Nonlinear Structures & Systems, Volume 1: Proceedings of the 40th IMAC, A Conference and Exposition on Structural Dynamics 2022 (Conference ... Society for Experimental Mechanics Series) 3031040856, 9783031040856

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Conference Proceedings of the Society for Experimental Mechanics Series

Matthew R. W. Brake Ludovic Renson Robert J. Kuether Paolo Tiso   Editors

Nonlinear Structures & Systems, Volume 1 Proceedings of the 40th IMAC, A Conference and Exposition on Structural Dynamics 2022

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.

Matthew R. W. Brake • Ludovic Renson • Robert J. Kuether • Paolo Tiso Editors

Nonlinear Structures & Systems, Volume 1 Proceedings of the 40th IMAC, A Conference and Exposition on Structural Dynamics 2022

Editors Matthew R. W. Brake Department of Mechanical Engineering Rice University Houston, TX, USA Robert J. Kuether Sandia National Laboratories Albuquerque, NM, USA

Ludovic Renson Department of Mechanical Engineering Imperial College London London, UK Paolo Tiso Department of Mechanical and Process Engineering ETH Zürich Zürich, Switzerland

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-031-04085-6 ISBN 978-3-031-04086-3 (eBook) https://doi.org/10.1007/978-3-031-04086-3 © The Society for Experimental Mechanics, Inc. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Nonlinear Structures & Systems represents one of nine volumes of technical papers presented at the 40th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held February 7–10, 2022. The full proceedings also include volumes titled Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace and Dynamic Environments Testing; Topics in Modal Analysis & Parameter Identification; and Data Science in Engineering. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Nonlinearity is one of these areas. The vast majority of real engineering structures behave nonlinearly. Therefore, it is necessary to include nonlinear effects in all of the steps of the engineering design: both in the experimental analysis tools (so that the nonlinear parameters can be correctly identified) and in the mathematical and numerical models of the structure (in order to run accurate simulations). In so doing, it is possible to create a model representative of the reality which, once validated, can be used for better predictions. Several nonlinear papers address theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust computational algorithms) as well as experimental techniques and analysis methods. There are also papers dedicated to nonlinearity in practice where real-life examples of nonlinear structures will be discussed. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Houston, TX, USA London, UK Albuquerque, NM, USA Zürich, Switzerland

Matthew R. W. Brake Ludovic Renson Robert J. Kuether Paolo Tiso

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Contents

1

Scattering from a Bi-stable Elastica Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domenico Tallarico, Bart Van Damme, Andrea Bergamini, Natalia V. Movchan, and Alexander B. Movchan

1

2

An Investigation of Complex Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Verhoeven, D. J. Ewins, M. H. M. Ellenbroek, X. Yao, and D. Di Maio

5

3

Investigating the Potential of Electrical Connection Chatter Induced by Structural Dynamics . . . . . Benjamin Dankesreiter, Manuel Serrano, Jonathan Zhang, Benjamin R. Pacini, Karl Walczak, Robert Flicek, Kelsey Johnson, and Ben Zastrow

19

4

Ensemble of Numerics-Informed Neural Networks with Embedded Hamiltonian Constraints for Modeling Nonlinear Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David A. Najera-Flores and Michael D. Todd

27

5

System Identification of Geometrically Nonlinear Structures Using Reduced-Order Models . . . . . . . . Mohammad Wasi Ahmadi, Thomas L. Hill, Jason Z. Jiang, and Simon A. Neild

31

6

Indirect Reduced-Order Modelling of Non-conservative Non-linear Structures . . . . . . . . . . . . . . . . . . . . . . . Evangelia Nicolaidou, Thomas L. Hill, and Simon A. Neild

35

7

Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads on Finite Element Structures Based on Stress Trial Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lukas Koller and Wolfgang Witteveen

8

Nonlinear Modelling of an F16 Benchmark Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Péter Zoltán Csurcsia, Jan Decuyper, Balázs Renczes, and Tim De Troyer

9

Mathematical Model Identification of Self-Excited Systems Using Experimental Bifurcation Analysis Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. H. Lee, D. Barton, and L. Renson

39 49

61

10

Shape Optimisation for Friction Dampers with Stress Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Denimal, R. Chevalier, L. Renson, and L. Salles

65

11

Design of Flap-Nonlinear Energy Sinks for Post-Flutter Mitigation Using Data-Driven Forecasting Jesús García Pérez, Amin Ghadami, Leonardo Sanches, Guilhem Michon, and Bogdan Epureanu

75

12

Tribomechadynamics Challenge 2021: A Multi-harmonic Balance Analysis from Imperial College London . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Lasen, L. Salles, D. Dini, and C. W. Schwingshackl

79

Experimental Proof of Concept of Contact Pressure Distribution Control in Frictional Interfaces with Piezoelectric Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Lasen, D. Dini, and C. W. Schwingshackl

83

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Contents

Experimental Observations of Nonlinear Damping of Additively Manufactured Components with Internal Particle Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthew Postell, Daniel Kiracofe, Onome Scott-Emuakpor, and Tommy George Jr.

87

15

Data-Driven Reduced-Order Model for Turbomachinery Blisks with Friction Nonlinearity . . . . . . . . Sean T. Kelly and Bogdan I. Epureanu

97

16

Determination of Flutter Speed of 2D Nonlinear Wing by Using Describing Function Method and State-Space Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Güne¸s Kösterit and Ender Cigeroglu

17

Application of Geometrically Nonlinear Metamaterial Device for Structural Vibration Mitigation 109 Kyriakos Alexandros Chondrogiannis, Vasilis Dertimanis, and Eleni Chatzi

18

Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams with Spectral Chebyshev Technique and Harmonic Balance Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Demir Dedeköy, Ender Cigeroglu, and Bekir Bediz

19

Experimental Characterization of Superharmonic Resonances Using Phase-Lock Loop and Control-Based Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Gaëtan Abeloos, Martin Volvert, and Gaëtan Kerschen

20

On Modelling Statistically Independent Nonlinear Normal Modes with Gaussian Process NARX Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Max D. Champneys, Gerorge Tsialiamanis, Timothy J. Rogers, Nikolaos Dervilis, and Keith Worden

21

Non-linear Kinematic Damping in Phononic Crystals with Inertia Amplification . . . . . . . . . . . . . . . . . . . . . 149 Bart Van Damme, Marton Geczi, Leonardo Sales Souza, Domenico Tallarico, and Andrea Bergamini

22

Mitigation of Nonlinear Structural Vibrations by Duffing-Type Oscillators Using Real-Time Hybrid Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A. Mario Puhwein and Markus J. Hochrainer

23

Approximate Bayesian Inference for Piecewise-Linear Stiffness Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Rajdip Nayek, Mohamed Anis Ben Abdessalem, Nikolaos Dervilis, Elizabeth J. Cross, and Keith Worden

24

Experimental Model Update for Single Lap Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Simone Gallas, Hendrik Devriendt, Jan Croes, Frank Naets, and Wim Desmet

25

Data-Driven Identification of Multiple Local Nonlinear Attachments Installed on a Single Primary Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Aryan Singh and Keegan J. Moore

26

Supervised Learning for Abrupt Change Detection in a Driven Eccentric Wheel . . . . . . . . . . . . . . . . . . . . . 185 Samuel A. Moore, Dean Culver, and Brian P. Mann

27

Bolt-Jointed Structural Modelling by Including Uncertainty in Contact Interface Parameters . . . . . 193 Nidhal Jamia, Hassan Jalali, Michael I. Friswell, Hamed Haddad Khodaparast, and Javad Taghipour

28

Parameter Estimation of Jointed Structures Using Alternating Frequency-Time Harmonic Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Javad Taghipour, Nidhal Jamia, Michael I. Friswell, Hamed Haddad Khodaparast, and Hassan Jalali

29

A Novel Test Rig for the Validation of Non-linear Friction Contact Parameters of Turbine Blade Root Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Daniel J. Alarcón Cabana, Jie Yuan, and Christoph W. Schwingshackl

30

A Study on Data-Driven Identification and Representation of Nonlinear Dynamical Systems with a Physics-Integrated Deep Learning Approach: Koopman Operators and Nonlinear Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Abdolvahhab Rostamijavanani, Shanwu Li, and Yongchao Yang

Contents

31

Data-Driven Nonlinear Modal Analysis: A Deep Learning Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Shanwu Li and Yongchao Yang

32

Higher-Order Invariant Manifold Parametrisation of Geometrically Nonlinear Structures Modelled with Large Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Alessandra Vizzaccaro, Andrea Opreni, Loic Salles, Attilio Frangi, and Cyril Touzé

33

Application of Black-Box NIXO to Experimental Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Michael Kwarta and Matthew S. Allen

34

Reliable Damage Tracking in Nonlinear Systems via Phase Space Warping: A Case Study. . . . . . . . . . 241 He-Wen-Xuan Li and David Chelidze

35

Nonlinear Modes of Cantilever Beams at Extreme Amplitudes: Numerical Computation and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Marielle Debeurre, Aurélien Grolet, Pierre-Olivier Mattei, Bruno Cochelin, and Olivier Thomas

36

Stability and Convergence Analysis of the Harmonic Balance Method for a Duffing Oscillator with Free Play Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

37

A Physics-Based Modeling Approach for the Dynamics of Bolted Joints: Deterministic and Stochastic Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Nidish Narayanaa Balaji and Matthew R. W. Brake

38

A Review of Critical Parameters Required for Accurate Model Updating of Geometrically Nonlinear Dynamic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

39

Magnetic Excitation System for Experimental Nonlinear Vibration Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 271 Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

40

Predicting Nonlinearity in the TMD Benchmark Structure Using QSMA and SICE . . . . . . . . . . . . . . . . . 281 Drithi Shetty, Kyusic Park, Courtney Payne, and Matthew S. Allen

41

Evolution of the Dynamics of Jointed Structures Over Prolonged Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Nidish Narayanaa Balaji, Scott Alan Smith, and Matthew R. W. Brake

42

On the Use of Variational Autoencoders for Nonlinear Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Thomas Simpson, George Tsialiamanis, Nikolaos Dervilis, Keith Worden, and Eleni Chatzi

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Chapter 1

Scattering from a Bi-stable Elastica Arm Domenico Tallarico, Bart Van Damme, Andrea Bergamini, Natalia V. Movchan, and Alexander B. Movchan

Abstract In this contribution, we study the transient scattering of waves from a structural bi-stable element modelled as a Euler elastica. The non-linear elastic element has localized inertia at its end points, each of which is connected to a semiinfinite straight rod. The focus is on longitudinal waves reflected and transmitted into the semi-infinite rods by the non-linear structural interface. The derivation of the transient governing ordinary differential equations for the reflected and transmitted amplitudes is outlined. The key point is the assumption that the semi-infinite rods are linear elastic, with non-linearity embedded into special transient transmission conditions for the scattering amplitudes. The governing equations are solved using standard numerical integration techniques. From a physical point of view, we focus on the effect of the bi-stability on the transmission resonances, identified in the linearized regime. Special attention is given to the up- and downconversion of the frequency content of scattered amplitudes as a function of the amplitude of a harmonic impinging wave. Keywords Scattering · Non-linear dynamics · Euler elastica

1.1 Introduction Geometric non-linear mechanisms and instability – traditionally associated with failure and strength limits of materials [1] – have recently allowed encoding diode-like, reversible, wave phenomena [5] and quasi-static configuration recovery and deployment [3, 4] for soft robotics applications. The geometric non-linearity of buckled beams has also been associated with the onset of inherent non-linear damping of structural vibrations [7]. The seminal work [2] initiated the efficient modelling of static and time-harmonic response of microstructured interfaces. In a similar spirit, in [6], the authors considered the transient scattering of longitudinal waves from a discrete interface featuring a non-linear elastica-like load-displacement response. The present contribution builds upon the models established in [6], with a special focus on bi-stability and its effect on the multi-frequency spectrum of the scattering amplitudes.

1.2 Background In [6], the authors proposed the reduced governing equations for the transient scattering of longitudinal elastic waves in a bar from a massless elastica with clamped ends. Figure 1.1a shows a schematic representation of the model, comprising two semi-infinite bars joined by a Euler elastica. It is assumed that the mass of the non-linear interface is concentrated at the end of the elastica. This is pivotal in reducing the non-linear partial differential equations to a more tractable system of non-linear ODEs for the reflected and transmitted wave amplitudes R(t) and T(t). The system of ODEs for the aforementioned scattering amplitudes is as follows:

D. Tallarico () · B. Van Damme · A. Bergamini Laboratory for Acoustics/Noise Control, Swiss Federal Laboratories for Material Science and Technology (EMPA), Dübendorf, Switzerland e-mail: [email protected]; [email protected]; [email protected] N. V. Movchan · A. B. Movchan Department of Mathematical Sciences, University of Liverpool, Liverpool, UK e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_1

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Fig. 1.1 Panel (a) is a schematic representation of the geometry. Panel (b) represents the exact interaction force F as a function of the compression level χ of a clamped-clamped elastica. The inset is a magnification on the post-buckling regime

R  (τ ) = u0 cos (τ ) +

 ES  v u0 sin (τ ) + R  (τ ) − F (u0 cos (τ ) + R (τ ) ; T (τ )) m− vω ESω

T  (τ ) = u0 cos (τ ) +

 ES   v T (τ ) + F (u0 cos (τ ) + R (τ ) ; T (τ )) , m+ vω ESω

(1.1)

is the radian where u0 is the amplitude of the time-harmonic incident wave; τ = − ωt, with t the time variable and ω √ frequency of the incident wave; m+ (m− ) is the mass concentrated at the right (left) end of the elastica; v = E/ρ is the longitudinal wave speed in the bar (E and ρ being Young’s modulus and the mass density of the bar, respectively); and S is the cross-section area of the bar. In Eq. (1.1) we have introduced F(u− ; u+ ), the load-displacement relation of the elastica with clamped ends. In this context, the force represents the non-linear interaction between the masses. Figure 1.1b illustrates the functional dependence of the force on the compression level χ of the elastica. A critical length scale ucr = χ cr  exists above which the elastica buckles, introducing a bi-stability in the transient problem. In the following section, we analyse the transient scattering solution for R(t) and T(t) as the amplitudes of the impinging longitudinal wave is varied from a subcritical to a supercritical regime. We focus on the frequency of the impinging wave for which a transmission resonance in the linearized regime exists, i.e.:  1/2 2 ω2 − ω2 ω+ − 0 ωT = , 2 ω0 ω+

(1.2)

Ee 2 = 4 ω2 = 2Ee S , with m = m = m and E is Young’s modulus of the elastica. In order for Eq. (1.2) where ω0 = vE , ω+ − + e − m to be real, the condition Ee >  S E2 /(2 m v2 ) has to be satisfied.

1.3 Analysis The non-linear system (1.1) can be tackled with a Runge-Kutta algorithm, e.g. by using the MATLAB(R) “ODE45” function. We discretize the time variable t ∈ [0, Tmax ] according to the frequency of the incident wave, by using 20 time-steps per period T = 2π/ω, with Tmax = 100 T. The initial conditions of the system are in principle arbitrary. Given the fact that the system is excited at the transmission resonance frequency in Eq. (1.2), it is reasonable to assume the initial condition   R(0) = R (0) = T (0) = 0 and T(0) = u0 , which heuristically emulates the transmission resonance. In order to gain insight into

1 Scattering from a Bi-stable Elastica Arm

3

Fig. 1.2 FFT of the scattering coefficients at various amplitudes of the incident field. Panel (a and b) refer to R(t) and T(t), respectively

the multiple frequency generation phenomenon, it is useful to expand the load-displacement relation around a post-buckling precompression (PBPC) χ = χ eq , which leads to:            

  4F1 χeq C1 χeq F2 χeq − C2 χeq F1 χeq   δχ + 4   δχ 2 + O δχ 3 , F χeq , δχ = 4 Feq + 3 C1 χeq C1 χeq

(1.3)

where Fi (χ eq ) and Ci (χ eq ), i = {1, 2}, are given functions of the PBPC, whose detailed expression is rather involved and is here omitted for simplicity and Feq = F(χ eq ) (see inset of Fig. 1.1b). A comparison between the exact load-displacement relation (black solid line) and the approximate relation in Eq. (1.3) (red dashed line) is given in the inset of Fig. 1.1b. The fact that the load-displacement relation admits an expansion in (odd and even) powers of δχ gives insights on the up- and downconversion phenomenon as is evident from Fig. 1.2, where we represent the fast Fourier transform (FFT) of the time series resulting from the solution of Eq. (1.1). Figure 1.2a, b refer to the FFT of the reflected and transmitted amplitudes, respectively. We observe that, for incident wave amplitudes such that u0 < ucr , the reflection coefficient is low, whereas the transmission coefficient admits a single resonance around the resonance frequency in Eq. (1.2). For u0 > ucr , up to 10 ucr , both reflection and transmission spectra show a plethora of harmonics at integer multiples of the frequency of the incident wave ωT . This consists with the polynomial form of Eq. (1.3).

1.4 Conclusion We have quantified the amplitude-dependent, multi-frequency spectra of the scattering amplitudes from a buckling clampedclamped elastica with concentrated masses. The semi-analytical model shall be further refined and extended by including a distribution of point masses, multiple Euler elasticae and/or higher-order buckling modes. Acknowledgements The presented work has been initiated under an Empa Internal Research Call scheme as project number 5213.00171.100.01. The authors gratefully acknowledge the funding that made this work possible.

References 1. Bažant, Z., Cedolin, L.: Stability of Structures. World Scientific, Singapore (2010) 2. Bigoni, D., Movchan, A.B.: Statics and dynamics of structural interfaces in elasticity. Int. J. Solids Struct. 39, 4843–4865 (2002)

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3. Bosi, F., Misseroni, D., Dal Corso, F., Neukirch, S., Bigoni, D.: Asymptotic self-restabilization of a continuous elastic structure. Phys. Rev. E. 94, 063005 (2015) 4. Jin, L., Khajehtourian, R., Mueller, J., Rafsanjani, A., Tournat, V., Bertoldi, K., Kochmann, D.: Guided transition waves in multistable mechanical metamaterials. Proc. Natl. Acad. Sci. 117(5), 2319–2325 (2020) 5. Librandi, G., Tubaldi, E., Bertoldi, K.: Programming nonreciprocity and reversibility in multistable mechanical metamaterials. Nat. Commun. 12, 3454 (2021) 6. Tallarico, D., Movchan, N., Movchan, A.: Scattering from a non-linear structural interface. J. Mech. Phys. Solids. 136, 103687 (2020) 7. Van Damme, B., Hannema, G., Sales Souza, L., Weisse, B., Tallarico, D., Bergamini, A.: Inherent non-linear damping in resonators with inertia amplification. Appl. Phys. Lett. 119, 061901 (2021)

Chapter 2

An Investigation of Complex Mode Shapes C. Verhoeven, D. J. Ewins, M. H. M. Ellenbroek, X. Yao, and D. Di Maio

Abstract This paper presents an investigation of complex mode shape analysis caused by non-linear damping. Nowadays, most academics are accustomed to complex mode shapes, which are a characteristic of most axisymmetric structures. The topic was deeply investigated during the 1980s, sparking the sharpest debates about their physical existence or not. However, after nearly three decades, one question still stands, do we know all about complex mode shapes? This paper takes the dust off this topic again and explores how complex eigenvectors arise when the percentage frequency separation between two mode shapes is the same order of magnitude as the percentage damping. The difference between the past and present investigations relates to the non-linear damping that might arise from joint dynamics under various vibration amplitudes. Hence, the new research question is about the investigation of amplitude-dependent damping on the modal complexity. Why bother? There are several engineering applications in both space and aerospace where axisymmetric structures and joint dynamics can impair the numerical analysis that is currently performed. This paper does not offer any solutions but does expand the research on an unsolved challenge by identifying the questions posed. Keywords Complex modes · Non-proportional damping · Non-linear · MSC · MCF

2.1 Introduction Mathematically a mode shape is defined as a manifestation of eigenvectors that describe the relative displacement of two or more elements in a mechanical system. These mode shapes are real if these elements move through the equilibrium at the same time. If this is not the case, the mode shapes are considered to be complex. The existence of complex mode shapes has been known for several decades. Research on this topic has been conducted, but not as much compared to other fields within structural dynamics. Possible reasons for this are the fact that highly complex modes do not often occur, with the exception of axisymmetric structures. Only when two natural frequencies are close and the system contains non-proportional damping can highly complex modes occur [1]. In practice, when analysing a structure using a finite element method, the damping is often not included in detail but may be represented by simple proportional damping. This latter case means that the eigenvectors will all be real. This is an issue if the experimental analysis has complex eigenvectors due to the fact that in reality there is non-proportional damping.

C. Verhoeven · M. H. M. Ellenbroek Department of Applied Mechanics, Mechanical Engineering, University of Twente, Enschede, Netherlands e-mail: [email protected]; [email protected] D. J. Ewins Imperial College London, London, UK e-mail: [email protected] X. Yao Department of Applied Mechanics, Mechanical Engineering, University of Twente, Enschede, Netherlands School of Aviation Engineering, Civil Aviation Flight University of China, Deyang, China e-mail: [email protected] D. Di Maio () Department of Applied Mechanics, Mechanical Engineering, University of Twente, Enschede, Netherlands Department of Mechanical Engineering, University of Bristol, Bristol, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_2

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This non-proportionality can have many sources, e.g. bolted connections or added dampers at specific locations. To make comparison between the FE model and the experimental possible, the complex mode shapes are converted to real mode shapes. F. Buffe et al. [2] introduced non-proportional damping, and thus complexity, into the FE model which increased the accuracy of the simulation. It appears that often complex modes are considered an undesirable complication rather than as a blessing. However some attempts have been made to make use of complex mode shapes. E. Lofrano et al. [3] proposed a method to detect damage. One basic assumption is that damage results in a local increase of damping making the damping matrix non-proportional and thus resulting in complex mode shapes. Detecting this complexity in the mode shapes provides information about the damage done to system. The layout of this paper is as follows. First an overview of the methods used will be given. After that, a simple model will be used to show the requirements for complex mode shapes. Four cases are discussed, proportional and non-proportional damping for separated modes and close modes. The modal parameters are calculated as well as multiple quantifiers for complexity. Finally the importance of the location of the non-proportional damping will be discussed. It will also be shown that the non-linear damper results in a complex operating deflection shape.

2.2 Methodology Modal analysis allows us to obtain the linear normal mode shapes corresponding to the natural frequencies [1]. This paper uses this theory to obtain and investigate both real and complex mode shapes. Different types of damping exist which require different analytical approaches to solve. The first part of this paper is based around a system containing structural damping. Classical structural damping is chosen which depends on the displacements. This means there are no velocity terms in the equation of motion (EOM) which makes solving it easier. The EOM of a structural damped system is as follows: [M] x¨ (t) + [K + iD] x (t) = f (t)

(2.1)

where [M] is the mass matrix, [K] the stiffness matrix and [D] the structural damping matrix which are all of size 4 × 4. x¨ (t) is the acceleration vector of the DOFS, x(t) the displacement vector and f(t) the force vector which are all 4 × 1 in size. Eigenvalues and eigenvectors can be computed by solving the eigenvalue problem: [M] λ2  = + [K + iD] 

(2.2)

where the eigenvectors  represent the unscaled mode shapes and the eigenvalues represent the natural frequencies and loss factor [4]:   λ2n = ω2n 1 + iηn

(2.3)

Besides the standard modal constants, three complexity quantifiers are used. Imregun and Ewins ([5]) have defined multiple parameters, called modal complexity factors (MCF), to quantify the amount of complexity in a system. MCF I focusses on the difference between the phases of the eigenvector components. MCF II takes into account the magnitude of the eigenvectors as well as their phase angles. Figure 2.1 shows how the MCF II is computed in a 4 DOF system. The final quantifier for complexity is called the mode shape complexity (MSC) [6]. This parameter is based on the strain energy of the system and can be calculated as follows: MSCn =

min (Un ) max (Un )

(2.4)

where Un is the strain energy during an oscillation of mode n. A switch will be made from structural damping to viscous damping for investigating the effect of non-linearity. This switch is necessary since quadratic structural damping does not exist. The EOM of the viscous non-linear model is as follows: [M] x¨ (t) + [C] x˙ (t) + [CNL ] sign (˙x (t)) x˙ (t)2 + [K] x (t) = f (t) where [CNL ] is build up using the damping constant(s) of the non-linear damper.

(2.5)

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Fig. 2.1 The MCF II is the area spanned by the eigenvector tips divided by the area of half a circle, where the radius of this circle is equal to the magnitude of the largest eigenvector

The equation will be solved using time integration. To achieve this the EOM has to be transformed to the state space: ⎡ ⎤

x

x˙ 0 1 0 0 ⎣ x˙ ⎦ + f = x¨ − M−1 K −M−1 C −M−1 CNL • sign (˙x) − M−1 ˙x2

(2.6)

The Runga-Kutta time integration scheme can now be used to find the numerical solution.

2.3 Structural Damped MDOF System A MDOF spring damper system has been created with the purpose of exploring complex mode shapes. This system contains 4 masses and 7 springs and dampers and can be seen in Fig. 2.2. The masses are assumed to move only in horizontal direction which makes this a 4 DOF system. In the first part of this paper, the damping is defined as structural damping instead of the more conventional viscous damping. The advantage of structural damping is that it is easier to model. The EOM looks as follows: [M] x¨ (t) + [K + Di] x (t) = f (t)

(2.7)

where: [M] = diag (m1 , m2 , m3 , m4 ) −k 3 k 1 + k 2 + k 3 −k 2 ⎢ + k 0 k 2 4 [K] = ⎢ ⎣ sym. k 3 + k 5 + k 6

⎤ 0 ⎥ −k 4 ⎥ ⎦ −k 5 k4 + k5 + k7 ⎡ ⎤ −d 3 0 d 1 + d 2 + d 3 −d 2 ⎢ ⎥ 0 −d 4 d2 + d4 ⎥ [D] = ⎢ ⎣ ⎦ sym. d 3 + d 5 + d 6 −d 5 d4 + d5 + d7 ⎡

The vector x contains the displacements of the masses. The dot ( ˙ ) indicates that the time derivative of the variable is taken. The force vector f(t) is 0 since we are interested in the free vibration response. The values chosen for the masses are m1 = 1, m3 = 2.7, m4 = 4 kg. The stiffness constants are k1 = k3 = k4 = k5 = k7 = 106 , k2 = 1.82 • 106 , k6 = 0.7 • 106 N/m. m2 and the damping values are not defined, since these will change throughout this paper. One might wonder why k2 and k6 have such specific stiffness values. The reason for this is that with these stiffness values it is possible to get the natural

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Fig. 2.2 MDOF spring damper system

Fig. 2.3 Effect of m2 on natural frequency 2 and 3, undamped system

Fig. 2.4 Effect of m2 on natural frequency 2 and 3, damped system

frequencies of mode 2 and 3 close by adjusting m2 (see Fig. 2.3). For specific damping values, it is even possible to make the natural frequencies of mode 2 and 3 intersect (see Fig. 2.4). The undamped receptance frequency response function (FRF) for this system when m2 = 2.6 kg can be seen in Fig. 2.5. The four resonance frequencies can be clearly observed. The choice has been made to show only the response of mass 3.

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Fig. 2.5 Undamped forced response of mass 3

Fig. 2.6 Force response of mass 3, case I

2.3.1 Case I: Proportional Damping, Separated Modes A damped system is either proportionally damped or non-proportionally damped. The system is proportionally damped if the damping matrix is a linear combination of the mass and stiffness matrices: [D] = α[M] + β[K], where α and β can have any constant value. In the 4 DOF system, the damping values are chosen in such a way that [D] = 0.01[K], meaning that the system is proportionally damped. Mass 2 has a weight of 2.6 kg. The FRF of mass 3 can be seen in Fig. 2.6. It is clear that the peaks are less sharp compared to the undamped case. Solving the eigenvalue problem of this system results in the eigenvalues and eigenvectors. These eigenvalues and eigenvectors provide information about the resonance frequencies, damping and mode shapes of the system. The mode shapes can be seen in Fig. 2.7. The normalized eigenvectors can be seen in Fig. 2.8. The colours of the arrows correspond to the colours of the masses in Fig. 2.7. Note that the eigenvectors are completely real meaning that the phase difference between de DOFs is either 0◦ or 180◦ .

2.3.2 Case II: Non-proportional Damping, Separated Modes Non-proportionally damped systems are very common in practice. Non-proportional damping can have many different causes, e.g. hinges, damage to the structure and shock absorbers. To create a non-proportionally damped system in this study, the damping values are changed. They are defined as follows: [d1 d2 d3 d4 d5 d6 d7 ] = 10−2 ·

1

2 k1

3k2 12 k3 32 k4 12 k5 32 k6 12 k7



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Fig. 2.7 Mode shapes of the proportionally damped system

Fig. 2.8 Eigenvectors proportionally damped system

These values where obtained by trial and error. There are many more damping values which will result in highly complex mode shapes. Using these damping values results in the following eigenvectors (visualized in Fig. 2.9): ⎡

⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ 0.720 + 0.002i −0.364 + 0.065i −0.315 − 0.061i 0.999 − 0.001i ⎢ 0.991 − 0.009i ⎥ ⎢ − 0.830 + 0.170i ⎥ ⎢ − 0.013 + 0.000i ⎥ ⎢ − 0.218 − 0.003i ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ 1 = ⎢ ⎣ 0.773 − 0.005i ⎦ 2 = ⎣ 0.479 − 0.101i ⎦ 3 = ⎣ − 0.824 − 0.176i ⎦ 4 = ⎣ − 0.114 + 0.002i ⎦ 0.866 + 0.004i 0.405 − 0.081i 0.573 + 0.118i 0.023 + 0.000i The difference between proportionally and non-proportionally damped systems is that the eigenvectors of the nonproportionally damped system contain complex terms. These complex terms indicate a phase offset of the DOFs. The angle of the eigenvector with respect to the positive real axis in the Argand diagram describes the phase angle (see Fig. 2.9). So if the eigenvector contains complex terms, the phase angle is not 0◦ or 180◦ . This means that when observing the mode shapes of the system, the masses do generally not cross the equilibrium simultaneously which is known as a travelling wave. An important question that arises is how to quantify the amount of complexity in each mode shape. For this paper the choice is made to utilize the MCF I, MCF II and MSC. The higher the percentage, the more complex the system is. Table 2.1 shows the properties of the non-proportionally damped system with separated modes (Fig. 2.10).

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Fig. 2.9 Eigenvectors of the non-proportional damped system with separated modes Table 2.1 Properties of a non-proportionally damped system

Mode # 1 2 3 4

ω (Hz) 78 157 168 331

η 0.008 0.014 0.008 0.018

Complexity (%) MCF I MCF II 0 0 1 8 6 9 1 0

MSC 0 0 0 0

Fig. 2.10 FRF case II

2.3.3 Discussion About the Validity of MCF II One might be surprised by the fact that the MCF II for mode 2 and 3 are relatively high although the eigenvectors appear to be mostly in line with each other. This is caused by how the MCF II is calculated. Reference [5] suggested that first all eigenvectors should be converted to the [0◦ ,180◦ ] region, by making the imaginary part of the eigenvectors positive. If one then computes the area covered by the tips of the eigenvectors, the area can be quite large even if the eigenvectors were in line with each other. This is especially the case if the eigenvectors are under an angle of 45◦ and − 135◦ . One could argue that a system in which the eigenvectors are in line with each other should not be considered complex, since the masses pass the equilibrium simultaneously. It is up to the reader to decide how trustworthy the complexity quantifiers are.

2.3.4 Case III: Proportional Damping, Close Modes So far we have seen that, for proportionally damped systems, the eigenvectors are completely real and for non-proportional damped system the eigenvectors contain small complex parts. However in both situations, the modes were well separated ( ωn > 10). Mode 2 and 3 can be brought closer by changing mass 2 from 2.6 kg to 2.037 kg. Figure 2.11 shows the eigenvectors of the proportionally damped system. It is clear that all the eigenvectors are still real, meaning the mode shape is a standing wave. Even for higher levels of damping, when the loss factor (η) is 1, the eigenvectors are still completely real. One might start to wonder if it is mathematically possible that the eigenvectors contain an imaginary component if the damping is proportional. It is actually possible for very specific settings that complexity will occur in the eigenvector. However the vectors will still be exactly 0 or 180◦ apart (Fig. 2.12).

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Fig. 2.11 Eigenvectors of the proportional damped system when mode 2 and 3 are close

Fig. 2.12 FRF case III

Fig. 2.13 Eigenvectors of the non-proportional damped system when mode 2 and 3 are close

2.3.5 Case IV: Non-proportional Damping, Close Modes For the non-proportionally damped system, the eigenvectors contain large imaginary components as can be seen in Fig. 2.13. This shows that for mode 2 and 3 the motion of the individual masses is out of phase with respect to each other. This complexity can also be seen when looking at the modal complexity factors in Table 2.2 and Fig. 2.14.

2.3.6 Effect of the Closeness of the Natural Frequencies on the Complexity In the previous section, it is shown that the non-proportional damping will result in complex modes. However, the importance of two of the system’s natural frequencies being close for generating complex modes is not yet clear. Adjusting mass 2 changes the closeness of the natural frequencies of mode 2 and 3. This will be used to obtain more insight in how the closeness of the natural frequencies influences the complexity. At a mass of 2.00058 kg, the natural frequencies are equal. The mass can be lowered or raised to increase the difference between natural frequencies of mode 2 and 3. Figure 2.15 shows

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Table 2.2 Properties of damped systems with close modes Damping Proportional

Non-proportional

Mode # 1 2 3 4 1 2 3 4

ω (Hz) 81 167.9 168.3 336 81 168.0 168.2 336

η 0.01 0.01 0.01 0.01 0.009 0.013 0.009 0.019

Complexity (%) MCF I MCF II 0 0 0 0 0 0 0 0 0 0 50 30 38 42 1 0

MSC 0 0 0 0 0 13 13 0

Fig. 2.14 FRF case IV

a)

b)

Fig. 2.15 MCFs and MSC as a function of ω3–2 (m2 ). (a) Mode 2. (b) Mode 3

the MCFs and MSC when mass 2 (m2 ) and thus the difference between ω3 and ω2 changes. For the MCFs it is difficult to find clear patterns, but in general, MCF I and MCF II become smaller when the natural frequencies get further apart. The MSC, on the other hand, shows a distinct bell shape. The MSC shows a maximum when the natural frequencies are 0.017 Hz apart. What is also interesting to observe is that the MSC curve is equal (difference < < 1%) for mode 2 and 3. Although the ratio between the maximum and minimum strain energy of mode 2 and 3 is equal, the strain energy of mode 2 is not the same as that of mode 3 (Fig. 2.16).

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Fig. 2.16 Effect of m2 on the MSC for different damping values

Fig. 2.17 Effect of m4 on the MSC for different damping values

The MSC is chosen to compare the effect of different damping levels. The non-proportional damping matrix is multiplied with different factors to observe the influence of the damping amplitude on the MSC. For each different damping setting, the MSC values were computed for a range of m2 values, which controls the closeness between ω3 and ω2 . The results can be seen in Fig. 2.17. It can be observed that a low damping results in a high narrow bell shape. Increasing the damping flattens out this bell shape. Furthermore the peak tends to move slightly to the left when the damping is increased. One might wonder if the peak would reach 100% if the damping would be lowered. Unfortunately for this case, lowering the damping prevents the crossing of the natural frequencies from mode 2 and 3. Another question is if this bell shape behaviour always looks like in Fig. 2.17. The answer is no. If one tunes ω3–2 by changing m4 instead of m2 , the bell curves look different (see Fig. 2.17). Instead of one peak around ω3–2 = 0, there are two peaks, one at each side. These peaks also differ in height.

2.4 Effect of Non-linear Damper Location From this section onwards, the structural damping will be replaced by viscous damping. Time integration is used to calculate the behaviour of the system.

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Fig. 2.18 FRF for different locations of the non-linear damper

Fig. 2.19 FRF for different force amplitudes. The vertical lines represent the natural frequencies of the undamped system

To determine if the location of a NL damper has a significant effect, a quadratic NL damper is added. The location if this NL damper changes from position 1 till 7. The damping constant is 40 N/m2 s−2 for the NL damper and 2 m/s for the linear dampers. The amplitude of the harmonic excitation force is 10 N. For each of the seven setups, the FRFs are calculated. Figure 2.18 shows the FRF for different locations of the NL damper. It clearly shows that the location of the NL damper is of importance. When using a sine sweep with a range of 50 till 300 Hz as excitation profile, the energy dissipated by the NL damper is 16 times higher when this damper is located at location 2 instead of location 6.

2.5 Complexity Due to Non-linear Damping A NL damper with a damping constant of 40 N/m2 s−2 is placed at location 4. The other dampers have a damping constant such that [D] = 10−6 [K], meaning that the non-proportionality in the damping is a result of the NL damper. In Fig. 2.19 it can be observed how two separated peaks merge when the force increases and a NL damper is present. Since the system is non-linear, there are no linear normal modes. However, one can observe the operating deflection shape (ODS). Figure 2.20 shows the ODS of the system at the undamped natural frequencies for mode 2 and 3. It is clear that the behaviour is complex. Figure 2.21 shows the ODS of the system when there is no NL damping but only proportional

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a)

b)

Fig. 2.20 ODS for different excitation frequencies. The system contains a NL damper as is described in this section. The angle θ reprents the phase difference between the excitation and the movement of the mass. The length of the arrows represents the magnitude of the motion. (a) 168.1 Hz. (b) 169 Hz

a)

b)

Fig. 2.21 ODS for the system without a NL damper, meaning the system is proportionally damped. (a) 168.1 Hz. (b) 169 Hz

damping. One might expect the system to show no complex behaviour; however it can be seen that there is also complexity when the system is proportionally damped. Even though the mode shapes are real, the ODS is still complex since both close modes have an influence on the ODS. If the natural frequencies would be far apart, the ODS would be almost completely described by a single mode, and thus it would not be complex.

2.6 Conclusion and Future Work The requirements for complex modes have been explored using a theoretical 4 DOF spring-damper system. This study confirms that proportional damping is impossible to obtain complex modes. Highly complex modes can be achieved if the damping is non-proportional and the natural frequencies are close. Using the MSC it has been shown that the optimal frequency difference of two close modes depends on the damping, the mass and possibly more parameters. This paper mainly focused on the basis of complex mode shapes. An unanswered question is whether or not the presence of complex mode shapes has an effect on the damping. Experimental research will be required to answer this question.

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References 1. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. Research Studies Press, London (2000) 2. F. Buffe et al.: A complex mode approach for the validation of FE models with structural damping (2014) 3. Lofrano, E., Paolone, A., Ruta, G.: Dynamic damage identification using complex mode shapes. Struct. Control. Health Monit. 27 (Sept. 2020). https://doi.org/10.1002/stc.2632 4. He, J., Fu, Z.-F.: 6 – Modal analysis of a damped MDoF system. In: He, J., Fu, Z.-F.. (eds.) Modal Analysis, pp. 123–139. Oxford: ButterworthHeinemann, (2001). isbn: 978-0-7506-5079-3. https://doi.org/10.1016/B978-075065079-3/50006-1 5. Imregun, Ewins, D.J.: Complex modes-origins and limits, pp. 496–504 (1995) 6. Koruk, H., Sanliturk, K.Y.: A novel definition for quantification of mode shape complexity. J. Sound Vib. 332(14), 3390–3403 (2013). issn: 0022-460X. https://doi.org/10.1016/j.jsv.2013.01.039

Chapter 3

Investigating the Potential of Electrical Connection Chatter Induced by Structural Dynamics Benjamin Dankesreiter, Manuel Serrano, Jonathan Zhang, Benjamin R. Pacini, Karl Walczak, Robert Flicek, Kelsey Johnson, and Ben Zastrow

Abstract When exposed to mechanical environments such as shock and vibration, electrical connections may experience increased levels of contact resistance associated with the physical characteristics of the electrical interface. A phenomenon known as electrical chatter occurs when these vibrations are large enough to interrupt the electric signals. It is critical to understand the root causes behind these events because electrical chatter may result in unexpected performance or failure of the system. The root causes span a variety of fields, such as structural dynamics, contact mechanics, and tribology. Therefore, a wide range of analyses are required to fully explore the physical phenomenon. This paper intends to provide a better understanding of the relationship between structural dynamics and electrical chatter events. Specifically, electrical contact assembly composed of a cylindrical pin and bifurcated structure were studied using high fidelity simulations. Structural dynamic simulations will be performed with both linear and nonlinear reduced-order models (ROM) to replicate the relevant structural dynamics. Subsequent multi-physics simulations will be discussed to relate the contact mechanics associated with the dynamic interactions between the pin and receptacle to the chatter. Each simulation method was parametrized by data from a variety of dynamic experiments. Both structural dynamics and electrical continuity were observed in both the simulation and experimental approaches, so that the relationship between the two can be established. Keywords Nonlinear dynamics · Model order reduction · Electrical chatter · Modal analysis · Contact mechanics

3.1 Introduction Structural dynamics typically focuses on the natural response of a system with the goal of evaluating the systems survivability while in operation [1]. Modern systems typically contain electrical components, and while they may not be critical to the systems mechanical survivability, they do perform critical functions that allow the system to operate as it was designed. Also, inherent in the design of circuits and connectors is the ability to maintain electrical continuity, which can heavily rely on the electromechanical features [2]. One example of such a feature is a pin-receptacle connection, as shown in Fig. 3.1. In a dynamic environment, the resistance between these two components can become sufficiently large such that the electrical energy can no longer be adequately transferred. This phenomenon is referred to as chatter [3]. The chatter is undesirable because it increases the resistance between the electrical contact and the pin, resulting in an electrical discontinuity. The root cause of the chatter phenomenon is still widely unknown and is an undeveloped area of research

B. Dankesreiter Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, USA e-mail: [email protected] M. Serrano Department of Aerospace and Mechanical Engineering, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected] J. Zhang Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA e-mail: [email protected] B. R. Pacini () · K. Walczak · R. Flicek · K. Johnson · B. Zastrow Sandia National Labs, Albuquerque, NM, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_3

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Fig. 3.1 Pin-receptacle geometry

spanning from multiple fields such as tribology, contact mechanics, structural dynamics, and many more. The focus of the current research is to improve the understanding of electrical chatter using structural dynamics analyses. Currently there are limitations in computing chatter behavior of small electrical contacts embedded in components using finite element models [4]. Reduced-order models (ROM) have been developed of such electrical contact subassemblies to evaluate the chatter behavior of the contacts during vibration and shock environments. Johnson et al. [4] provides a model validation of a reduced-order model created by Lacayo and Brake that attempts to predict the chatter behavior of a small assembly. They performed testing of the electrical contact in fluids of varying viscosities that will help characterize the effect of the fluid on the contact for an improved ROM. The work showed that the addition of fluid induces a frequency softening effect as well as separates the complimentary in-phase and out-of-phase bending modes of the system which could be crucial when creating models to predict chatter events in a system. This research topic has been explored by a previous Nonlinear Mechanics and Dynamics (NOMAD) team in the summer of 2019. This team developed a testbed to induce, record, and investigate electrical chatter in a pin receptacle contact assembly. Electrical chatter was induced in the contact assembly using sine sweeps and band-limited random excitation. The testbed was fully characterized because of time constraints and resource limitations. However, the test hardware was too complex to fully explore the relationship between pin-receptacle motion and electrical chatter. In addition to experimental work, the NOMAD 2019 team generated a high-fidelity linear model of the pin and receptacle that was only valid when the two were in contact [5]. Zastrow et al. [5] utilized Sierra Solid Mechanics finite element code [6] to solve for the dynamic response of the contact pair implicitly. The nature of the Sierra Solid Mechanics simulation required a great deal of computational resources, and therefore it was only reasonable to simulate between one and three milliseconds of chatter events, which is typical for this type of study. Three milliseconds are long enough to capture individual instances of chatter, but far too short to describe typical mechanical environments, which last on the order of seconds or minutes. This limited time range necessitated the need to build a more lightweight model of nonlinear contact events that could simulate longer times with less computational cost. This report provides a model validation for a reduced-order model (ROM), created using the Hurty/Craig-Bampton reduction, that attempts to predict the chatter behavior of a small subassembly (pin-receptacle). The study focuses on investigating what effects the structural dynamics of the pin-receptacle connection have on electrical chatter. The research objectives for the NOMAD 2021 Institute were to develop a reduced-order Hurty/Craig-Bampton model, experimentally measure the force-contact resistance relationship, and integrate those measurements into the reduced-order model to create a multi-physics model directly predicting electrical contact resistance. The paper is organized as follows. First, the background and modeling of the pin-receptacle are shown in Sec. 3.2. In Sec. 3.3, a parametric study (1) examining the development of a reduced-order Hurty/Craig-Bampton model, (2) measuring the force-contact resistance relationship, and (3) verifying the reduced-order model against a high-fidelity model is carried out. Finally, Sec. 3.4 summarizes the conclusions and future work.

