would be performed before every actual force limited test. Therefore, the exact difference between the sensitivity update factors would be known for each load cell regardless of the axis of excitation. Unfortunately, this is likely impractical for many actual test articles because fixtures typically are not designed to be tested in any more than the required three orthogonal axes.

9.5 Conclusion The advent of force limiting in environmental vibration testing greatly increased a test engineer’s ability to adequately excite a structure to its in-flight load levels. The artificially high interface force levels caused by test article resonances were able to be controlled by measuring and limiting on these forces. Technological advancements over the past 40 years have led to this technique being deployed in many vibration labs across the country. A few notable advances include the development of compact tri-axial piezo electric load cells and multi-channel response limiting capable controllers. While force-limited testing has made great strides in recent history, the test engineer must be cognizant of its existing flaws and shortcomings. This chapter discusses common pitfalls in modern force-limited testing and some guidelines to avoid them. Some complications are unavoidable, but they can be partially remedied if properly accounted for in test planning and execution. There are two major challenges discussed in this chapter – (1) these load cells are now calibrated for the in-line force sum, but not calibrated in off axis directions and (2) the results may be slightly less accurate when compared to analytical models. Unfortunately, there is no industry standard for what components should be included in the “measured” mass of a test article. This measured mass is critical to performing in situ calibration, which relies on the comparison of measured mass to the apparent mass and adjustment of the load cell sensitivities to account for any differences. A future study researching which components of the load cell and mounting hardware should be included in the test article’s measured mass would be beneficial to the industry.

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Lastly, it is common practice in force-limited testing to sum multiple load cells to calculate total force in the test axis. During in situ calibration, it is important to consider how applying a blanket sensitivity adjustment factor decreases the accuracy of individual load cell readings. Each individual load cell should have its own respective adjustment factor; however, after the summing has occurred, this is not possible. Using individual load cell readings for model validation or applying additional test limits now requires relying on slightly inaccurate force readings. Future studies, as described above, should aim to quantify the ramifications of performing in situ calibration on load cells that are required for either multiple axis of force limiting or are needed for comparison to analytical models. With this information, test engineers will have a better understanding of how these pitfalls affect a test and its accompanying results. Quantification of these uncertainties will allow the test engineer to better inform their customer of negative ramification of in situ calibrations and ultimately provide improved insight into the testing.

References 1. Scharton, T.D.: Analysis of dual control vibration testing. Annual Technical Meeting (1990) 2. Scharton, T.D.: Cassini spacecraft force limited vibration testing. Sound Vib. (1999) 3. Mcnelis, M.E.: Benefits of Force Limiting Vibration Testing. NASA/TM 1999-209382 4. NASA-HDBK-7004C: Force Limited Vibration Testing, 30 Nov 2012 5. Staab, L.D.: Application of the Semi-Empirical Force-Limiting Approach for the CoNNeCT SCAN Testbed, NASA/TM 2012-217627 6. Irvine, T.:An Introduction to Frequency Response Functions. 11 Aug 2000