3.2 Hurty/Craig-Bampton Overview An alternative to computationally expensive finite element analysis (FEA), which still maintains accuracy with respect to nonlinear contact and displacements at the nodes of interest, is the Hurty/Craig-Bampton (HCB) transformation method [7, 8]. HCB transformations are well understood to be highly effective for nonlinear solutions to dynamic contact. HCB reduction creates an efficient model by projecting the equations of motion onto a subset of fixed-interface modes and static constraint modes. The reduced model then allows for nonlinear elements to be attached to the boundary degrees of freedom which are represented by constraint modes. The multi-degree of freedom system is described in matrix form as follows:

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where M is the mass matrix, K is the stiffness matrix, vector f is the nonlinear forces applied to an element structure of the system, vector x is the displacement vector, and the subscripts i and b refer to the interior and boundary nodal coordinates. This creates a transformation matrix TCB , given as follows:  x=

xi xb



=

φik ψˆ ib 0 Ibb



qk xb

 = TCB q

where vector q is a vector of the fixed-interface mode coordinates. This transformation is used to derive the reduced HCB model of the following form: T Mˆ CB q¨ + Kˆ CB q + TCB



fN L,i (q) fN L,b (q)



 T = TCB

0 f (t)



Using the HCB transformation suite integrated into Sierra/SD, mass and stiffness matrices are produced in terms of the defined HCB degrees of freedom, allowing the reduced system to be further analyzed using various solution methods in both the time and frequency domain.

3.3 Reduced-Order Model for the Pin-Receptacle Connection To perform the Hurty/Craig-Bampton reduction, certain parameters regarding the model must be prescribed a priori, namely, the number of fixed-interface modes, the selection of nodes to be placed in the boundary set, and the contact interaction between the pin and receptacle surfaces. The first 20 fixed-interface modes of the full fidelity structure were used. Seven nodes were placed in the interface, four of which were subjected to fixed boundary conditions. The remaining three nodes (and corresponding nine degrees of freedom) were chosen specifically as shown in Fig. 3.2, one node on the outer surface of the receptacle arm, one node on the inner surface of the receptacle arm, and one node on the surface of the pin. These nodes were chosen specifically because the contact interaction is expected to happen between the inner node of the arm and the pin node, and the outer node on the arm may be used to displace the arm as part of a preloading step. It should be noted that these nodes are all in the same out-of-page plane. For the HCB reduction, the initial contact formulation between the two contacting nodes was a linear penalty spring. However, different nonlinear contact formulations are explored below.

Fig. 3.2 CB Interface node locations

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Fig. 3.3 Fits of polynomial, rational, piecewise linear, and exponential contact force models to high-fidelity force-displacement data

The system matrices are of size 29 × 29 (20 mode shapes, 3 × 3 = 9 DOF’s from interface nodes not subject to BC’s). These matrices are imported into MATLAB, and the system is propagated forward in time using a Newmark-Beta ODE Solver [9]. Four different contact formulations were used in the nonlinear ROM to attempt to replicate the results from the full fidelity FEM in Zastrow et al. [5]. These model forms are listed below along with nominal formulations: • Polynomial models of the form Fc = K0 + K1 x + K2 x2 + . . . + Kn xn . n n−1 +···+a x 2 +a x+a n−1 x 2 1 0 . • Rational models of the form Fc = banxxm +a +bm−1 x m−1 +···+b2 x 2 +b1 x+b0 m  m1 x + b1 x < a . • Piecewise linear models of the form Fc = m2 x + b2 x ≥ a • Exponential models of the form Fc = ax exp (bx). Our analysis of these contact models begins with a baseline: the contact force displacement data gathered from the highfidelity simulations. The coefficients for each of the above models were determined by performing a least squares fit between the model form and the data. The results from this fit are shown below in Fig. 3.3. All the contact model forms fit the high-fidelity data well, with all R2 values above 0.92. However, the time history data at the end of the simulation varied widely. Zastrow et al. [5] performed a similar one-arm drop test on a pin-receptacle whose results are shown below in Fig. 3.4. The time responses for a one-arm drop test simulation with the reduced-order model and various contact formulations are shown in Fig. 3.5. It can be immediately observed that there are several differences in the behavior of the contact nodes with different contact models. In the piecewise linear model, the two contact nodes remain in contact and no longer continue to separate from each other after approximately 2 ms. This feature is not seen on other models. In the rational model, as soon as the two nodes come in contact, the receptacle node immediately bounces away from the pin node. This is in direct contrast to the polynomial, exponential, and piecewise linear models, where the receptacle and pin remain in contact for a small duration before separating again. The significant variation in the responses indicates that the choice of contact model is critical even if candidate models all accurately reproduce the force displacement curve obtained via experiment or other simulation. Although the reduced-order model simulations do not exactly match the parameters of Zastrow et al., a qualitative comparison can still be made between the two results. Zastrow et al. used an augmented Lagrange contact model with Coulomb friction in their simulations. This model behaves most similarly to the piecewise linear model used in the reducedorder model simulations in the sense that upon initial contact, the receptacle and pin stay in contact for a short duration before separating again.

3 Investigating the Potential of Electrical Connection Chatter Induced by Structural Dynamics

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Fig. 3.4 Time history of pin and receptacle displacements [5]

Fig. 3.5 Time response of contacting nodes for a drop test with different ROM contact force models

In addition, the vibration amplitudes decay similarly for the two models. Between 1 and 2 ms in Zastrow’s simulation, the pin and receptacle remain very closely in contact and do not bounce against each other. In addition, the overall amplitude of oscillation is decreasing. Of the four models tested in this study, only the piecewise linear shows this decay. This seems intuitively accurate, as it follows a Newtonian notion of physics. It is important to highlight the drastic reduction in computing time achieved simply by using the reduced-order model. Using the parameters described above, SIERRA/SD took 5 minutes to generate the reduced system matrices on a basic 8core workstation. The same system then took approximately 25 min (as a high estimate) to propagate forward 3 ms using

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MATLAB. In total, the reduced-order model required approximately 30 min to completely perform a simulation. In stark contrast, the high-fidelity model used by Zastrow took 4 days to simulate on Sandia’s high-performance computers. Although the high-fidelity model outputs significantly more information than the reduced-order model, this reduction in computing time is certainly nontrivial and makes it practical to simulate multiple forcing functions and various initial conditions.

3.4 Atomic Force Microscope/Optical Profiler Measurements In addition to an efficient model which describes the structural dynamics, another important part of the pin-receptacle model is the relationship between contact force and electrical contact resistance. This relationship is the key to extending the structural dynamics simulation into a multi-physics simulation which outputs electrical contact resistance directly. Barber et al. developed a relationship between contact conductance and the incremental stiffness of rough bodies [10]. The following equation describes the electrical contact resistance, R, as a function of either the voltage, V, and current, I, or the resistivity, ρ, and total current flux, Q. Re =

(ρ1 + ρ2 ) (V2 − V1 ) = I Q

Additionally current flux can be related to incremental stiffness,

dF dw

using a composite modulus of elastic properties, M.

dF = MQ dw Finally, these two equations can be equated to derive a general electrical conductivity, C. C≡

1 dF 1 = Re M (ρ1 + ρ2 ) dw

For the pin and receptacle to be considered rough bodies, the statistical roughness parameters such as root mean squared roughness, kurtosis, skewness, and size of asperity must be measured. With these values, the incremental stiffness can be determined, as well as other characteristics such as real contact area. To find these parameters, a sample of the pin and receptacle were studied using atomic force microscopy. A 20 × 20 square micron sampling area on the contacting surface of both samples was scanned using an MFP3D atomic force microscope from Asylum Research in AC imaging mode with a standard AC 160 probe. Because of the challenges obtaining experimental electrical chatter data, it is difficult to validate these physical considerations with respect to the dynamic model. Therefore, it will be applied to future work on this subject.

3.5 Conclusion The Hurty/Craig-Bampton numerical method was employed to develop a reduced-order model (ROM) for predicting chatter in a small electrical system. Polynomial, rational, piecewise linear, and exponential contact models were implemented in the ROM. It was observed that in each case the contact model fits the high-fidelity data well. However, the time history data at the end of the simulations varied widely. This implies that the choice of contact model is critical. Comparison to a similar one-arm drop test performed by Zastrow showed that while none of the four models precisely matched the high-fidelity simulation, the piecewise linear contact model kept the important qualitative features of the high-fidelity simulation. Unfortunately, these results also imply that a solid conclusion about the most accurate model cannot be reached without additional investigation. While atomic force microscopy was utilized to get surface roughness measurements, additional parameters are required for the consideration of a detailed multi-physics model. The authors see several future avenues of exploration for this project and, more generally, for research in electrical chatter. Most importantly, future work could validate both computational models with experiments. Specifically, exciting the pin-receptacle structure with several vibration conditions to test the computational models in all environments would be

3 Investigating the Potential of Electrical Connection Chatter Induced by Structural Dynamics

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beneficial, in addition to tuning the reduced-order models to improve their match to the high-fidelity models (recall that only the time histories were directly compared in this work). Comparing the frequency content of the responses by means of a time frequency analysis will be useful, as analysis of the frequency content is likely a better comparison of the two models than simply their time histories. Additionally, future studies should incorporate more accurate models of electrical contact resistance in a multi-physics simulation. This task will require additional measurements of physical samples and calculation of additional parameters. Acknowledgments The authors acknowledge the NOMAD Research Institute at Sandia National Laboratories as well as the team mentors for making this work possible. This research was conducted at the 2021 Nonlinear Mechanics and Dynamics (NOMAD) Research Institute supported by Sandia National Laboratories and hosted by the University of New Mexico. This manuscript was authored by National Technology and Engineering Solutions of Sandia, LLC, under contract no. DE-NA0003525 with the US Department of Energy/National Nuclear-Security Administration. 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.

References 1. Collins, I., Hossain, M., Dettmer, W., Masters, I.: Flexible membrane structures for wave energy harvesting: A review of the developments, materials, and computational modelling approaches. Renew. Sust. Energ. Rev. 151, 111478 (2021) 2. Dubey, K.A., Mondal, R.K., Kumar, J., Melo, J.S., Bhardwaj, Y.K.: Enhanced electromechanics of morphology-immobilized co-continuous polymer blend/carbon nanotube high-range piezoresistive sensor. Chem. Eng. J. 389, 124112 (2020) 3. Yuan, H., Dai, H., Wei, X., Ming, P.: Model-based observers for internal states estimation and control of proton exchange membrane fuel cell system: A review. J. Power Sources. 468, 228376 (2020) 4. Johnson, K. M.: Characterization of a Small Electro-Mechanical Contact Using Non-conventional Measurement Techniques (2017) 5. Zastrow, B.G., et al.: Investigation of electrical chatter in bifurcated contact receptacles. In: Proceedings of the 66th IEEE Holm Conference, San Antonio, TX, October 24–27 (2021) 6. Sierra Structural Dynamics Development Team Sierra/SD – User’s Manual – 4.56. SAND2020–3028, 202 7. Hurty, W.: On the dynamic analysis of structural systems using component modes. 1st annual meeting (1964) 8. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968) 9. Subbaraj, K., Dokainish, M.A.: A survey of direct time-integration methods in computational structural dynamics-II. Implicit methods. Comput. Struct. 32(6), 1387–1401 (1989) 10. Barber, J.R.: Bounds on the electrical resistance between contacting elastic rough bodies. Proc.R. Soc. Lond. A. 459(2029), 53–66 (2003). https://doi.org/10.1098/rspa.2002.1038

Chapter 4

Ensemble of Numerics-Informed Neural Networks with Embedded Hamiltonian Constraints for Modeling Nonlinear Structural Dynamics David A. Najera-Flores and Michael D. Todd

Abstract Data-driven machine learning models are useful for modeling complex structures based on empirical observations, bypassing the need to generate a physical model where the physics is not well known or readily otherwise model-able. One disadvantage of purely data-driven approaches is that they tend to perform poorly in regions outside the original training domain. To mitigate this limitation, physical knowledge about the structure can be embedded in the model architecture via the model topology or numerical constraints in the formulation. For large-scale systems, relevant physics, such as the system-state matrices, may be expensive to compute. One way around this problem is to use scalar functionals, such as energy, to constrain the network to operate within physical bounds. We propose a neural network framework based on Hamiltonian mechanics to enforce a physics-informed structure to the model. The Hamiltonian framework allows us to relate the energy of the system to the measured quantities (e.g., accelerations) through the Euler-Lagrange equations of motion. In this work, the potential, kinetic energy, and Rayleigh damping terms are each modeled with a multilayer perceptron. Auto-differentiation is used to compute partial derivatives and assemble all the relevant equations, including computing the generalized inertia matrix by forming the Hessian of the kinetic energy with respect to the generalized coordinates. Moreover, a Bayesian approach is used to estimate model-form error to predict domain shifts in the data and enable model correction. The network incorporates a numerics-informed loss function via the residual of a multistep integration term, allowing the ensemble of networks to be time-integrated with new initial conditions and an arbitrary external force after it has been trained. The approach is demonstrated on simple exemplars, such as a two degree-of-freedom (DOF) damped oscillator with cubic nonlinearities. Keywords Nonlinear dynamics · Hamiltonian mechanics · Euler-Lagrange · Neural networks

4.1 Introduction Data-driven machine learning models are useful for modeling complex structures based on empirical observations, bypassing the need to generate a physical model where the physics is not well known or readily otherwise model-able. In this work, we impose a mathematical structure on a data-driven approach to improve robustness of data-driven models and reduce the need for large amounts of training data. We use an ensemble of neural networks constrained according to Hamiltonian mechanics to ensure that the networks are learning physically interpretable representations. The Hamiltonian framework allows us to relate the energy of the system to the measured quantities (e.g., accelerations) through the Euler-Lagrange equations of motion. This work demonstrates the approach by predicting the response of a two degree-of-freedom (DOF) damped oscillator with cubic nonlinearities.

D. A. Najera-Flores Department of Structural Engineering, University of California San Diego, San Diego, CA, USA ATA Engineering, Inc., San Diego, CA, USA e-mail: [email protected] M. D. Todd () Department of Structural Engineering, University of California San Diego, San Diego, CA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_4

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4.2 Neural Network Architecture The framework is composed of an ensemble of neural networks that are communicating with each other. Figure 4.1 shows the general architecture of networks that was used. We begin by assuming that the observations may not be in an inertial coordinate system and that, in general, they do not correspond to a set of generalized coordinates. The initial coordinates are denoted by r, and the generalized coordinates are denoted by q. An autoencoder is used to learn coordinate transformations between these two sets of coordinates. The encoder is first used to map the initial coordinates to the generalized coordinates, and a decoder is used to map them back to the original reference frame. After the transformation is learned, the latent space functions as the generalized coordinate system. During this step, it would be possible to reduce the number of dimensions but that has not been explored yet. For now, the dimension of r and q is assumed to be the same. Once in latent space, three separate networks are used to learn the energy terms: the potential (U) and kinetic energy (T) terms and the Rayleigh damping (I) are each modeled with a multilayer perceptron (MLP), as shown below. ˙ , U (q) = UN N (q, q)

˙ , T (q) = TN N (q, q)

˙ I (q) = RN N (q, q)

˙ − UN N (q). ˙ = TN N (q) The Lagrangian term is obtained from the potential and kinetic energy: L (q, q) From this, the equations of motion are enforced in the canonical coordinate system by first obtaining the system’s momenta for each DOF and constructing the system’s Hamiltonian. ˙ = p · q˙ − L (q, q) ˙ H (p, q) The generalized force term is derived from d’Alembert’s principle of virtual work [1], which results in projections of the force in the original coordinate system onto the partial derivative of the displacements in the original coordinate system with respect to the generalized displacements. Qnc =

N i=1

Fi ·

∂DN N i ∂qk

The Euler-Lagrange equations of motion are then enforced by including the residual as a loss term:  LossEL =

d dt



∂L ∂ q˙k

 −

∂L ∂I + − Qnc ∂qk ∂ q˙k

Fig. 4.1 Ensemble of neural network architecture with Hamiltonian constraints

2

4 Ensemble of Numerics-Informed Neural Networks with Embedded. . .

29

Fig. 4.2 Phase portrait (left) and Lagrangian (center) time history showing the comparison between actual response (color) vs. neural network prediction (black). Nonlinear forced-response curves for varying excitation amplitudes (right)

Furthermore, the network also enforces a time-derivative check by incorporating a multistep integration term, allowing the ensemble of networks to be time-integrated with new initial conditions and an arbitrary external force after it has been trained. LossF =

⎧ s ⎨ ⎩

j =0

 αj

q k+j q˙ k+j

 −h

s  j =0

 βj

q˙ k+j q¨ k+j

⎫   ⎬2 t  k+j  t k+j ⎭

4.3 Results and Discussion The approach is demonstrated on a two DOF damped oscillator with cubic nonlinearities. Each of the MLPs had four hidden layers, each with eight hidden units. A swish activation function was used for all the networks. The ensemble of networks was all trained simultaneously using a single response realization to a random force. The ensemble of networks was implemented using Tensorflow 2.5 [2]. Figure 4.2 shows the masses’ response to a new random excitation, along with the predictions for the Lagrangian. The response was obtained by performing direct time integration on the trained network with scipy.integrate, and the results are presented after 6000 time steps to show that the response does not degrade with time. The trained network was then subjected to sinusoidal excitation at varying amplitudes. The response at steady state was obtained via direct time integration until transients decayed. The steady-state response was then plotted as a function of excitation amplitude to obtain nonlinear forced-response curves, which are shown in Fig. 4.2 (right). As illustrated, the network was able to capture the stiffening effect typical of the cubic spring system.

4.4 Conclusion This work introduced an ensemble of neural networks that has been constrained by the mathematical and physical structure of Hamiltonian mechanics. Preliminary results show that the approach is robust and does not suffer from excessive error accumulation. It has also been found that limited data may be sufficient for training the network to be used outside of its training regime. These results are a direct consequence of the physical and numerical constraints, including the time integration residual loss term. Future work will extend the method to larger systems and to be fully Bayesian for use in a probabilistic reasoning framework to inform structural health diagnosis and prognosis. Acknowledgments This work was funded by Sandia National Laboratories; Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

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References 1. Cline, D.: Variational Principles in Classical Mechanics: Revised Second Edition. River Campus Libraries (2019) 2. Abadi, M., et al.: TensorFlow: Large-scale machine learning on heterogeneous systems. Software available from tensorflow.org (2015)

Chapter 5

System Identification of Geometrically Nonlinear Structures Using Reduced-Order Models Mohammad Wasi Ahmadi, Thomas L. Hill, Jason Z. Jiang, and Simon A. Neild

Abstract System identification of engineering structures is an established area in the structural dynamics research community. It is often used to characterise certain physical properties of a structure using the data measured from it. For structures exhibiting nonlinear behaviour, physics-based approaches are used where a form of nonlinearity is synthesised and parameters are estimated using the data, or probabilistic approaches are investigated to tackle the model uncertainty of structures. However, to build reliable models, the estimated parameters from the measurement data must reflect the true underlying physics of the structure. Therefore, Reduced-Order Models (ROMs) can be used as the surrogate models, where the nonlinear parameters of the ROMs are having a meaningful relation with the physical parameters of the system. In this work, we propose nonlinear system identification in the context of using some recently developed ROMs which account for the kinetic energy of unmodelled modes. It is shown how ROMs may be used to represent low-order, accurate models for system identification. Identification of a nonlinear system with strong modal coupling is demonstrated, using simulated data, while the estimated ROM response shows good convergence with that of full order system. Similarly, the estimated parameters match with those of directly computed ROM. Keywords Structural dynamics · Nonlinear dynamics · System identification · Reduced-order models

5.1 Introduction To simulate the nonlinear behaviour of a structure, a reliable model is required; this often requires nonlinear system identification to be performed. In this context, measurements are taken from a physical structure and are used to construct a mathematical model. Although system identification is well-established for linear systems, nonlinear system identification is still undergoing significant developments. Here we consider system identification of geometrically nonlinear systems. Mainly, system identification consists of detection, characterisation and parameter estimation. Parameter estimation can be classified into main categories as: time and frequency-domain methods, time-frequency analysis, linearisation, modal methods, black-box modelling and numerical model updating, [1]. Developments have also been made in quantifying uncertainties in parameters. This can mainly be due to numerous candidate models which can be fitted into a set of measured data [1]. This has driven attention towards probabilistic approaches such as Bayesian framework[1]. These approaches can be appealing as they allow for optimum model to be found from a set of candidate models. Although, for more complex structures (i.e. structures with high number of degrees-of-freedom) synthesising candidate models based on prior knowledge can be a very biased process. Also, the number of nonlinear parameters can drastically increase. This cannot only make the parameter estimation process a challenging task but also increase the uncertainty in the confidence bound and distributions. More recently Reduced-Order Models (ROMs) have been investigated as surrogate models for complex structures to accurately represent the desired behaviour of a structure through a smaller sized set of second order differential equations. The usage of ROMs can reduce the computational costs when simulating the response of a structure [2, 3]. Thus, in this work we bridge between system identification and ROMs which can bring a two-fold benefit: first that it will make the system identification process easier and more reliable; secondly, it will bring computational benefits when used to simulate the response of the structure.

M. W. Ahmadi () · T. L. Hill · J. Z. Jiang · S. A. Neild Faculty of Engineering, University of Bristol, Bristol, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_5

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In the next section, identification process of a nonlinear system is illustrated. This is followed by the results section and final remarks given in the conclusions at the end.

5.2 System Identification Here, we consider identification of a geometrically nonlinear structure—specifically a cantilever beam with spring attached at the free end, as shown in Fig. 5.1. In this beam the lower frequency bending modes of the system are statically coupled with higher frequency transverse modes. This coupling happens due to the membrane stretching in this cantilever type beam. We use some recently developed ROMs which accounts for this coupling [4]. The system is modelled in Abaqus Finite-Element package with parameters L = 0.3 m, Lsp = 0.1 m, Ksp = 20 Nm−1 , cross sectional area of A = 0.025 m × 0.001 m, Young’s modulus E = 205 GPa and Poisson’s ratio v = 0.3. Note that the spring is unstretched when its length is Lsp . The FE model resulted in 1440 DOFs with the first mode being in the bandwidth of interest with ω1 = 57.815 rad s−1 . The model considered for this system is an inertially compensated ROM (ICROM) [4] as shown in Eq. 5.1, while without the 2nd and 3rd terms it would be a standard ROM (SROM). In this Equation the dynamics of governing modes (•r ) is represented and the coupled modes are accounted for in the inertial compensation terms. q¨r +

gi =

 ∂gi 2 ∂qr N 

q¨r +

N   ∂gi  ∂ 2 gi  2 2 q ˙ + ω q + γn qrn = 0 r r r ∂qr ∂qr2

(5.1)

n=2

βi,n qrn

(5.2)

n=2

where q represents the response of the system, ωr is the natural frequency in (rad s−1 ), γ being the nonlinear parameters of up to Nth order. gi is the coupling function for (i) spanning over coupled modes, with β as the coupling coefficient, shown as Eq. 5.2. For SROM without the inertial terms in Eq. 5.1, a simple linear regression can be applied to estimate the nonlinear parameters γ using the response data of the system. However, the ICROM has to be further developed to be treated as an inverse problem. We consider both the nonlinearity and coupling functions of up to 5th order. By considering a single-mode ICROM (r = 1) and including rest of the modes (i = [2 to 1440]) in the coupling function, the IC terms are derived as outlined in Eq. 5.3.  ∂gi 2 ∂q1

q¨1 +

 ∂gi  ∂ 2 gi  2 q˙1 =(4q¨1 q12 + 4q˙12 q1)A¯ + (12q¨1 q13 + 18q˙12 q12 )B¯ + (q¨1 q14 + 2q˙12 q13 )C¯ + . . . ∂q1 ∂q12 (4q¨1 q15 + 10q˙12 q14 )D¯ + (2q¨1 q16 + 6q˙12 q15 )E¯ + (40q¨1 q17 + 140q˙12 q16 )F¯ + . . .

(5.3)

¯ (25q¨1 q18 + 100q˙12 q17 )G The above expression representing the inertial terms is substituted into the Eq. 5.1 and applied in a least squares (LSQ) manner to estimate the unknown parameters. Response data is extracted from the FE model with 0.5% damping ratio for the first mode. An initial condition is applied on the system, such that the structure is given a tip displacement of (1/3)L which is sufficient to activate nonlinearity. The force is removed and the decaying response of the system is captured. Using the resonant decay method (RDM) proposed in [5] the first backbone curve of the system is measured as shown in Fig. 5.2. In L

Lsp

K sp

Fig. 5.1 Cantilever beam schematic

5 System Identification of Geometrically Nonlinear Structures Using Reduced-Order Models

33

0.01

Q1

0.008 0.006 0.004

Full Model 5 th order SROM ID

0.002 0

5 th order ICROM ID

58

59

60

61

Fig. 5.2 The first backbone curve of full model, identified SROM and identified ICROM

the LSQ problem we use the Fourier components of the response data, where certain harmonics within the interest range are included in the computation.

5.3 Results The estimated parameters are used to reconstruct both the 5th order SROM and ICROM, then using numerical continuation technique, in the MATLAB based toolbox—continuation core (COCO) [6], the backbone curves of identified models are constructed and compared with that of full order model in Fig. 5.2. Backbone curve plots are shown in Fig. 5.2, where the best match with the full model is for ICROM, while the SROM is diverging from the full model at almost all levels of amplitude. The identified ICROM perfectly matches the full model backbone at low amplitude levels, however, slight inaccuracies can be because the full model Backbone is constructed from decay response which contains small error due to damping present in the data [5]. This shows the importance of choosing the right model in the identification. It also has to be declared that up to 5th harmonics are included in the identification as higher harmonics’ contribution are getting negligible and found to have minimum effect on the response.

5.4 Conclusion Identification of a nonlinear cantilever type structure has been addressed. It was shown how ROMs may be used in the system identification context, while they can be of manifold advantage. The identification has been applied to an FE model using two types of ROMs while their results were compared. The ICROM has resulted in better response and parameters compared to the SROM. This is due to the significant inertial coupling between the modes of the system, which cannot be accounted for in an SROM. Including additional degrees-of-freedom in SROM can potentially improve the response but will result in additional parameters and computational complexity. Acknowledgments Authors acknowledge the support of EPSRC grant EP/R006768/1.

References 1. Noël, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017) 2. Gordon, R.W., Hollkamp, J.J.: Reduced-order models for acoustic response prediction of a curved panel. In: Collection of Technical Papers AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (April), pp. 1–14 (2011)

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3. Hollkamp, J.J., Gordon, R.W.: Reduced-order models for nonlinear response prediction: implicit condensation and expansion. J. Sound Vibr. 318(4–5), 1139–1153 (2008) 4. Nicolaidou, E., Hill, T.L., Neild, S.A.: Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy. Proc. R. Soc. A Math. Phys. Eng. Sci. 476(2243), 20200589 (2020) 5. Londoño, J.M., Neild, S.A., Cooper, J.E.: Identification of backbone curves of nonlinear systems from resonance decay responses. J. Sound Vibr. 348, 224–238 (2015) 6. Dankowicz, H., Schilder, F.: Recipes for Continuation. Society for Industrial and Applied Mathematics: Philadelphia (2013)

Chapter 6

Indirect Reduced-Order Modelling of Non-conservative Non-linear Structures Evangelia Nicolaidou, Thomas L. Hill, and Simon A. Neild

Abstract Engineering structures are often designed using finite element (FE) models. Performing non-linear dynamic analysis on high-fidelity FE models can be prohibitively computationally expensive, due to the very large number of degrees of freedom. Non-linear reduced-order modelling allows the salient dynamics of the FE model to be captured efficiently in a smaller, computationally cheap reduced-order model (ROM). Recent developments in indirect reduced-order modelling techniques enable ROMs to be developed efficiently, accurately, and robustly, for a wide range of structures. Nevertheless, these methods are applicable to conservative systems and are unable to capture the effects of, for example, damping and external forcing. In this work, we show how indirect reducedorder modelling methods can be extended to non-conservative non-linear structures, which offers invaluable insight into the behaviour of the system. We demonstrate the proposed method using a simple oscillator. Keywords Reduced-order model · Geometric non-linearity · Non-conservative · Finite element model · Structural dynamics

6.1 Introduction In this work, we study the dynamics of elastic structures oscillating at large amplitudes, giving rise to geometric non-linearity. We focus on methods for deriving parametric ROMs of geometrically non-linear structures in a non-intrusive, or indirect, manner [1–3]. Such methods do not require knowledge of the equations of motion (EOMs) of the full-order model and can be applied to structures modelled using commercial FE software. These methods are typically used to compute the backbone curves of a structure, i.e. loci of periodic responses of the equivalent conservative system. Whilst these are closely related to the forced/damped response, it is often useful to simulate the non-conservative dynamics explicitly. Here, we show how non-conservative terms can be properly incorporated into the reduced dynamics, enabling the computation of forced responses.

6.2 Background The full-order EOMs of the geometrically non-linear structure can be written as follows: M¨x + Kx + fx (x) = Fx (x, x˙ ) ,

(6.1)

where x is the vector of generalized displacements, M and K are the linear mass and stiffness matrices, fx (x) is the vector of non-linear restoring forces, and Fx (x, x˙ ) is a vector of any non-conservative forces acting on the structure; these may be a result of a dissipative process and/or external forcing. The system can be reduced by assuming that its dynamics can be captured by a small subset of its linear normal modes, i.e. x = r r + θ(r), where r is the vector of reduced coordinates, r is the matrix containing the reduced modeshapes in its columns, and θ is a vector of quasi-static coupling functions which

E. Nicolaidou () · T. L. Hill · S. A. Neild Department of Mechanical Engineering, University of Bristol, Bristol, UK e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_6

35

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E. Nicolaidou et al.

can be approximated using a series of static solutions of the full-order model. This is the idea underpinning the implicit condensation and expansion with inertial compensation (ICE-IC) method, discussed in [2]. From this, the reduced dynamics of the conservative system can be derived as follows: r¨ +



∂θ ∂r

T

¨+ M ∂θ ∂r r



∂θ ∂r

T

∼

M ∂∂r2θ r˙ r˙ + r r + f r (r) = 0, 2

(6.2) ∼

where r contains the squares of the natural frequencies of the reduced modes in its main diagonal and f r is the vector of reduced non-linear restoring force functions, which can be approximated using the existing static solution dataset.

6.3 Method and Results When the non-conservative terms are taken into account, it can be shown that the reduced dynamics can be written as follows: r¨ +



∂θ ∂r

T

¨+ M ∂θ ∂r r



∂θ ∂r

T

∼ 2 M ∂∂r2θ r˙ r˙ + r r + f r (r) = r +

∂θ ∂r

T



Fx r r + θ, r +

∂θ ∂r

  r˙ .

(6.3)

In order to demonstrate our proposed method, we use a simple, analytical model of a mass connected to two linearly elastic springs, as shown in Fig. 6.1a. The mass is free to move in the x − y plane. When the system is in equilibrium, the springs are unstretched and perpendicular to each other. The system is subject to harmonic forcing and Rayleigh damping, i.e. Fx (x, x˙ ) = P cos t − C˙x, where P is the vector of forcing amplitudes,  is the forcing frequency, C = αM + βK is the linear damping matrix, and α and β are the mass- and stiffness-proportional Rayleigh damping coefficients, respectively.

Fig. 6.1 (a) Schematic diagram of the simple oscillator used to demonstrate our proposed method and (b) backbone curves and forced-response curves of the full- and reduced-order model, shown in the projection of forcing frequency against amplitude of the first (reduced) mode

6 Indirect Reduced-Order Modelling of Non-conservative Non-linear Structures

37

Here, we compare our proposed ROM (Eq. 6.3) with a naive extension of the conservative ICE-IC ROM, where only the non-conservative terms corresponding to the reduced modes are propagated from the full- to the reduced-order model, i.e.: r¨ +



∂θ ∂r

T

¨+ M ∂θ ∂r r



∂θ ∂r

T

∼

M ∂∂r2θ r˙ r˙ + r r + f r (r) = Tr P cos t − Tr Cr r˙ . 2

(6.4)

The parameters used here are m = 0.1 kg,  = 0.1 m, k1 = 10 N m−1 , k2 = 1000 N m−1 , α = 0, and β = 10−3 s, such that the natural frequencies are well separated, i.e. ω1 = 10 rad s−1 and ω2 = 100 rad s−1 . The modal damping ratios are ζ 1 = 0.5% and ζ 2 = 5%. We construct a ninth-order ROM based on the first mode, using the ICE-IC method. Its backbone curve (black line) is found to be in good agreement with that of the full-order model (dashed grey line), as shown in Fig. 6.1b. From this, we compute the forced response of the ROM using our proposed method (green line), as well as the naive approach (blue line), with P = [5 0]T × 10−3 N. The corresponding forced response of the full-order system is also shown (dashed red line). It can be seen that the “naive” ROM severely underestimates the amount of damping present in the system. This is to be expected, as it is inherently unable to capture the non-conservative behaviour of the heavily damped second mode. On the other hand, our proposed ROM incorporates the damping and forcing acting on the condensed mode, into the reduced dynamics, and is able to accurately approximate the forced response of the full-order system.

6.4 Conclusion In this work, we have shown how the non-conservative dynamics of geometrically non-linear systems can be efficiently and accurately captured using non-intrusive reduced-order modelling techniques. This development enables forced-response curves to be computed, without requiring any additional data or analysis. We have demonstrated our proposed method using a simple oscillator as a motivating example and have found good agreement between the full- and reduced-order model. The method can straightforwardly be applied to any finite element model constructed using commercial software. Acknowledgements The authors acknowledge the support of the EPSRC via a DTP studentship and program grant EP/R006768/1.

References 1. Mignolet, M.P., Przekop, A., Rizzi, S.A., Spottswood, S.M.: A review of indirect/non-intrusive reduced order modeling of nonlinear geometric structures. J. Sound Vib. 332(10), 2437–2460 (2013) 2. Nicolaidou, E., Hill, T.L., Neild, S.A.: Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy. Proc. R. Soc. A. 476(2243), 20200589 (2020) 3. Nicolaidou, E., Melanthuru, V.R., Hill, T.L., Neild, S.A.: Accounting for quasi-static coupling in nonlinear dynamic reduced-order models. J. Comput. Nonlinear Dyn. 15(7) (2020)

Chapter 7

Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads on Finite Element Structures Based on Stress Trial Vectors Lukas Koller and Wolfgang Witteveen

Abstract An established approach to reduce the effort for the numerical time integration of Finite Element (FE) structures is given by model order reduction (MOR) via subspace projection. These techniques lead to a significant decrease of the necessary degrees of freedom (DOF) by using proper deformation trial vectors. However, if nonlinear loads are applied on distributed regions of the FE structures surface, the computation of these forces is based on physical state-information of all involved nodes. To avoid this dependency, Hyper-Reduction (HR) methods provide a suitable framework to compute the nonlinearity with a reduced number of DOF too. In this contribution, the HR of the nonlinear surface load is based on stress trial vectors, which can be either determined in conjunction with the deformation trial vectors for the MOR (a priori) or as a result of given solution snapshots of the nonlinearity under consideration (a posteriori). In both cases, the stress trial vectors span a subspace, which is combined with a problem formulation via the calculus of variations and a procedure for a reduced selection of integration points (e.g., empirical cubature method). As a result, an HR approach is obtained that allows a more efficient evaluation of the acting nonlinear loads. A numerical comparison of an a priori and an a posteriori subspace is made by using a planar crank drive mechanism, where an elastohydrodynamic (EHD) contact is considered between the piston and the cylinder liner. Keywords Model order reduction · Hyper-reduction · Nonlinear surface loads · Multibody simulation · Elastohydrodynamic lubrication

7.1 Introduction The consideration of nonlinear and distributed surface loads (e.g., contacts) within a dynamic simulation can be realized via a finite element (FE) modeling of the involved structures. Here, an increasing level of detail inevitably results in an increase of the involved degrees of freedom (DOF). In research and industry, more and more advanced analyses are performed to improve the physical validity of simulations. Therefore, not only computing power but also more sophisticated and effective computational methods are required to keep the development times low. For this purpose, the use of model order reduction (MOR) is recommendable when dynamic simulations of FE structures are performed. Assuming that the deformations of the components follow a linear characteristic, an ansatz like that of CraigBampton [1] can be used to drastically reduce the dimension of the given problem. Thereby, the equations of the full DOF model are projected into a subspace via a properly chosen reduced basis (RB), which is intended to describe the deformation behavior of the high-fidelity model. An overview about several methods is given in [2] or [3], for instance. The effectiveness of the latter mentioned MOR methods is reduced when nonlinearities (e.g., geometric, contact,. . . ) have to be taken into account to obtain a more realistic analysis. Since the consideration of such features usually requires both an evaluation that includes all involved DOF and a subsequent projection into the reduced subspace, their presence can significantly reduce the performance of the time integration procedure. To decrease the computational burden even in such cases, the so-called Hyper-Reduction (HR) methods can be applied. Similar to the MOR techniques, they also target a reduced description of the given nonlinearity by means of a lower dimensional approximation. Reference is made to [4–7], where several HR approaches are discussed.

L. Koller () · W. Witteveen Study Program “Mechanical Engineering”, University of Applied Sciences Upper Austria, Wels, Austria e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_7

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L. Koller and W. Witteveen

This contribution deals with an HR strategy for a more efficient computation of nonlinear and state-dependent surface loads, which was recently published by the authors in [8]. Here, one of the key ideas is to use the fundamental properties of stress trial vectors (also called stress modes), which are superimposed to describe the arising surface loads in a modal manner, see [9] for a detailed description. As discussed in [8], the stress modes can have two different origins. On the one hand, an a priori determination in conjunction with the RB for the deformations is possible. In this case, the resulting stress modes depend on the physical characteristics of the FE structure and represent a generally valid space. On the other hand, snapshots of the nonlinearity can be gathered during a dynamic simulation, followed by a subsequent extraction of the most relevant information via the proper orthogonal decomposition (POD) [10], for instance. The created trial vectors are then referred to as a posteriori stress modes and enable a more problem-specific description. Combining one of these stress mode types with a problem formulation via the calculus of variations yields dimensionally reduced equations, whose complexity is further decreased by means of a reduced numerical integration scheme. Here, the empirical cubature method (ECM) of Hernández et al. [11] is suggested for this task. Finally, the resulting system of equations to describe the nonlinear surface load is in the dimension of the stress modes used and, moreover, requires only a subset of all involved DOFs for its construction because of the ECM procedure.

7.2 Model Order Reduction and Surface Load Reconstruction To approximate the deformation of a linear FE a reduction via projection (e.g., Craig-Bampton [1]) can be applied. "m ! structure, Thereby, the m invariant (n × 1) trial vectors ϕ i i=1 must adequately represent the FE models deformation behavior at the n DOF. These trial vectors, also called modes, span an RB, which can be linearly superimposed with m variant scaling factors q = {qi }m i=1 . Mathematically, this concept results in u ≈ ϕ 1 q1 + . . . + ϕ m qm = q,

(7.1)

 where the deformations are defined by the (n × 1) vector u and the (n × m) matrix  = ϕ 1 . . . ϕ m contains the single mode shapes. The equations of motion (EOM) for an FE model are given as ˙ , Mu¨ + Ku = f L + f N L (u, u)

(7.2)

with the state-independent (n × n) mass matrix M and stiffness matrix K. Substituting Eq. (7.1) in Eq. (7.2) and a multiplication with  from the left lead to ˙ = f L + f N L (q, q) ˙ . Mu¨ + Ku =  f L +  f N L (q, q)

(7.3)

Here, the (m × m) mass matrix M =  M and (m × m) stiffness matrix K =  K define the dimensionally downsized invariant terms of the EOM. On the right hand side of Eq. (7.3), the reduced (m × 1) linear force vector f L =  f L and the ˙ =  f N L (q, q) ˙ are obtained due to the subspace projection, reduced (m × 1) nonlinear force vector f N L (q, q) ˙ must be known before the multiplication with  . whereby f N L = f N L (q, q) ˙ contributes In the context of a frictionless contact situation, solely the (c × 1) vector of nodal pressures p = p(u, u) to the nonlinear force f N L = f N L (p). Since p must be determined based on the state-information on all c nodes in the contact region, the effort for its computation grows with an increasing contact area or a finer meshing. To overcome this dependence on the nodal discretization, a modal representation of the acting surface load is suggested in [8, 9]. Similar to the deformations in Eq. (7.1), a linear combination of stress modes is proposed for an approximation of the (c × 1) vector of normal stresses σ n . By equating the contact pressure with these negative normal stresses p =! −σ"n , a modal description is r obtained due to the action-reaction law. Accordingly, a suitable set of r normal stress modes ψ ni i=1 can be used together with r scaling factors z = {zi }ri=1 to calculate the pressure p by means of p = −σ n = −ψ n1 z1 − . . . − ψ nr zr = − n z.

(7.4)

The principle of superposition can also be applied to approximate the derivatives of the pressure, yielding pj =

∂p ∂σ n =− = −ψ nj,1 z1 − . . . − ψ nj,r zr = − nj z, ∂j ∂j

(7.5)

7 Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads. . .

41

where j ∈ {x, y} applies in terms of Cartesian coordinates and the (c × 1) vector ψ j,i = ∂ψ i /∂j specifies the derivative of   ψ i with respect to j . The stress modes are grouped in the (c × r) matrices n = ψ 1 . . . ψ r and nj = ψ j,1 . . . ψ j,r , whereby a priori or a posteriori mode shapes can be utilized. A detailed description of these two types is provided in [8], which is why only the basic properties are addressed at this point. A priori stress modes do not require any information from dynamic simulations, which include the nonlinearity in an unreduced manner, but are computed according to the idea of modal stress recovery [12, 13]. Here, it is assumed that each deformation mode ϕ i causes a corresponding stress distribution. Due to the absence of tangential stresses in the contact n region, only the normal stresses at the c nodes are collected in the (c × 1) normal fore-stress mode ψ i . Hence, the (c × m)   n

matrix of a priori fore-stress modes = ψ 1 . . . ψ m is obtained as an intermediate result, which is then further processed to n via the application of POD. The necessity for this treatment can be traced back to the origin of . This matrix is  obtained via an orthonormalization of  = s v l with respect to the mass matrix, where the (n × s) static mode matrix s and the (n × v) vibrational mode matrix v describe the global deformations of the structure. The (n × l) local mode matrix l is intended to approximate the mechanical behavior directly in the contact area. Although the matrix  can be interpreted as a mixture of the ingredients of , it is assumed that the local modes in l are mainly responsible for the n relevant content in . This presumption is underlined by the fact that when POD is applied, a steep decline is observable after l singular values. Thus, r = l stress modes are stacked in n . A possible workflow for the computation of l and n further details on the properties of and n are given in [9]. A posteriori stress modes origin from distributions of the surface load, which are collected during a dynamic simulation. The number of gathered solution snapshots of the is quantified by f time-steps, yielding the (c × f )  contact nonlinearity n n matrix of a posteriori fore-stress modes = p1 . . . p f . Similar to the a priori modes, also contains redundant information, which is why POD is used to extract the r relevant a posteriori stress modes in this case too. Besides a decrease in the singular values, another objective criterion for the determination of r is given in [8]. As a result, the normal stress modes in n only cover the history of the preceding dynamic simulation with the corresponding pressure distributions ! "f pi i=1 . Hence, the created normal stress space might become less accurate if future states of the system differ significantly from those of the snapshot collection. However, if minor or moderate state changes occur, the usage of the more problemspecific a posteriori stress modes should result in a smaller number r when compared to the generally valid a priori modes.

7.3 Hyper-Reduced Description of the Surface Load The theory behind the applied HR approach is based on the calculus of variations and the use of the surface load reconstruction via stress modes. For a detailed description of the different concepts and an in-depth derivation, the interested reader is referred to [8], which is why only the essential ideas are briefly outlined hereafter. If a planar problem in the field of contact mechanics is considered, a cost functional can be defined as # ∂p , (7.6) J (p) = F (p, px )d with px = ∂x  where Cartesian coordinates are used to describe the one-dimensional (1D) contact region . Here, p = p(x) is the unknown pressure with its derivative px = px (x). According to Lanczos [14], an extremum, or at least stationary value of J (p), is given if the Euler–Lagrange equation ∂ ∂F − ∂p ∂x



∂F ∂px

 =0

(7.7)

is satisfied. The function F = F (p, px ) needs to be determined according to the chosen contact law, which is specified via g(p, px ) = 0. Using this notation, the following rule for the selection of the problem-describing function F is obtained: ∂F ∂ g(p, px ) = − ∂p ∂x



∂F ∂px

 =0

(7.8)

If Eqs. (7.4) and (7.5) are used for a modal representation of the nodal pressure and its derivative in x-direction, the contact law changes to g(p, px ) ≈ g(− n z, − nx z) = g(z) = 0. The rigorous application of this approximation also affects the problem-specific function F (p, px ) ≈ F (− n z, − nx z) = F (z) and the cost functional in Eq. (7.6), yielding

42

L. Koller and W. Witteveen

# J (z) =

(7.9)

F (z)d. 

By applying an optimization strategy (e.g., Quadratic Programming (QP) [15]), a vector z can be determined that lets J = J (z) become stationary. Thus, a solution for the contact problem, which is posed under consideration of the Euler– Lagrange equation, is provided. Since the focus of the HR method is on dynamic simulations with FE structures, a discrete representation is used instead of a continuous one: J (z) ≈

c 

Am Fm (z) =

m=1

1

z Pz + z k 2

(7.10)

Accordingly, the integral over  in Eq. (7.9) is replaced by a discrete sum that considers all c nodes in the contact region, including the scalar value Am to describe the area around the mth node. The problem-specifying variables P and k are computed based on Fm (z) = F (xm , z). Due to the characteristics of the QP and the surface load reconstruction via stress modes, the symmetric (r × r) matrix P is defined as P=

c 

Am (−Ψ nx (xm )) (−Ψ nx (xm ))FP (xm ) =

m=1

c 

ˆ m )FP (xm ), Am P(x

(7.11)

m=1

where the (1 × r) vector Ψ nx (xm ) describes the content of all r stress mode derivatives at one discrete node xm . These ˆ m ) = (−Ψ nx (xm )) (−Ψ nx (xm )). The structure vectors are required to form the symmetric and invariant (r × r) matrix P(x of Eqs. (7.10) and (7.11) presumes that solely the terms with a quadratic dependency on z contribute to FP (xm ). In contrast, the terms with a linear dependency on z are grouped in the scalar expression Fk (xm ), which is used to compute the (r × 1) vector

k=

c 

Am (−Ψ n (xm )) Fk (xm ).

(7.12)

m=1

In this case, the (1 × r) vector Ψ n (xm ) contains the mode information at the mth node. As a consequence of this procedure, the dimension of P and k correlates with the number of stress modes used. As typically r  c applies, the number of equations to calculate the pressure is reduced as a result of the surface load reconstruction. Although P and k can already be used to calculate z, all c nodes in the contact region are needed for the construction of the dimensionally downsized mathematical expressions. According to [8], this problem represents a Semihyper-Reduction (SHR) and is also addressed under this term within this work. However, instead of a description in the full physical contact domain with g(z) = 0, the HR approach yields a reduced formulation G(z) = 0, where g(z) ≈ G(z) = 0 should be valid to obtain reasonable results. To meet this requirement, the application of a reduced numerical integration scheme is proposed in [8]. More precisely, the ECM [11] is suggested to create P and k with a smaller number of integration points and weights. The application of this idea onto k results in kG =

c 

Am (−Ψ n (xm )) Fk (xm ) ≈

m=1

ck 

ωk,g (−Ψ n (x k,g )) Fk (x k,g ),

(7.13)

g=1 c

c

k k with the ck integration points ξ k = {x k,g }g=1 and the associated integration weights ωk = {ωk,g }g=1 , which are gathered in the set K = {ξ k , ωk }. Due to the symmetry of P, some adaptions are made in [8], which are intended to maintain the special structure of this matrix. Therefore, only the upper triangular matrix of P is approximated. This in turn affects the underlying ˆ which is why the terms matrix P,

⎡

 P = P1

P11 . . . ⎢ .. ... Pr = ⎣ . 0

⎤ ⎡ ⎤ P1r Pˆ11 (x) . . . Pˆ1r (x)   ⎢ .. ⎥ .. ⎥ , P(x) .. ˆ = Pˆ 1 (x) . . . Pˆ r (x) = ⎣ . . ⎦ . ⎦ Prr 0 Pˆrr (x)

(7.14)

7 Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads. . .

43

are introduced. Based on these modifications, the integration points and weights are separately determined for each (u × 1) column vector P u , u = 1, . . . , r of P, yielding the following statement for the reduced numerical integration scheme: P G,u =

c  m=1

Am Pˆ u (xm )FP (xm ) ≈

cPu 

u ˆ ωP,g P u (x uP,g )FP (x uP,g )

(7.15)

g=1 c

Pu and the integration weights ωuP = Accordingly, for every uth vector P G,u , the cPu integration points ξ uP = {x uP,g }g=1 u }cPu are computed. These are assembled in P u = {ξ u , ωu }, resulting in the total set P = {P 1 , . . . , P r } for all r {ωP,g P g=1 P columns. Finally, PG ≈ P is obtained by mirroring the upper triangular matrix of PG around the main diagonal. Since the ECM algorithm requires training data for the calculation of the integration nodes and weights [11], snapshots are collected during a preceding simulation. If, for instance, a posteriori stress modes are considered, the states and pressures of the same training-simulation can be used for both tasks. Further details on the ECM itself can be found in [11]. Moreover, reference is made to [8], where the determination of P and K or PG and k G is thoroughly explained.

7.4 Application in the Field of Elastohydrodynamics The lubrication film is modeled by means of the Reynolds equation [16], which is evaluated at the discrete points {xm }cm=1 within the elastohydrodynamic (EHD) contact region. Since a planar crank drive is used for the numerical investigations, the dimension in y-direction is omitted. As a consequence of this specification, the problem-defining equation g(p, px ) = −6ηU hx + 12ηh˙ −

∂ 3  h px = 0 ∂x

(7.16)

is obtained, with x as 1D coordinate, t for the time and p for the pressure. The variable h denotes the gap height, with its derivatives hx = ∂h/∂x and h˙ = ∂h/∂t. In addition, the dynamic viscosity of the lubricant is given by η and U is the relative velocity between the involved parts. It is worth mentioning that the numerical analyses in this contribution are performed with a dimensionless form of Eq. (7.16). To keep the notation simple, only the dimensional version is outlined and reference is made to [8] for more detailed information about the nondimensionalization procedure. To become a dimensionally reduced representation via utilizing the characteristics of stress modes and the calculus of variations, the problem-specifying function F is identified according to the condition in Eq. (7.8). Applying this idea onto Eq. (7.16) yields        1 1 F = −6ηU hx + 12ηh˙ p + h3 px2 ≈ −6ηU hx + 12ηh˙ −Ψ n z + h3 z (Ψ nx ) Ψ nx z. 2 2

(7.17)

The latter equation implies that g(p, px ) ≈ g(z) = 0 holds true, and therefore, the reliance on the nodal pressure values p and px is replaced by a description with the modal scaling factors in z. The terms with a linear dependency on z are summarized in ˙ m) Fk (xm ) = −6ηU (xm )hx (xm ) + 12ηh(x

(7.18)

and those which are quadratically dependent on z are collected in FP (xm ) = h(xm )3 ,

(7.19)

where the function argument xm is written out to emphasize the nodal discretization. In combination with the normal stress modes in n , the latter two equations yield a complete definition of the problem-specific terms P and k. In addition to this requirement, an adequate solution strategy is needed to take into account the boundary conditions for solving the Reynolds equation. On the one hand, the Gümbel boundary condition (GBC) [17] is utilized to avoid unphysical negative pressures. On the other hand, zero-pressures are prescribed at the boundaries of the contact region. Moreover, the properties of the a priori and a posteriori normal stress modes must also be considered when selecting a strategy. A possible approach to satisfy the aforementioned constraints in terms of a priori stress modes is given by the method of Lagrange multipliers, which allows a solution of Eq. (7.10) by means of

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L. Koller and W. Witteveen



P˜ −C

e −Ce 0

z −k˜ = . λ 0

(7.20)

Since the approximations PG ≈ P and k G ≈ k must be valid to obtain meaningful results for the HR description, the terms P˜ ∈ {P, PG } and k˜ ∈ {k, k G } are introduced for a more general formulation. The position of the zero-pressure boundaries is specified via the (e × r) matrix Ce , holding the stress mode information at the e boundary nodes in its rows. As the a posteriori stress modes automatically inherit the zero-pressure boundaries from the pressure snapshots, a solution of Eq. (7.10) can be determined via ˜ ˜ = −k. Pz

(7.21)

By evaluating either Eq. (7.20) or Eq. (7.21), the vector z is obtained and can then be used to reconstruct the nodal pressures with p = − n z. If necessary, at this point, negative values in the pressure distribution are set to zero because of the GBC.

7.5 Numerical Investigation of the Different Stress Modes For the numerical investigation of the HR approach in combination with the different stress mode types, the planar crank drive system from [8] is used, see Fig. 7.1a. In this example, only the piston is considered as a flexible structure, where the corresponding FE model is depicted in Fig. 7.1b. The contact region between the piston and the rigid cylinder liner consists of c = 802 nodes, which are evenly distributed over the top and bottom piston skirts. The orthonormalized deformation mode base  is built upon s = 1 static-, v = 10 vibrational- and l = 25 local mode shapes. In this case, the number of local modes is determined by means of a convergence analysis, see [8] for further details. Moreover, in this work, the dynamic viscosity of the lubricant is defined as η = 2.5e–7 MPa · s and a constant rotational speed of nC = 3100 rpm is set for the motion of the crankshaft. Since in the present example, the number of relevant a priori normal stress modes corresponds with the number of local deformation modes, the relation r = l = 25 is chosen. In order to obtain comparable results, the same amount of a posteriori stress modes is extracted out of the collected pressure snapshots. Based on these types of stress modes, the nonlinear surface load is described in two different ways. Firstly, the SHR method, where all nodes in the contact area are employed to create P and k, is applied to analyze just the dimensionality reduction in the resulting system of equations. Secondly, the proposed HR procedure yields an additional decrease in the computational complexity, since PG and k G only rely on a subset of the involved nodal area. For this investigation, the number of required integration nodes and weights is set to ck = 35 for k G and a constant number of cPu = 45 is defined for all u = 1, . . . , r columns to calculate the approximation PG ≈ P. Based on these specifications and the snapshot data for the system’s states, the ECM greedy-algorithm (see [11]) returns suitable nodes and weights for the reduced numerical integration. However, to highlight the main differences between a priori and a posteriori stress modes, cycles with different loading cases are analyzed. For this purpose, the whole dynamic simulation is divided into different phases:

Fig. 7.1 Description of the numerical example: (a) Illustration of the planar crank drive mechanism [8] and (b) finite element representation of the 2D piston [8]

7 Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads. . .

45

1. Run-in process: The simulation is performed until fairly convergent system states are obtained. During this period of time, no gas force is applied to the piston head. 2. Cycle for training data collection: Depending on the method to describe the nonlinear surface load, after phase 1, a full cycle without gas force is used to gather snapshots of pressures and system states. It is to mention that the length of one cycle is specified as 720◦ crank angle (CA). Subsequently, the a posteriori stress modes and/or the sets P and K for the HR procedure can be computed. These are then immediately used in the following cycles. 3. Control cycle without gas force: Compared to the previous phases, the loads on the dynamic system are not modified, which is why more or less the same response is expected. 4. Control cycle with gas force: In a certain range of CA degrees, an additional gas force is applied on the piston head. This provokes a change in the states of the system, through which the properties of a priori and a posteriori stress modes can be analyzed in greater detail. In order to rate the quality of the results when using a computational procedure based on stress modes, a reference solution of the Reynolds equation is calculated with the established finite difference method (FDM). Here, the measures $ Rp,i =

p

i p i c

%&

d

l=1 p i p i

, Rp = % & d

i=1 p R,i p R,i

· 100%

(7.22)

are used to quantify the error between the reference pressure pR,i = pR (ti ) of the FDM and a pressure p S,i = pS (ti ), which relies on a computation with stress modes. These values are regarded to evaluate the absolute error pi = pS,i − pR,i at the ith time-step, whereby i = 1, . . . , d applies. Whilst the measure Rp,i holds at a particular point in time, the value for Rp describes a period. Note that c is the number of nodes in the contact area.

7.6 Results and Discussion To assess the results in phase 3 of the dynamic simulation, Fig. 7.2 illustrates the error measure Rp,i for the different methods. Instead of directly using the time, the error value is plotted against the associated CA degrees of the cycle under consideration. In the remainder of this section, the approaches with a priori stress modes are denoted as SHR-APRIO and HR-APRIO, whereas these with a posteriori ones are abbreviated as SHR-APOST and HR-APOST. Although the errors are generally quite small, a better result quality is obtained when a posteriori stress modes are used. This tendency can be traced back to the fact that the underlying pressure snapshots were gathered under similar conditions, since the load case in phase 3 is identical to phase 2. Therefore, it seems plausible that the a posteriori modes are more tailored to this type of system loading. The exemplary selected pressure distributions in Fig. 7.3a and b also reveal a good agreement between the FDM and the methods based on stress modes. According to the error in Fig. 7.2, the SHR and HR methods provide similar results, which

Fig. 7.2 Error measure over the CA degrees for phase 3

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Fig. 7.3 Exemplary pressure distributions in phase 3: (a) 216◦ CA and (b) 320◦ CA Table 7.1 Error measure Rp in phase 3 and 4 depending on the method and type of stress mode Error measure Rp (%) Phase 3 Phase 4

SHR-APRIO 1.032 1.300

SHR-APOST 0.517 1.796

HR-APRIO 1.032 1.301

HR-APOST 0.537 1.800

Fig. 7.4 Error measure over the CA degrees for phase 4

is why only the curves for the HR are plotted. The above-discussed results are underlined by a more compact evaluation in Table 7.1, where the first row contains the error measure Rp for each method. Here, the values for SHR-APOST and HR-APOST confirm the more problem-specific characteristic of the a posteriori stress modes. Nevertheless, all analyzed computational approaches showed a good agreement with the reference solution of the FDM. Compared to phase 3 and the snapshot collection in phase 2, a variation in the states of the system is introduced in phase 4. To illustrate the occurrence of the applied gas force, the dashed black line in Fig. 7.4 shows its normalized curve. The error values in this figure emphasize the more general attributes of the a priori stress modes. As can also be seen in Table 7.1, the Rp values for SHR-APRIO and HR-APRIO are not influenced as much as those for SHR-APOST and HR-APOST when the gas force is added. This trend in the error valuesindicates a shortcoming of a posteriori stress modes, which apparently become less accurate if states outside the underlying training data appear. However, the selected pressure distributions in Fig. 7.5a and b still correspond quite good with the reference solution, although small deviations are emerging for HR-APOST. Another noteworthy finding is the close similarity between the SHR and the HR, regardless of which type of stress mode is used. This implies that the reduced numerical integration via the ECM is not as sensitive to state changes as the a posteriori modes. Table 7.1 underlines this statement as the Rp values for both the SHR and the HR agree well for phase 3 and 4. Accordingly, the quality of the stress modes is crucial for the entire reduction procedure. In cases, where only small deviations in the loading of the system occur,

7 Hyper-Reduced Computation of Nonlinear and Distributed Surface Loads. . .

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Fig. 7.5 Exemplary pressure distributions in phase 4: (a) 216◦ CA and (b) 320◦ CA

the use of a posteriori stress modes might yield sufficiently accurate results. If greater variations of the states are evoked by the applied forces, the a priori stress modes are expected to provide a more generally valid subspace. However, presuming a suitable stress mode base, the proposed HR strategy allows an accurate computation of the given nonlinear surface load by means of a dimensionally reduced system of equations, which only requires a subset of the overall nodes in the contact region for its construction.

7.7 Conclusion This work deals with an HR approach for the reduced computation of nonlinear surface loads. The theoretical framework is built upon the calculus of variations, a reduced numerical integration scheme, and stress modes. The application of these concepts yields a mathematical problem, the dimension of which is equal to the number of stress modes used and, moreover, loses the dependence on the full physical information in the contact region. In this contribution, the characteristics of two different stress mode types are discussed and analyzed. A priori stress modes result from linear FE calculations with the flexible FE structure and therefore relate to its mechanical properties. This origin yields a generally valid space, which does not rely on the states or loading of the dynamic simulation. This property is emphasized by the numerical investigation, where the quality of the results does not change seriously when the underlying states are varied. A contrary behavior is observed for a posteriori stress modes. These are based on snapshots of pressure distributions that are gathered in a preceding simulation run. Here, a more significant increase in the error values is detectable, if the states of the system are noticeably altered compared to those of the training data. However, in cases where slighter changes in the states or the loading are expected, the a posteriori stress modes can provide a more problem-specific subspace for the modal description of the nonlinear surface load. Nevertheless, for the given planar crank drive mechanism, the use of a priori stress modes is recommendable, especially in cases where different load cases or rotational speeds are specified.

References 1. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968) 2. Noor, A.K.: Recent advances and applications of reduction methods. Appl. Mech. Rev. 47(5),125 (1994) 3. Qu, Z.-Q.: Model Order Reduction Techniques: With Applications in Finite Element Analysis. Springer: London (2004) 4. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010) 5. Negri, F., Manzoni, A., Amsallem, D.; Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015) 6. Farhat, C., Chapman, T., Avery, P.: Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int. J. Numer. Methods Eng. 102(5), 1077–1110 (2015)

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7. Rutzmoser, J.B., Rixen, D.J.: A lean and efficient snapshot generation technique for the hyper-reduction of nonlinear structural dynamics. Comput. Methods Appl. Mech. Eng. 325, 330–349 (2017) 8. Koller, L., Witteveen, W., Pichler, F., Fischer, P.: A general hyper-reduction strategy for finite element structures with nonlinear surface loads based on the calculus of variations and stress modes. Comput. Methods Appl. Mech. Eng. 379, 113744 (2021) 9. Koller, L., Witteveen, W., Pichler, F., Fischer, P.: Semihyper-reduction for finite element structures with nonlinear surface loads on the basis of stress modes. J. Comput. Nonlinear Dyn. 15(8), 081004 (2020) 10. Volkwein, S.: Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Available at: https://www.math.uni-konstanz.de/ numerik/personen/volkwein/teaching/POD-Book.pdf (2013). Accessed 13 Oct 2021 11. Hernández, J.A., Caicedo, M.A., Ferrer, A.: Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Comput. Methods Appl. Mech. Eng. 313, 687–722 (2017) 12. Fischer, P., Witteveen, W., Schabasser, M.: Integrated MBS-FE-durability analysis of truck frame components by modal stresses. In: Adams User Meeting Rome 2000. (2000) 13. Tobias, C., Eberhard, P.: Stress recovery with Krylov-subspaces in reduced elastic multibody systems. Multibody Syst. Dyn. 25(4), 377–393 (2011) 14. Lanczos, C.: The Variational Principles of Mechanics. Dover Books on Physics. Dover Publications: Newburyport (2012) 15. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer Science+Business Media LLC: New York (2006) 16. Hamrock, B.J. , Schmid, S.R., Jacobson, B.O.: Fundamentals of fluid film lubrication. Mechanical Engineering, vol. 169, 2nd edn. Dekker: New York (2004) 17. Bartel, D.: Simulation von Tribosystemen: Grundlagen und Anwendungen. Vieweg+Teubner | GWV Fachverlage GmbH: Wiesbaden (2010) ISBN: 978-3-8348-9656-8. https://doi.org/10.1007/978-3-8348-9656-8

Chapter 8

Nonlinear Modelling of an F16 Benchmark Measurement Péter Zoltán Csurcsia, Jan Decuyper, Balázs Renczes, and Tim De Troyer

Abstract Engineers and scientists want mathematical models of the observed system for understanding, design, and control. Many mechanical and civil structures are nonlinear. This paper illustrates a combined nonparametric and parametric system identification framework for modelling a nonlinear vibrating structure. First step of the process is the analysis: measurements are (semiautomatically) preprocessed, and a nonparametric best linear approximation (BLA) method is applied. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Second, based on the information obtained from the BLA process, a linear parametric (state-space) model is built. Third, the parametric model is used to initialize a complex polynomial nonlinear state-space (PNLSS) model. The nonlinear part of a PNLSS model is manifested as a combination of high-dimensional multivariate polynomials. The last step in the proposed approach is the decoupling: transforming multivariate polynomials into a simplified, alternative basis, thereby significantly reducing the number of parameters. In this work a novel filtered canonical polyadic decomposition (CPD) is used. The proposed methodology is illustrated on, but of course not limited to, a ground vibration testing measurement of an F16 aircraft. Keywords MIMO systems · Nonlinearity · Decoupling · System identification · Ground vibration testing

8.1 Introduction This paper illustrates a combined nonparametric and parametric system identification framework for modelling nonlinear vibrating structures. The proposed methodology is illustrated on – but not limited to – a ground vibration testing (GVT) measurement of an F16 aircraft. Many mechanical and civil structures are inherently nonlinear. The problem lies in the fact that there are many different types of nonlinear systems; each of them behaves differently; therefore modelling is very involved, and universally usable design and modelling tools are not available. For these reasons the nonlinear systems are often approximated with linear systems, because its theory is user friendly and well understood. For an overview of the nonlinear modelling techniques, we refer to [1–3]. The first step is the data analysis: measurements are (semi-automatically) preprocessed, and a nonparametric best linear approximation (BLA) method is performed. The nonparametric best linear approximation (BLA) framework will be used as a first step [4]. The outcome of the BLA analysis results in a nonparametric frequency response function together with noise and nonlinear distortion estimates. The considered nonlinear model is a polynomial nonlinear state-space (PNLSS) model which consists of the classical linear state-space part (initialized by BLA FRF) and a nonlinear extension. The PNLSS model structure is flexible, as it can capture many different nonlinear dynamic behaviors. However, it suffers from the issue that even for a moderate complexity problem, there are an excessive number of parameters needed. To overcome this issue, a filtered canonical polyadic decomposition-based decoupling is applied: transforming multivariate polynomials into a simplified representation, thereby reducing the number of parameters.

P. Z. Csurcsia () · J. Decuyper · T. De Troyer Department of Engineering Technology (INDI), Vrije Universiteit Brussel (VUB), Elsene, Belgium e-mail: [email protected] B. Renczes Department of Measurement and Information Systems (MIT), Budapest University of Technology and Economics (BME), Budapest, Hungary © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_8

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In this work multisines are considered as excitation signals. The advantage of the multisines is that (1) there is no problem with spectral leakage/transient, (2) they result in high-quality frequency response functions (FRFs), and (3) they provide easy-to-understand information about the nonlinearities. The numeric results of the nonparametric (and partly the parametric) work are obtained by the use of the SAMI (simplified analysis for multiple input systems) toolbox [5]. The initial PNLSS models are obtained by the freely available PNLSS toolbox [6]. In this work, however, we focus on high-level understanding, instead of the usage of the toolbox or elaborating formulas. This paper is organized as follows. Section 8.2 briefly describes the considered systems and the main assumptions applied in this work. Section 8.3 discusses the nonparametric BLA estimation framework. The parametric polynomial nonlinear state-space model is elaborated in Sect. 8.4. In Sect. 8.5 the description and analysis of the GVT experiments of an aircraft are given. Conclusions can be found in Sect. 8.6.

8.2 Basics The dynamics of a linear multiple input, multiple output (MIMO) system can be nonparametrically characterized in the frequency domain by its frequency response matrix (FRM) [7] G at discrete frequency index k, which relates the inputs U to outputs Y as follows: Y (k) = G(k)U (k)

(8.1)

In this work are BIBO (bounded-input, bounded-output) stable physical systems [8]. For the sake of simplicity, the frequency indices will be omitted, and it is assumed to understand each quantity at frequency index k. This system represented by G is linear when the superposition principle is satisfied in steady state, i.e.: Y = G (a + b) U = GaU + GbU = (a + b) GU

(8.2)

where a and b are scalar values. If G does not vary, for any a, b (and excitation), then the system is called linear time invariant (LTI). On the other hand, when G varies with a and b (and the variation depends also on the excitation signal – e.g., level of excitation, distribution, etc.), then the system is called nonlinear. Because time-varying systems are often misinterpreted as nonlinear systems, it is important to mention that when G varies over the measurement time, but at each time instant the principle of superposition is satisfied, then the system is called linear time-varying (LTV) [9–11]. In this work we consider nonlinear time-invariant stable (damped) mechanical (vibrating) systems, and the output of the underlying system has the same period as the excitation signal (i.e., the system has PISPO behavior: period in, same period out [12]). Further, it is assumed that the excitation signal is known (measured precisely); the actuator of the system is linear.

8.3 Design of Experiment 8.3.1 Multisine Excitation and Detection of Nonlinearities In modern system identification, special excitation signals are available to assess the underlying systems in a user-friendly, time-efficient way [13]. To avoid any spectrum leakage, to reach full nonparametric characterization of the noise, and to be able to detect nonlinearities, a periodic signal is needed. The best signal that satisfies the desired properties is the user-friendly multisine signal (see Fig. 8.1) which looks like Gaussian white noise, behaves like it, but is not noise. The random-phase (uniformly distributed) multisine is a sum of harmonically related sinusoids. The amplitude distribution of a random-phase multisine is approximately normal (it approaches a Gaussian distribution as the number of harmonics tend to infinity). Note that this signal is also known as pseudo-random (multisine) signal [12].

8 Nonlinear Modelling of an F16 Benchmark Measurement

Gaussian noise

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periodic noise

time

random multisine

time

frequency

frequency

time

frequency

Fig. 8.1 Different excitation signals in time and in frequency domain

8.4 Best Linear Approximation 8.4.1 Theoretical Structure and the Basic Assumptions The best linear approximation (BLA) has been widely used in the last decades to efficiently estimate FRFs [13]. The BLA of a nonlinear system is an approach of modelling that minimizes the mean square error between the true output of a nonlinear system and the output of the linear model. The proposed BLA technique makes use of the knowledge that the excitation signal has both stochastic and deterministic properties. In this work the excitation signal is a random-phase multisine signal, and it is assumed to be measured precisely. In each measurement there are m different random realizations of multisines, and each of the realizations is repeated p times. The BLA estimate consists of several components. Figure 8.2 shows the theoretical structure of the considered BLA estimator. GLinear is the linear (transfer function) component of BLA. This component is phase coherent: harmonic-wise randomphase rotation in the input excitation would result in a proportional phase rotation at the output. In case of noncoherent behavior, the input-phase rotation would result in a random-phase rotation at the output. Please note that significant part of the nonlinearities is noncoherent. When many input-phase rotations are performed, the random output rotations can be seen as an additional (nonlinear) noise source (GS ) next to the ordinary measurement noise (GE ) – assumed to be additive i.i.d. normal distributed with zero mean with a finite variance. The unmodelled nonlinearities results in a bias term (Gbias ). The usage of periodic excitation reduces the effects of the measurement noise GE . The usage of multiple random-phase realizations reduces the level of noncoherent nonlinearities.

8.4.2 Two-Dimensional Averaging When the BLA estimation framework is applied, the observed system is excited by random-phase multisines. In this work there are m different realizations of the multisine excitation signal; each realization is repeated p period times. The considered steady-state model in frequency domain at frequency bin index k is given by: [m][p] [m][p] −1 ˆ BLA + G ˆ S[m] + G ˆE ˆ [m][p] = Yˆ measured U[m] = G G

(8.3)

The steady-state signals are obtained in this work by discarding some periods at the beginning of each realization block. In order to estimate the underlying system, one has to average over p periods of repeated excitation signal and over the m different realizations of the excitation signal [12] (see Fig. 8.3).

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Fig. 8.2 The theoretical structure of the best linear approximation

Fig. 8.3 Evaluation of BLA estimate with the help of 2D averaging

First, let us average over the p periods of a realization. If p is sufficiently large, then (considering the law of large numbers [m][p] ˆE converges to zero, so that this term and the distribution properties of the observation noise) the expected value of G is eliminated. In other words, averaging over repeated blocks results in an improvement of the SNR. Because the stochastic ˆ S[m] do not vary over the repetition of the same realization, we have to average over the m different nonlinear contributions G ˆ S[m] “half-stochastic” nonlinear noise source converges to zero, so that term is realizations. If m is sufficiently large, then G eliminated. After the 2D averaging, the BLA estimate is obtained. The estimate of the noise sample variance σˆ 2GE is calculated from the averaged sample variance of each realization. The ˆ [m] . total variance of the FRM σˆ 2 is calculated from the sample variance of each different partial BLA estimates G ˆ BLA G

The difference between the total variance and the noise variance is an estimate of the variance of the stochastic nonlinear 

− σˆ 2GE . contributions σˆ 2N L ≈ σˆ 2Gˆ BLA Using the proposed 2D averaging technique, the influence of the noise and nonlinear contribution can be decreased, and the final result is the BLA FRF estimate. In [7] it has been suggested to choose the number of periods more than one (p ≥ 2) and the different realizations more than six (m ≥ 7). Detailed calculations of the proposed 2D approach can be found in [14].

8.5 Polynomial Nonlinear State-Space Model A polynomial nonlinear state-space (PNLSS) model consists of the classical linear state-space part and the nonlinear extension part where auto and cross terms of the input and states are considered; see Fig. 8.4.

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Fig. 8.4 The theoretical structure of PNLSS model

It is based on the work of [15, 16]. It estimates and simulates polynomial PNLSS models from measured data. The PNLSS model structure is flexible, as it can capture many different nonlinear dynamic behaviors (hysteresis, nonlinear feedback, etc.). The model structure can also easily deal with multiple inputs. The method has already been successfully applied in a large range of applications (mechanical, electronical, electrochemical). Because of the state-space representation, it is suitable for control and simulation. The main concern is that the PNLSS is extremely sensitive to unseen input distribution, and it might produce unstable simulation output. The foundation of these issues can be seen when looking at the lower level: the kernel of these models is a high-dimensional multivariate polynomial nonlinear function. A possible solution to this problem is discussed next.

8.6 Decoupling Decoupling aims at transforming generic multivariate nonlinear functions into decoupled functions. The decoupled structure is characterized by the fact that the relationship is described by a number of univariate functions of intermediate variables. Decoupling is designed to post-process multivariate nonlinearities which emerge naturally in a large number of dynamical models. The objective is to achieve model reduction while gaining insight into the nonlinear mapping [17]. Given a generic nonlinear function: q = f (p)

(8.4)

with q ∈ Rn and p ∈ Rm , the idea is to introduce an appropriate linear transformation of p, denoted V, such that in this alternative basis, univariate functions may be used to describe the nonlinear mapping. The rationale behind the method is that classical regression tools, e.g., a polynomial basis expansion, not necessarily result in a sparse representation. By allowing a rotation toward a more favorable basis, a more efficient representation can be obtained. The decoupled function is then of the following form: 

f (p) = Wg V T p

(8.5)

where the ith function is gi (zi ) with zi = viT p, emphasising that all functions are strictly univariate. The number of univariate functions, denoted r, is a user choice which can be used to control the model complexity (r may be larger or smaller than n). It plays a crucial role since it will determine whether the implied equivalence of (5) can be attained. A second linear transformation W maps the function back onto the outputs. The matrices then have the following dimensions: V ∈ Rm x r and W ∈ Rn x r . The decoupled structure is represented graphically in Fig. 8.5. Decoupled functions have a number of attractive features. The fact that the nonlinearity is captured by a set of univariate functions enables to easily visualize the relationship. This may lead to insight. Moreover, decoupled functions are often a much more efficient parameterization of the nonlinearity, resulting in a significant reduction in the number of parameters. In this work, the filtered CPD approach of [18] is used to decouple the multivariate polynomial present in the state equation, Eζ (x, u) in Fig. 8.3. The algorithm links the original function to a decoupled function on the basis of its first-order derivative information. The method relies on the underlying diagonal structure of the Jacobian, which is a consequence of using univariate functions gi . Applying the chain rule, one obtains the Jacobian of (5): J  = W diag ([h1 (z1 ) . . . hr (zr )]) V T

(8.6)

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Decouple

f(p)

Fig. 8.5 Illustration of the decoupling technique

J0

=

=

Fig. 8.6 Illustration of the decoupling technique. Center: a collection of evaluations of the Jacobian of the decoupled function, stacked in the third dimension. Left: corresponding third-order tensor. Right: extracting the diagonal plane reveals a diagonal tensor decomposition

[19] found that the underlying diagonality of the Jacobian can be exploited in the decoupling process. It was suggested to construct a third-order tensor, J , out of evaluations of the Jacobian of the known function, f(p), and compute a diagonal decomposition, denoted J  , such that J ≈ J  . The tensor decomposition is depicted in Fig. 8.6. The decomposition returns three matrix factors: both the required linear transformation matrices W and V, together with a third matrix H which stores nonparametric estimates of the first-order derivative of the functions gi . In [18], the decomposition was modified by introducing finite difference filters. This allows for the Jacobian tensor to be   decomposed into the more convenient factors {W, V, H }, where H directly stores evaluations of the univariate functions gi . The method of [18] can be summarized in three steps: 1. Evaluate the Jacobian of the known function, J, in a number of operating points and stack the matrices into a three-way array, i.e., the Jacobian tensor J ∈ Rn×m×N .  2. Factor J into {W, V, H } by computing a filtered diagonal tensor decomposition (F-CPD).  3. Retrieve the functions, gi , by parametrizing the nonparametric estimates stored in H . The filtered CPD approach is a generic tool which can be used to retrieve decoupled functions, regardless of the function family. Irrespective of the size of the function in terms of m and n, the decoupling procedure boils down to solving a thirdorder tensor decomposition. Only in a final parameterization step, an appropriate basis function is selected for the univariate branches. An additional advantage of the filtered CPD is that the method no longer relies on the uniqueness properties of tensor decomposition. The result is that meaningful decompositions, pointing toward decoupled functions, can be obtained for a user chosen value of r, enabling to control the model complexity. The results illustrate that the nonlinear functions found in PNLSS models may typically be replaced by decoupled functions with a low number of univariate branches. This alludes to the fact that nonlinear dynamical systems are in many cases driven by a low number of internal nonlinearities.

8.7 Experimental Illustration 8.7.1 F-16 Measurement This section concerns the ground vibration testing measurement campaign of a decommissioned F-16 aircraft with two dummy payloads mounted at the wing tips; see Fig. 8.8. The detailed description of the measurement and benchmark data are openly accessible [20]. The right wing is excited by a shaker using combined multisines: odd multisines with skipping one random bin within each group of four successively excited odd lines. This sparse grid is used to detect even and odd nonlinear contributions. The sampling frequency is 400 Hz, the period length 16,384 samples. The reference (voltage), input (force), and output

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Fig. 8.7 F-16 ground vibration testing measurement campaign. The right wing is excited by a shaker. Major part of the nonlinearities is related to the payload connection

Fig. 8.8 Periodicity at the payload connection. The left figure shows the block repetitions (accelerometer data). The right figure shows the differences between the last block minus every block in dB scale. Observe in the second figure the fast decay in the first few seconds (this is the transient). The grayed area refers to the automatically detected transient (delay) block which will be discarded during the processing of the measurement. Please note that the automated transient check involves all available signals

(acceleration) signals are measured. The range of excitation is between 1 and 60 Hz; there are 3 periods and 9 realizations per excitation level. There are 3 different input levels measured at 12.2, 49.0, and 97.1 N RMS. In the 1–15 Hz band, the aircraft possesses about 10 resonance modes. The first few modes below 5 Hz correspond to rigid body motions of the structure. The first flexible mode around 5.2 Hz corresponds to wing bending deformations. The mode involving the most substantial nonlinear distortions is the wing torsion mode located around 7.3 Hz: the mounting interface of the payload features nonlinearities in stiffness and damping, due to clearance and friction. Therefore, the models are estimated for the frequency range of interest (4.5 . . . 15 Hz) at the payload connection (Fig. 8.7).

8.7.2 Data Processing The data processing is fully automated by the toolbox: segmentation of data, trends (such as the mean/offset values) from the individual segments are removed [13]; the transient is analyzed. Figure 8.8 shows the visualization of toolbox transient checkup routine. The left side of the figure shows the acceleration (output) measurement at the payload connection. In order to determine the length of the transient (i.e., the number of delay blocks), the last block (period) – assumed to be nearly in steady state – is subtracted from every preceding block. Because the transient decays as an exponential function, the differences are shown in logarithmic scale [13].

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Fig. 8.9 The measured reference (voltage) and input (force) signals are shown. Darker shades refer to higher excitation levels. Thick gray shades refer to the signals. Thin gray shades refer to noise estimates. Orange shades (on signal measurements) refer to the odd distortions. Blue shades (on signal measurement) refer to the even distortions

Fig. 8.10 The output (acceleration) measurement shown at the payload connection. Darker shades refer to higher excitation levels. Thick gray shades refer to the signals. Thin gray shades refer to noise estimates. Orange shades (on signal measurements) refer to the odd distortions. Blue shades (on signal measurement) refer to the even distortions. Observe that the higher the excitation, the more the dominance of odd nonlinear distortions

8.7.3 Reference and Input Signals The reference and input signals are shown in Fig. 8.9, the low, medium, and high level of excitation. The left figure shows the generated reference (voltage) signals and their noise estimates. This measurement has very good quality (SNR is greater than 75 dB); the even and odd nonlinear distortions are hidden in the noise. The excitation forces (see right figure) are measured with approximately 50 . . . 80 dB SNR. It is interesting to point out that the highest input signal is 18 dB higher than the lowest one, but the SNR is decreased with around 8 dB. This indicates the presence of (weak) nonlinearities at the excitation system.

8.7.4 Payload Measurement Further, in order to simplify the analysis, the output and FRF are shown at the payload connection only. The output (acceleration) measurements are shown in Fig. 8.10. As can be seen, the SNR is around 40 . . . 50 dB at the resonances. At the higher excitation level, the SNR decreased with approximately 10 dB w.r.t the lowest-level excitation.

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Fig. 8.11 The FRF estimation shown at the payload-right wing connection. Darker shades refer to higher excitation levels. Thick gray shades refer to the FRFs. Thin gray shades refer to noise estimates. Red shades refer to the nonlinear distortions

It can also be observed that the resonances have been shifted, which is also a further indication of nonlinearities. This is due to the fact that at higher level of excitation we have dominant odd distortions, which usually manifest in changing resonance locations and shapes (it is the so-called hardening or softening stiffness nonlinear effect). Please note that even distortions usually manifest as excessive noise on the measurement.

8.7.5 FRF Analysis Figure 8.11 shows the FRFs at low and high level of excitation. This is the classical approach, FRFs at multiple levels of excitation are compared with each other. It is interesting to point out that despite the fact that the high excitation is only 18 dB higher than the lowest-level excitation, it can be clearly observed that FRFs at different levels differ a lot from each other, for instance, shifting resonances and varying damping. This clearly indicates the presence of nonlinearities. The usage of the proposed multisines allows us to obtain noise and nonlinearity-level estimations as explained earlier. For instance, when looking at the high-level excitation case, the most dominant resonance (around 7.3 Hz) has an approximate SNR of 43 dB and an SNLR (signal-to-nonlinearity ratio) of 15 dB. This means that at that resonance the main error source is the nonlinearity.

8.7.6 Post-Processing The next step is the parametric post-processing of the data. In order to reduce the computational needs, nine realizations were taken from low and high level. In each realization the periods were averaged (and the resulting data are split into): • Estimation dataset: 4–4 realizations (to build a model). • Validation dataset: 4–4 realizations (to validate a model). • Test dataset: 1–1 realization (to compare different approaches).

8.7.7 Parametric BLA A parametric (state-space) model is built based on low-level nonparametric BLA (FRFs and noise) estimates. In order to determine the parametric model order (i.e., number of states), a cross-validated model order scanning method is used between

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Fig. 8.12 Parametric and nonparametric FRFs at driving point estimated at the low excitation level Table 8.1 Overview of the results Model Nonparametric BLA trained on low level Parametric BLA trained on low level Full PNLSS PNLSS with 2 branches – optimized

Relative RMS error on test data Low level High level 1.001 2.335 0.162 1.132 0.082 0.400 0.070 0.088

Number of parameters 427 169 169 + 6552 169 + 29

orders 1 and 15 in the frequency range of interest. The best fit (see Fig. 8.12.) w.r.t. cross-validation error was found with a model order of 12, resulting in a total of 169 parameters to be estimated (see Table 8.1).

8.7.8 PNLSS Model and Decoupling A PNLSS model is initialized with the help of the previously obtained parametric low-level BLA model. In order to reduce the complexity of the problem and the computational needs, only matrix E (state nonlinearities) is considered with secondand third-order multivariate cross terms resulting in more than 6000 parameters. The PNLSS model is obtained using an optimization routine with 40 iteration steps [15] trained on the 4–4 low- and high-level realizations of the estimation dataset. Applying the F-CPD method to the function of the nonlinear part of the PNLSS model Eζ (x, u) reduces the number of nonlinear parameters from 6552 to 29. Decoupling method has been applied to obtain one univariate branch based on the full PNLSS model. The resulting reduced-order model was post-optimized.

8.7.9 Discussion of Results The performance of the parametric and nonparametric BLA and PNLSS and decoupled models is detailed in Table 8.1. A model fitting on a high-level data segment is shown in Fig. 8.13. Observe that the worst results are obtained with the nonparametric BLA FRF model: the relative RMS error is always above 100% (at high level above 200%). The nonparametric model involved more than 400 parameters (FRF data points). The reduced-order BLA SS model with 169 parameters provides already acceptable fit for low level of excitation (around 16% error), whereas at high level of excitation the error level is still above 100%. This is expected since the linear models were estimated only at the low excitation level. The PNLSS model provides already a good fit (with maximum error of 40%) at the cost of increased number of parameters (more than 6000 parameters). The best results are obtained with the decoupling technique, with less than 200 parameters; the relative error is maximum 8.8%.

8 Nonlinear Modelling of an F16 Benchmark Measurement

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Fig. 8.13 Fitting illustration on a segment of the test data

8.8 Conclusions In this work a novel, semiautomated semi-parametric approach was developed to provide a user-friendly modelling tool for nonlinear benchmark data. The results turned out to be useful for modelling the nonlinear ground vibration testing of the aircraft because: • It required minimal user interaction, and no expert-user was needed. • The input, output, and transfer function measurements were nonparametrically characterized. • High-dimensional PNLSS models were successfully decoupled resulting in a very compact powerful model. Acknowledgments This work was funded by the Strategic Research Program SRP60 of the Vrije Universiteit Brussel and by the Flemish fund for scientific research FWO under license number G0068.18 N.

References 1. Kerschen, G., Worden, K., Vakakis, A., Golinval, J.-C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006) 2. Worden, K., Tomlinson, G.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. Institute of Physics Publishing, Bristol (2001) 3. Schoukens, J., Ljung, L.: Nonlinear system identification: a user-oriented road map. IEEE Control. Syst. Mag. 39(6), 28–99 (2019) 4. Csurcsia, P.Z., Peeters, B., Schoukens, J.: User-friendly nonlinear nonparametric estimation framework for vibro-acoustic industrial measurements with multiple inputs. Mech. Syst. Signal Process. 145 (2020) 5. Csurcsia, P.Z., Peeters, B., Schoukens, J., Troyer, T.D.: Simplified analysis for multiple input systems: a toolbox study illustrated on F-16 measurements. Vibration. 3(2), 70–84 (2020) 6. Schoukens, J.: Prof. Dr. Johan Schoukens Website. Vrije Universiteit Brussels (2018) [Online]. Available: http://homepages.vub.ac.be/ ~jschouk/. Accessed 1 Jan 2020 7. Pintelon, R., Schoukens, J.: System Identification: a Frequency Domain Approach, 2nd edn. Wiley-IEEE Press., ISBN: 978-0470640371, New Jersey (2012) 8. Ljung, L.: System identification: Theory for the User, 2nd edn. Prentice-Hall., ISBN: 9780136566953, New Jersey (1999) 9. Csurcsia, P.Z., Lataire, J.: Nonparametric estimation of time-variant systems using 2D regularization. IEEE Trans. Instrum. Meas. 65(5), 1259–1270 (2016) 10. Csurcsia, P.Z., Schoukens, J., Kollár, I.: Identification of time-varying systems using a two-dimensional B-spline algorithm. In: 2012 IEEE International Instrumentation and Measurement Technology Conference, Graz, Austria (2012) 11. Csurcsia, P.Z., Schoukens, J., Kollár, I.: A first study of using B-splines in nonparametric system identification. In: IEEE 8th International Symposium on Intelligent Signal Processing, Funchal, Portugal (2013) 12. Csurcsia, P.Z.: Static nonlinearity handling using best linear approximation: an introduction. Pollack Periodica. 8(1) (2013) 13. Schoukens, J., Pintelon, R., Rolain, Y.: Mastering System Identification in 100 Exercises. Wiley., ISBN: 978047093698, New Jersey (2012)

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14. Alvarez Blanco, M., Csurcsia, P.Z., Janssens, K., Peeters, B., Desmet, W.: Nonlinearity assessment of mimo electroacoustic systems on direct field environmental acoustic testing. In: International Conference on Noise and Vibration Engineering, Leuven (2018) 15. J. Paduart, Identification of nonlinear systems using Polynomial Nonlinear State Space models, Belgium: PhD thesis (2008) 16. J. Decuyper, Nonlinear state-space modelling of the kinematics of an oscillating circular cylinder in a fluid flow, Belgium: PhD thesis (2017) 17. Decuyper, J., Tiels, K., Schoukens, M.R.A.J.: Retrieving highly structured models starting from black-box nonlinear state-space models using polynomial decoupling. In: Mechanical Systems and Signal Processing, vol. 146, (2021) 18. Decuyper, J., Tiels, K., Weiland, S., Schoukens, J.: Decoupling Multivariate Functions Using a Non-parametric (Filtered-CPD) Approach. Padova, Italy (2021) 19. Dreesen, P., Ishteva, M., Schoukens, J.: Decoupling multivariate polynomials using first-order information. {SIAM} J. Matrix Anal. Appl. 36(2), 864–879 (2015) 20. Noël, J.P., Schoukens, M.: F-16 aircraft benchmark based on ground vibration test data. In: 2017 Workshop on Nonlinear System Identification Benchmarks, Brussels, Belgium (2017)

Chapter 9

Mathematical Model Identification of Self-Excited Systems Using Experimental Bifurcation Analysis Data K. H. Lee, D. Barton, and L. Renson

Abstract Self-excited vibrations can be found in many engineering applications such as flutter of aerofoils, stick-slip vibrations in drill strings, and wheel shimmy. These self-excited vibrations are generally unwanted since they can cause serious damage to the system. To avoid such phenomena, an accurate mathematical model of the system is crucial. Selfexcited systems are typically modelled as dynamical systems with Hopf bifurcations. The identification of such non-linear dynamical system from data is much more challenging compared to linear systems. In this research, we propose two different mathematical model identification methods for self-excited systems that use experimental bifurcation analysis data. The first method considers an empirical mathematical model whose coefficients are identified to fit the measured bifurcation diagram. The second approach considers a fundamental Hopf normal form model and learns a data-driven coordinate transformation mapping the normal form state-space to physical coordinates. The approaches developed are applied to bifurcation data collected on a two degree-of-freedom flutter rig and the two methods show promising results. The advantages and disadvantages of the methods are discussed. Keywords Self-excited systems · Hopf bifurcation · Hopf normal form · Parameter estimation · Data-driven model · Control-based continuation

9.1 Introduction Self-excited systems exhibit periodic responses without any oscillating input. Self-excited oscillations can have catastrophic consequences, such as a plane crash due to wing flutter. Therefore, it is often essential to accurately model and predict the response of self-excited systems. Self-excited systems are usually modelled using parameter-dependent differential equations that capture changes in the system’s states over a so-called bifurcation parameter. A typical scenario leading to oscillatory responses in a self-excited system is the Hopf bifurcation. A Hopf bifurcation is a critical point where the system’s equilibrium changes stability, and limit cycle oscillations (LCOs) are generated from the critical point. This paper will discuss two different mathematical modelling approaches to capture the LCOs of self-excited systems based on experimental bifurcation analysis data. The latter were collected using control-based continuation (CBC) as CBC allows to measure both stable and unstable LCOs of the system [1]. The two modelling methods are demonstrated on the experimental data collected on a fluttering aerofoil (Fig. 9.1). The first approach is based on a mechanistic model, i.e. a model constructed from physical principles. To determine the parameters of this model, we use centre manifold reduction and normal form theory to predict the bifurcation diagram of the model. Model parameters are then optimized to minimize the difference between model-predicted and experimentally measured bifurcation diagrams. The second approach is based on a phenomenological model and uses machine learning (ML) to establish a transformation from this simple model to the coordinates of the real system. We define the ML model as a prediction of observables made

K. H. Lee () · D. Barton Department of Engineering Mathematics, University of Bristol, Bristol, UK e-mail: [email protected]; [email protected] L. Renson Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_9

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Fig. 9.1 Flutter rig. (a) Schematic, (b) physical system in Bristol’s wind tunnel facility Fig. 9.2 Comparison between measured and computed (mechanistic model) amplitudes of the LCOs. Red circles correspond to unstable LCOs measured using CBC, blue circles are stable LCOs (also measured using CBC), and the blue line is the numerical continuation of the model

from the centre manifold. The reduced dynamics on the centre manifold captures the bifurcation structure of the data, and the observables are trained using neural networks to predict the time series accurately.

9.2 Modelling Using a Mechanistic Model The unsteady flutter model is the basis mechanistic model of our modelling approach. To estimate the parameters of this model, we use a two-stage identification approach where the parameters of the linearized model are identified first and then the parameters of the non-linear part are identified. We identify the linearization using the small amplitude free-decay response by minimizing the prediction error of the state-space model [2]. In the second stage, the non-linear part of the model is identified by parametrizing the amplitude of the LCOs using the centre manifold reduction and simplest normal form of the Hopf bifurcation [3] (see Fig. 9.2). Results show a very good agreement between measured and predicted LCOs, especially in the unstable region where the assumption of the mechanistic model is valid. For larger oscillation amplitudes, model predictions deteriorate. It is thought to result from the simplistic aerodynamic model considered.

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Fig. 9.3 Comparison of the bifurcation diagram of the ML model and the measured data

9.3 Modelling Using a Phenomenological Model: The Hopf Normal Form Topologically equivalent dynamical systems are transformable to each system using invertible coordinate transformation. We can use this mathematical framework while building the model. The idea is to use a modified Hopf normal form – subcritical Hopf normal form added with the quintic non-linear term – to capture the bifurcation structure of the experiment. The LCO in coordinates of normal form is a circle, and the two-dimensional observation vector transforms this circle to a measured closed curve. The mapping between the centre manifold and the observable is trained using a neural network (see Fig. 9.3). The oscillation speed is trained to minimize the prediction of time-series response using the differential equation solver equipped with a machine learning package [4]. The trained model can predict the bifurcation diagram and the time series of the LCOs.

9.4 Conclusion In this research, we show two different modelling approaches of a dynamical system with Hopf bifurcations. The first approach is using a mechanistic model and identifying the unknown parameters to match the bifurcation diagram. The second approach uses a Hopf normal form and ML techniques to train a mapping between the centre manifold and the observations. The first approach provides more physical insight than the second approach, while the second approach provides more modelling flexibility and hence accuracy since a mechanistic model is not required. Acknowledgement L.R. acknowledges the financial support of the Royal Academy of Engineering with the Research Fellowship #RF1516/15/11.

References 1. Barton, D.A.W., Mann, B.P., Burrow, S.G.: Control-based continuation for investigating nonlinear experiments. J. Vib. Control. 18(4), 509–520 (2012) 2. Ljung, L.: System Identification Toolbox: User’s Guide. MathWorks Incorporated, Natick (1995) 3. Yu, P., Leung, A.Y.T.: The simplest normal form of Hopf bifurcation. Nonlinearity. 16(1), 277 (2002) 4. Saeed, M.M.: Ordinary Differential Equation Neural Networks: Mathematics and Application using Diffeqflux. jl. (2019).

Chapter 10

Shape Optimisation for Friction Dampers with Stress Constraint E. Denimal, R. Chevalier, L. Renson, and L. Salles

Abstract Friction dampers are classically used in turbomachinery for bladed discs to control the levels of vibrations at resonance and limit the risk of fatigue failure. It consists of small metal components located under the platforms of the blades, which dissipate the vibratory energy through friction when a relative displacement between the blades and the damper appears. It is well known that the shape of such component has a strong influence on the damping properties and should be designed with a particular attention. With the arrival of additive manufacturing, new dedicated shapes for these dampers can be considered, determined with specific numerical methods as topological optimisation (TO). However, the presence of the contact nonlinearity challenges the use of traditional TO methods to minimise the vibration levels at resonance. In this work, the topology of the damper is parametrised with the moving morphable components (MMC) framework and optimised based on meta-modelling techniques: here kriging coupled with the efficient global optimisation (EGO) algorithm. The level of vibration at resonance is computed based on the harmonic balance method augmented with a constraint to aim directly for the resonant solution. It corresponds to the objective function to be minimised. Additionally, a mechanical constraint based on static stress analysis is also considered to propose reliable damper designs. Results demonstrate the efficiency of the method and show that damper geometries that meet the engineers’ requirements can be identified. Keywords Nonlinear vibrations · Topological optimisation · Kriging · Friction damping · Resonance mitigation

10.1 Introduction The design of aircraft turbines is of major importance in the aeronautic industry, as they are subjected to numerous loadings: thermal, vibrational, stress, etc. More particularly, a vibration analysis must be carried out to limit the phenomenon of high cycle fatigue that could lead to dramatic accidents. Due to the high modal density of turbines, all resonances cannot be avoided, and so the level of vibrations at resonance must be controlled. One solution consists of the introduction of damping in the system. For such applications, dry friction damping is now a classical solution. It can be introduced in different locations of the bladed disc, namely, at the tip of the blades, under the platforms of the blades, etc. For high pressure turbine, a classical solution is the introduction of underplatform dampers [1]. It consists of small metal pieces, introduced under the platforms of adjacent blades, that are maintained in position due to the centrifugal loading during operation. When the blades vibrate, a relative displacement appears between the latter and the damper. It generates friction and so energy is dissipated. Due to this dissipation, vibrations are damped. This friction contact is the origin of a nonlinearity in the system dynamics. It is established that the shape of the contact surface has a strong impact on the damping efficiency [2]. With the coming of additive manufacturing, the question of the interest of topological optimisation (TO) for friction damper is natural.

E. Denimal () · R. Chevalier Univ. Gustave Eiffel, Inria, COSYS/SII, I4S, Rennes, France e-mail: [email protected]; [email protected] L. Renson Dynamics group, Imperial College London, London, UK e-mail: [email protected] L. Salles Skolkovo Institute of Science and Technology, Moscow, Russia e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_10

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TO for continuum structures consists in determining the optimal distribution of the material over a given domain and for given boundary conditions [3]. The topology of a component is defined by the location of the limit between the interior and the exterior and the location of inner holes. These limits and these inner holes are optimised to minimise an objective function with respect to constraints. The first family of classical approaches to solve TO problems is the density-based methods. An initial mesh of the system is constructed and the material density of each mesh element is then optimised. The updating process in the optimisation is based on the sensitivity of the objective function to the density variation in each element. A second family of methods exists where the geometry is described implicitly with a level-set function (LSF) and then propagated by solving the Hamilton-Jacobi equation using the shape sensitivities of the LSF. Both methods require the sensitivities of the objective function to the element densities or to the shape; if not known analytically, they might be too expensive to compute numerically. A recent approach, called moving morphable components, proposes to parametrise the LSF with a few parameters [4], making possible to use more standard optimisation methods and more particularly global optimisation algorithms that are well adapted for expensive objective function and/or when the gradients are difficult to determine [5–7]. This method has already proven to be efficient to deal with the optimisation of friction dampers for nonlinear vibrations [6, 7]. Previous works were only focused in the minimisation of the vibration amplitude at resonance. However, numerous geometries presented very thin parts and so were not reliable. The interest of considering a static stress constraint in the optimisation to avoid such geometries is investigated in the present work. First, the system under study is briefly presented. Second, the moving morphable component framework is presented, as well as the optimisation process. The computation of the nonlinear response with the harmonic balance formulation augmented with a phase constraint is introduced, as well as the computation of the static stress constraint. Finally, results are analysed and the choice of the static stress constraint discussed.

10.2 System Under Study The mechanical system under study is a 2D system that simulates the dynamic behaviour of two blades of a high-pressure turbine [8]. It is composed of two blades represented by two beams with platforms and represented in Fig. 10.1. The blades are connected to a basis that represents the disc. The damper is located between the two blades, under the platforms. In realworld conditions, the damper is maintained in position due to the centrifugal loading. Close to resonance, when the blades vibrate, a relative displacement exists between the damper and the platforms, and so friction appears. It dissipates energy and so the vibrations are damped. In this study, the blades are excited at the base of blade 1 (see arrow in Fig. 10.1) with an amplitude of 8 N, and output displacements are observed at the top of blade 1 (green point in Fig. 10.1). A finite element model (FEM) for the blades is constructed in Abaqus; it is composed of 3324 8-node bi-quadratic plane strain elements. The structure is made of steel with a Young modulus of 197GPa and a density of 7800 kg/m3 . First bending modes of the blades can be in-phase (IP) or out-of-phase (OOP) and takes place at 246.73 Hz and 247.51 Hz, respectively. They are obtained without the damper and

Fig. 10.1 System under study (a) – in-phase (b) and out-of-phase (c) bending mode of the blades

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mode shapes are represented in Fig. 10.1. An initial FEM for the damper is constructed and is composed of 3604 elements with the same material than the blades. The meshes are constructed to ensure the matching at contact points. The damper mesh will be updated during the optimisation process, as explained later on. Contact surfaces are discretised and a node-to-node contact modelling approach is adopted [9]. For each contact element, the 2D contact model is composed of one Jenkins element and one normal spring to ensure normal load variations. Each element is characterised by four parameters: the friction coefficient μ, the normal contact stiffness kn , the tangential contact stiffness kt , and the initial normal preload N0 . Three contact states are possible, namely, separation, stuck condition and slip condition. For the rest of the study, the friction coefficient is taken equal to μ = 0.5, and it is assumed that the contact stiffnesses are equal in the normal and tangential directions and depend on N0 . The hypothesis is made that the initial contact pressure is homogeneous and depends directly on the centrifugal loading Cf and the number of contact point nc0 . As an illustration, for a full damper, 51 contact points are present on each contact line; the normal preload is N0 = 9.8987N and kn = kt = 20000 N/m.

10.3 Damper Parametrisation and Optimisation Process 10.3.1 Geometry Parametrisation The damper geometry is described implicitly with a level-set function (LSF) , defined on the design space D [3]. For a point of coordinates (x, y),  takes positive values if the point is in the material domain  and negative values if it is in the void domain and is equal to 0 on the boundary. The MMC framework proposes to decompose this global LSF in the assembly of nMMC several local LSF  i , i = 1, . . . , nMMC , defined ' explicitly based on a few parameters [4]. Each local LSF defines a material domain i and the final material domain is  = i i . More concretely, each local LSF  i is defined as:  i (x, y) = −

    m cos θi x − x0,i + sin θi y − y0,i Li 2

 +

    m − sin θi x − x0,i + cos θi y − y0,i ti 2

 −1

where (x, y) are the coordinates of a point, θ i the inclination of the ith component, Li its length, ti its thickness, and (x0, i , y0, i ) the position of its centre. m is an even number, equal to 6 here. The different parameters are illustrated in Fig. 10.2a for a component, and in Fig. 10.2b its LSF is represented (negative values have been set to 0 for the sake of readability). By assembling different components, expanding them, shrinking them and moving them, one can describe complex geometries. This approach has two advantages: the topology is parametrised and the number of parameters is low. This makes possible the use of more traditional optimisation approaches as gradient-free ones. The connectivity of the different local LSF is checked, and unconnected geometries are removed during the initialisation or penalised in the optimisation. The global LSF is then mapped on the damper mesh, which will be explained latter.

Fig. 10.2 MMC parameters (a) and corresponding LSF (b)

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10.3.2 Efficient Global Optimisation Algorithm The optimisation consists in the minimisation of the displacements at resonance at the blade tip, considering a constraint on the static stress distribution in the damper. The optimisation process is detailed in this section; details for the computation of the objective and constraint functions are given afterwards. Considering the non-convexity of the considered problem and the numerical difficulty to compute the gradients of the functions, a gradient-free optimisation approach is adopted, based on the EGO algorithm that exploits kriging meta-models of the objective and constraint functions coupled with an updating process [5–7, 10].  An initial set of Ninit input points P = p(1) , . . . , p(Ninit ) is generated; it corresponds to different damper geometries. For each of them, the objective function and the constraint function are evaluated: the vibration amplitude at resonance (denoted upeak ) and the maximum of stress (denoted σ max ) are computed. Based on these sets of inputs and outputs, two kriging meta-models are created: one for the objective function and another for the constraint function. Considering a given criterion, these kriging meta-models are exploited to find the new point to be added in the learning set. The objective function and the constraint are evaluated for this new geometry, and the learning set is extended with this new point. The iterative process is stopped after a maximum number of iterations chosen by the user. The choice of the criterion is important as it must balance exploration and accuracy in the area where the solution could be. A classic criterion is the expected improvement (EI) for the unconstraint problem, defined as [10]:

∼



⎛

EI [p] = upeak − upeak (p)  ⎝ (best)

(best)

∼

upeak − upeak (p) s (p)

⎞

⎛

⎠ + s (p) φ ⎝

∼

(best)

upeak − upeak (p) s (p)

⎞ ⎠

∼

(best)

where upeak is the best solution found so far, upeak (p) is the kriging meta-model prediction of the objective function at p at the current iteration, s(p) is the predicted standard deviation at p, (.) is the normal distribution of the normal law, and φ(.) is the normal law density function. The new point that should be added is the point that maximised the EI. For each configuration, the connectivity of the components is checked first. If it is unconnected, the EI is set to a negative value to avoid these configurations. In the case of a constraint optimisation problem, the EI is penalised to take into consideration the probability that the ∼ constraint will  alsobe satisfied. The kriging surrogate model at p of the constraint is denoted g and follows a normal law of ∼ ∼2

parameters μ, s

∼

. Yet

∼

g−μ ∼ s

∼ N (0, 1); thus the feasibility probability, denoted P(p), is [10]:

⎛ ∼ ∼  ⎞ # ∼ 

∼ g−μ μ 1 ⎝ ⎠ ≤ −∼ = √ P g (p) ≤ 0 = P ∼ 2π s s

∼ ∼

−μ/ s −∞

⎛

⎞ s exp ⎝− ⎠ ds 2 ∼2

The constrained expected improvement is then equal to: CEI (p) = EI [p] ∗ P (p) The new point to be added to the learning set is the point that maximises this criterion. It is solved by using a genetic algorithm coupled with a gradient evaluation of the CEI [11]. In this case, the best point taken in the computation of the EI is the point that minimises the objective function and that satisfies the constraint.

10.3.3 Nonlinear Analysis Because of the contact nonlinearity, the vibration amplitude at resonance is obtained from a nonlinear analysis. The equation of motion of the system is: M¨x(t) + C˙x(t) + Kx(t) + Fnl (x(t), x˙ (t)) = Fe (t)

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with M, C, and K the mass, damping, and stiffness matrices, respectively, x the vector of retained DOF, Fnl the vector of nonlinear forces, and Fe the vector of the excitation force. The mass and stiffness structural matrices are obtained from a Craig-Bampton reduction of the full system matrices. Retained DOF corresponds to the contact nodes, excitation DOF (for the platform), output node (for the platform), and a set of modes, so the size of the matrices will change as the number of contact nodes will change during the optimisation. The number of modes is kept constant for numerical reasons and high values used to ensure the accuracy of the reduced system for all cases (12 modes for the platform and 30 for the damper). A Rayleigh damping matrix of 0.2% is adopted for each component to ensure numerical convergence even when the friction damping is low. For the full damper case, the size of the problem is 454. Before the Craig-Bampton reduction, the LSF is projected onto the initial mesh of the damper. To avoid localised modes, void elements are removed. The problem is solved with the well-established harmonic balance method (HBM) [12], where the periodic solution x is (S) approximated by a truncated Fourier expansion of cosine coefficients x(C) k and sine coefficients xk with Nh harmonics and grouped in the vector q. The EOM in the Fourier domain that must be solved is: ∼

∼

J1 (q, ω) = Z (ω) q + Fnl (q) − Fe = 0 To avoid the computation of the full-frequency response function with a continuation algorithm, the resonance is directly obtained by adding a phase constraint to the previous equation [13]; it translates the idea that at resonance the phase φ between the excitation and the output is equal to π2 : J2 (q, ω) = φ −

π =0 2

The angular frequency becomes an unknown of the problem and one must find α = [q, ω] that satisfies [J1 (q, ω), J2 (q, ω)] = 0. It is solved with a trust region dogleg algorithm.

10.3.4 Static Stress Analysis Previous works have raised a concern about optimised geometries that have extremely thin parts and so not able to ensure the reliability of the damper in working conditions. For this reason, a constraint is added on the static stress distribution over the damper. The LSF must first be carefully projected on the initial mesh in order to avoid localised stress peak at the domain boundaries. For an element that is at the boundary, an intermediary density di is allocated defined as [3–5]: μi =

1 × Si

# H o  (x, y) dxdy (x,y)∈elemi

where Si is the surface of the element i, H the Heaviside function, and  the global LSF. Then, the Young modulus of the element is modified and equals Ei = (1 − μi )Eerstaz + μi E0 , where E0 is the Young modulus of the material and Eersatz the Young modulus of the ersatz material (taken equal to 10−3 E0 ). For void elements, the Young modulus is set to the ersatz Young modulus. Similarly, the density is also updated. The stress computation is done in Abaqus. The damper is subjected to different loadings, namely, the centrifugal loading (i) and the contact forces. The centrifugal loading of the element i is CF = mi di ω2 , with mi the mass of the element and di the distance to the rotation centre (taken as the distance to the centre of the base here). Contact forces are decomposed as a normal component N and a tangential one T. The von Mises stress constraint limit is taken equal to σ lim = 100MPa. It has proved to be a good threshold to remove geometries with thin parts or with sharp angles. As an illustration, example of geometries that do not satisfy the stress constraint (and so that are removed) is given in Fig. 10.3.

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Fig. 10.3 Example of damper geometries that do not satisfy the stress constraint (elements with density inferior to 0.5 are removed for the sake of readability)

10.4 Results 10.4.1 Optimisation Parameters In this work, five components are employed to describe the damper geometry. A few assumptions are made to take into consideration physical and engineering constraints but also to reduce the number of optimisation parameters. First, one component is set to a horizontal position at the top of the damper with a small thickness (about one element thickness) to seal the platforms. Second, the damper is assumed to be symmetric about its central axis. Third, to ensure the existence of the contact between the damper and the platform, the centre of one component must be on the contact line (i.e. vertical and horizontal positions of one component are dependent). With these different choices, the optimisation problem is of size 9. The objective is to minimise the objective function, which is the vibration amplitude at resonance obtained from the nonlinear analysis, under a static stress constraint. To improve the efficiency of the algorithm, the objective function is modified. Instead of optimising the vibration amplitude at resonance (denoted upeak ), the inverse of the opposite of the latter is optimised, i.e. fpeak = − 1/upeak . This is motivated by the idea that the vibration amplitudes are small and this transformation tends to spread the objective function values, and so it makes the optimisation more robust.

10.4.2 Optimisation Results As a first illustration of the results, the evolution of the objective function and of the constraint versus the iteration number as well as the objective function versus the maximum stress is given in Fig. 10.4. In Fig. 10.4a, blue points denote configurations for which the constraint is satisfied and red points configuration for which the constraint is not satisfied. The black dots are the best current minimum that satisfy the constraint. The blue diamond is the best configuration that satisfies the constraint and the green square the best configuration without constraint consideration. In Fig. 10.4c, red points denote points that are in the initial DoE and black points configurations obtained during the optimisation part. One can see that in the initial design, the objective function takes values between −1.5 mm−1 and 0 mm−1 and about half the geometries satisfy the constraint. As soon as the optimisation starts, the objective function takes value between −2 mm−1 and -1.5 mm−1 most of the time, which demonstrates the ability of the algorithm to identify quickly more efficient damper geometries. Optimal damper geometries that reduce the vibration amplitude at resonance but also verify the stress constraint are identified iteratively. At the end, the best geometry that satisfies the constraint gives a vibration amplitude at resonance of 0.55 mm and a maximal stress of 95.3 MPa. It is represented in Fig. 10.5a. The damper geometry that has the highest damping properties gives a vibration amplitude of 0.52 mm but has much higher level of stress (equal to 206.4 MPa). It is represented in Fig. 10.5b. Considering Fig. 10.4c where the objective function versus the maximum stress is given, one can see that geometries associated with the lowest level of vibrations at resonance also have higher level of stress and a Pareto front is observable. Without stress consideration, the optimal geometry tends to have a shape composed of two thin arms. With this shape, the damper mass is reduced, and the contact surface is increased which tends to maximise the number of contact points experiencing stick-slip. However, it also gives higher level of stress in the damper. By adding a stress constraint, the material is more located on the top of the damper. The global shape can be considered as similar (two thin arms) but with thicker arms that gives a better stress distribution in the damper. Different damper geometries are represented in Fig. 10.6. Each geometry is identified with a coloured square that refers to results given in Fig. 10.4. Geometries of the first line are geometries from the initial DoE. One can see they have various and

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Fig. 10.4 (a) Evolution of the objective function versus iteration number – (•) best current minimum, (• ) configuration that satisfies the constraint limit, (• ) configuration that doesn’t satisfy the constraint limit (b) Evolution of the constraint function versus iteration number – (- - - ) constraint limit (c) Pareto front between the objective function and the constraint function – (• ) initial DoE (♦ ) best configuration that satisfies the constraint (■ ) best configuration without constraint consideration () damper geometries represented in Fig. 10.6

Fig. 10.5 (a) Best geometry that satisfies the stress constraint and (b) best geometry that doesn’t satisfy the stress constraint – dashed lines represent the limit of the full damper, material domain in grey (elements with a low density are removed for the sake of readability)

unusual shapes. Some of them have thin parts (see the yellow one for example) or thin junctions between the left and right parts on the damper that are at the origin of stress concentration (see the orange one). Geometries on the second lines are the different geometries that correspond to the best current minimum satisfying the constraint function. If strong variations in terms of shape are observed at the beginning, the algorithm converges quickly to the final geometry as the two last geometries are very close to the final one. Finally, geometries on the last line are displayed to illustrate the large variety of geometries that are tested during the optimisation and how the compromise between damping properties and stress constraint might be difficult to meet. Indeed, the light blue and red geometries verify the stress constraint but have low damping efficiency. The blue one has low damping efficiency and presents high level of stress. Finally, the pink one is located near the constraint

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Fig. 10.6 Different damper geometries tested during the optimisation Table 10.1 General characteristics of the different damper geometries Full damper Best with constraint Best without constraint

Mass ratio 1 0.1095 0.1090

Resonance freq. (Hz) 403.57 378.71 374.74

Amplitude at resonance (mm) 1.63 0.55 0.52

Max. stress (MPa) 51.42 95.3 206.4

Table 10.2 Contact characteristics of the different damper geometries Full damper Best with constraint Best without constraint

N0 (N) 9.8987 2.2292 1.6428

Nb contact points 102 52 70

Nb points in stick-slip 5 38 52

Nb points in impact 1 14 18

Nb points in stuck 96 0 0

limit and so has satisfactory damping properties and acceptable stress levels. It demonstrates that the approach is able to go all over the design space but is also able to optimise finely the final geometry. To compare with the full damper geometry, the principal characteristics are summarised in the Table 10.1. Both optimal geometries represent a drastic reduction of the damper mass (about 90% mass reduction compared to the full damper). The optimal geometry that satisfies the stress constraint presents a frequency shift of about 25 Hz at resonance compared to the full damper case, and a shift of 29 Hz is observed for the other geometry. In terms of damping, both geometries present a substantial reduction of the vibration amplitude at resonance for the blade as it is reduced by 67% and 68%, respectively. So, both geometries have similar dynamic properties, and finally their main difference relies in the static stress. As the dynamic properties are strongly related to the contact point status (stick-slip, impact or separation), the contact information is summarised in the Table 10.2. Between the two geometries, the contact surface varies a lot (70 contact points for one case and 52 contact point for the other case), and so the initial contact normal loading is strongly impacted (see variations of N0). However, the ratio of contact points in stick-slip remains the same and is about 75%. Despite the difference in terms of geometry, the ratio of contact points that dissipate energy is similar, which explains the good damping properties of the optimal geometry despite its smaller contact surface and higher contact normal load (which delays the entering of stick-slip for contact points). In comparison, for the full damper, the number of points in stick-slip condition represents only 5% of the number of contact points, which explains the lower damping properties of the latter.

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10.5 Conclusion and Perspectives Constrained topological optimisation for nonlinear vibrations due to contact friction has been presented here. The general approach is based on the MMC framework and the EGO algorithm for the optimisation. Kriging surrogate models of the objective function and of the constraint function are constructed. The shape of an underplatform damper has been optimised to minimise the vibration amplitude at resonance with the consideration of a constraint on the static stress. By adding this constraint, non-reliable geometries with thin parts are eliminated during the optimisation. The final optimal geometry that satisfies the stress constraint has a mass distribution over the design space that is more localised on the top of the damper compared to the geometry that just minimises the nonlinear resonance. However, despite their shape difference and the difference in the contact surface, both geometries have a ratio of contact points in stick-slip that is equal and have no points in stuck condition. Future works will be dedicated to the extension to 3D and experimental validation. Acknowledgements E. Denimal and L. Salles have received funding from Rolls-Royce and the EPSRC under the Prosperity Partnership Grant CornerStone (EP/R004951/1). L. Renson has received funding from the Royal Academy of Engineering (RF1516/15/11). E. Denimal has received funding from Rennes Metropole. Rolls-Royce, the EPSRC, the Royal Academy of Engineering, and Rennes Metropole are gratefully acknowledged.

References 1. Krack, M., Salles, L., Thouverez, F.: Vibration prediction of bladed disks coupled by friction joints. Arch. Comput. Methods Eng. 24(3), 589–636 (2017) 2. Denimal, E., Wong, C., Salles, L., Pesaresi, L.: On the efficiency of a conical Underplatform damper for turbines. J. Eng. Gas Turbines Power. 143(2), 021020 (2021) 3. Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013) 4. Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J. Appl. Mech. 81(8) (2014) 5. Raponi, E., Bujny, M., Olhofer, M., Aulig, N., Boria, S., Duddeck, F.: Kriging-assisted topology optimization of crash structures. Comput. Methods Appl. Mech. Eng. 348, 730–752 (2019) 6. Denimal, E., El Haddad, F., Wong, C., Salles, L.: Topological optimization of under-platform dampers with moving morphable components and global optimization algorithm for nonlinear frequency response. J. Eng. Gas Turbines Power. 143(2), 021021 (2021) 7. Denimal, E., Renson, L., Salles, L.: Topological optimisation of friction dampers for nonlinear resonances mitigation. Nodycon. 2021 (2021) 8. Pesaresi, L., Salles, L., Jones, A., Green, J.S., Schwingshackl, C.W.: Modelling the nonlinear behaviour of an underplatform damper test rig for turbine applications. Mech. Syst. Signal Process. 85, 662–679 (2017) 9. Petrov, E.P., Ewins, D.J.: Advanced modelling of underplatform friction dampers for analysis of bladed disc vibration. Turbo Expo: Power for Land, Sea, and Air. 42401, 769–778 (2006) 10. Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998) 11. Mebane Jr., W.R., Sekhon, J.S.: Genetic optimization using derivatives: the rgenoud package for R. J. Stat. Softw. 42(1), 1–26 (2011) 12. Detroux, T., Renson, L., Masset, L., Kerschen, G.: The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Comput. Methods Appl. Mech. Eng. 296, 18–38 (2015) 13. Renson, L., Hill, T.L., Ehrhardt, D.A., Barton, D.A.W., Neild, S.A.: Force appropriation of nonlinear structures. Proc. Math. Phys. Eng. Sci. 474(2214), 20170880 (2018)

Chapter 11

Design of Flap-Nonlinear Energy Sinks for Post-Flutter Mitigation Using Data-Driven Forecasting Jesús García Pérez, Amin Ghadami, Leonardo Sanches, Guilhem Michon, and Bogdan Epureanu

Abstract Flutter instabilities might lead to structural failure in aeroelastic systems. Recently, a flap control surface turned into a nonlinear energy absorber (flap-NES) has been introduced to suppress post-flutter oscillations. This nonlinear absorber not only benefits from aerodynamic damping but also acts in a broad range of frequencies. However, the parameters of the NES must be chosen carefully to avoid adverse effects. The optimal design of the flap-NES demands a comprehensive study on the effect of its parameters on the flutter speed and post-flutter dynamics of the aeroelastic system where traditional approaches require considerable analytical and computational efforts. In this work, we study the optimization of a flapNES to control flutter instabilities of a typical pitch-plunge section. To facilitate efficient optimization and exploration of the design space, we employ genetic algorithm in conjunction with a bifurcation forecasting method, i.e., a data-driven nonlinear stability analysis algorithm developed to construct the bifurcation diagram using limited number of trajectories collected in the pre-flutter regime. Results show that an optimal design on the flap-NES system results in an increased critical flutter speed, reduced post-flutter limit cycle oscillation amplitudes, and improved nature of the bifurcation diagram from subcritical to supercritical. Keywords Flutter · Nonlinear energy absorber · Data-driven prediction · Nonlinear dynamics · Optimization

11.1 Introduction Increasing structural flexibility of wings allows more aerodynamic and weight-efficient airframes [1]. However, this might lead to aeroelastic instabilities, which aggravate the aircraft’s performance and, in the worst-case scenario, can cause the failure of the structure. Among these phenomena, flutter arises from the coupling between a fluid and a structure. This instability creates self-excited vibrations also referred to as limit cycle oscillations (LCO) that increase the fatigue of the structure and limit the flight envelope. To date, several solutions have been suggested to overcome and control these oscillations. These solutions can be classified in active and passive control strategies. Passive methods do not require an external source of energy, and several include nonlinear energy sinks [2]. Recent studies have shown that the energy of the main system can be irreversibly transferred to the NES and dissipated through damping. This phenomenon is referred to as targeted energy transfer (TET) or energy pumping [2]. This nonlinear elastic approach for the flap control surface was recently introduced to create a flap-NES that reduces the LCO amplitude and delays flutter speed. This innovative solution was studied experimentally by Escudero et al. [3]. However, choosing parameters to achieve optimal flutter mitigation requires a design optimization procedure. To achieve this goal, analyzing the bifurcation diagram for preselected parameters and optimizing their values are required. Recently, a novel data-driven technique, called the bifurcation forecasting

J. García Pérez Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Université de Toulouse, ICA, CNRS, ISAE-Supaéro, Toulouse, France e-mail: [email protected] A. Ghadami · B. Epureanu Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]; [email protected] L. Sanches · G. Michon Université de Toulouse, ICA, CNRS, ISAE-Supaéro, Toulouse, France e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_11

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Fig. 11.1 Schematic of 3-DOF airfoil with a flap-NES

Fig. 11.2 Forecasted bifurcation diagram of plunge amplitude for supercritical (left) and subcritical (right) with the exact diagram

method, has been introduced to construct the bifurcation diagram of nonlinear systems using a limited number of system measurements in the pre-bifurcation regime [4] and is used in this study as an alternative to the traditional approaches to speed up the optimization process. In this work, we present a foundational study of flap-NES-based flutter control of a typical pitch-plunge section, consisting of a 3-DOF system with nonlinear stiffness. The flap-NES parameters are optimized with a genetic algorithm for two objective functions, namely, the flutter speed and the post-flutter response magnitude. Nominal systems that exhibit supercritical or subcritical bifurcations without the flap-NES are considered. In both cases, optimization results show that optimal solutions provide an increased flutter speed and a reduced LCO amplitude, and they favor supercritical bifurcations to subcritical. The aeroelastic model of a 3-DOF airfoil with a flap-NES is shown in Fig. 11.1, where the three degrees of freedom are plunge h, pitch α, and control surface deflection β. The flap-NES parameters are ch , rβ , xβ (geometric), χβN (damping), and kβN (nonlinear stiffness).

11.2 Forecasting Bifurcations Identifying the post-flutter regime for nonlinear systems is challenging because it requires complex computations as well as accurate models that are hard to identify. Thus, alternatives to traditional methods are required. The bifurcation forecasting technique is a data-driven computationally efficient technique that identifies the flutter speed, flutter type, and the post-flutter limit cycle amplitudes using a limited number of measurements of the system response only in the pre-flutter regime. This approach significantly speeds up the optimization process and is based on the critical slowing down (CSD) phenomenon observed in nonlinear aeroelastic systems as they approach a Hopf (flutter) instability [5]. The closer the flutter boundary is, the longer it takes for the system to recover its initial equilibrium state after a perturbation occurs in the pre-flutter regime [5]. Figure 11.2 shows forecasted bifurcation diagrams for the nondimensional plunge ξ for two different bifurcations (supercritical and subcritical) of the 3-DOF airfoil with a flap-NES. The forecasted results are compared to the exact diagrams computed with MATCONT. It is observed that the data-driven approximated diagrams match well the reference diagrams in both supercritical and subcritical cases. This approach is used in the optimization process to evaluate the performance of a designed NES instead of using costly traditional nonlinear stability analysis approaches.

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Fig. 11.3 Optimal bifurcation diagram (red dashed) for supercritical (left) and subcritical bifurcations (right) and the nominal reference system (black solid)

11.3 Optimization Results In this section, we discuss optimal values of NES parameters for aeroelastic tailoring of two different 2-DOF systems. The difference between the two systems arises from the structural nonlinearity of the airfoil, and it defines the type of LCOs that occur. In the first case, we consider a 2-DOF airfoil that exhibits a supercritical bifurcation before optimization. The goal of this first study is to design an NES to increase the linear flutter speed and to reduce the plunge LCO amplitudes. In the second case, we consider a 2-DOF system that exhibits a subcritical bifurcation before optimization. The aim of the second study is to show that an optimally designed NES can change the nature of the bifurcation from subcritical to supercritical and achieve a flutter speed higher than the initial linear flutter speed and higher than the lowest speed where LCOs occur (i.e., the saddle-node bifurcation of cycles observed in the subcritical case). Supercritical bifurcations are less dangerous compared to subcritical ones because the limit cycle amplitudes increase gradually as velocity increases without jumping from zero to a large amplitude. The non-sorting genetic algorithm NSGA-II genetic algorithm is chosen to optimize the multiple variable functions with 5 NES design parameters. Multiple runs of the GA are performed and compared to confirm the convergence of the method. The bifurcation diagrams for pitch and plunge shown in Fig. 11.3 (black, solid line) correspond to one of the designs on the Pareto curve. The bifurcation diagrams for the initial 2-DOF system are shown as well (red, dashed line). For the first case of an initially supercritical system (left), we can observe that the addition of a flap-NES allows to increase critical flutter and reduce LCO amplitude for both DOFs, reducing the plunge amplitude by up to 73%. For the second case of an initially subcritical system (right), results show that the flap-NES not only increases the critical flutter speed over the lowest speed where LCOs occur, but it also changes the nature of the bifurcation to supercritical.

11.4 Conclusions This study was focused on optimal design of parameters for a flap-NES added to a pitch-plunge airfoil. An optimal NES for a pitch-plunge airfoil exhibiting supercritical flutter resulted in an increased flutter speed and a considerable reduction of the LCO amplitude for both pitch and plunge. In addition, an optimal NES for a pitch-plunge airfoil exhibiting a subcritical flutter

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changed the type of the flutter instability from subcritical to a supercritical one and also increased the flutter speed. Changing the bifurcation from subcritical to supercritical is important because subcritical bifurcations are dangerous because they allow the sudden onset of large amplitude LCOs even below the linear flutter speed. The optimization approach presented unveils the performance of flap-NESs to increase safety and improve performance. Further research should be devoted to the development of an online/adaptive optimization of NES for varying flying conditions.

References 1. Nguyen, N.T., Ting, E., Lebofsky, S.: Aeroelastic analysis of wind tunnel test data of a flexible wing with a Variable Camber Continuous Trailing Edge Flap (VCCTEF). In: 56th AIAA/ASCE/AHS/ASC structures, structural dynamics, and materials conference, AIAA SciTech Forum (January 2015) 2. Gendelman, O., Manevitch, L.I., Vakakis, A.F., M’Closkey, R.: Energy pumping in nonlinear mechanical oscillators: part I. J. Appl. Mech. 68(1), 34–41 (2000) 3. Fernandez Escudero, C.: Passive Aeroelastic control of aircraft wings via nonlinear oscillators. PhD Polytechnique Montréal (May 2021) 4. Lim, J., Epureanu, B.I.: Forecasting a class of bifurcations: theory and experiment. Phys. Rev. E. 83(1), 016203 (January 2011) 5. Ghadami, A., Epureanu, B.I.: Bifurcation forecasting for large dimensional oscillatory systems: forecasting flutter using gust responses. J. Comput. Nonlinear Dyn. 11(061009) (July 2016)

Chapter 12

Tribomechadynamics Challenge 2021: A Multi-harmonic Balance Analysis from Imperial College London M. Lasen, L. Salles, D. Dini, and C. W. Schwingshackl

Abstract This work presents the approach and results of the Dynamics Group at Imperial College in face of the Tribomechadynamics 2021 challenge. The challenge encourages to obtain the best blind prediction of a benchmark structure so that a transversal comparison, among the groups working in nonlinear studies, is done. The approach of the Dynamics Group consists in predicting the behaviour due to friction nonlinearities at the location where more energy dissipation is observed. The results show a slight softening in the contact with an overall shifting of the linear frequency of 2.6% and a damping increase of about 1.5% with respect to the linear damping. The effect of the contact is modest, given the lack of dissipated energy and the fact that geometric nonlinearities are not considered throughout this study. Keywords Joints · Tribomechadynamics · Harmonic balance · Nonlinear · FE modelling

12.1 Introduction As part of the 2021 joints community activity, the Tribomechadynamics team introduced a joint modelling challenge to bring the community together during COVID-19 times. The aim of the challenge is to assess the capability to produce blind prediction of a jointed structure. For this purpose, the challenge provides industry standard CAD drawings of a bolted structure, guidelines about the bolt loads, and the excitation of the structure. No further information is provided to let the participants decide on the best way forward. This paper summarises the approach taken by the Dynamics Group at Imperial College London to tackle the problem, purely focusing on the friction nonlinearity and ignoring the presence of the geometric nonlinearity. The approach is based on a detailed discretisation of the nonlinear contact interface with Jenkins elements and the harmonic balance method (Imperial’s in-house solver FORSE) to find the steady-state response of the system under base excitation. The paper will discuss the chosen approach in detail, discuss Imperial’s results, and provide recommendations on how to improve the modelling of such bolted joints.

12.2 Background This computational campaign includes three steps. First, a finite element (FE) model is built in Abaqus, and static, modal, and dynamic analyses are carried out. Second, a pre-processing step reads the simulation’s output and writes the input files for FORSE [1–3]. Finally, force controlled nonlinear simulations are computed, and the information is post-processed to obtain the results in the format requested for the challenge. The FE model and static results are displayed in Fig. 12.1. To simplify things, it was decided to remove the base of the support and focus on the plate holder only. The mesh is comprised of quadratic brick elements (C3D20). The simplified support (in green) has 30066 nodes and 6064 elements and the panel (in red), 12881 nodes and 2432 elements. A series of concentric circles around each bolt is included in the mesh to capture the pressure spread due to the pressure cone.

M. Lasen () · L. Salles · D. Dini · C. W. Schwingshackl Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_12

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Fig. 12.1 (a) FE model mesh: support (green), panel (red). (b) Pressure distribution (CPRESS) of the right side of the panel

Front Interface Back Interface

Fig. 12.2 Nonlinear mesh, indicating nodes on the front (higher z-values) and back (lower z-values) interfaces of the refined edge

A bolt load of 9900 N is applied using the Bolt Load tool available in Abaqus [4]. The contact conditions are ‘hard contact’ in the normal direction and ‘penalty’ friction formulation, with μ = 0.67, in the tangential direction. An intermediate step between the FE simulations and FORSE simulations is required. This step consists of the extraction of the data from the static, modal, and dynamic simulations, to build the corresponding input files for the in-house software FORSE. To ensure converged results, a single row of nonlinear nodes in the x-direction was added consecutively at 0.5 mm from the edge, starting from 1 up to 5 rows, until the nodes in the last row are stuck and therefore not dissipating energy. Figure 12.2 shows this refinement and the status at resonance of the 5 rows, 180 nonlinear nodes in total.

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Fig. 12.3 (a) Frequency-amplitude dependency and (b) damping-amplitude dependency, due to different excitation forces

12.3 Analysis With the final mesh in place, the nonlinear simulations are carried out from 2N until 0.000001N (linear case) of excitation in the middle node of the panel, in the z-axis. From these simulations the FRFs are extracted, and a linear modal fitting tool is used to extract the corresponding amplitude and damping at resonance. Since the linear tool is not really appropriate for a nonlinear FRF, an adjustment is applied by using a linear formulation for x/F at resonance [5] to adjust the amplitude of the fitted FRF to the predicted nonlinear values. It must be noted that this approach is not ideal but will provide a rough guideline for the damping due to friction in the joint. Figure 12.3 displays frequency and damping dependency on amplitude. Figure 12.3a shows a reduction of 2.6% in the resonance frequency with respect to the linear case at 118.3Hz, with a rapid decay as the nonlinear nodes transition from the stuck to the sliding case. Figure 12.3b shows the damping dependency on amplitude. It should be noted that a very low linear damping ratio of 1E-3% was used in the simulation to not overpower the frictional effects. For cases close to linear, the damping stays around the expected modal damping. As the load increases, the damping very quickly reaches a peak value of 3.2E-3%, between 0.05N and 0.1N, after which the damping stabilises around 2.5E-3% since more nonlinear nodes detach from the contact, therefore no longer being able to dissipate energy. Overall, this is considered very low damping, highlighting in the authors opinion that the joint does not lead to much damping.

12.4 Conclusion A computational study, in answer to the open Tribomechadynamics challenge 2021, is presented here by the Dynamics Group at Imperial College London. A progressive approach is taken to increase certainty that the nonlinear behaviour in the panel-support interfaces is fully captured. The predictions show that a small amount of friction damping occurs and therefore the contact exhibits some frequency shift and damping variation with respect to the linear case, as more nodes transition from a stuck to sliding condition. By gluing all nodes that are not expected to contribute to the damping, this approach assures that only the elements that would dissipate energy are considered, and no extra solving time is added to include the effect of permanent stuck nodes. Therefore, it is relatively fast to solve; in this case 180 nonlinear nodes were required to capture the behaviour of the edge, and a wall time of approximately 6 mins was required to solve each nonlinear response.

References 1. Salles, L., Blanc, L., Gouskov, A., Jean, P., Thouverez, F.: Dual Time Stepping Algorithms With the High Order Harmonic Balance Method for Contact Interfaces With Fretting-Wear. (April 2014). https://doi.org/10.1115/1.4004236. 2. Petrov, E.P.: Explicit Finite Element Models of Friction Dampers in Forced Response Analysis of Bladed Disks. (2008). https://doi.org/10.1115/ 1.2772633 3. Pesaresi, L., Armand, J., Schwingshackl, C.W., Salles, L., Wong, C.: An advanced underplatform damper modelling approach based on a microslip contact model. J. Sound Vib. 436, 327–340 (December 2018). https://doi.org/10.1016/j.jsv.2018.08.014

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4. Smith, M.: ABAQUS/Standard User’s Manual, Version 6.9. Dassault Systèmes Simulia Corp (2009). Accessed 23 September 2021. [Online]. Available: https://www.research.manchester.ac.uk/portal/en/publications/abaqusstandard-users-manual-version-69(0b112d0e-5eba-4b7f-9768cfe1d818872e)/export.html. 5. Ewins, D. J.: Modal Testing: Theory, Practice and Application, 2nd edn, p. 576 (2009). Accessed 23 September 2021. [Online]. Available: https://www.wiley.com/en-us/9780863802188.

Chapter 13

Experimental Proof of Concept of Contact Pressure Distribution Control in Frictional Interfaces with Piezoelectric Actuators M. Lasen, D. Dini, and C. W. Schwingshackl

Abstract Complex machinery holding several components in assembled structures requires a multitude of joints which inevitably develop frictional contacts under different dynamics conditions. Those conditions have been studied from a passive control point of view, predesigning the shape of the contact that will face later frictional circumstances. That passive approach can be extended into an active form of control. This paper proposes a novel concept of active manipulation of the contact shape and consequently the contact pressure distribution developed in frictional interfaces. This concept is initially tested with computational simulations that show sufficient confidence in the concept as to allow its continuation with the development of a physical prototype and experiments. The proposed system has been manufactured as a proof of concept, and here we present the setup along with the experimental measurements that will be carried out with it. Keywords Pressure distribution · Friction · FE modeling · Piezoelectric · Joints

13.1 Introduction Joints in assembled structures dissipate energy through friction in the contact interfaces. This dissipation has been studied extensively in the last three decades to reduce vibration amplitudes and hence extend the life span of critical components [1]. Various frictional parameters have been shown to influence the nonlinear dynamic behavior in joints, including normal load distribution, contact stiffness, and friction coefficient. While the latter two are normally fixed in an assembly, the normal load distribution can be optimized for better performance. This is most easily achieved by a priori designing the interface geometry to provide an optimal load distribution [2, 3] or by locally controlling the normal load [4, 5]; however not much attention has been put on how this load distributes in the contact interface. This paper proposes a somewhat more adventurous concept to control the contact interface geometry and hence the pressure distribution, during vibration via a set of active piezo actuators with the aim of modifying the energy dissipation and stiffness of the contact interface. Special attention is placed on the preliminary experimental analysis to prepare the adequate setup for the dynamics study.

13.2 Methods The model consists of three actuators under a clamped thin plate and a punch on top of the plate. The punch is fixed in the vertical direction; therefore, the total load comes directly from the actuators. The piezoelectric actuators are activated in different patterns: 010 (only the middle one is activated), 101 (only the lateral ones are activated), and 111 (all activated). This actuation produces different contact pressure distributions in the plate-punch interface, and thus it changes the hysteresis of the contact and the dynamics of the assembly. Initially the new concept is tested in quasi-static simulations using the commercial finite element software Abaqus; see Fig. 13.1a. Due to the large computational time associated with the execution of detailed models of the piezoelectric elements, the piezoelectric stack is simplified to a single spring with the short-circuit stiffness of the piezoelectric material, and the voltage

M. Lasen () · D. Dini · C. W. Schwingshackl Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_13

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Fig. 13.1 (a) Finite element model. Contact pressure distribution patterns (b) 010 and (c) 111

Ed kt

Fig. 13.2 Hysteresis loops for 010 pattern at different voltages, showing contact stiffness (kt) and energy dissipation (Ed)

is applied as an equivalent excitation force at the base of the spring. The tip of the piezoelectric elements is explicitly modeled with a curvature of radius, r = 5 mm. The plate is clamped on the lateral sides, from which the tangential force is extracted. The punch is fixed in the vertical direction, and thus the normal force can be extracted from its grounded connection. The tangential excitation is applied in the horizontal direction of the punch. The punch and actuators’ s tips are rigid bodies, and the plate is an elastic component made of steel (E = 200 GPa, ν = 0.3). The mesh is highly densified in the plate-punch interface to capture sliding in detail. Figure 13.1b shows the contact pressure distributions extracted from the plate-punch interface after the statically applied patterns 010 and 111, where a “1” in the pattern means that the piezoelectric is actuated at 100 V, with an equivalent base excitation, and “0” means no activation at all. It can be observed how Hertzian-like contacts develop at the interface as a product of the curvature in the actuators tip on the lower surface of the plate. The maximum pressure obtained at each Hertzian-like contact patch is the same since the actuators’ s tips are sufficiently apart from each other.

13.3 Results and Discussion After the static loading step, the reciprocating tangential load is applied by the punch, and consequently the hysteresis loops can be extracted from the contact. The hysteresis loops are extracted at different equivalent voltages, and the corresponding tangential contact stiffness and dissipated energy per cycle are computed from the loop geometry as shown in Fig. 13.2 [6]. The exercise in Fig. 13.2 is extended for all the patterns, and the summary of results is presented in Fig. 13.3. It can be observed how stiffness and dissipated energy grow as more voltage is applied; this is a consequence of the increase in the contact area induced by the actuation of the piezoelectric elements. Since the punch is locked in the vertical position, more load forces the plate to comply with the flat surface of the punch. Once confidence in the proposed concept is achieved, the experimental campaign is started. This paper aims to show the new system and all its manufactured components and to discuss the measurements that will be performed to prove the effectiveness of the idea and the viability of achieving active control of interfacial response via piezoelectric actuation. The setup of the concept is installed in the 1D friction rig at Imperial College London, given its capabilities to drive high-frequency dynamics analyses and the corresponding measurements to obtain friction hysteresis loops [6]. Figure 13.4 displays the original rig and the new modified component. The modification in Fig. 13.3b consists of a new static arm, capable of holding the three actuators (PICMA® P-882.11) and a clamped plate. The selection of these actuators is due to their compact design, fast response, and high load range. The actuators have curved tips in both ends to avoid shear stresses

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Fig. 13.3 (a) Contact stiffness. (b) Dissipated energy

Moving Arm Static Arm

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Fig. 13.4 (a) Original 1D friction rig. (b) Proposed concept setup

into the fragile piezoceramic [7]. Also, to restrain movement in the transversal plane, lateral 3D-printed holders are used to clamp only the metallic tip at the base, to avoid any extra constrain in the extension of the piezoceramic. Before placing the arm in its final position, the actuators are placed in position thanks to the holder, and then the plate is aligned with the tips of the piezoelectric actuators and finally clamped down with bolts tightening the clamps from underneath. This is a particularly important operation as the tolerances are such that excessive tightening might compromise the actuation by pre-straining the piezoelectric components. After this the actuators can operate their three different patterns, with the corresponding voltage amplifier (PI: E-663.00). Finally, for the dynamics loading, the shaker excites the moving arm that rigidly holds the punch. Measurements can then be taken to study the system response.

13.4 Conclusion A new concept of shape and contact pressure distribution control is introduced, initially via a set of computational simulations that offered enough confidence to move into a physical realization of the concept. The new setup has been manufactured and arranged so that the required experiments to obtain the hysteresis loops can be performed. Preliminary measurements with pressure films and optical microscopes will be carried out to check for the contact in the punch-plate interfaces as well as the plate-actuator tip interface. After this, the hysteresis loops will be extracted from the tangential force recorded on load cells next to the static arm, and the corresponding relative displacement will be captured using a high-speed camera, which will focus on the region in the proximity of the punch-plate interface [8]. Acknowledgments Matias Lasen thanks the funding from the President’s PhD Scholarship provided by Imperial College London.

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References 1. Gaul, L., Nitsche, R.: The role of friction in mechanical joints introduction and review of literature (2001). Accessed: 25 Apr 2019. [Online]. Available: http://www.asme.org/about-asme/terms-of-use 2. Koh, K.-H., Griffin, J.H.: Dynamic behavior of spherical friction dampers and its implication to damper contact stiffness. J. Eng. Gas Turbines Power. 129(2), 511 (Apr. 2007). https://doi.org/10.1115/1.2436547 3. Panning, L., Popp, K., Sextro, W., Götting, F., Kayser, A., Wolter, I.: Asymmetrical Underplatform Dampers in Gas Turbine Bladings: Theory and Application, pp. 269–280 (2008). https://doi.org/10.1115/gt2004-53316 4. Dupont, P., Kasturi, P., Stokes, A.: Semi-active control of friction dampers. J. Sound Vib. 202(2), 203–218 (May 1997). https://doi.org/10.1006/ JSVI.1996.0798 5. Gaul, L., Lenz, J., Sachau, D.: Active damping of space structures by contact pressure control in joints. Mech. Struct. Mach. 26(1), 81–100 (1998). https://doi.org/10.1080/08905459808945421 6. Fantetti, A., et al.: The impact of fretting wear on structural dynamics: experiment and simulation. Tribol. Int. 138, 111–124 (Oct. 2019). https:/ /doi.org/10.1016/J.TRIBOINT.2019.05.023 7. Aggogeri, F., et al.: Design of piezo-based AVC system for machine tool applications. Mech. Syst. Signal Process., 1–12 (2011). https://doi.org/ 10.1016/j.ymssp.2011.06.012 8. Brøns, M., Kasper, T.A., Chauda, G., Klaassen, S.W.B., Schwingshackl, C.W., Brake, M.R.W.: Experimental investigation of local dynamics in a bolted lap joint using digital image correlation. J. Vib. Acoust. Trans. ASME. 142(5) (Oct. 2020). https://doi.org/10.1115/1.4047699

Chapter 14

Experimental Observations of Nonlinear Damping of Additively Manufactured Components with Internal Particle Dampers Matthew Postell, Daniel Kiracofe, Onome Scott-Emuakpor, and Tommy George Jr.

Abstract Additive manufacturing (AM) offers unique capabilities to incorporate the effects of different mechanics. Traditionally, AM parts are fabricated to reduce weight while maximizing strength. More recently, it has been observed that AM can also be used to fabricate components with internal particle dampers through the process of leaving a small pocket of unfused powder during the printing process. This small pocket of unfused powder can assist in eliminating unwanted vibrations in parts without the post-processing involved with traditional particle dampers. Previous works have reported on the damping capability of various AM designs. Those works focused on a narrow range of excitation amplitudes, over which approximately linear behavior was observed. This work reports on a study of damping capability over a much wider range of excitation amplitudes. Over this wider range, nonlinear behavior was observed. These results indicate certain pocket configurations and modes result in a system regime such that the response amplitude does not exceed a threshold value, up to a certain system excitation. This trend directly translates into an increasingly damped system as excitation increases to a certain level. Other configurations and modes result in a typical single DOF oscillator FRF, while the damping is dependent on the amplitude of the base excitation. Through further experiments, it was determined that there are certain ranges of base excitation that result in consistent nonlinear responses while other ranges result in a response similar to that of a linear system. These results suggest that a system intentionally designed to operate in scope of the nonlinear region may have significantly higher vibration reduction than was thought possible based on previous works. Keywords Nonlinear · Particle damper · Additive manufacturing

14.1 Introduction Additive manufacturing (AM) applications in aerospace are a growing field in recent years. One application of AM is to combine multiple separate parts into one single part, thus saving weight. However, eliminating interfaces between parts also eliminates the structural damping at those interfaces. A classic conventional example of this is the blisk, which is a machined part that combines both the fan/compressor blades and the disk housing into a single component. Combining the two reduces part count and weight at the cost of significantly reduced damping. The vibrations caused by reduced damping can result in a shorter life span, frequent maintenance, and high cycle fatigue failures. Recent proposals to incorporate damping to AM parts are through the installation of particle dampers [1, 2]. Particle dampers have been known for many years. A typical particle damper installation method involves machining a sealable pocket inside a component where sand or glass beads fill the crevice [3, 4]. Such a method can be difficult to install in conventionally machined parts whereas it is very easy to apply to an AM part: a pocket of unfused feedstock powder can be left during the printing process, creating a particle damper. Prior works have reported on the development and characterization of such additively manufactured parts with integrated dampers [2, 5–9]; however, these works were limited to linear approximations of the dynamics. In this work, we attempt to characterize the nonlinear dynamics of these components.

M. Postell () · D. Kiracofe Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected] O. Scott-Emuakpor · T. George Jr. Air Force Research Laboratory, Wright-Patterson AFB, OH, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_14

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14.2 Experimental Setup A series of cantilever beams of different geometries were printed out of gas atomized Inco 718 at the Air Force Research Lab. Details to the specifics of the manufacturing of the beams can be found in [8], and the overall schematic of all printed beams can be found in Reference [10]. Two representative beams were chosen for this work. These will be referred to as the thick and thin beams. The geometry and dimensions are summarized in Fig. 14.1. Experiments on the beams were set up for both a closed-loop and open-loop base excitation, with the majority of tests using a closed-loop setup. This is done by mounting an accelerometer on the shaker head which tells the controller how much acceleration the system is observing. The controller then adjusts the voltage so that the acceleration of the shaker head is constant to what the user specifies. An open-loop base excitation system is different in that there is no need for an accelerometer to communicate with the controller—the user dictates what voltage the shaker would be driven by. For the experiments run, base excitation was the preferred method as opposed to applying a force to the tip with a shaker and stinger, as the later can cause mass loading, added damping, and introduce additional system resonances. Since both the beam and clamp are being driven at a constant acceleration, a 27 kN Unholtz-Dickie electrodynamic shaker was used. A Polytec PSV-500 scanning laser vibrometer was used with the laser 2–3 mm from the tip of the beam. The beam was clamped between two heavy blocks of metal and clamped using three bolts. The overall setup of the experiment is shown in Fig. 14.2. It should be noted that this system setup is not a perfect boundary condition: The measured mode shapes compared to a theoretical fixed-free beam differ in natural frequency and amplitude, as seen in Fig. 14.3.

Fig. 14.1 Beam geometry and dimensions

Fig. 14.2 Experimental setups. Overall setup is shown on the left, and a zoom-in on the specimen is shown on the right

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Fig. 14.3 Comparison of experimental and theoretical (clamped-free) mode shapes, showing that the experimental boundary condition is not a perfect clamp. Black dashed line indicates the location of the pocket of unfused powder

Lastly, there is a known shaker system resonance around 2500 Hz, which does coincide with the 3rd mode of the thick beam. It is believed that the effect of the system mode has no observable effect on the results of the beams, as the 3rd mode results are similar between the thick and thin beam, which have a resonance frequency of around 2500 Hz and 1800 Hz, respectively. The test procedure itself is relatively simple. Once the experiment is set up, a frequency sweep between mode 2 or mode 3 bandwidths is run. The selected sweep rate was 3 minutes per sweep in both up and down directions. A series of tests were run with varying base accelerations and data was overlayed to observe any potential nonlinear trends.

14.3 Nonlinear Behavior As particle dampers are known to be highly nonlinear [11], it was expected that some form of nonlinear response would be observed on the beams. There were three key observations from the tests. First, the damping was nonlinear; in other words there was a nonlinear relationship between the excitation amplitude and the response amplitude. Second, for some tests, the frequency response function (FRF) exhibited a saturation or flat-top phenomena (as opposed to a classic SDOF oscillator FRF). Finally, jump phenomena were observed, with differences between sweep up and sweep down, which is an indication of bistable regions. These observations will be detailed in the following sections.

14.3.1 Amplitude Dependency If a system response is linear, one would expect the response to scale with the excitation—if the system is driven at 2x the force, the response should be 2x. For the tested beams, in some circumstances, the response did not increase linearly. For example, in Fig. 14.4a, going from 0.01 g base excitation to 0.02 g base excitation, the excitation has doubled, but the response has increased by only ~15%. Coincidentally, this low increase also appears around excitations where flat-top behavior occurs, as indicated in Fig. 14.4b. Further details to this flat-top behavior are discussed in Sect. 14.3.2. The responses from all beams tested indicated some forms of amplitude dependency. In other words, the level of damping is dependent on the level of excitation. Various levels of acceleration were tested, as low as 0.005 g’s and as high as 1 g base excitation. The shaker has sufficient force to test higher levels; however, the increased thermal energy within the particle damper approaches levels where particle fusion can start to occur, thus irreversibly changing the system [9]. Other researchers have examined the nonlinear damping of particle dampers [12–15]. Liu et. al. [14]ran tests on a particle damper and collected FRFs at first resonance. FRFs were measured from various response levels, from 0.1 g up to 40 g. Their

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Fig. 14.4 Legends represent base excitation. Closed-loop sweep down responses for (a) thick beam mode 2, (b) thick beam mode 3, (c) thin beam mode 2, (d) thin beam mode 3

findings indicated a rise in damping at lower response levels as well as a shifting of resonance frequency, up to around 16 g’s response, where further results were unchanging. Another work also identified an amplitude dependency in a particle damper system: In his dissertation, Abel [15] observed a particle damped system’s response as input amplitude was increased, and noticed the system damping increased rapidly at low excitations while at higher excitations the system damping reduced inversely. All of these prior works utilized tens or hundreds of particles on the order of millimeters in diameter. The system considered here has tens of millions of particles on the order of 25 μm in size. A goal of the present work is to determine if these different systems would exhibit similar behavior. Figures 14.4a–d show the response of the vibrometer at the tip for varying levels of base excitation. They are broken up into 2nd and 3rd resonance frequencies for both beams tested. It can be observed that both beams, despite having a difference in geometry, experience very similar trends in terms of amplitude responses; mode 2 responses mostly resemble classic single degree of freedom responses, while mode 3 has some jump phenomena, as well as some flat-top behavior that deviates significantly from a linear SDOF response (labeled in Fig. 14.4b). Normalizing the response of the vibrometer with respect to the base excitation can help identify whether certain base excitations induce a linear or nonlinear response. When normalizing the laser output to the specific base excitation, or a FRF in g/g, a linear response would appear superimposed with another linear response. However, the resulting Fig. 14.5a–d shows amplitude-response curves that do not superimpose one another. In fact, there appears to be two regions of a linear response: one occurs at high base excitations while the other occurs at very low excitations. The middle ground excitation responses sitting between the two high-low excitations appear to be a transition region that is nonlinear.

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Fig. 14.5 Results of vibrometer response/control accelerometer output. (a) thick beam mode 2, (b) thick beam mode 3, (c) thin beam mode 2, (d) thin beam mode 3

Taking the results from the FRF, we can map out a trendline of the response versus the base excitation in Fig. 14.6. When looking at the peak amplitude versus the base excitation on a log-log scale, a trendline is formed that highlights a clear transitional, nonlinear region (shown in gray) between two regions that begin to flatten out at very low or higher excitations. Due to system limitations, higher and lower excitation datapoints were not able to be collected; however, the trendlines imply that the system is linear at those low and high excitations. For example, in Fig. 14.5b, the response for 0.01 g and 0.02 g excitation are nearly on top of each other, and the response for 0.5 g and 1 g are also nearly on top of each other. However, the intermediate excitation amplitudes show different responses. As this behavior is present in both modes and two different geometric beams, the results from these experiments indicate that this behavior is due to the particle damper itself and not a result of the experiment procedure or test pieces.

14.3.2 Flat-Top Behavior While determining the effects of the amplitude dependency of the nonlinear system, there were certain excitation levels where the system response flattened out around the third resonance frequency (see Fig. 14.4b). Second resonance showed no indication of this behavior. This flattening of the response curve occurred within the linear to nonlinear transition and did not exceed a specific amplitude response. Only when the base excitation was high enough did the flat-top behavior die out. An attempt to highlight the differences between what a theoretical SDOF FRF might look like when compared

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Fig. 14.6 Amplification of Thick Beam (red) and Thin Beam (blue). Nonlinear region is marked with a gray box

Fig. 14.7 Experimental response in flat-top region versus a theoretical SDOF response

to the experimental FRF is shown in Fig. 14.7. Interestingly, these two responses appear similar up to a point where the experimental system transitions into this flat-top region. Further observations in Fig. 14.8 indicate this flat-top behavior occurring after what appears to be a jump in response: the system starts off appearing like a normal SDOF FRF, then jumps to a much flatter region. Given that the excitation and motion of the particles are in the same direction as gravity (i.e., vertical), it was hypothesized that the transition regions in the FRF may correspond to the particles starting to lift off of the bottom of the pocket. In other words, for very low excitations, the accelerations of the particle would not exceed 1 g, and thus mostly stay in contact with the bottom wall and never contact the top wall. For very high excitations, the particles should lose contact with the top wall and also contact the top wall. To investigate this hypothesis, we attempted to quantify what acceleration the particles were moving at. Given the response at the tip, a response at the pocket can be inferred by multiplying the tip response by the mode shape at the pocket location. By doing so, we can observe that the flat-top behavior occurred when the pocket response was between 4 and 6 g’s for both beams. This is somewhat higher than we had expected based on the hypothesis of the particles lifting off the bottom wall. It may correspond to the point where the particles start to contact the top wall. This will be discussed further in Sect. 14.4.

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Fig. 14.8 (a) open-loop system response (b) closed-loop system response. Both results are from a frequency sweep down

14.3.3 Open-Loop Versus Closed-Loop Controller In order to verify that the behaviors observed were not an artifact of the closed-loop shaker controller, a separate experiment was conducted with an open-loop control: a constant voltage was supplied to the shaker amplifier instead of controlling the voltage based on an accelerometer. This type of controller allows the system excitation, in terms of base acceleration, to vary with frequency. To maintain consistency between the two systems, a similar sweep rate was used. The open-loop system had a sweep rate of 0.7 Hz/sec and the closed-loop had a sweep rate of 0.78 Hz/sec. The results of this experiment, shown in Fig. 14.8, indicate similar behaviors observed in mode 3 responses. There is a difference in amplitude value and where the transition regions occur; however, this may be due to run-to-run variability in the system (see Sect. 14.3.5) rather than a difference caused by the controller.

14.3.4 Sweep Up Versus Sweep Down The direction of a frequency sweep should be accounted for when observing nonlinear behavior. Because there appears to be jump phenomena (for example in Fig. 14.8), a sweep up should be run to investigate bistable behavior. The results of the sweep up versus sweep down experiments are shown in Fig. 14.9. For the frequency sweep up, there is a jump down on the left of the resonance. For the frequency sweep down, the jump is on the right side and is a jump down for lower amplitudes, but a jump up for higher amplitudes. These results seem to indicate the presence of two stable states: one with a higher damping and one with lower damping. Adjustments to both system’s sweep rate show no impact to the relative frequency and response level—the nonlinearities occur at the same instance. Therefore, we can assume this behavior to be consistent and a result of the particle damper.

14.3.5 Repeatability Validating repeatability for a system with nonlinear amplitude responses ensures that the results are indeed a direct result of the dynamics of the particle damper system and not an artifact of the controller or other outside parameters. To test repeatability, the beams were taken off the clamp and reinstalled, realigned, re-torqued, and remeasured 7 times over 4 days. From the tests run, the overall results do not change; both the amplitude dependency and the flat-top behavior were present qualitatively in every single trial as seen in Fig. 14.10a, and repeated sweep data is shown in Fig. 14.10b. However, it should be noted that though the trends did not change, the level at which the transition point occurred in the flat-top behavior did vary from run to run. For example, in Fig. 14.11, for one set of experiments the 0.1, 0.2, and 0.5 g curves all have flat tops with

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Fig. 14.9 Comparison between sweep up (solid) and sweep down responses (dashed). System phase is shown in lower subplot

Fig. 14.10 (a) Multiple iterations of reinstallation of Thin beam, (b) Repeated amplitude sweep of one trial (solid) versus another trial

an amplitude around 8 g, whereas for the other set, the same curves have a flat-top around 13 g. Another behavior observed here is the jump phenomena appearing only at higher amplitudes for one data set, and for all amplitudes for the other data set. Other observations include a minimal shift in frequency (+/− 0.1%) and a greater shift in amplitude response varying by excitation. These variations could indicate that these behaviors are dependent on the boundary condition’s effect on the system response. One possible cause is due to the non-rigid clamp and shaker head, which makes it difficult to measure the true base acceleration at both modes. Further observations into this theory were made: using a scanning laser vibrometer, the dynamics of the shaker head was mapped out and from that it was found that the shaker head and clamp are not rigid near mode 3.

14.4 Future Works A discrete element method (DEM) model has been created to simulate the response of these particle dampers [10]. In Reference [10], the model was compared to experimental results for the higher end of the base excitation range where the responses mostly appeared linear (i.e., 0.5 g for mode 3). A good comparison was found. Comparisons over the full range of base excitations including the nonlinear behavior have not yet been conducted. The trends obtained here will be useful in validating the results of the simulation model – if the simulation can accurately estimate damping factors of various beam

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Fig. 14.11 Repeatability of nonlinear behavior of an identical beam. Experiment parameters are identical, except the dashed lines were taken one week after solid lines. Amplitude responses between the two tests quantitatively differ, but are qualitatively similar

geometries, and if the model can estimate similar nonlinear trends, then it adds more belief and justification to the results of the model. Additionally, if the model can accurately predict the experiment, then the model can be used to explore the causes of the flat-top behavior and what is possibly changing about the internal behavior of the particles.

14.5 Conclusions Nonlinear behavior is difficult to both identify consistently as well as experimentally. This paper serves as an introduction and attempt to quantify said nonlinear behavior for additively manufactured structures with integrated particle dampers. Of the geometries tested, three instances of nonlinear behavior were determined: a flat-top saturation phenomenon, nonlinear amplitude relationship between base excitation and damping levels, and jump phenomenon. These results indicate that the damping from the particle damper is dependent on excitation amplitudes. Knowing this, it can be extremely useful in designing a damper for a component that undergoes specific excitation or cannot exceed a certain response amplitude. In order to accurately design such a component, there has to be a better understanding on the specific cause of the amplitude dependency as well as how to tune the system parameters (i.e., pocket size) for the best excitation value. Acknowledgments The authors would like to thank the personnel and researchers at the Turbine Engine Fatigue Facility at Wright-Patterson Air Force Base, especially John Hollkamp, Philip Johnson, and Dino Celli for their assistance in experimental setup and data analysis.

References 1. Goldin, A., Scott-Emuakpor, O., George, T., Runyon, B., Cobb, R.: Structural Dynamic and Inherent Damping Characterization of Additively Manufactured Airfoil Components. American Society of Mechanical Engineers Digital Collection. American Society of Mechanical Engineers, New York (2021) 2. Scott-Emuakpor, O., George, T., Runyon, B., Sheridan, L., Holycross, C., OHara, R.: Assessing Manufacturing Repeatability of Inherently Damped Nickel Alloy Components via Forced-Response Testing. American Society of Mechanical Engineers Digital Collection. American Society of Mechanical Engineers, New York (2019) 3. Panossian, H.: Non-obstructive particle damping: New experiences and capabilities. In: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference,10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference. American Institute of Aeronautics and Astronautics (2008). eprint: https://arc.aiaa.org/doi/pdf/10.2514/6.2008-2102 4. Panossian, H.V.: Structural damping enhancement via non-obstructive particle damping technique. J. Vibr. Acoust. 114(1), 101–105 (1992) 5. Scott-Emuakpor, O., Beck, J., Runyon, B., George, T.: Determining unfused powder threshold for optimal inherent damping with additive manufacturing. Additive Manufact. 38, 101739 (2021)

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6. Scott-Emuakpor, O.E., George, T., Beck, J., Runyon, B.D., OHara, R., Holycross, C., Sheridan, L.: Inherent damping sustainability study on additively manufactured nickel-based alloys for critical part. In: AIAA Scitech 2019 Forum. American Institute of Aeronautics and Astronautics. eprint: https://arc.aiaa.org/doi/pdf/10.2514/6.2019-0410 7. Scott-Emuakpor, O., Schoening, A., Goldin, A., Beck, J., Runyon, B., George, T.: Internal geometry effects on inherent damping performance of additively manufactured components. AIAA J. 59(1), 379–385 (2021) American Institute of Aeronautics and Astronautics 8. Scott-Emuakpor, O., Beck, J., Runyon, B., George, T.: Validating a multifactor model for damping performance of additively manufactured components. AIAA J. 58(12), 5440–5447 (2020). American Institute of Aeronautics and Astronautics 9. Scott-Emuakpor, O., Sheridan, L., Runyon, B., George, T.: Vibration Fatigue Assessment of Additive Manufactured Nickel Alloy With Inherent Damping. American Society of Mechanical Engineers Digital Collection. American Society of Mechanical Engineers, New York (2021) 10. Kiracofe, D., Postell, M., Scott-Emuakpor, O., Runyon, B., George Jr., T.: Discrete element method simulations of additively manufactured components with integrated particle damper. In: ASME Turbo Expo 2021 (2021) 11. Sanchez, M., Manuel Carlevaro, C.: Nonlinear dynamic analysis of an optimal particle damper. J. Sound Vibr. 332(8), 2070–2080 (2013) 12. Wong, C., Rongong, J.: Control of particle damper nonlinearity. AIAA J. 47(4), 953–960 (2009). American Institute of Aeronautics and Astronautics. eprint: https://doi.org/10.2514/1.38795 13. Zhang, K., Chen, T., Wang, X., Fang, J.: Rheology behavior and optimal damping effect of granular particles in a non-obstructive particle damper. J. Sound Vibr. 364, 30–43 (2016) 14. Liu, W., Tomlinson, G.R., Rongong, J.A.: The dynamic characterisation of disk geometry particle dampers. J. Sound Vibr. 280(3), 849–861 (2005) 15. Abel, J.T.: Development of a CubeSat Instrument for Microgravity Particle Damper Performance Analysis. PhD Thesis, California Polytechnic State University, San Luis Obispo, California (2011)

Chapter 15

Data-Driven Reduced-Order Model for Turbomachinery Blisks with Friction Nonlinearity Sean T. Kelly and Bogdan I. Epureanu

Abstract Modern turbomachinery contains integrally bladed disks, or blisks, which are nominally cyclic symmetric structures manufactured as a single piece. Unlike traditional disks with inserted blades, blisks lack contact interfaces and thus have very low internal damping. Additionally, due to imperfections in material properties and geometry variations among sectors, called mistuning, energy localization can occur during operation creating amplified response amplitudes and stresses and greater risk of high cycle fatigue failure. To reduce vibration amplitudes, nonlinear friction damping via contact interfaces can be introduced. One such method is to add a friction ring damper to the underside of the disk portion within a groove, which is held in place by centrifugal forces. Due to blisk finite element models often containing millions of degrees of freedom, modeling the nonlinear dynamics of these systems necessitates the use of reduced-order models to be computationally feasible. Thus, physics-based nonlinear reduced-order models have been developed to predict the nonlinear dynamic behavior of blisks with friction interfaces. However, data-driven methods for predicting nonlinear blisk dynamics have remained largely unexplored. Here, we introduce a novel data-driven reduced-order model for predicted blisk dynamics based on two feed-forward neural networks. These networks are based on sector-level data, allowing for significantly fewer simulations and/or experiments needed for training data generation. Unlike previous physics-based methods, this approach does not use linear or nonlinear modal information. This approach is validated for a lumped-mass model representative of a blisk with a friction ring damper and small stiffness mistuning subject to a traveling-wave excitation of the type seen during turbomachinery operation. Keywords Data-driven · Nonlinear dynamics · Blisks · Turbomachinery · Neural networks

15.1 Introduction Integrally bladed disks, or blisks, are critical components often used within the compressor stages of modern turbomachinery. Unlike bladed disks with separate disk and blade components in which blades are inserted in to the disk, blisks lack any inherent contact interfaces which can serve to dampen blade vibration and dissipate energy during operation. This can result in significantly higher response amplitudes during operation when subject to a traveling-wave excitation like that experienced during operation [1]. In practice, vibration amplitudes are further increased by sector-to-sector deviations from nominal geometry and material properties, called mistuning, which result from manufacturing tolerances and cannot be known a priori [2]. To dampen vibration particularly at system resonances, contact interfaces can be introduced via damping mechanisms such as a friction ring damper [3] or shrouds [4]. These damping mechanisms introduce nonlinearities which significantly increase computational expense. While the cyclicity of the system can be exploited for model reduction if all sectors are nominally identical, mistuning destroys this cyclic symmetry. Thus, to predict blisk dynamics in operation with mistuning present, many simulations with stochastically varying mistuning are often performed for Monte Carlo probabilistic analyses [5]. However, for high-fidelity finite element models, individual simulations even for linear systems can remain computationally expensive and for nonlinear systems may be infeasible. Thus, physics-based reduced-order models (ROMs) for mistuned blisks with and without nonlinearities have been developed [6–8]. For nonlinear systems, these are often projection-based and consider bases that span the nonlinear and linear system dynamics and/or consider frequency-domain methods such as the harmonic balance method (HBM). However, these models cannot directly include experimental data

S. T. Kelly () · B. I. Epureanu Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_15

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and remain purely physics-based, and to our knowledge, little work exploring data-driven approaches for modeling nonlinear blisk dynamics has been done to date. We present a novel data-driven ROM based on two feed-forward neural networks (FFNNs) for predicting the response behavior of a mistuned blisk with friction interfaces. In particular, we study the case of a blisk with a friction ring damper. Unlike previous physics-based approaches, this method does not require any projection basis nor any linear or nonlinear modal information. Additionally, because a sector-level formulation for both networks is considered, this allows for significantly fewer nonlinear simulations required for generating training data. The proposed approach is validated using a lumped mass mode (LMM) representative of a blisk with a friction ring damper and small stochastic blade-alone stiffness mistuning.

15.2 Background and Methodology The cyclic LMM used for validation mimics a blisk with a friction ring damper, with energy dissipated via friction between the disk and damper. A Coulomb friction model is used, which considers local relative displacements between pairs of disk and damper masses [5]. This model consists of N sectors, which are nominally cyclic symmetric but contain random stiffness mistuning applied to the blade-alone springs. A general sector i is shown in Fig. 15.1, consisting of blade component (masses mib0 , mib1 , mib2 ), disk component (masses mic1 , mic2 , mic3 , mic4 , mic5 ), and damper component (masses mid1 , mid2 , mid3 ). An external traveling-wave excitation is applied, with the force Fi on sector i applied at the blade tip mass mib2 . To apply mistuning, many previous physics-based ROMs often consider variations in the cantilever-blade frequencies parameterized as variations in the blade-alone Young’s modulus [2, 7]. Here, we consider an analogous mistuning in the stiffness values of i i the blade-alone springs, which are perturbed from nominal values k b0 and k b1 . Mistuning values, denoted by δ i , are given by δ i = σ ri , where σ is the mistuning magnitude and ri is a random value generated from a normal distribution with mean i i i = i = 0 and standard deviation 1. The perturbed blade-alone stiffnesses are kb0 (1 + δi ) k b0 and kb1 (1 + δi ) k b1 , which are indicated in Fig. 15.1. FFNNs can broadly be considered function approximators, which serve to “learn” a mapping between a set of inputs to a set of outputs [9]. Here, a solution framework similar to that developed previously by the authors in [10] is employed, consisting of two sector-level FFNNs: a coupling network (NNc ) and sector response network (NNs ). However, rather than using real and imaginary response data from linear mistuned systems like in [10], we replace these instead with harmonic coefficients from solving this system using the HBM with the alternating-frequency time (AFT) scheme [11]. We do not cover this procedure in detail here, but for the purposes of this approach, note that HBM assumes all responses, and forces are steady state and periodic in time and can be represented as a sum of linearly independent harmonic functions which are integer multiples of the excitation frequency [5]. The coefficients of these functions are what we use as inputs and outputs to both FFNNs. Following a similar procedure as in [10], the coupling network for a sector i takes as input the harmonic

Fig. 15.1 General sector i for LMM mimicking blisk with nonlinear friction ring damper

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Fig. 15.2 Example actual and predicted nonlinear blade tip responses for 16-sector LMM

coefficients from neighboring sector interface masses from sectors i + 1 and i − 1 along with the parameters: j-th forcing frequency ωj , real and imaginary forcing Fi, R and Fi,I , mistuning δ i , contact stiffness kc , and microslip parameter ρ = μF0 /F where μ is the coefficient of friction, F0 is the normal preload, and F is the traveling-wave forcing amplitude. The coupling network outputs the harmonic coefficients of the internal sector masses. The sector response network for a sector i takes as input the harmonic coefficients from the internal sector masses along with all parameters input to the coupling network and outputs the harmonic coefficients of the blade tip mass mib2 . We use a coupling procedure based on the coupling network discussed in detail in [10], and we obtain responses for all interface masses, which are subsequently input into the sector response network to obtain blade tip responses for all sectors.

15.3 Results The mistuned LMM shown in Fig. 15.1 using N = 16 sectors is considered and subjected to a traveling-wave engine-order (EO) four excitation. A normal preload F0 = 1000 N and friction coefficient μ = 0.2 are used and held constant over a frequency range containing system resonances. The actual and predicted normalized (by forcing amplitude F) nonlinear blade tip response max and min envelopes along with the mean blade tip response across all blades are shown in Fig. 15.2 for an example test case. As shown, the nonlinear max envelope for smaller response amplitudes follows the mistuned stick curve (i.e., when damper is always stuck); however from 642 to 651 Hz, the nonlinear response shows a clear reduction relative to both the linear slip (i.e., when damper is always in slip) and stuck cases. The predicted values follow their respective curves with good accuracy over the entire frequency range, also accurately predicting the nonlinear reduction (NL Red.) and the amplification factor (AF). Here, the AF is defined as the ratio of the maximum linear tuned response without the damper over the entire frequency range to the maximum mistuned nonlinear response over the entire frequency range.

15.4 Conclusions A novel data-driven ROM based on sector-level FFNNs was presented. This approach leverages the HBM with AFT to consider steady-state periodic solutions, employing a similar methodology as presented in [10] now using harmonic

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coefficients. This approach was validated using an LMM mimicking a mistuned blisk with a nonlinear friction ring damper subject to traveling-wave excitation. As shown in Fig. 15.2, the maximum and minimum normalized blade tip response amplitudes are accurately predicted for an example test case over a range of excitation frequencies containing system resonances. Acknowledgments The authors would like to gratefully acknowledge the generous support offered by the GUIde 6 Consortium toward their investigation and research in this topic.

References 1. Jones, K.W., Cross, C.J.: Traveling wave excitation system for bladed disks. J. Propuls. Power. 19(1), 135–141 (Jan. 2003). https://doi.org/ 10.2514/2.6089 2. Castanier, M.P., Pierre, C.: Modeling and analysis of mistuned bladed disk vibration: current status and emerging directions. J. Propuls. Power. 22(2), 384–396 (Mar. 2006). https://doi.org/10.2514/1.16345 3. Baek, S., Epureanu, B.: Reduced-order modeling of bladed disks with friction ring dampers. J. Vib. Acoust. 139(6), 061011 (Aug. 2017). https://doi.org/10.1115/1.4036952 4. Mitra, M., Zucca, S., Epureanu, B.I.: Effects of contact mistuning on shrouded blisk dynamics. In: Volume 7A: Structures and Dynamics, p. V07AT32A026, Seoul, South Korea (Jun. 2016). https://doi.org/10.1115/GT2016-57812 5. Mitra, M., Epureanu, B.I.: Dynamic modeling and projection-based reduction methods for bladed disks with nonlinear frictional and intermittent contact interfaces. Appl. Mech. Rev. (Mar. 2019). https://doi.org/10.1115/1.4043083 6. Madden, A., Epureanu, B.I., Filippi, S.: Reduced-order modeling approach for blisks with large mass, stiffness, and geometric mistuning. AIAA J. 50(2), 366–374 (Feb. 2012). https://doi.org/10.2514/1.J051140 7. Mitra, M., Zucca, S., Epureanu, B.I.: Adaptive microslip projection for reduction of frictional and contact nonlinearities in shrouded blisks. J. Comput. Nonlinear Dyn. 11(4), 041016 (May 2016). https://doi.org/10.1115/1.4033003 8. Saito, A., Epureanu, B.I.: Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators. J. Sound Vib. 330(14), 3442–3457 (Jul. 2011). https://doi.org/10.1016/j.jsv.2011.02.018 9. Aggarwal, C.C.: Neural networks and deep learning, vol. 1, 1st edn. Springer (2018) 10. Kelly, S.T., Lupini, A., Epureanu, B.I.: Data-driven modeling approach for mistuned cyclic structures. AIAA J., 1–13 (Apr. 2021). https:// doi.org/10.2514/1.J060117 11. Cameron, T.M., Griffin, J.H.: An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J. Appl. Mech. (1989). https://doi.org/10.1115/1.3176036

Chapter 16

Determination of Flutter Speed of 2D Nonlinear Wing by Using Describing Function Method and State-Space Formulation Güne¸s Kösterit and Ender Cigeroglu

Abstract Flutter is a phenomenon that occurs in wings or platelike structures as a result of aerodynamical forces when a certain flow speed, i.e., flutter speed, is reached. Flutter results in severe vibrations which eventually leads to fatigue failure of the wing. Many solutions are suggested against flutter phenomena. Wings or platelike structures under the effect of flowing air may contain nonlinearities due to connections or materials used. In this paper, effect of different structural nonlinear elements on the flutter speed is studied by using a 2D wing model. Aerodynamic lift and moment acting on the airfoil is obtained by utilizing Theodorsen’s unsteady aerodynamics which is only applicable to subsonic flow. In this paper, modified Theodorsen model for a 2D wing is used. To solve the flutter equation, several methods are suggested in the literature. Methods like k method and p-k method assume harmonic vibration in the generalized coordinates resulting in an eigenvalue problem. These methods are applied to linear systems. When nonlinearities are present in the system, numerical time marching solutions to differential equations are used; however, they are very costly in terms of computational time. In this study, state-space approach is utilized to obtain the flutter speed in frequency domain by using describing function method (DFM). The nonlinear system of differential equations is converted into a nonlinear eigenvalue problem utilizing state-space approach from which the flutter speed resulting in unstable solutions is obtained. Nonlinear eigenvalue problem obtained can be solved iteratively without time marching methods. This method of finding flutter speed is computationally much faster than solving nonlinear flutter problems in time domain. Free play nonlinearity is a frequently observed nonlinearity where there exists a gap at both sides of the wing after which it is restricted by stiffnesses. Piecewise linear stiffness is a symmetric nonlinearity similar to free play where finite stiffness exists instead of a zero stiffness in between the gap. Softening cubic stiffness is a nonlinearity where stiffness of the structure decreases as the amplitude of the vibration increases. In this study, free play (gap nonlinearity), piecewise linear stiffness, and cubic stiffness nonlinearities acting on the rotational degree of freedom are considered in the case studies. Results obtained for these nonlinearities are presented and compared with each other. Keywords Unsteady aerodynamics · Describing function method · Flutter speed · Nonlinear wings · Frequency domain

16.1 Introduction Determination of flutter speed is a critical aeroelastic analysis in the design of air vehicles. Wings that are experiencing flutter are bound to fail due to fatigue; therefore, flutter should always be avoided. In order to do so, the speed at which flutter occurs, i.e., flutter speed, should be identified to prevent occurrence of flutter during flight. To calculate flutter speed, many methods and procedures are available. However, due to linkages or other mechanical parts present in the system, the structural equations may contain nonlinearities which make the flutter speed determination computationally a very expensive and complex process. In this paper, a frequency domain approach is used to determine flutter speed of a 2D aeroelastic model which have structural nonlinear elements attached to the torsional degree of freedom. There exist many aerodynamic models in literature which do not require CFD solutions. In this paper Theodorsen’s unsteady aerodynamics [1, 2] are used. This theory is a simple yet effective theory for subsonic flows. For supersonic flows, piston theory [3] can be used. The procedure described in this paper can be applied to other aerodynamics theories. Linear flutter problems can be solved by utilizing many methods. The most common methods are p, k(g), and p-k methods [2, 4]. These methods require to build up flutter equations and then solve them by using flutter determinant. The drawback of

G. Kösterit · E. Cigeroglu () Department of Mechanical Engineering, Middle East Technical University, Ankara, TR, Turkey e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_16

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these methods is that they focus on the flutter speed determination rather than the solution of differential equations; hence, mode shape information is not obtained. Most of the time, this is not required in flutter speed determination, but it is required in coupled solvers. In this paper, state-space approach is used to solve the flutter equations. State-space formulation is using the same principle as the p-k method. P-k method uses simplified artificial damping introduction to the system where state space directly focuses on solving the differential equations and eigenvalue problem. State space is preferred in this paper due to its simplicity and its ability to give information on mode shapes of the system. Determination of flutter speed for nonlinear wings are most of the time performed in time domain which is a very costly and slow process. In this paper, describing function method is used to determine flutter speed of a nonlinear 2D aeroelastic model. The nonlinear elements are attached to the torsional degree of freedom, and variation of flutter speed for different vibration amplitudes is obtained.

16.2 Modelling 16.2.1 Structural Modelling of the Airfoil A sketch of a 2D airfoil is given in Fig. 16.1, 2 DOF airfoil model. h is the translational generalized coordinate and θ is the rotational generalized coordinate. Points P and Q represent, respectively, the reference point (where h is measured) and the aerodynamic center. kh and kθ are the structural stiffness for h and θ generalized coordinates. Equation of motion for this simple 2D model can be obtained as follows: mh¨ + mbx θ θ¨ + kh h = −L,

(16.1)

IP θ¨ + mbx θ h¨ + kθ θ = M.

(16.2)

where m is airfoil mass, b is half chord of the wing, xθ is a parameter to calculate relative mass, IP is moment of inertia, L is lift from the aerodynamic forces, and M is moment from aerodynamic forces. The dots over the generalized coordinates represent the time derivative.

16.2.2 Aerodynamic Forces Aerodynamic forces can be obtained through many different theories and methods. The most common method is to obtain the aerodynamic forces as a function of structure deformation which requires a two-way coupling between a fluid solver and structural solver at every time increment. This approach is very expensive due to the coupled solution at each time increment. Using aerodynamic theories such as piston theory, Theodorsen’s theory, and Peter’s theory decreases the computational time since they don’t require expensive CFD simulations. Moreover, these theories can be used in a frequency domain solution. These advantages come with a compromise of accuracy and application range of the theories used. Every aerodynamic theory comes with certain assumptions, and if these assumptions are not violated, they result in faster determination of aerodynamic forces. Fig. 16.1 2 DOF airfoil model

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In this paper, modified Theodorsen’s unsteady flow theory is used [2]. Theodorsen’s theory is first published in 1935 for an airfoil with a control surface. In this paper, the modified version given by Hodges and Pierce, 2011 [2], is used. This theory assumes very thin plate, incompressible flow, and experiencing small harmonic oscillations, which is used to obtain the lift and moment created by the flow in this paper. According to [2] aerodynamic lift and moment can be obtained as follows:     1 L = 2πp∞ U bC(k) h˙ + U θ + b − a θ˙ + πp∞ b2 h¨ + U θ˙ − ba θ¨ , 2 M = −πp∞ b3

(16.3)

    1 1¨ 1 a − θ¨ + b + a L. h + U θ˙ + b 2 8 2 2

(16.4)

where C(k) is defined as: C(k) =

H1 (2) (k)

(16.5)

H1 (2) (k) + iH0 (2) (k)

Hn (2) = Jn (k) − iY n (k).

(16.6)

Note that Eqs. (16.1) and (16.2) are homogenous differential equations. Substituting Eqs. (16.3) and (16.4) into Eqs. (16.1) and (16.2) and writing in matrix form, the following relation is obtained: Mq¨ + Cq˙ + Kq = 0.

(16.7)

M, C, and K are mass, damping, and stiffness matrices which are defined as follows: ⎡ ⎢ ⎢ M=⎢ ⎣ ⎡ ⎢ ⎢ C=⎢ ⎣

mb2 + πp∞ b4 mb2 xθ +

πp∞ b4 − 2



⎤

mb2 xθ − b4 aπp∞ 

1 + a πp∞ b4 IP + πp∞ b4 2



1 a − 8 2



 +



1 + a πp∞ b4 a 2

⎥ ⎥ ⎥, ⎦

 1 − a + πp∞ b3 U 2      

1 1 3 3 1 + a −b + a 2πp∞ U C(k) − a + πp∞ U + πp∞ b3 U − 2πp∞ U b C(k) 2 2 2 

2πp∞ U b3 C(k)

2πp∞ U b3 C(k)

⎡

⎤ ⎥ ⎥ ⎥, ⎦

(16.8)

⎤ 2πp∞ U 2 b2 C(k) ⎢ ⎥   ⎥, K=⎢ ⎣ ⎦ 1 2 2 + a 2πp∞ U C(k) 0 kθ − b 2 kh b 2

q = (h/bθ )T .

16.2.3 State-Space Formulation Flutter is the instability in the system resulting in unbounded solutions. One way of identifying these unstable solutions is to convert the set of second-order differential equations of motion into a set of first-order differential equations, i.e., to write them in state-space form and determine the eigenvalues of the coefficient matrix. Note that the solution of the differential equation system is not required. Just the stability characteristic of the equations is important.

0 M M C

 

  q¨ −M 0 q˙ + = 0, q˙ 0 K q

(16.9)

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A˙y + By = 0,

(16.10)



0 M −M 0 A= , B= . M C 0 K

(16.11)



To solve the differential equation given in Eq. (16.10), harmonic solution should be assumed. Harmonic solution can be assumed since this is a constant coefficient linear homogeneous differential equation. The harmonic solution may have an exponentially decaying solution or exponentially increasing solution. Therefore, q and y can be written as follows: ˆ λt , q(t) = qe

(16.12)

y = yˆ eλt .

(16.13)

(Aλ + B) yˆ eλt = 0,

(16.14)

−A−1 Bˆy = yˆ .

(16.15)

Substituting Eq. (16.13), gives Eq. (16.14):

This eigenvalue problem can be solved to obtain eigenvalues and eigenvectors. Note that the real part of the eigenvalues gives information on the divergence characteristic of the system. Positive real part for an eigenvector shows that the solution has an exponentially increasing term which is the flutter problem. Imaginary part of the eigenvectors shows the harmonic frequency of the system. The eigenvalue problem given in (16.15) results in four different complex eigenvalues. Two of these eigenvalues result in negative oscillation frequency which is physically impossible. Therefore, these two eigenvalues should be neglected, and the remaining two eigenvalues should be considered in the determination of the flutter speed each of which correspond to a vibration mode of the system. The speed at which one of these eigenvalues return a positive real part is the flutter speed. For flow speeds higher than this value, the flutter should occur.

16.2.4 Nonlinear Wing Model In this paper, nonlinear elements are attached to the rotational degree of freedom. Nonlinearity can be introduced to linear equation of motion as follows: A˙y + By + fn (y) = 0.

(16.16)

The nonlinear vector function, fn (y), can be any nonlinear element. Assuming a single harmonic solution for y, nonlinear internal forcing vector can be written as follows by using describing function method [8]. fn (y) =  · y,

(16.17)

where  is the nonlinearity matrix. Since the nonlinearity in this study is connected to the rotational degree of freedom, nonlinearity matrix can be written as follows: ⎡

0 ⎢0 =⎢ ⎣0 0

0 0 0 0

0 0 0 0

⎤ 0 0⎥ ⎥. 0⎦ ν

(16.18)

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ν is the describing function of the nonlinear element and it can be defined as follows: i ν= πA

#

2π

fN (A sin (ψ)) e−iψ dψ,

(16.19)

0

where fN (x) is the nonlinear function, Ais the amplitude of the assumed harmonic solution, and ψ = ωt. Substituting Eq. (16.17) into Eq. (16.16), eigenvalue problem for the nonlinear system can be obtained as follows: A−1 [B + ] yˆ = yˆ .

(16.20)

It should be noted that nonlinearity matrix is a function of displacement amplitude; hence, in order to obtain the flutter speed, one needs to know the amplitude of angular deflection of the airfoil. Therefore, flutter speeds are obtained as a function of amplitude of angular displacement. The piecewise linear stiffness is a stiffness element where the stiffness value differentiates between two linear values. In this paper nonlinearity used has a higher stiffness value for higher deflections. The describing function for this nonlinearity can be defined as follows: ⎧ ⎫ k1 A < δ⎬ ⎨   %    2 ν = 2(k1 −k2 ) , (16.21) ⎩ π sin−1 Aδ + Aδ 1 − Aδ + k2 A > δ ⎭ here, k1 and k2 are the initial stiffness and secondary stiffness and δ is the gap amount where secondary stiffness becomes active. Free play nonlinearity is a special case of piecewise linear stiffness nonlinearity where k1 is zero. This is a common nonlinearity in wings which is caused by a gap between structural connection points. Cubic softening stiffness nonlinearity is an accurate representation of an elastic element. Elastic elements are assumed to have a linear force displacement relationship. However, as the displacement increases, increase in the force decreases resulting in softening behavior. This nonlinearity is selected for representation of such behavior. The describing function for softening cubic stiffness is as follows: 3 ν = k − kc A2 , 4

(16.22)

where k is the linear stiffness and kc is the cubic stiffness.

16.3 Results Procedure described in previous section is applied on an example case with same parameters given by Irani, S., and S. Sazesh, 2013 [5], and parameters are given in Table 16.1. Firstly, the linear system is solved. Linear system is solved using incremental speed increase, and the two eigenvalues obtained are plotted with real and imaginary parts in Fig. 16.2, real part of the eigenvalue as a function of speed, and Fig. 16.3, imaginary part of the eigenvalue as a function of speed. Note that, the eigenvalues with negative imaginary parts are neglected. The second mode is where flutter occurs when real part crosses 0. This method gives flutter speed to be 21.04. In reference [5] the flutter speed is found to be 21.02 which shows that this procedure agrees well with existing literature. Table 16.1 Parameters for the test case

b p∞ a xθ Ip kh kθ m

0.1 m 1.225 kg/m3 −0.41 0.15 0.00147kgm2 590 N/m 11.45 nm/rad 0.49kg

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Fig. 16.2 Real part of the eigenvalue as a function of speed

Fig. 16.3 Imaginary part of the eigenvalue as a function of speed

Fig. 16.4 Flutter speed as a function of deformation for free play and piecewise linear stiffness nonlinearity

Next, this procedure is applied to wings with different nonlinear elements attached to the rotational degree of freedom. As explained in the previous sections, flutter speed is calculated for different deformation amplitudes. For each deformation amplitude, the flutter speed obtained is given in Figs. 16.4 and 16.5. For some nonlinearities, as deformation increases the flutter speed increases; for some as deformation increases the flutter speed decreases. It is due to the nature of the nonlinearity present.

16.4 Conclusion In this paper flutter speed is obtained for wings with nonlinear connection elements by considering a two degrees of freedom airfoil model using Theodorsen unsteady aerodynamics. Flutter equations are solved by using state-space approach, and nonlinear elements are modelled by utilizing describing function method.

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Fig. 16.5 Deformation vs flutter speed plot for softening cubic stiffness

The advantage in using the state-space method over widely used methods such as p-k or k method is the ease of distinction of the modes. In p-k method Laplace parameter p is analyzed which does not give information on mode shapes. Hence, this makes the distinction of the translational and rotational modes not possible. On the other hand, state-space method gives frequencies with corresponding mode shapes which makes it possible to identify which mode is critical for flutter and which structural parameters should be modified in order to prevent flutter in this particular mode. The method described in this paper to obtain flutter speed of wings with nonlinear connection elements is computationally very advantageous over time integration methods which are widely used in the literature. Time integration methods require significant computational power to calculate the flutter speed, since the flutter speed determination process requires several solutions for different flow speeds. On the other hand, the flutter speed can be obtained very quickly in frequency domain by the use of describing function method, similar to the calculation of frequency response functions. For free play nonlinearity, flutter speed is constant until the amplitude in rotational degree of freedom equals to δ value. As the amplitude increases further, stiffness comes into contact with the system and introduces additional stiffness. This additional stiffness increases the structural rigidity and as a result of which flutter speed increases as well. It should be noted that as the amplitude increases the nonlinear element behaves like a linear stiffness. Therefore, after a certain amplitude, flutter speed saturates and does not change. Piecewise linear stiffness nonlinearity is very similar to free play nonlinearity for which the gap is replaced by an elastic spring. Until the amplitude equals to δ, initial linear stiffness is connected to the system; hence, flutter speed is constant. When the amplitude becomes larger than δ, flutter speed increases similar to free play nonlinearity. As the deformation amplitude increases further, flutter speed becomes constant again, similar to the free play nonlinearity. For softening cubic stiffness nonlinearity, flutter speed decreases as the amplitude increases. As the stiffness element softens, structural rigidity of the overall system decreases resulting in flutter to occur at a lower speed.

References 1. Theodorsen, Theodore: Report no. 496, general theory of aerodynamic instability and the mechanism of flutter. J. Franklin Inst. 219(6), 766–767 (1935). https://doi.org/10.1016/s0016-0032(35)92022-1 2. Hodges, D.H., Pierce, G.A.: Introduction to Structural Dynamics and Aeroelasticity. Cambridge University Press (2011) 3. Ashley, H., Zartarian, G.: Piston theory-a new aerodynamic tool for the aeroelastician. J. Aeronaut. Sci. 23(12), 1109–1118 (1956). https:// doi.org/10.2514/8.3740 4. Hassig, H.J.: An approximate true damping solution of the flutter equation by determinant iteration. J. Aircr. 8(11), 885–889 (1971). https:// doi.org/10.2514/3.44311 5. Irani, S., Sazesh, S.: A new flutter speed analysis method using stochastic approach. J. Fluids Struct. 40, 105–114 (2013). https://doi.org/10.1016/ j.jfluidstructs.2013.03.018 6. Price, S., et al.: The aeroelastic response of a two-dimensional airfoil with bilinear and cubic structural nonlinearities. In: 35th Structures, Structural Dynamics, and Materials Conference (1994). https://doi.org/10.2514/6.1994-1546 7. Ding, Q., Wang, D.-L.: The flutter of an Airfoil with cubic structural and aerodynamic non-linearities. Aerosp. Sci. Technol. 10(5), 427–434 (2006). https://doi.org/10.1016/j.ast.2006.03.005 8. Tanrikulu, O., et al.: Forced harmonic response analysis of nonlinear structures using describing functions. AIAA J. 31(7), 1313–1320 (1993). https://doi.org/10.2514/3.11769

Chapter 17

Application of Geometrically Nonlinear Metamaterial Device for Structural Vibration Mitigation Kyriakos Alexandros Chondrogiannis, Vasilis Dertimanis, and Eleni Chatzi

Abstract One of the major challenges encountered by civil structures throughout their life cycle pertains to exposure to dynamic loadings, with earthquakes representing an extreme case of such a load. The frequency content of a dynamic excitation is of primary importance not only because of the potential resonance it induces but also due to limitations that arise for the capabilities of vibration mitigation devices. A recently emerging technology for civil structure applications includes the development of metamaterial configurations. These are structures, which are formed by periodic arrangement of a fundamental component design, termed the unit cell. They can offer impressive filtering properties within specific frequency ranges, the so-called bandgaps. Considering the low-frequency content of earthquake excitation, a design that features a bandgap in the lower-frequency range is required. In this study, the potential of a geometrically nonlinear design for vibration mitigation purposes is investigated for lowering of the corresponding bandgap. The system consists in the periodic arrangement of nonlinear unit cells, each including a triangular arch configuration, which under large displacement considerations can produce not only geometrically nonlinear behavior but also negative stiffness effects. Analytical derivations result to the determination of the amplitude-dependent bandgap of the system. The proposed configuration is attached to a target structure subjected to protection. An assessment on the capabilities of the device toward this direction was performed via numerical analyses, revealing considerable effectiveness. Acceleration response and energyrelated measures are considered in the evaluation of the system’s performance. An additional potential, which the proposed configuration can offer, refers to the applicability of the system for retrofitting purposes of existing structures. It is concluded that the system can offer significant vibration mitigation capabilities, while further study and development of the design, taking into consideration constructability limitations, can lead to an efficient passive vibration absorption device. Keywords Metamaterials · Vibration mitigation · Geometric nonlinearities · Structural protection · Negative stiffness

17.1 Introduction Vibration mitigation devices target the counteraction of the effects of adverse dynamic loading, with particular focus on earthquake excitation. A broadly exploited solution in this direction is that of the tuned mass damper [1]. A linear attachment to a primary system leads to significant vibration attenuation capabilities at specific frequencies, which under proper adjustment can be tuned to match the natural frequency of the protected structure. Extension of this concept leads to the study of nonlinear attachments, investigating nonlinear targeted energy transfer phenomena [2]. Recent developments in vibration absorption devices study the effect of metamaterial configurations [3–5]. These are formations of repeating patterns of a fundamental design, which is called the unit cell. They can lead to extraordinary filtering properties within a specified frequency range, called the bandgap [6, 7]. These stop bands can be created by Bragg scattering phenomena, related to the periodicity of the lattice, or by local resonance effects, utilizing similar principles to the tuned mass damper [8, 9]. Several studies investigate the application of linear metamaterial designs for structural protection. Realistic application studies include designs in the form of barriers, inserted between the soil and a protected structure, or foundations that support the weight of the structure [10, 11].

K. A. Chondrogiannis () · V. Dertimanis · E. Chatzi Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, Zurich, Switzerland e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_17

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In an attempt to explore the full potential that metamaterial configurations can offer by widening the bandgap range, nonlinear behavior is investigated. Geometrically nonlinear behavior, negative stiffness elements, and non-smooth phenomena in the form of impacts are being studied in the literature as potential solutions for optimizing the designs [12– 16]. In the works of Chen et al. [17, 18] and Al-Shudeifat et al. [19], negative stiffness elements are investigated in the form of shallow arch configurations for nonlinear composite structures and energy sink applications. The current study utilizes triangular arch designs that produce geometrically nonlinear behavior, in a metamaterial configuration for vibration mitigation purposes.

17.2 Nonlinear Unit Cell In the current work, geometrically nonlinear behavior is considered for metamaterial development. This fundamental configuration consists of a rigid support of mass m, to which a triangular arch is attached, connected to mass μ, as depicted in Fig. 17.1. It is formed by two identical linear springs of stiffness kn in triangular arrangement. As the tip of the arch undergoes large displacements, geometrically nonlinear behavior occurs, which leads to negative stiffness phenomena. Successive cells are elastically interconnected with linear springs of stiffness k, while a dashpot element is considered both at the connection of adjacent cells and between the masses of each cell, with damping coefficients c and cn , respectively.

17.3 Geometrically Nonlinear Behavior By examining the nonlinear configuration of the system depicted in Fig. 17.2, it is interesting to focus on the nonlinear equilibrium path that it is created. Although the behavior of the system is clearly nonlinear, as inferred from the equilibrium path in Fig. 17.2, elastic behavior is present throughout the deformation. A set of symmetric equilibrium points is identified for δ = 0 and δ = 2H, representing the two symmetric positions, where the springs are at their natural length. Additionally, a third equilibrium point can be pinpointed for v(t) = H, which is unstable and corresponds to the position where the two springs are aligned. It is observed that the system is able to not only produce nonlinear behavior but further exhibits negative stiffness phenomena, for specific displacement values, which is a result of its bistable nature. The higher the initial height value H the more intense the negative stiffness phenomena, as inferred from the slope of the nonlinear path in Fig. 17.2. The exact nonlinear force-displacement relation can be calculated in Eq. (17.1). This function of δ can be approximated with a third-order polynomial, after using Taylor series expansion about v = H, according to Eq. (17.2) [19]. ,  ⎞ -  L 2  δ 2 1 − Hδ + 1 − H ⎠% P (δ) = 2 · kn · L ⎝1 − . H  2  L 2  L +1 + 1− H ⎛

H

where L =

√

L2 + H 2

Fig. 17.1 Schematic representation of the geometrically nonlinear unit cell

 δ 2 H

(17.1)

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Fig. 17.2 (a) Triangular arch configuration, (b) nonlinear equilibrium path

P (δ) ≈ k1 · (δ − H ) +k2 · (δ − H )3 , k1 = −2 · kn · k2 =

kn L L3

L L

−1

(17.2)

17.4 Dispersion Relation An important aspect that needs to be identified in a metamaterial design is the dispersion relation. This provides a useful indication with respect to the frequency ranges that are filtered by the configuration revealing the bandgap of the system. In order to identify this relation, it is crucial to consider an infinite lattice of identical unit cells, as shown in Fig. 17.3. This consideration enables the assumption of periodic solutions for the oscillation of the cells [12], as in Eq. (17.3).   uj ±1 (t) = U1 eiωt + U 1 e−iωt e±iq  iωt  νj ±1 (t) = V1 e + V 1 e−iωt e±iq ν(t) = u(t) − y(t)

(17.3)

of mass m and μ of the j-th cell, respectively, q is the reduced wave number, ω is where uj (t) and yj (t) are the displacements √ the angular frequency, and i = −1. After the application of Eq. (17.2) for the approximation of the nonlinear interaction between mass m and μ for each unit cell, the equations of motion of the system read:    mu¨ j (t) + k uj (t) − uj −1 (t) − νj −1 (t) + k1 νj (t) + k2 νj (t)3 = 0     μ u¨ j (t) − ν¨ j (t) + k uj (t) − νj (t) − uj +1 (t) − k1 νj (t) − k2 νj (t)3 = 0

(17.4)

Substitution of Eq. (17.3) into Eq. (17.4) yields the dispersion relation of the system as follows:     2k k1 + 3k2 V1 V 1 − (m + μ) k + k1 + 3k2 V1 V 1 ω2 + mμω4 cos(q) =   2k k1 + 3k2 V1 V 1

(17.5)

Focusing on the above relation, it is important to analyze its dependence on the relative oscillation amplitude between the nonlinearly connected masses of each cell. This is included in the ansatz of v in Eq. (17.3) for V1 and V 1 coefficients and appears in Eq. (17.5), in the form of V1 V 1 (V1 V 1 = |V1 |2 ). Solution of Eq. (17.5) for discrete frequencies is possible after defining the value of |V1 | and therefore V1 V 1 parameter. Figure 17.4 depicts the solution of the dispersion relation of the infinite lattice. In contrast to linear systems, this is a three-dimensional representation that contains the additional parameter of the oscillation amplitude. The acoustic and optical branches appear in the solution, which form the limits of the corresponding bandgap. It is observed that the bandgap shifts into the frequency domain depending on the oscillation

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Fig. 17.3 Schematic representation of the infinite lattice, required for the calculation of the analytical dispersion relation

Fig. 17.4 Dispersion curves of the infinite lattice. Evolution of the bandgap with respect to the oscillation amplitude is denoted with green color. (m = 2, k = 103, μ = 1, kn = 2 ·103, H = 0.15, L = 0.5)

amplitude, as well as changes in width. For lower amplitudes, the triangular arch oscillates at small angles, thus establishing the system to be more flexible and therefore lowers the bandgap in the frequency range. For higher oscillation amplitudes, where the triangular arch experiences larger angles, the system becomes stiffer with a result of increasing the opening frequency of the bandgap. Additionally, for these high oscillation amplitudes, the width of the bandgap is increased, as the optical branch shifts significantly to higher frequencies.

17.5 Numerical Validation For the validation of the dynamic properties of the nonlinear system, numerical analyses have been performed. For this purpose, a finite lattice of N identical unit cells has been considered, as shown in Fig. 17.5. The configuration of Fig. 17.5 has been simulated numerically in MATLAB® environment, with the use of the ode45 function (AbsTol = 1·10−10 , RelTol = 1·10−7 ). For this purpose, the system is brought to a nonlinear state-space form:   z˙ (t) = g z(t), xg (t)  T z = u1 , u˙ 1 , y1 , y˙ 1 | . . . |uN , u˙ N , yN , y˙ N where uj (t) and yj (t) are the displacements of mass m and μ, respectively, of the j-th unit cell.

(17.6)

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Fig. 17.5 Schematic representation of a finite lattice, used for the numerical estimation of the dispersion relation

Fig. 17.6 The employed input sine-sweep excitation in the frequency range 0.1–20.0 Hz

The equations of motion of the first unit cell are formed as follows:       mu¨ 1 + c u˙ 1 − x˙ g + cn u˙ 1 − y˙ 1 + k u1 − xg + Fnl1 = 0     my¨ 1 + c y˙ 1 − u˙ 2 + cn y˙ 1 − u˙ 1 + k (y1 − u2 ) − Fnl1 = 0

(17.7)

For the i-th unit cell, the equations of motion are formed as follows:   mu¨ i + c (u˙ i − u˙ i−1 ) + cn u˙ i − y˙ i + k (ui − ui−1 ) + Fnli = 0     my¨ i + c y˙ i − u˙ i+1 + cn y˙ i − u˙ i + k (yi − ui+1 ) − Fnli = 0

(17.8)

For the last (N-th) unit cell, they are formed as follows:   mu¨ N + c (u˙ N − u˙ N −1 ) + cn u˙ N − y˙ N + k (uN − uN −1 ) + FnlN = 0   my¨ N + cn y˙ N − u˙ N − FnlN = 0

(17.9)



 

j j j j j j where Fnl = Fnl xrel = P xrel , xrel = uj − yj , and for P xrel , the exact relation of Eq. (17.1) is used. For the analyses in this section, the following parameters are considered, while damping is not included (c = 0, cn = 0): m = 2, μ = 1, k = 1 · 103 , kn = 2 · 103 , L = 0.5, H = 0.15, N = 64 The configuration is excited by base excitation xg , which follows a sine-sweep function in the frequency range 0.1–20.0 Hz for a time period of 50 seconds, as shown in Fig. 17.6. In order to estimate the dispersion relation of the setup, a two dimensional fast Fourier transform in time and in space is applied to the output states [20]. In this set of analyses, the reference output state was considered to be the displacement of mass m of each unit cell. Figure 17.7 shows the numerical estimations of the dispersion curves in comparison to the analytical approach. The two methods result in good agreement for both the acoustic and the optical branch. Furthermore, the shifting of the dispersion curves, and the corresponding widening of the bandgap, in the frequency domain, depending on the oscillation amplitude is evident in both the analytical and numerical results.

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Fig. 17.7 Comparison of the numerical against the analytically computed dispersion curves. The contour plot corresponds to the numerical analyses and dashed lines to the analytical solutions. (a)−|V1 | = 1.6H, (b)−|V1 | = 2.1H. (m = 2, k = 103 , μ = 1, kn = 2·103 , H = 0.15, L = 0.5)

Fig. 17.8 Structure to metamaterial coupling

17.6 Application of the Metamaterial Device for Vibration Mitigation In this section, the effect of the metamaterial configuration is evaluated with respect to its vibration mitigation capabilities. For this purpose, the model in Fig. 17.8 is studied. It consists of a reference structure founded on a raft foundation. The basement is laterally connected to the metamaterial configuration, while at the bottom the shear connection as a result of soil-structure interaction is considered. A simplification of this setup results in the model in Fig. 17.9. The shear connection between the basement and the soil is replaced by an elastic connection in the horizontal direction, while its stiffness KSSI is calculated following the work of Gazetas [21]. Vertically propagating shear waves are considered in this study. In that respect, in the simplified model, the excitation xg is applied to both ends of the setup, acting in the horizontal direction, as shown in Fig. 17.9. The parameters of the metamaterial configuration that were used are the following: m = 2.25Mgr, μ = 1.125Mgr, k = 45 · 106 kN/m, kn = 3400kN/m, cn = 10Mgr/s, L = 0.5m, H = 0.05m while the damping ratio in the connection between the cells was set to 5%. The protected structure corresponds to an equivalent of a five-story building with the following parameters: M = 50Mgr, Mbase = 24.4Mgr, K = 6040kN/m

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Fig. 17.9 Simplified model considering soil-structure interaction, in the form of its stiffness and damping properties KSSI and CSSI , respectively, acting in the horizontal direction

while 5% damping is considered. Finally, the stiffness properties of the soil-structure interaction is calculated, according to the work of Gazetas [21], as KSSI = 462349 kN/m, with 5% damping. This calculation corresponds to soil properties vs = 150m/s, ρ = 1700 kg/m3 , v = 0.5, considering undrained conditions. For the evaluation of the system’s performance in terms of mitigation of vibrations, numerical analyses have been performed for the nonlinear configuration, according to the previous section. The equations of motion of the system need to be adjusted for the final (N-th) unit cell, compared to the previous section as follows:   mu¨ N + c (u˙ N − u˙ N −1 ) + cn u˙ N − y˙ N + k (uN − uN −1 ) + FnlN = 0     my¨ N + c y˙ N − x˙ base + cn y˙ N − u˙ N + k (yN − xbase ) − FnlN = 0

(17.10)

For the basement, the equations of motion are formed as follows:     Mbase x¨ base + c x˙ base − y˙ N + k (xbase − yN ) + CSSI x˙ base − x˙ g +     + KSSI xbase − xg + C x˙ base − X˙ + K (xbase − X) = 0

(17.11)

Finally, for the top mass, the equations of motion are formed as follows:   M X¨ + C X˙ − x˙ base + K (X − xbase ) = 0

(17.12)

where xbase and X are the displacements of the basement and top mass, respectively. The state-space vector of Eq. (17.6) is updated in order to include the added degrees of freedom: T  z = u1 , u˙ 1 , y1 , y˙ 1 | . . . |uN , u˙ N , yN , y˙ N |xbase , x˙ base |X, X˙

(17.13)

As a first step, harmonic base excitation is applied in the form of ground acceleration x¨ g (t) = A sin (ωt), where a constant acceleration amplitude A is considered for all analyses. These were applied for 30 seconds, for discrete frequencies. The maximum response amplitude was recorded and plotted versus the corresponding excitation frequency, as shown in Fig. 17.10. The results are further compared to the response of the unprotected structure, where soil-structure interaction is considered. It is observed that the nonlinear system is able to reduce the acceleration response at the top mass for a frequency range >1 Hz, compared to the unprotected configuration. As the number of unit cells that are included in the metastructure is increased, the acceleration mitigation effect is more significant for frequencies >1 Hz. The inclusion of one unit cell results in slight reduction of the accelerations around the resonance frequency, with respect to the unprotected case. The inclusion of more cells in the lattice affects a wider range of frequencies, as observed in Fig. 17.10. Interestingly, the reduction of acceleration amplitudes at resonance frequency is not improved for a five-cell lattice compared to the two cell. The most prominent effects for the longer lattice are focused in frequencies ranging between 1.0 and 1.6 Hz. However, it can be pointed that in a frequency range 1 Hz and amplification for the lower range is intensified for an increasing number of unit cells. Subsequently, in the application of such a structure for

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Fig. 17.10 Frequency response function of the absolute acceleration of mass M, protected by the nonlinear configuration (A = 10 m/s2 ). Comparison to the response of the unprotected structure and an equivalent linear metamaterial

Fig. 17.11 (a) Artificial input acceleration time history, (b) comparison to the EC8 elastic spectrum (soil type D, agR = 0.36 g, γ i = 1)

vibration mitigation purposes, where an excitation of rich frequency content is applied, there needs to be a balance between the favorable properties of the setup within the bandgap and the unfavorable ones outside of it. To counteract this issue, a factor that can be crucial is the number of unit cells that comprise the metamaterial lattice, where the smaller the number of cells, the lower the amplification outside the bandgap regions. The effectiveness of the system is further studied on the vibration mitigation potential on the target structure against a realistic input earthquake time history. For this purpose, a modification was performed to the JMA earthquake record [22], so that it meets the elastic spectrum of EC8 [23], according to Ferreira et al. [24], the time history and response spectrum of which are depicted in Fig. 17.11 and is applied to the system of Fig. 17.9 as ground excitation xg . Figure 17.12 displays the acceleration response time history of three individual systems for the earthquake input excitation. The configurations that are studied include the unprotected setup with the consideration of soil-structure interaction, the nonlinear setup of Fig. 17.9, and an equivalent linear system, the two latter consisting of two unit cells each. In the equivalent linear system, the nonlinear element is replaced by a linear relation, the stiffness of which is determined by the tangential stiffness of the triangular arch at the stable equilibrium points. It is observed that both metamaterial configurations are capable of reducing significantly the recorded accelerations at the protected structure, compared to the unprotected case. In terms of the maximum acceleration on the primary mass, the linear system offers a reduction of 16%, while the nonlinear system enhances this reduction to 20%. The effect of the metamaterial design is quantified more clearly by evaluating energybased measures, which can offer additional indication of damage [25]. For this purpose, the calculation of the cumulative total energy (kinetic+potential) of the primary structure is performed as follows: #

t

Ecum (t) =

EK (τ ) + EP (τ ) dτ

0

where EK and EP are the kinetic energy and potential energy, respectively.

(17.14)

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Fig. 17.12 (a) Acceleration response of the primary mass. (b) Cumulative energy of the primary mass. All metamaterial configurations consist of two unit cells

The comparison of the results depicted in the energy plot of Fig. 17.12 reveals the efficacy of the metamaterial designs. Comparing the nonlinear to the linear application, a further reduction of the total energy is achieved. The linear setup results to a reduction of 29%, compared to the unprotected case, while the nonlinear device improves this reduction to 36%, indicating great potential for further development. It is important to consider that for the investigated setup of Fig. 17.8, the primary structure is strongly connected to the ground via the soil-structure interaction properties. Therefore, any reduction of the output response is seriously more challenging than the case where the protected system is completely detached from the ground. This application can be found useful for retrofitting of existing structures, where the application of vibration mitigation measures below the building is not preferable.

17.7 Conclusions This study investigates the dynamic properties of a geometrically nonlinear metamaterial device. The dispersion relation of the system is both analytically and numerically investigated. The two approaches are in agreement in terms of the predicted dispersion curves, while the amplitude dependence of the relation has been identified. Therefore, the created bandgap is calculated, indicating the frequency ranges where wave propagation is prohibited. The effectiveness of the configuration is investigated on a reference structure with the consideration of soil-structure interaction. The frequency response of the design is calculated numerically, revealing the beneficial properties that it can offer to the protected structure in terms of vibration mitigation, within the created bandgap. Finally, the response of the protected structure is studied under the application of an earthquake record, which was modified in order to meet the elastic spectrum of EC8, containing the resonance frequency of the protected structure. The results were compared to the conventional unprotected structure and to an equivalent linear system, revealing the significant effect of both the linear and nonlinear metamaterial inclusions, as well as the contribution of nonlinearity in the design. Great potential can be identified for retrofitting purposes of existing structures, where the application of vibration mitigation measures below the foundation of the structure is not possible. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No INSPIRE-813424.

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Chapter 18

Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams with Spectral Chebyshev Technique and Harmonic Balance Method Demir Dedeköy, Ender Cigeroglu, and Bekir Bediz

Abstract In this paper, nonlinear forced vibrations of uniform and functionally graded Euler-Bernoulli beams with large deformation are studied. Spectral and temporal boundary value problems of beam vibrations do not always have closedform analytical solutions. As a result, many approximate methods are used to obtain the solution by discretizing the spatial problem. Spectral Chebyshev technique (SCT) utilizes the Chebyshev polynomials for spatial discretization and applies Galerkin’s method to obtain boundary conditions and spatially discretized equations of motions. Boundary conditions are imposed using basis recombination into the problem, and as a result of this, the solution can be obtained to any linear boundary condition without the need for re-derivation. System matrices are generated with the SCT, and natural frequencies and mode shapes are obtained by eigenvalue problem solution. Harmonic balance method (HBM) is used to solve nonlinear equation of motion in frequency domain, with large deformation nonlinearity. As a result, a generic method is constructed to solve nonlinear vibrations of uniform and functionally graded beams in frequency domain, subjected to different boundary conditions. Keywords Nonlinear vibration · Functionally graded beam · Forced response · Harmonic balance method

18.1 Introduction Nonlinear vibration analysis of beams plays an important role in the design of many engineering structures, especially those experiencing dynamic loads such as airplane wings, wind turbines and jet engine blades, electronic boards, etc. Many failures in these structures can be predicted through nonlinear vibration analysis of beams. To prevent failures, without constructing costly analysis models, it is possible to change the design of the structure (e.g., geometry, material property, etc.). Nonetheless, the complexity of the nonlinear problem and the deficiency in the literature increase the need for research in nonlinear beam vibrations. In large deformation nonlinearity, the deformations higher than the thickness of the beam result in stretching force which induces nonlinear behavior. This nonlinearity is reflected in the differential equations of motion. The resulting nonlinear differential equations can be solved by time domain or frequency domain methods. For instance, Chakrapani et al. [1] and Swain et al. [2] studied the force vibration of composite beams in time domain. Similarly, Liao-Liang Ke et al. [3] worked on free vibration of geometrically nonlinear composite beams. However, frequency domain methods are computationally very efficient compared to time domain methods to obtain the steady-state response. Harmonic balance method (HBM) and describing function method (DFM) are the most common frequency domain methods used to obtain the steady-state response of nonlinear systems. For example, H. Youzera et al. [4] implemented HBM in the solution of forced vibration of symmetric laminated composite beams.

D. Dedeköy · E. Cigeroglu () Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey e-mail: [email protected] B. Bediz Mechatronics Engineering Program, Sabanci University, Istanbul, Turkey Middle East Technical University, Ankara, Turkey e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_18

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In this paper, HBM is employed to convert the nonlinear differential equations of motion into nonlinear algebraic equations. The resulting set of nonlinear algebraic equations is solved by using Newton’s method with arclength continuation. Several case studies are performed on uniform and functionally graded beams. Simply supported, fixed-fixed, and fixedpinned boundary conditions are considered in the case studies. Deflections at the middle point of the beam are presented as a function of frequency for different mechanical properties and different external excitations.

18.2 Theory In the SCT, Chebyshev polynomials are used to spatially discretize the beam. Transverse displacement function of a beam can be expressed by Chebyshev series expansion as follows:

y(x) =

∞ 

(18.1)

αk Tk (x).

k=0

where Tk (x) is Chebyshev polynomials of the first kind which can be given as follows [5]:

 Tk (x) = cos k cos−1 (x) f or k = 0, 1, 2 · · · .

(18.2)

The displacement function of the beam can be represented by sampled points at certain increments for numerical calculations. If the sampling point number is selected the same as the number of Chebyshev polynomials, there occurs a oneto-one mapping between the sampled points and Chebyshev coefficients α k [5]. For N number of Chebyshev polynomials, N number of Gauss-Lobatto points are used for sampling spatial domain, which are defined as follows:  pk = cos

(k − 1) π N −1

 .

(18.3)

The relation between sampled displacement function of the beam and the Chebyshev expansion coefficients can be written as follows: a = F y

(18.4)

where  F is an N x N forward transformation matrix. Additionally, backward transformation matrix is defined which is the inverse of forward transformation matrix. Derivative and integral of the any function and vector constructed by Chebyshev polynomials can be obtained as follows:

#

l2

y(n) = Qn y

(18.5)

y(x)dx = vT a.

(18.6)

l1

Here Qn is the derivative matrix with respect to order n. v is the definite integral vector. Derivation of the Qn and v according to spectral Chebyshev method is given in reference [6]. Equation of motion of a Bernoulli beam is written as follows:

∂2 ∂ 2 y (x, t) ∂ 2 y (x, t) = f (x, t) . E(x)I + ρA ∂x 2 ∂x 2 ∂t 2

(18.7)

18 Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams. . .

121

Boundary conditions for this equation can be written in a generic way such as: βij 3 y + βij 2 y + βij 1 y + βij 0 y = αij (t)

(18.8)

Here βs are the constants of the spatial part of the boundary condition, whereas αs are the constants of the temporal part. Both can be written in vector form. The i and j indices correspond to the boundary location (0 and L, i = 1,2) and the number of the boundary condition (j = 1,2). When boundary conditions change, the derivation of equation does not change; only these matrices change. An important step in imposing boundary conditions is expressing y by using projection matrices [6] as follows: y = Pz + Rα

(18.9)

This technique makes the problem solvable for z which only satisfies homogenous boundary conditions, where y satisfies all the boundary conditions [18.6]. Calculation of P and R matrices are given in reference [6]. In order to obtain the approximate solution, Galerkin’s method is applied. Imposing projection matrices into Eq. 18.8, the residual from approximation is written as  .. ..  φ = m Pz + Rα − Q4 (Pz + Rα) − f

(18.10)

 where m = ρA EI and f is normalized by dividing it with EI. To minimize this residual, the inner product of weighted residuals must vanish, i.e.: #

l2

θ (x)φ(x)dx = θT Vφ = 0

(18.11)

l1

The inner product of any two functions θ (x) and φ(x) can be constructed by Chebyshev polynomials by using the inner product matrix (V) which is described in Appendix. Including derivative matrices, boundary projection matrices, and applying Galerkin’s method, Eq. (18.7) can be written as follows: 

   ρA ρA .. .. T T T P VPz + P VQ4 Pz = P Vf − PT VRα − PT VQ4 Rα EI EI

(18.12)

In Eq. (18.12), mass and stiffness matrices and the forcing vector can be obtained as follows:  M= .

 ρA PT VP, K = PT VQ4 P, f∗ = PT Vf EI

(18.13)

..

Since for basic boundary conditions α, α, and α are 0, f term is simplified to the form given in Eq. (18.13). When a mechanical property varies along the beam’s longitudinal direction, the related inner product matrix changes. As a study, Young’s modulus is considered to be varying along the beam’s length such as: E (x) = (E2 − E1 ) x/L + E1 .

(18.14)

With respect to this variation, the E value in the equation of motion is not a scalar anymore but a function. By using chain rule, applying Galerkin’s method and imposing boundary projection matrices to the equation of motion, the following equation is obtained:

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   .. I PT VE Q4 + 2VE Q3 + 2VE Q2 Pz + ρAPT VPz = PT Vf

(18.15)

Here, VE is the inner product matrix with respect to varying Young’s modulus. Calculation of VE , VE , and VE is given in Appendix. With respect to Eq. (18.15), new mass and stiffness matrices and the forcing vector of the system become as follows:  M=

    ρA 1 T T   ∗ P VP, K = P VE Q4 + 2VE Q3 + 2VE Q2 P, f = PT Vf I I

(18.16)

18.2.1 Application of Harmonic Balance Method The equation of motion of a Euler-Bernoulli beam with geometric nonlinearity is given as follows: ∂ 4y ∂ 2y EI 4 + ρA 2 = ∂x ∂t Here the term

EA 2L

/ l2 ∂y 2 l1

∂x

dx

∂2y ∂x 2



EA 2L

#

l2  ∂y 2 l1

∂x

 dx

∂ 2y + f (x, t) . ∂x 2

(18.17)

comes due to the stretching effect occurring along the beam. This nonlinear

phenomenon is studied by many researchers with different solution techniques [7–10]. The nonlinear term in Eq. (18.17) can be written by using Chebyshev technique as follows: 

EA 2L

#

l2  ∂y 2 l1

∂x

 dx

∂ 2y EA T = v  F (Q1 Pz)2 Q2 Pz 2 2L ∂x

(18.18)

For the beam with variation of its Young modulus along length, nonlinear term can be defined as below, with respect to the system parameters defined at Eq. (18.16): 

EA 2L

#

l2  ∂y 2 l1

∂x

 dx

∂ 2y VE A T v  F (Q1 Pz)2 Q2 Pz = 2 2L ∂x

(18.19)

Equation (18.17) can be written in the following form: 

 ρA A T .. PT VPz + PT VQ4 P = v  F (Q1 Pz)2 Q2 Pz + PT Vf. EI 2I L

(18.20)

A single harmonic solution is assumed as follows: ⎧ ⎫ ⎪ ⎪ zs1 sin θ + zc1 cos θ ⎪ ⎪ ⎨ ⎬ . . z= ⎪ ⎪ . ⎪ ⎪ ⎩ ⎭ zsn sin θ + zcn cos θ

(18.21)

The z vector is placed into Eq. (18.20) and coefficients of the similar terms are balanced to determine the unknowns. The nonlinear term in the equation can be concluded to the form given below with the help of trigonometric relations.

18 Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams. . .



⎫ ⎧ ψs1 sin θ + ψc1 cos θ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ A T . v  F (Q1 Pz)2 Q2 Pz z = vL ⎪ ⎪ . 2I L ⎪ ⎪ ⎭ ⎩ ψsn sin θ + ψcn cos θ

123



(18.22)

where vL = 2IAL vT  F and ψs1 , ψc1 are the numerical expressions arising from harmonic balance method. With the addition of Eq. (18.21), nonlinear equation of motion can be converted into a set of nonlinear algebraic equations in frequency domain for sine cosine terms of the displacement as the unknowns. These nonlinear algebraic equations can be solved by using Newton’s method with arclength continuation [11].

18.3 Case Study Solving the eigenvalue problem with the system matrices defined in Eq. 18.13, the first five natural frequencies of a EulerBernoulli beam subjected to pinned-pinned, fixed-fixed, and fixed-pinned boundary conditions are given in Table 18.1. Additionally, exact solutions are calculated for the problem and given in the table. In these results, material of the beam is aluminum with the following properties E = 71 GPa, ρ = 2770 kg/m3 , w = 0.03 m (width), L = 1 m, and h = 0.01 m (thickness). The first five natural frequencies of the functionally graded beam are obtained with the previous parameters that are given, except E1 and E2, which are taken as 85.2 GPa and 28.4 GPa (Table 18.2). As expected, the natural frequencies of the beam decreased, since overall stiffness of the beam is reduced due to the new variation of Young modulus. Frequency response of the beams defined above (uniform and functionally graded) is solved with geometric nonlinearity. A sinusoidal force is applied at the midpoint of the beam and the transverse deflection at the midpoint is obtained for different excitation forcing amplitudes. A viscous damping coefficient of 0.03 is considered for the whole system by assuming proportional damping. The response was given as normalized amplitude which is displacement divided by force (Fig. 18.1, 18.2 and 18.3). Frequency response plot for functionally graded beam (E1 = 85.2 GPa, E2 =28.4 GPa.) with fixed-pinned boundary conditions is given below (Fig. 18.4). Additionally, a frequency response plot for a varying Young modulus beam with different scenarios is given in Fig. 18.5 (with 6 N force applied). As expected, if Young modulus at the end of the beam decreases, the overall stiffness of the beam also decreases. Hence, the natural frequency of the beam becomes lower and the deflection of the beam increases.

18.4 Conclusion In this paper a generic method is proposed to solve nonlinear vibrations of uniform and functionally graded beams. The method generates a fast solution for the problem in the frequency domain. If one is after the frequency response of any point along the nonlinear beam subjected to different boundary conditions, the method yields efficient solutions that are not computationally expensive. As case studies, the frequency response of the midpoint deflections of uniform and functionally graded beams subjected to different basic boundary conditions is presented.

1st 2nd 3rd 4th 5th

Natural frequencies (rad/s) Pinned-pinned Exact SCT with 9 solution polynomials 144.244 144.244 576.9769 576.994 1298.197 1298.677 2307.907 2452.392 3606.105 4012.955 SCT with 13 polynomials 144.244 576.977 1298.198 2307.957 3606.596

Fixed-fixed Exact solution 326.98 901.347 1767 2920.96 4363.407 SCT with 9 polynomials 326.985 901.647 1770.323 3242.881 5165.42 SCT with 13 polynomials 326.985 901.348 1767.003 2921.322 4365.938

Table 18.1 First 5 natural frequencies of uniform beam subjected to different boundary conditions

Exact solution 225.336 730.239 1523.586 2605.423 3975.749

Fixed-pinned SCT with 9 polynomials 225.337 730.309 1525.876 2811.163 4601.434

SCT with 13 polynomials 225.337 730.236 1523.58 2605.539 3977.551

124 D. Dedeköy et al.

1st 2nd 3rd 4th 5th

Natural frequencies (rad/s) Pinned-pinned SCT with 9 polynomials 126.63 503.519 1133.394 2070.412 3672.316 SCT with 13 polynomials 126.63 503.158 1130.656 2008.442 3139.349

Fixed-fixed SCT with 9 polynomials 282.558 781.206 1543.647 2682.509 4767.871 SCT with 13 polynomials 282.557 781.171 1533.353 2536.243 3799.932

Table 18.2 First 5 natural frequencies of a beam with varying Young modulus subjected to different boundary conditions Fixed-pinned SCT with 9 polynomials 203.332 641.88 1339.53 2341.85 4423.264

SCT with 13 polynomials 203.332 641.876 1331.586 2272.009 3468.51

18 Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams. . . 125

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2

10-3 3N 6N 9N

1.8

Normalized Amplitude (m/N)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 100

150

200

250

300

Frequency (rad/s) Fig. 18.1 Mid-point normalized deflection of the uniform beam with pinned-pinned boundary conditions

1.4

10-3 3N 6N 9N

Normalized Amplitude (m/N)

1.2

1

0.8

0.6

0.4

0.2

0 260

280

300

320

340

360

380

Frequency (rad/s) Fig. 18.2 Mid-point normalized deflection of the uniform beam with fixed-fixed boundary conditions

400

420

18 Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams. . .

10-3

1.6

3N 6N 9N

1.4

Normalized Amplitude (m/N)

127

1.2 1 0.8 0.6 0.4 0.2 0 140

160

180

200

220

240

260

280

300

320

340

Frequency (rad/s) Fig. 18.3 Mid-point normalized deflection of the uniform beam with fixed-pinned boundary conditions

1.8

10-3 3N 6N 9N

Normalized Amplitude (m/N)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 140

160

180

200

220

240

260

280

Frequency (rad/s) Fig. 18.4 Mid-point deflection of the functionally graded beam with fixed-pinned boundary conditions

300

320

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Normalized Amplitude (m/N)

1.5

10-3 E1=71, E2=71 GPa E1=71, E2=110 GPa E1=71, E2=50 GPa E1=71, E2=25 GPa

1

0.5

0 140

160

180

200

220

240

260

280

300

320

340

360

Frequency (rad/s) Fig. 18.5 Mid-point deflection of the functionally graded beam with fixed-pinned boundary conditions and different cases

Appendix Calculation of Inner Product Matrix Values ofany two functions f(x) and g(x) at N Gauss-Lobatto points are written as fN and gN . Product of interpolated functions has order of 2 N. f2N = S2 fN

(A.1)

S2 is constructed as follows: S2N =  B2N [IN ; ON ]  FN

(A.2)

Here B2N is the 2 N x 2 N backward transformation matrix. IN and ON are the N x N dimensional identity and zero matrices. The inner product of f(x) and g(x) can be written as follows: #

l2 l1

f(x)g(x)dx = fT V g = fT2N vd,2N g 2N

(A.3)

Here vd, 2N is a matrix whose diagonal has the elements of multiplication vT2N  F2N . Then the inner product matrix is written as follows: V = ST2N v d,2N S2

(A.4)

When the differential equation has variable coefficients, a weighted inner product is defined with respect to a weighting function γ(x). In the problem given in case study, γ(x) is the variation of the Young modulus distribution, E(x). Since there is a weighting function, the inner product has order of 3 N. Consequently, the inner product and inner product matrix can be described as follows:

18 Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams. . .

#

l2 l1

f (x)g(x)E(x)dx = fT3N VE gN VE = ST3N vd,3N Ed,3N S3

129

(A.5)

(A.6)

where vd,3n and Ed,3N are 3 N x 3 N matrices whose diagonals have the values of f3N and E3N The first and second derivative (with respect to x) of E(x) are E (x) and E(x). While finding the inner product matrix as described above, if E (x) is used, then VE  (x) is obtained. Similarly, if E(x) is used, then VE (x) is obtained.

References 1. Chakrapani, S.K., Barnard, D.J., Dayal, V.: Nonlinear forced vibration of carbon fiber/epoxy prepreg composite beams: theory and experiment. Compos. Part B. 91, 513–521 (2016) 2. Ke, L.-L., Yang, J., Kitipornchai, S.: Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos. Struct. 92(3), 676–683 (2010) 3. Swain, P.R., Adhikari, B., Dash, P.: A higher-order polynomial shear deformation theory for geometrically nonlinear free vibration response of laminated composite plate. Mech. Adv. Mater. Struct. 26(2), 129–138 (2017) 4. Youzera, H., Meftah, S.A., Challamel, N., Tounsi, A.: Nonlinear damping and forced vibration analysis of laminated composite beams. Compos. Part B. 43(3), 1147–1154 (2012) 5. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, United States (2013) 6. Yagci, B., Filiz, S., Romero, L.L., Ozdoganlar, O.B.: A spectral-Tchebychev technique for solving linear and nonlinear beam equations. J. Sound Vib. 321(1–2), 375–404 (2009) 7. Woinowsky-Krieger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17(1), 35–36 (1950) 8. Evensen, D.A.: Nonlinear vibrations of beams with various boundary conditions. AIAA J. 6(2), 370–372 (1968) 9. Singh, G., Venkateswara Rao, G., Iyengar, N.G.R.: Re-investigation of large-amplitude free vibrations of beams using finite elements. J. Sound Vib. 143(2), 351–355 (1990) 10. Singh, G., Sharma, A.K., Venkateswara Rao, G.: Large-amplitude free vibrations of beams—a discussion on various formulations and assumptions. J. Sound Vib. 142(1), 77–85 (1990) 11. Cigeroglu, E., Samandari, H.: Nonlinear free vibration of double walled carbon nanotubes by using describing function method with multiple trial functions. Physica E. 46, 160–173 (2012)

Chapter 19

Experimental Characterization of Superharmonic Resonances Using Phase-Lock Loop and Control-Based Continuation Gaëtan Abeloos, Martin Volvert, and Gaëtan Kerschen

Abstract Experimental characterization of nonlinear structures usually focuses on fundamental resonances. However, there is useful information about the structure to be gained at frequencies far away from those resonances. For instance, nonfundamental harmonics in the system’s response can trigger secondary resonances, including superharmonic resonances. Using the recently introduced definition of phase resonance nonlinear modes, a phase-locked loop feedback control is used to identify the backbones of even and odd superharmonic resonances, as well as the nonlinear frequency response curve in the vicinity of such resonances. When the backbones of two resonances (either fundamental or superharmonic) cross, modal interactions make the phase-locked loop unable to stabilize some orbits. Control-based continuation can thus be used in conjunction with phase-locked loop testing to stabilize the orbits of interest. The proposed experimental method is demonstrated on a beam with artificial cubic stiffness exhibiting complex resonant behavior. For instance, the frequency response around the third superharmonic resonance of the third mode exhibits a loop; the fifth superharmonic resonance of the fourth mode interacts with the fundamental resonance of the second mode; and the second superharmonic resonance of the third mode exhibits a branch-point bifurcation and interacts with the fourth superharmonic resonance of the fourth mode. Keywords Control-based · Superharmonic · Characterization

19.1 Introduction The design of mechanical structures with nonlinear behavior is a challenging task. It is often necessary to build reliable models upon physical experiments. Usually, post-processing the experimental data is a long procedure. To accelerate the design iterations, characterization methods relying on control have been proposed that identify nonlinear modes or nonlinear responses without the need for a priori knowledge about the experiment. Two prominent methods are phase-locked loop (PLL) testing [1] and control-based continuation (CBC) [2]. They control two different experimental parameters, the phase lag and the amplitude of the response’s first harmonic, respectively. Until now, PLL testing and CBC were used exclusively to characterize the primary resonance of structures, i.e., the resonance of the first harmonic. However, important information about the structure can be obtained through higher harmonics of the response. Typically, nonlinear modes can enter into resonance when the structure is excited at a fraction of the resonance frequency. These superharmonic resonances are included in the recent definition of phase resonance nonlinear modes (PRNMs) [3]. This work applies PLL testing to a structure possessing an artificial nonlinear stiffness presented in [4], to identify the backbone curves of superharmonic resonances and the response curve in their vicinity. In the presence of modal interactions, the PLL has been shown incapable of stabilizing certain unstable responses [5]. To improve the stabilization, PLL testing is coupled with online CBC as implemented in [4].

G. Abeloos () · M. Volvert · G. Kerschen Department of Aerospace & Mechanical Engineering, University of Liège, Liège, Belgium e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_19

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19.2 Methods The structure is excited by a signal that is a sine wave at steady-state f (t) = p sin(t). This work focuses on periodic & H responses x of the structure; they can be truncated to NH Fourier coefficients: x(t) = N k=1 xˆ k sin(kt + φk ). The value φk is the phase lag of the kth harmonic. The phase lag of each harmonic is estimated at every moment using adaptive filtering, as proposed in [4]. When applying PLL testing, the excitation frequency is output by a PI controller as proposed in [1]. / t The controller’s input is the difference between a phase target φ ∗ and the phase lag φk : (t) = κp (φ ∗ − φk (t)) + κi 0 (φ ∗ − φk (τ )) dτ . The controller, depending on its gains, cannot stabilize every response. To improve stabilization, the PLL can be coupled with CBC. The excitation signal is output by a PD controller whose input is the difference a reference signal  between  /t ∗ whose frequency is determined by the PLL and the response as proposed in [2]: f (t) = kp x sin( 0 (τ ) dτ ) − x(t) +   /t kd dtd x ∗ sin( 0 (τ ) dτ ) − x(t) . The amplitude x ∗ of the reference signal determines the amplitude of the excitation. Frequency response curves (FRCs) are obtained with PLL testing by keeping the excitation amplitude p constant and performing a sweep on φ ∗ . PRNMs are obtained with PLL testing coupled with CBC by keeping φ ∗ = −π/2 for odd superharmonic resonances (including primary resonances) [3] or φ ∗ = −3π/4 and φ ∗ = −7π/4 for even superharmonic resonances [3] and performing a sweep on x ∗ . PRNMs are defined upon the observation that phase lag drops by a value π when the frequency response crosses a resonance. However, it is observed that φ1 dropping by a value π entails that φ3 drops by a value 3π and φ5 by a value 5π . The odd harmonics therefore interfere with each other. A similar observation is made between even harmonics. Such interference is of no concern when the superharmonic resonances are well-separated but become important in the case of modal interaction. To counteract the interference of lower harmonics, the backbone of a 3rd superharmonic resonance is obtained by keeping φ3 −3φ1 = −π/2 and the backbone of a 5th superharmonic resonance by keeping φ5 −φ3 −(5−3)φ1 = −π/2. Further analytical work is needed to confirm these assumptions and find similar expressions for even superharmonic resonances.

19.3 Results By placing four equally spaced accelerometers from one quarter of the beam’s length to its tip, the mode shape corresponding to the beam’s response can be identified up to the fourth mode. The mode shape is decomposed in each harmonic. Each superharmonic resonance is associated with a mode, e.g., if the 5th harmonic follows the 4th mode shape, the superharmonic resonance is dubbed “H5M4.” Figure 19.1 shows all the identified features. In plain black curves, FRCs at different excitation amplitudes p in N, identified using PLL testing by sweeping φ1 near H1M2, φ2 near H2M3, φ3 near H3M3, and φ5 near H5M4. The dashed black curves are obtained by an open-loop sine sweep to complete the FRC at p = 1 N. Plain blue curves in Fig. 19.1 show the PRNMs identified using PLL testing coupled with CBC and the phase lags controlled at the displayed values. In the presence of modal interaction, the PRNM does not always correspond to resonant responses, and resonances can occur at other phase lags. Imposing the phase lag φ1 for the 1st harmonic, φ3 − 3φ1 for the third, and φ5 − φ3 − (5 − 3)φ1 for the fifth leads to the identification of more consistent superharmonic resonance backbones. They are shown in dash-dotted orange curves in Fig. 19.1.

19.4 Conclusion This work is, to our knowledge, the first to identify nonlinear superharmonic resonance backbones and frequency responses with control-based methods. It is also the first one to propose a coupling of PLL testing and CBC to improve the stabilization of responses when the PI controller of the PLL is not sufficient, e.g., in the presence of modal interaction. Further work on the definition of PRNMs is envisioned to include modal interactions.

19 Experimental Characterization of Superharmonic Resonances Using Phase-Lock Loop and Control-Based Continuation

4.5

10

133

-4

H5M4

4

H1M2

3.5

H4M4

Displacement [m]

H3M 3 3

3

= - /2

5

= -3 /2

p=4 3.5

1 2

1.5

0.75

1

2.5

0.25

5

= - /2

0.5 0 24

H2M3

3

0.5

1

= -3 /4

p=2 1.5

3

2

5

2

p=4

2.5

= - /2

26

28

30

32

34

36

38

40

42

44

Frequency [Hz] Fig. 19.1 Frequency response of the beam around its second primary resonance for different excitation amplitudes p in N obtained by PLL testing (plain black curves) and uncontrolled swept sine excitation (dashed black curves); PRNMs of resonances H1M2, H3M3, H5M4, and H2M3 obtained by PLL testing coupled with CBC (plain blue lines); backbone curves of the same resonances obtained by PLL testing coupled with CBC (dash-dotted orange curves)

Acknowledgments G.A. is funded by the FRIA grant of the Fonds National de la Recherche Scientifique (F.R.S.-FNRS). He gratefully acknowledges the financial support of the F.R.S.-FNRS.

References 1. Peter, S., Leine, R.I.: Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation. Mech. Syst. Signal Process. 96, 139–158 (2017) 2. Barton, D.A., Sieber, J.: Systematic experimental exploration of bifurcations with noninvasive control. Phys. Rev. E 87, 052916 (2013) 3. Volvert, M., Kerschen, G.: Phase resonance nonlinear modes of mechanical systems. J. Sound Vibr. 511, 116355 (2021) 4. Abeloos, G., Renson, L., Collette, C., Kerschen, G.: Stepped and swept control-based continuation using adaptive filtering. Nonlinear Dyn. 104, 3793–3808 (2021) 5. Givois, A., Tan, J.J., Touzé, C., Thomas, O.: Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification. Meccanica 55(3), 481–503 (2020)

Chapter 20

On Modelling Statistically Independent Nonlinear Normal Modes with Gaussian Process NARX Models Max D. Champneys, Gerorge Tsialiamanis, Timothy J. Rogers, Nikolaos Dervilis, and Keith Worden

Abstract Linear modal analysis has provided a robust and eminently useful framework for the analysis of structural dynamic systems. Considerable attention has been directed towards the development of a nonlinear variant of modal analysis that is effective in the presence of nonlinearities. Thus far, essentially two approaches to constructing nonlinear normal modes (NNM) have gained traction. The first was that of Rosenberg, whereby the modes are defined in terms of synchronous motions of the structure. The second is the geometrically more general approach of Shaw and Pierre, wherein modes are defined on invariant manifolds of the phase space of the system. A recent third approach from Worden and Green proposes a statistical definition of a nonlinear normal mode. Under this framework, the modal coordinates are defined by latent directions in the configuration space that result in statistically uncorrelated time series. This paper examines the properties of the NNMs generated by this framework, by applying techniques from nonlinear system identification (NLSI). Both linear and nonlinear time domain models are fitted to the physical and modal coordinates of a two degree-of-freedom simulated system with a cubic nonlinearity. It is demonstrated in this work, that the NNMs generated are able to decompose the system into independent functionals, and that the modal transformation generalises to lower excitation levels, while maintaining excellent reconstruction of the physical displacements. Keywords Nonlinear normal modes · Time series analysis · Gaussian process NARX models

20.1 Introduction Linear modal analysis is a mainstay in the practical analysis of linear dynamic systems. The framework has a number of desirable properties that permit analysis of linear dynamics over a wide array of applications. The principal concept of linear modal analysis is the specification of invariant, or modal, properties that are sufficient to entirely describe the underlying linear functional, the equations of motion (EOM). To the dynamicist, these are familiar as the mode shapes, natural frequencies, damping ratios and frequency response functions. The specification of these quantities has been attempted in myriad ways, using approaches ranging from direct derivation from the EOM [1], to sophisticated statistical methods that glean the parameters from measured data [2]. Once specified, these parameters give rise to the other great advantage of modal analysis, that of linear decoupling. Given proportional (or Raleigh type) damping, the dynamics of a generic multi-inputmulti-output (MIMO) linear system can be exactly expressed as a linear combination of single-input-single-output (SISO) oscillators and vice versa. This modal decomposition can be expressed as, y = u,

u =  −1 y

(20.1)

M. D. Champneys () Industrial Doctorate Centre in Machining Science, Advanced Manufacturing Research Centre with Boeing, University of Sheffield, Sheffield, UK Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected] G. Tsialiamanis · T. J. Rogers · N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_20

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where y are the physical displacements, u are the modal displacements and  is the modal matrix. This decoupling permits a greatly simplified analysis as each mode can be analysed independently and tested against prospective inputs individually, without concern about interaction with other modes. The overall response of the system can be found as a simple linear sum of such analyses—this is the principal of superposition. The presence of nonlinearity in structural dynamics is inescapable in the modern engineering landscape. As the demands of engineering structures become even more complex, composite materials, complex geometries and loading conditions outside linear-elastic regions are increasingly part of cutting-edge design. Regrettably, these conditions lead to nonlinearity. Nonlinearity breaks linear modal analysis in more than one place. The invariant modal properties cease to be invariant and often come to depend on the excitation energy of the system. Furthermore, a linear decoupling into SISO oscillators is no longer possible. Superposition is no longer possible. It is without wonder that much attention has been given to the extension of modal analysis to the nonlinear case. Although unproven, it is difficult to imagine that every positive attribute of modal analysis can be preserved in the nonlinear case and so a pragmatic approach is required. In [3] it is argued that the engineer can view a nonlinear normal mode in one of two ways. 1. A (local or global) coherent motion of a structure; this generalises the idea of the linear mode shape. 2. A decomposition into lower-order dynamical systems, the motions of which correspond to the ‘modes’; this generalises the idea of a modal decomposition. Approaches corresponding to both viewpoints can be found in the literature. The first, from Rosenberg [4], generalises the idea of the linear mode shape by allowing an arbitrary function to describe the displacements of the system given a single displacement. The Rosenberg ansatz is therefore, yi = fi (y1 )

∀i = 2, . . . , D

(20.2)

where D is the number of degrees-of-freedom. In practice, any displacement can be selected for the ‘input’ displacement, but y1 is used here without loss of generality. The Rosenberg framework has desirable properties, including reduction to the linear case in the absence of nonlinearity, but also comes with some significant drawbacks. Neither non-conservative forces nor internal resonances are natively supported, although there have been contributions to address these shortcomings [5]. For a more detailed description of the method, the reader is directed to the reference [6]. A geometrically more general approach was later proposed by Shaw and Pierre in [7]. In the extended framework, the idea of a coherent motion is extended to the phase space of the nonlinear system, where the displacements and velocities of the system response are permitted to depend functionally on a single displacement-velocity pair. The ansatz is now, yi = fi (y1 , y˙1 ) y˙i = hi (y1 , y˙1 )

(20.3)

This approach retains the properties of the Rosenberg ansatz but extends the approach naturally to non-conservative forces. The fundamentally linear notion of a planar mode shape in the configuration space has been generalised to the phase space and is now an invariant manifold. These curved manifolds in the phase space retain many of the desirable properties of a nonlinear modal analysis. If motion is initiated on such a manifold, then it remains on that manifold for all time. This is akin to the idea that no linear mode can excite any others. Secondly, the manifolds are shown by Shaw and Pierre to be tangent to the underlying linear modes (whereby all nonlinear terms in the EOM are deleted) at the equilibrium point. This ensures that the Shaw-Pierre NNM is equivalent to the linear mode in the limit of zero nonlinearity. Following the second viewpoint of [3] leads to a somewhat more practical view of NNMs. Here, the ethos is to retain the properties of linear modal analysis that facilitate simpler analyses. Many techniques are applicable to this problem including nonlinear embedding methods [8], polynomial decoupling methods [9] and statistical decoupling approaches such as proper orthogonal decomposition [10]. It is argued here that the metrics of success for a NNM with these motivations are, 1. The ability of the decomposition to decompose the dynamics into independent modal dynamics (preferably SISO) wherein the dynamics are simplified. 2. The extent to which the original dynamics can be recovered from the decomposition (nonlinear superposition).

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Fig. 20.1 Overview of the new NNM approach

Reference [3] proposes a third class of NNM, based on a nonlinear decomposition into uncorrelated time series. The motivation for this NNM is that often during analysis, the engineer does not have access to the underlying EOM of their system. An output-only approach is therefore desirable. The fundamental idea behind this NNM can be stated simply. Generalising the linear idea of decomposition, the framework specifies a static map f from the physical displacements to a new coordinate system within which the nonlinear modal displacements are uncorrelated time series. As an additional inductive bias, the framework seeks to retain the idea of orthogonality that is present in linear modal analysis (the mode shapes are an orthogonal basis), by requiring that the forward and inverse transformations are conformal and locally preserve orthogonality. Figure 20.1 depicts the approach graphically. In the figure, dotted lines are functionals and solid lines represent static maps. As can be seen, the framework also specifies the approximate inverse map f −1 , permitting an approximate nonlinear superposition. Following a machine learning approach, f is learnt from data via an objective function which penalises the pairwise correlations of the modal displacements and encourages orthogonality of the transformation. The original work [3] adopts a truncated multinomial expansion for f and a Gaussian process (GP) form for f −1 . This approach is inspired by the power series solutions used by Shaw-Pierre when substituting their ansatz into the equations of motion. However, this approach was shown in [11], to lead to bloated models that led to poor performance of the heuristic optimisation methods used to tune the parameters of the multinomial. Despite this, the polynomial approach showed good performance on a range of nonlinear single-input-multiple-output (SIMO) systems comprising both simulated and experimental datasets. A recent work adopts an alternative neural architecture for f and f −1 [12]. In this work, a cycle-consistent generative adversarial network (cycle-GAN) is used. The cycle-GAN architecture consists of two generator models (feed-forward neural networks) representing f and its inverse. These models are trained adversarially against discriminator models that encourage the generators to map to a target distribution. For additional details on adversarial training and GANs, the reader is directed to the seminal paper [13]. Thus far, the performance of these NNMs have been judged by a mixture of qualitative and quantitative measures. When first proposed, the quality of the simplification was judged ‘by-eye’ to be the extent to which the modal displacements produced solitary peaks in their power-spectral densities (PSD). In [11], this was extended to a convolution-based approach, and in [12], to an approach based on the inner-product. The quality of the reconstruction can of course be evaluated by a normalised mean-square error (NMSE) metric. Thus far, little attention has been given to the precise nature of the dynamics that are present in the NNMs. This paper seeks to address this issue by utilising techniques from nonlinear system identification (NLSI). The approach here is to model the dynamics of the nonlinear functional x → u and compare them to equivalent models of the physical dynamics x → y, to look for more concrete evidence of simplification. The structure of this paper is as follows. The next section summarises the specification and numerical simulation of a nonlinear dataset for the present study. The third section describes the nonlinear modal analysis, conducted via the cycleGAN approach. The fourth section describes the NLSI approach and presents the fitted models. The final section presents some discussion of the results and introduces promising avenues for further investigation.

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20.2 A Simulated Nonlinear Duffing Equation The nonlinear system under consideration in the present study will be a two degree-of-freedom. Duffing-type equation comprised of a single cubic stiffness between the ground and the first degree-of-freedom. The EOM of this system is given by, M y¨ + C y˙ + Ky + K3 y3 = x(t)

(20.4)

where, M, C, K, K3 are square parameter matrices corresponding to the nonlinear system depicted in Fig. 20.2. In order to produce the nonlinear time series data, the equations of motion are integrated forward in time by a fixed-step fourth-order Runge–Kutta method. For excitation, a white Gaussian signal with a mean of zero and standard deviation of 20 is used. The input signal is additionally filtered onto the interval [0, 50] Hz, such that all modes of vibration are excited, but the input signal remains well below the sampling frequency of 50 Hz. The excitation is specifically selected to introduce a nonlinear response and evidence of this can be seen in the cubic hardening visible in Fig. 20.3. All model and simulation parameters are collected in Table 20.1.

20.3 Nonlinear Modal Analysis This section follows the approach of [12], in constructing the NNMs. A full presentation of the training scheme and gradient propagation are beyond the scope of the current paper and the author is directed to the relevant paper [15] for full detail. An overview of the procedure is depicted in Fig. 20.4. The structure of the cycle-GAN model here is essentially two generative adversarial networks (GAN) [13], tasked with learning the forward and inverse mapping to the modal coordinates. Within each GAN, there are two neural networks each possessed of a single hidden layer. Of these two networks, the first—the generator—learns the desired mapping, while the second—the discriminator—learns to classify according to the target distribution. Here, as in [12], the targeted distribution

Fig. 20.2 Schematic of the two-DOF system used in this paper

Fig. 20.3 PSDs (estimated by the Welch method [14]) for the nonlinear and underlying linear system, the nonlinear data shows a clear hardening via shifts in ‘resonance’

20 On Modelling Statistically Independent Nonlinear Normal Modes with Gaussian Process NARX Models Table 20.1 Parameters for the nonlinear benchmark system

Parameter m c k k3 [ω1 , ω2 ] fs fc σx Ns

Description Mass (Kg) Viscous damping coefficient (Ns/m) Linear stiffness (N/m) Cubic stiffness (N/m3 ) Linear natural frequencies (Hz) Sampling frequency (Hz) Lowpass cut-off frequency (Hz) Excitation standard deviation (N) Points sampled

139 Value 1 20 1 × 104 5 × 109 [15.9, 27.5] 500 50 20 1 × 105

Fig. 20.4 Overview of the cycle-GAN approach

for the modal coordinates is a pair of independent Gaussian distributions. The target distribution for the physical coordinates is learnt from the measured samples. The adversarial training of the GANs ensures that the generator models learn the required mappings, while respecting the target distributions. Adversarial training leads to a number of loss functions for the cycle-GAN that are optimised simultaneously by a gradient descent approach. The first are the adversarial losses, computed as in the original cycle-GAN paper, y

L1 = E[log Df (y)] + E[log(1 − Df (f (u)))] Lu1 = E[log Df −1 (u)] + E[log(1 − Df −1 (f −1 (u)))]

(20.5)

where E is the expectation operator, f (resp. f −1 ) are the forward and inverse modal transformations and Df (resp. Df −1 ) are the discriminator models trained to classify the target modal and physical distributions. The next objective function is the reconstruction loss, L1 = y − f (f −1 (y))2 y

ˆ 2 Lu1 = uˆ − f −1 (f (u))

(20.6)

where the uˆ are samples from independent Gaussian distributions. Finally, the inductive bias of orthogonality is introduced by a third loss term L3 that ensures that the inverse modal transformation u → y is conformal. For additional detail on the orthogonality enforcement assembly, the interested reader is directed to [12]. The networks are trained over a number of epochs consisting of 2048 samples per epoch using an ADAM [16] stochastic gradient descent algorithm. The loss from a given training epoch is given by, y

y

L = λ1 (L1 + Lu1 ) + λ2 (L1 + Lu1 ) + λ3 L3 where the λi are weights in the objective function. In [12] and here, values of λ1 = 1, λ2 = 10 and λ3 = 1 are used.

(20.7)

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Table 20.2 Network parameters for the cycle-GAN network

Parameter nr ne ns e nl hl hm ns Whann Wo

Description Training repeats Training epochs Training examples per epoch Number of hidden layers in f and f −1 Min. hidden nodes in f and f −1 Min. hidden nodes in f and f −1 Hidden node step size Samples per Welch segment Welch segment overlap

Value 10 5000 2048 1 10 200 10 2048 0

Fig. 20.5 PSD of physical and modal displacements

Following the proposing authors, in order to remove any dominant linear correlations, the input displacements are first transformed by a PCA decomposition. Transforming the data in this way frees the neural architecture to learn any nonlinearity that might be required for f without having to learn all the linear transformations as well. The same structure is of course present in the inverse mapping. The network structures and parameters are detailed in Table 20.2. The number of hidden nodes in both the forward and inverse mappings is set to the same value and training is repeated over a range of values to ensure the best possible chance of learning a good decomposition. For this paper, values between 10 and 200 are considered with an increment of 10, with 10 repeats per increment. All networks, across all repeats, are trained for 5000 epochs. Once trained, an inner-product metric is used to select the best encoder and decoder pair. This metric ensures that the transformation gives good separation of peaks in the frequency domain. To simplify the computational complexity of the approach, the inner-product score is computed every 100 training epochs. The metric is defined over the power-spectral densities (PSD) of modal coordinates P u and is given by,

Linner =

ndof 

P ui · P uj 1 1 P ui  1P uj 1 i=1,j =i+1

(20.8)

In the above, the P ui and P uj are estimated by a Welch method [14]. This leads to the introduction of some additional hyperparameters arising from the window length and overlap size in the Welch method. The authors’ experience with the cycle-GAN approach has shown that too much noise in the PSD can lead to poor performance and so values of the parameters that promote smoothness are selected. In the present study, a Hamming window of length 211 samples is used with zero overlap. The PSDs of the modal coordinates are compared to those of the physical in Fig. 20.5. When judged visually, the peaks in the PSDs are well separated, indicating a good decomposition.

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20.4 Nonlinear System Identification Now that the y and u are established, attention can be turned towards analysis of the underlying dynamics. The approach in this work will be to conduct NLSI using both linear and nonlinear models. For the linear modelling auto-regressive models with exogenous inputs (ARX) will be used. ARX are a discrete time formulation where the model form is,

yt =

ny 

ai yt−i +

i=1

n x −1

bi ut−i

(20.9)

i=0

where nx and ny refer to the number of lagged inputs and outputs, respectively. For a given LAG structure, ARX models permit a vectorised formulation, a y=H b

(20.10)

where a is a vector of model weights and H is the Hankel matrix of lagged inputs and outputs at time t given by, ⎡ ⎢ ⎢ ⎢ H =⎢ ⎢ ⎢ ⎣

yt−1 yt−2 yt−3 .. .

yt−2 . . . yt−ny yt−3 . . . yt−ny −1 yt−4 . . . yt−ny −2

xt xt−1 xt−2 .. .

⎤ xt−1 xt−2 . . . xt−nx +1 xt−2 xt−3 . . . xt−nx ⎥ ⎥ ⎥ xt−3 xt−4 . . . xt−nx −1 ⎥ ⎥ ⎥ ⎦

(20.11)

xt−N +p−1

yt−N +p

where nx and ny refer to the number of lagged inputs and outputs, respectively, and N are the total number of temporal points. ARX models are completely defined by the model weights ai and bi which can be simply fitted to input data by solution of the linear equation in (20.10). In the present work, a singular-value decomposition (SVD) approach is used. For the nonlinear models, Gaussian process NARX models will be used. The model formulation is that of a generic nonlinear auto-regressive model with exogenous inputs (NARX) as, yt = N(yt−1 , yt−2 , . . . , yt−ny , xt , xt−1 , xt−2 , . . . , xt−nx +1 )

(20.12)

where N is a static nonlinear map. As before, NARX models can be expressed in terms of the Hankel matrix, y = N(H )

(20.13)

where N is a static nonlinear map from the Hankel matrix to the predictions. For the GP-NARX formulation it is assumed that N can be approximated by a Gaussian process. This formulation comes with a number of advantages over more traditional approaches, such as a polynomial expansion. Uncertainty quantification is handled naturally, and the non-parametric form gives excellent model flexibility. A full description of Gaussian processes and the GP-NARX model is beyond the scope of this paper and so only a brief overview is included here; for excellent references on the topics, the reader is directed to [17–19]. A Gaussian process (GP) is a essentially a regression over the space of functions. For a multi-input single-output (MISO) model the core of the GP is the regression formulation, yi = f (x) + i ,

i ∼ N (0, σn2 )

(20.14)

where f is a latent function that has not been observed directly, but rather observations y. The GP is formed by assuming a distribution over possible values of the latent function as,

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f (x) ∼ GP(m(x), k(x, x))

(20.15)

where k is a positive semi-definite covariance kernel k(x, x) → R+ and m(x) → R is the mean function. Several choices for both of these functions are applicable and their selection is often driven by domain knowledge [17]. In the case of dynamic systems, where variables oscillate about an equilibrium point, the zero-mean function m(x) = 0 is appropriate and so is adopted here. The GP formulation allows one to marginalise over all possible latent functions f . The result is a multivariate normal distribution over some observed data (X, y) and unobserved testing points (X∗ , y∗ ). 

y y∗





 K (X, X∗ ) K(X, X) + σn2 I ∼ N 0, K (X∗ , X) K (X∗ , X∗ ) + σn2

(20.16)

From this distribution, it is straightforward to calculate the posterior predictive distribution for unobserved variables y∗ as, y∗ ∼ N (m∗ (X∗ ), K ∗ (X∗ , X∗ ))

(20.17)

where the posterior mean prediction is given by, −1   y m∗ (X∗ ) = k X∗ , X K(X, X) + σn2 I

(20.18)

and the posterior variance is given by, −1     K ∗ (X∗ , X∗ ) = K X∗ , X K(X, X) + σn2 I K X, X∗ + σn2

(20.19)

Substituting the above into the formulation of a NARX model in (20.13) one has, y ∼ N (m∗ (H ∗ ), K ∗ (H ∗ , H ∗ ))

(20.20)

where H is the Hankel matrix as before. Taking the mean prediction of the distribution gives the prediction or one step ahead (OSA) prediction from the GP-NARX on some unseen lagged inputs H ∗ as, −1     y y∗ = m∗ H ∗ = K H ∗ , H K(H, H ) + σn2 I

(20.21)

A more rigorous test of the performance of the model is the simulation or model predicted output (MPO). To obtain this type of prediction, the model predictions are fed back into the model as inputs to the next time step. MPO prediction requires predictions at each time step and so the vectorised form of the equations above cannot be used. The prediction at time t for MPO prediction is given by, −1     yt∗ = m∗ h∗ = k h∗ , H K(H, H ) + σn2 I y

(20.22)

where h∗ are the lagged inputs and lagged predicted outputs yˆ from the current time step,  h∗ = yˆt−1 yˆt−2 . . . yˆt−ny xt xt−1 xt−2 . . . xt−nx +1

(20.23)

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MPO prediction raises some difficulties in the context of GP-NARX models. The major issue is that, by definition, the GP is not permitted to have uncertainty on the values of the inputs. This limitation makes it impossible to fully incorporate the uncertainty quantifications during MPO prediction and, in practice, leads to underestimates of the predicted variance. One approach to circumvent this issue is to use a Monte Carlo-based sampling technique [18, 19]. However, because uncertainty quantification is not the principal aim of the current investigation this has not been considered here. Practically, there are still a number of hyperparameters that must be addressed when fitting GP-NARX models. The first is the specification of the covariance function. A common choice for physical processes is the squared-exponential (SE) kernel. Although there are several potential choices for the covariance function the SE kernel is used here because of its tendency to produce smooth functions,   1   1 1  2  12 1 k x, x = σf exp − 2 x − x 2

(20.24)

where σf2 is the signal variance and the  is a (1 × (nx + ny )) vector of length scales. Utilising a vector of length scales rather than a single hyperparameter allows for greater model flexibility and is referred to as automatic-relevance detection (ARD) in the GP literature. These parameters, along with the noise variance σn2 and lag structure, are hyperparameters of the GP-NARX model and must be optimised to ensure a good model. The training approach for GP-NARX models trained in this paper is the following. Firstly, a training, validation and testing set of data are taken from the simulated and modal data each comprising of 103 points. The next step is to fix the lag structure. This is achieved by a grid search over the number of lags in the input nx and output ny , respectively. For each point in the grid, a GP-NARX model is trained on the training data and the remaining hyperparameters (σn2 , σf2 , ) are optimised. In order to alleviate some of the computational difficulties associated with such a brute-force approach, the optimisation of the hyperparameters during lag structure optimisation (LSO) is completed using an evidence framework for which the objective function is given by the negative log of the marginal likelihood, 2 2  1  1 2 2 JLSO (σn2 , σf2 , ) = − yT K(X, X) + σn2 I y − log 2K(X, X) + σn2 I 2 2 2

(20.25)

With GP-NARX models trained at each point on the grid, the lag structure is chosen to be the one that produces the lowest normalised mean-square error (NMSE) on the validation data. The NMSE metric is defined by,

JNMSE =

NJ  2 100  yi − yˆi 2 σy NJ

(20.26)

i

where NJ = Np − (nx + ny ). A visualisation of the resultant cost surfaces is given in Fig. 20.6. With the lag structure set, the models are re-trained using an NMSE loss function on the MPO predictions; this is done to encourage the models to learn the underlying dynamics of the data. All hyperparameter optimisations are completed using a population-based particle swarm algorithm [20]. The optimisation and training parameters are collected in Table 20.3. For each SISO functional mapping xi → yi and xi → ui both ARX and GP-NARX model are fitted using the approach described above. The resultant MPO predictions on the unseen testing data are collected in Figs. 20.7 and 20.8. The resultant NMSE scores and lag structures are given in Table 20.4.

20.5 Discussion As can be seen in Fig. 20.7 the NLSI of the physical displacements has been successful. The ARX models achieve good agreement with the data with NMSE scores of around 4–5% on the unseen testing data. This agreement can likely be attributed to the odd cubic nonlinearity admitting a good linear approximation by models that artificially stiffen the linear

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Fig. 20.6 GP-NARX NMSE surface during LSO training. NMSE scores exceeding 100 are excluded from the plot Table 20.3 Parameters used in the optimisation of the GP-NARX models

Parameter Sp Sg ntrain nval ntest 2 σRMS

Description Population size Number of generations Training points Validation points Testing points Regularisation noise (MPO training only)

Value 200 200 1000 1000 1000 1% RMS

stiffness terms. The GP-NARX fits are excellent (both less than 1%) and indicate that the underlying dynamics of the structure have been captured well. The GP-NARX model has outperformed the linear ARX model substantially in each case. In the modal coordinates, (Fig. 20.8), the situation is the same with the exception of the second modal coordinate u2 in which the GP-NARX NMSE score is slightly lowered. A possible reason for this is the lower absolute amplitude of the u2 displacements, and therefore a lower signal-to-noise ratio. It is promising to see that excellent SISO models can be fitted to the modal coordinates indicative of a good decoupling. In order to examine the generalisation of the modal transformation NLSI is performed again on data generated at two additional levels of excitation (σx = 8 and σx = 14). The same modal transformation is used to generate modal displacements ui for these excitation levels and NLSI is performed on these as well. Finally each model trained at a single excitation level is used to make predictions at both of the other excitation levels. The results are collected in Figs. 20.9 and 20.10. As might be expected, in the physical coordinates, the nonlinear models generalise very well between excitation levels, whereas the linear ARX models suffer from poor accuracy. This is more evidence of bias in the linear models. The modal coordinates have some more interesting structure. The u1 displacements exhibit similar signs of nonlinearity where the GP-NARX is generalising whereas the ARX models cannot. However, in the u2 displacements, the generalisation performance is very similar between the two models. The authors offer two explanations for this observation. The first is that the NLSI may be entirely limited by the poor signal-to-noise ratio in the u2 and so the GP-NARX makes a simplistic linear

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Fig. 20.7 Results from the NLSI: MPO predictions on unseen testing data for ARX and GP-NARX models compared to the simulated physical data

Fig. 20.8 Results from the NLSI: MPO predictions on unseen testing data for ARX and GP-NARX models compared to the modal data Table 20.4 Lag structures and MPO NMSE scores for the trained models on the unseen testing data

Target y1 y1 y2 y2 u1 u1 u2 u2

Model ARX GP-NARX ARX GP-NARX ARX GP-NARX ARX GP-NARX

nx 11 17 3 21 7 39 15 1

ny 13 22 14 34 9 38 13 26

MPO NMSE 4.08 0.0502 4.95 0.234 4.25 0.154 3.62 3.16

approximation. This seems unlikely given the flexibility of the GP-NARX model and the relatively good NMSE scores that are found by both model classes. The second potentially very interesting explanation is that the cycle-GAN has learned a static map that is able to decouple the SIMO nonlinear functional into SISO nonlinear and SISO linear functionals. Such a decoupling would represent a highly useful tool in the analysis of nonlinear structural dynamics. Overall, the nonlinear modal analysis conducted in this paper has been successful by both metrics defined in the introduction. Regarding decomposition, this can be evidenced by the good visual separation of the peaks in the modal spectra (Fig. 20.5), and by the low NMSE scores achieved by SISO models trained on the functional x → u. Regarding the second criteria, the reconstructed yˆ = f −1 (f (y)) are compared to the original displacements in Fig. 20.11. The reconstruction is excellent on both degrees-of-freedom and achieves NMSE scores of 0.0262% and 0.0133% on y1 and y2 , respectively, when evaluated on the entire 105 sampled points. In conclusion, the results of this paper are another promising indication that the statistically independent NNM offers a promising alternative to linear modal analysis in the presence of a common engineering analysis; nonlinear systems undergoing broadband excitation.

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Fig. 20.9 Cross-validation NMSE of GP-NARX and ARX models of physical displacements at different excitation levels

Fig. 20.10 Cross-validation NMSE of GP-NARX and ARX models of modal displacements at different excitation levels

Fig. 20.11 Modal reconstructions yˆ = f −1 (u) from the data trained and evaluated at σx = 20

Acknowledgments MDC would like to acknowledge the support of the EPSRC grant EP/L016257/1. G.T. would like to thank the EU’s Horizon 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement DyVirt (764547). KW would like to acknowledge support via the EPSRC Established Career Fellowship EP/R003625/1 and Programme Grant EP/R006768/1.

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References 1. Ewins, D.J.: Modal Testing: Theory, Practice and Application. Wiley, Hoboken (2009) 2. Peeters, B., De Roeck, G.: Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Signal Process. 13(6), 855–878 (1999) 3. Worden, K., Green, P.L.: A machine learning approach to nonlinear modal analysis. Mech. Syst. Signal Process. 84, 34–53 2 (2017) 4. Rosenberg, R.M.: The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 29, 7–14 (1962) 5. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23, 170–194 1 (2009) 6. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal Modes and Localization in Nonlinear Systems. Kluwer Academic Publishers, Alphen aan den Rijn (2001) 7. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vibr. 164(1), 85–124 (1993) 8. Dervilis, N., Simpson, T.E., Wagg, D.J., Worden, K.: Nonlinear modal analysis via non-parametric machine learning tools. Strain 55, e12297 (2018) 9. Decuyper, J., Dreesen, P., Schoukens, J., Runacres, M.C., Tiels, K.: Decoupling multivariate polynomials for nonlinear state-space models. IEEE Control Syst. Lett. 3, 745–750 (2019) 10. Feeny, B.F., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vibr. 211, 607–616 (1998) 11. Champneys, M.D., Worden, K., Dervilis, N.: Nonlinear modal analysis based on complete statistical independence. In:Nonlinear Vibrations, Localisation and Energy Transfer, vol. 160, p. 978 (2019). 12. Tsialiamanis, G., Champneys, M.D., Dervilis, N., Wagg, D.J., Worden, K.: On the application of generative adversarial networks for nonlinear modal analysis. Mech. Syst. Signal Process. 166, 108473, 3 (2022) 13. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative Adversarial Nets. Adv. Neural Inf. Process. Syst. 27 (2014) 14. Welch, P.D.: A direct digital method of power spectrum estimation. IBM J. Res. Develop. 5, 141–156, 4 (2010). 15. Zhu, J.Y., Park, T., Isola, P., Efros, A.A.: Unpaired image-to-image translation using cycle-consistent adversarial networks. In: IEEE International Conference on Computer Vision (2017) 16. Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. Preprint arXiv:1412.6980 (2014) 17. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006) 18. Rogers, T.J.: Towards Bayesian System Identification: With Application to SHM of Offshore Structures. PhD Thesis (2019) 19. Worden, K., Becker, W.E., Rogers, T.J., Cross, E.J.: On the confidence bounds of Gaussian process NARX models and their higher-order frequency response functions. Mech. Syst. Signal Process. 104, 188–223, 5 (2018) 20. Sun, J., Lai, C.H., Wu, X.J.: Particle swarm optimisation: Classical and quantum perspectives. Particle Swarm Optimisation: Classical and Quantum Perspectives, pp. 1–393, 4. CRC Press, Boca Raton (2016)

Chapter 21

Non-linear Kinematic Damping in Phononic Crystals with Inertia Amplification Bart Van Damme, Marton Geczi, Leonardo Sales Souza, Domenico Tallarico, and Andrea Bergamini

Abstract Mechanical metamaterials are structures designed to obtain a predefined dynamical property. A typical example are band gaps, frequency bands in which only evanescent waves exist. At these frequencies, the metastructure can be used as an efficient vibration damper or isolator, without the use of viscoelastic materials. The advantage of such a design is its full tunability, beyond the range of bulk properties of classical linear construction materials. In the quest for vibration isolators that are at the same time highly attenuating and stiff enough to carry high loads, phononic crystals can take advantage of inertia amplification. In these designs, the main motion direction is kinematically coupled to small masses with separate degrees of freedom. This coupling results in a high dynamic mass due to the additional inertia. If the geometrical relation between the degrees of freedom is non-linear, the equation of motion and wave equation become non-linear as well. More precisely, it introduces a damping term proportional to the square of the velocity. In our work, we use analytical and numerical models to confirm the existence of amplitude-dependent vibration damping of mechanical resonators with inertia amplification. Two types of resonators are investigated: one with masses undergoing only translational degrees of freedom and one where translation and rotation are coupled. The damping of each structure can be tuned by altering the geometry of the resonator, in particular the angle of the connectors between the individual masses. Furthermore, we show how the amplitude dependency affects the wave propagation in phononic crystals that exploit inertia amplification to lower the start frequency of the band gap. Keywords Non-linear dynamics · Phononic crystals · Damping

21.1 Introduction Low-frequency vibration isolation is a difficult task, since isolators are only effective at frequencies above the mass-spring resonance of the elastic support [1]. Achieving lower values requires either very soft supports or a higher mass of the vibrating structure, both criteria that might not be acceptable within the design specs. Moreover, materials with high viscous losses are typically too soft to support heavy vibrating structures in a safe way. Mechanical metamaterials are intensively investigated to achieve combinations of properties that are not available in bulk materials. For the specific goal of vibration isolation, there is an ongoing quest for metastructures with a low mass density but high static stiffness and achieving high elastic wave attenuation within a desired frequency range. Several approaches have shown promising results. The basic idea is typically to create a periodic wave guide with internal resonators. The combination of periodicity and coupled resonators leads to wide frequency bands in which no propagating waves exist, the so-called band gaps. These periodic structures that act on acoustic waves are often referred to as phononic crystals (PCs), because of the similarities with their optical counterpart. We have shown in the past that there is a lower limit to the band gap frequency of PCs built up from point masses [2]. However, the apparent dynamic mass can be increased using a mechanism called inertia amplification (IA) [3]. Coupling small masses with additional degrees of freedom, either translational or rotational, to the main energy propagation direction leads to a significant increase of the perceived mass. This evidently lowers the resonance frequency of a mass-spring system and lowers the band gap frequency. The coupling mechanism inevitably introduces a non-linear, amplitude-dependent damping in the resonators. In this work, we present initial numerical results of this damping term on the efficiency of the PC band gaps.

B. Van Damme () · M. Geczi · L. S. Souza · D. Tallarico · A. Bergamini Laboratory for Acoustics/Noise Control, Empa, Materials Science and Technology, Dübendorf, Switzerland e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_21

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21.2 Background We investigate a particular PC, which makes use of rotational inertia amplification as shown in Fig. 21.1. A compression of the overall structure is coupled to a rotation of the disks due to the slanted connectors. We have shown the efficiency of the rotational IA in previous publications [4], proving the existence of low-frequency complete band gaps for well-chosen designs. The angle of the connecting struts defines the inertia amplification factor, which increases as the struts become more vertical. In a recent paper [5], the equation of motion of a single unit cell of such a PC is derived: ∼ ∼ 2 2 F = M U¨ + C U˙ 2U˙ 2 + kU ∼

∼

where M and C are the apparent mass and damping coefficient. Assuming small deformations and not too steep connectors, they are given by their zeroth-order Taylor expansion: ∼

M=M+

M 2

2 sin θ0

∼

,C =

M cos2 θ0 2L sin4 θ0

in which M is the mass of a disk. These relations show that choosing a smaller angle θ 0 increases the apparent mass but also increases the damping coefficient. Moreover, the damping term in the equation of motion takes a quadratic form, similar to the effect of drag forces. This non-linearity results in amplitude-dependent damping, which increases with the displacement. Therefore, the normalized resonance curves of a single unit cell depend on the harmonic driving force. At higher driving forces, the velocity spectrum normalized by the input force spectrum becomes wider and flatter. The apparent damping therefore increases with amplitude. The dynamic behaviour of PCs is most often investigated through their dispersion relation. The imaginary part of the wave number for a certain wave type describes its spatial attenuation, and high imaginary values occur within band gaps. It is well known that material damping has a significant effect on the band gap efficiency. The standard way to calculate dispersion properties, namely, applying periodic boundary conditions on a modal analysis of a unit cell, is only valid in linear materials. Although some publications use this approach to investigate wave propagation in explicitly non-linear structures [6], detailed insights can only be gained from explicit transient simulations in the time domain. Our approach is to investigate the response of a two-cell PC to a short impact force with varying amplitude.

Fig. 21.1 PCs with varying strut angle implying different inertia amplification. Left: 20◦ , right: 30◦

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Fig. 21.2 Transfer functions (force-force) for 10 N and 100 N pulses on two PCs: 20◦ strut angle (left) and 30◦ strut angle (right)

21.3 Analysis The finite element model of the structure is created in Ansys 2020R2. The PC has a fixed boundary condition on one side and undergoes an impulsive force consisting of a half cosine period with frequency 2000 Hz. The total simulated response time is 0.5 s, with automatic time step definition. Large deflection effects are activated in order to allow for non-linear structural effects. Two PCs with varying strut angle are used to visualize the effect of increasing IA. The efficiency of the band gap is quantified by the transfer function between the reaction force spectrum at the fixed boundary and the input force spectrum. They are shown for two amplitude levels for both PCs in Fig. 21.2. The figures show a trend towards narrower band gaps at high-input amplitudes. The high attenuation frequency band ends at a lower frequency for both geometries. The lower frequency of the band gap is not affected by the force amplitude. The efficiency of the band gap is similar in both cases, although differences of between 5 and 10 dB can be seen.

21.4 Conclusion PCs based on IA to achieve low-frequency band gaps show amplitude-dependent dynamic effects due to the kinematics of the structure. The equation of motion contains a non-linear damping term due to the coupling of degrees of freedom within the same structure. This phenomenon cannot be captured by dispersion analysis, which is by default independent of the amplitude. Transient finite element simulations in the time domain show that the desired band gap gets narrower for increasing excitation amplitudes. This fundamental study implies that special care has to be taken when such PCs are used for the isolation of high-input forces, in which case the efficiency might be less good than predicted.

References 1. Cremer, L., Heckl, M.: Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies. Springer (2013) 2. Delpero, T., et al.: Structural engineering of three-dimensional phononic crystals. J. Sound Vib. 363, 156–165 (2016) 3. Yilmaz, C., Hulbert, G.M., Kikuchi, N.: Phononic band gaps induced by inertial amplification in periodic media. Phys. Rev. B. 76(5), 054309 (2007) 4. Bergamini, A., et al.: Tacticity in chiral phononic crystals. Nat. Commun. 10(1), 1–8 (2019) 5. Van Damme, B., et al.: Inherent non-linear damping in resonators with inertia amplification. Appl. Phys. Lett. 119, 061901 (2021) 6. Khajehtourian, R., Hussein, M.I.: Dispersion characteristics of a nonlinear elastic metamaterial. AIP Adv. 4(12), 124308 (2014)

Chapter 22

Mitigation of Nonlinear Structural Vibrations by Duffing-Type Oscillators Using Real-Time Hybrid Simulation A. Mario Puhwein and Markus J. Hochrainer

Abstract This work addresses the vibration reduction of nonlinear host structures by dynamic absorbers of Duffing type. For deriving the equations of motion, both host structure and absorber are modeled as nonlinear SDOF oscillators. The method of harmonic balance is applied to obtain an approximate analytical steady-state solution, which is used for parameter identification. Therefore, the frequency response function of either host structure or absorber is determined experimentally, and the dynamic system parameters are identified by a nonlinear least squares method. Since the focus of the work is on the absorber design, a generalization of Den Hartog’s equal-peak method renders the optimal parameters. These are strongly dependent on the operating point’s amplitude, and consequently, the optimization must be often repeated. To keep this effort minimal, DOE methods are applied in the test and design phase. After confirming a proper absorber design by numerical simulations, a physical model of the absorber is tested in a real-time hybrid simulation setup. Hence, the behavior of the nonlinear host structure is reproduced by a virtual simulation model coupled to the physical absorber using a transfer system. This method allows to focus on the physical absorber without the need to construct an expensive laboratory host structure model. Furthermore, all host structure parameters can be changed in the virtual model, without modifying the real-time hybrid simulation setup. Furthermore, the virtual host structure cannot be damaged or destroyed, and thus, all experiments can be repeated and optimized straight away. So far, all real-time hybrid simulation experiments are in good accordance with the theoretical predictions of this work and corresponding results already published in literature. Keywords Experimental nonlinear vibrations · Periodic forcing · Duffing oscillator · Magnetic forces

22.1 Introduction The performance requirements of flexible structures are continuously increasing, and consequently nonlinear dynamic system behavior occurs more often. Therefore, the mitigation of resonant oscillations of nonlinear structures is of practical importance [1] and the focus of the present study. Nonlinear systems exhibit a number of phenomena not observed in linear systems. In particular, a key property is the dependence of the resonant frequency on the vibration amplitude [2]. Because of their wider bandwidth, nonlinear vibration absorber can reduce disturbances in a broad frequency range [3]. This makes nonlinear vibration absorbers suitable candidates for vibration damping of nonlinear primary structures that perform better than linear systems [4]. However, it is known that the performance of nonlinear tuned vibration absorber (NLTVA) strongly depends on the vibration amplitudes [5], and this requires a careful adjustment of the absorber nonlinearities. Interestingly, this is similar to centrifugal pendulum vibration absorber in rotating machinery. Since pendulum-type vibration absorber have a natural frequency that scales with angular velocity, they can be tuned over a continuous range of rotor speeds, e.g., to follow an engine order [6–8, and]. Recently, Habib et al. [9] introduced a nonlinear similarity principle for NLTVA which is much less sensitive to the forcing amplitude. It states that the nonlinearity of absorber and structure should match. The well-established Den Hartog’s equalpeak method (EPM) [10, 11] can be generalized to nonlinear vibration regimes, at least up to the fusion of isolated resonance amplitudes [9]. Detroux et al. [12] investigated the dynamics involved during this fusion and how NLTVA parameters can be

A. M. Puhwein Technical University of Vienna, Vienna, Austria M. J. Hochrainer () University of Applied Sciences, Wiener Neustadt, Austria e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_22

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adjusted to prevent such undesired phenomena. In the absence of exact solutions, the analysis of nonlinear systems is usually carried out using approximation methods such as the harmonic balance method (HBM) [13], numerical simulations [14, 15], or the high-order normal form [2]. Thus, the quality of the results is directly dependent on the approximation approach. In order to show this effect, the HBM is extended by third harmonics components. In this work the theoretical results are confirmed using the real-time hybrid simulation (RTHS) methodology. Since the focus of the work is on the absorber parameter selection, the physical model consists of the absorber only; all other components are integrated in a virtual model which is simulated on a real-time PC. If the models are coupled correctly, the dynamic behavior is equivalent to that of a complete physical structure-absorber system tested in the laboratory. Saving a lot of resources is one main advantage of real time hybrid testing. Furthermore, the host structure can be reparametrized easily, experiments can be repeated without any difficulty, there is no risk of damaging or destroying the real structure, and experiments can be undertaken long before the physical building exists. Nevertheless, the real dynamic absorber is tested under very realistic conditions, there is no simplification due to unmodeled dynamics, and the parameter tuning can be studied and investigated on the real prototype. The present work aims to demonstrate that using the real-time hybrid simulation (RTHS) the applied generalized equalpeak method (GEPM) leads to very good vibration mitigation of nonlinear host structures, not only from a theoretical point of view but also during experimental testing. Similarly, the results confirm the correspondence of the cubic nonlinearities of host structure and absorber presented in [9] for a restricted forcing amplitude range.

22.2 Methods The experimental implementation of this work is entirely based on the real-time hybrid testing method, because the advantages already discussed make hybrid testing very attractive for absorber design. However, for proper behavior, the physical interfaces must be modeled correctly, because violation of physical principles might change and degrade the dynamic results substantially. Since resonance phenomena of the nonlinear structures are studied, the systems are only lightly damped. If a nonoptimal coupling further reduces the damping, the coupled structure might show instabilities, which do not exist in reality. This phenomenon requires special attention and is often due to an almost inevitable time delay in the transfer system. The transfer system converts the simulated structural response into a real physical motion, which acts as excitation of the dynamic absorber under test. Therefore an electrodynamic actuator is required; see Fig. 22.1a. It obtains the energy from a power amplifier, and inaccurate mechanical power transfer might accumulate and lead to system instabilities. To avoid this problem, the study is limited to steady-state vibrations only. The advantage of periodicity conditions from a control point of view is that time delays can be compensated completely, e.g., adaptive control mechanism like iterative learning control schemes [16]. However, when compared to transient vibration testing, the proposed steady-state study is more time-consuming but also more precise. Furthermore, stochastic vibrations and quasi-periodic solutions cannot be reproduced experimentally by the proposed setup. Another benefit of iterative learning control is that no mathematical model of transfer system or absorber is required for proper coupling, because the unmodeled or unknown dynamics is estimated by the adaptive control mechanism. Equally important to the physical excitation provided by the transfer system is

Fig. 22.1 (a) Model of a damper with nonlinear restoring forces caused by repelling magnets in combination with a linear elastic leaf spring. The absorber is excited by a feedback controlled shaker setup (b) schematic representation of a nonlinear oscillator with a nonlinear dynamic absorber

22 Mitigation of Nonlinear Structural Vibrations by Duffing-Type Oscillators Using Real-Time Hybrid Simulation

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Fig. 22.2 Selection of design points, (a) schematics of the CCD combined with the DM, [20] (b) design points to determine the design of experiment surface

the measurement of the vibration amplitudes with different types of transducers (displacement and accelerometers). Although there might be significant noise in the signals at low amplitudes, filtering must be avoided because it induces phase shifts and therefore time delays. Instead of applying commonly used causal filters, the signal processing is integrated in the adaptive control. From a real-time simulation point of view, the mechanical system is very slow, and consequently, there are no special computational and measurement hardware requirements for the real-time system. Without difficulty, it would be possible to implement much more complex structural model into the setup. The focus of this work is on the determination of the optimal linear and nonlinear parameters for a nonlinear absorber attached to a given host structure. As already mentioned, host structure and absorber are represented by a SDOF Duffing oscillators; see Fig. 22.1b. Given a host structure, the absorber parameter mass, linear stiffness, linear viscous damping, and nonlinear cubic stiffness have to be determined. For optimal tuning the mass ratio μ is chosen, and the linear parameters of the absorber are calculated with Den Hartog’s EPM [11]. To optimize the missing nonlinear stiffness parameter, the resulting host structure amplitude peaks in the FRF are minimized numerically according to Den Hartog’s EPM criterion. Since the investigation is not restricted to a single parameter set, variations of the absorber-structure mass ratio and the linear as well as the nonlinear building stiffness are necessary. This renders an optimization problem with three input parameters and the nonlinear stiffness as output. To avoid carrying out a complete calibration according to the time-consuming full factorial design (FFD) method, Witek-Krowiak et al. [17] suggest a combination of methods with less supporting points to determine a design of experiment surface (DOES). In this work, a modified combination of the central composite design (CCD) methods [18] and the Doehlert matrix (DM) [19] is used to find a DOES approximation with as little grid points as possible; see Fig. 22.2.

22.3 System Description In vibration analysis, complex structural models are often modally reduced, and if the investigation is focused on the fundamental resonance, a single degree of freedom system results. In case of geometrically nonlinear structures, the reducedorder model is a nonlinear oscillator. If the restoring force of the structure is described/approximated by a cubic polynomial, it forms a Duffing-type oscillator with the well-established equation of motion: mS x¨ S + cS x˙ S + kS xS + kS3 xS 3 = F cos ωt,

(22.1)

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where xS , mS , cS , kS , kS3 , F, and ω describe the host structure displacement, mass, damping, linear, and nonlinear stiffness as well as the excitation amplitude and frequency, respectively. If a dynamic absorber is attached to such a host structure, it should also be of Duffing type for effective vibration damping. Then the equations of motion of the coupled system, see Fig. 22.1b, are obtained by extending Eq. 22.1 to the following: mS x¨ S + cS x˙ S + kS xS + kS3 xS 3 = F cos ωt − FA ,

mA x¨ A + cA (x˙ A − x˙ S ) + kA (xA − xS ) + kA3 (xA − xS )3 = 0,

(22.2)

where xA , mA , cA , kA , and kA3 represent the absolute absorber displacement, mass, damping, linear, and nonlinear stiffness, respectively. FA denotes the absorber-structure interaction force: FA = mA x¨ A = −cA (x˙ A − x˙ S ) − kA (xA − xS ) − kA3 (xA − xS )3 .

(22.3)

A dimensionless representation is obtained by introducing the normalized time τ = ωnS t, with the natural host structure √ ∼ frequency ωnS = kS /mS , the structure damping coefficient ζ S = cS /(2mS ωnS ), and the nonlinear stiffness ratio √ α3 = 3kS3 /(4kS ); see [10]. Similarly, the absorber natural frequency and linear damping coefficient become ωnA = kA /mA , ζ A = cA /(2mA ωnA ). Given the absorber-structure frequency ratio λ = ωnA /ωnS , the mass ratio μ = mA /mS , the static deformation xS, stat = F/kS , the normalized excitation frequency γ = ω/ωnS , and the normalized displacements qS = xS /xS, stat , qA = [xS (t) − xA (t)]/xS, stat , the dimensionless coupled equations of motion become: 4 4 qS + 2ζS qS + qS + α3 qS3 + 2ζA λμqA + λ2 μqA + μβ3 qA3 = cos (γ τ ) , 3 3

4 4 qA + 2ζS qS + qS + α3 qS3 + 2ζA λ (μ + 1) qA + λ2 (μ + 1) qA + (μ + 1) β3 qA3 = cos (γ τ ) , 3 3

(22.4)

(22.5)

where according to [9]: ∼

α3 = α 3 f 2 ,

β3 =

2 3 xS,stat kA3 . 4 mA ωnS 2

(22.6)

To solve the equations of motion in the investigated frequency range, the single frequency HBM with qA ≈ A2 cos (γ τ ) + B2 sin (γ τ ) generally provides a very good approximation for low displacement amplitudes. When compared to high-accuracy methods (e.g., high-order HBM, high-order normal form, and numerical simulations), there are noticeable deviations for high amplitudes. Therefore, also the third harmonic is considered: qS = A1 c1 + B1 s1 + C1 c3 + D1 s3 , qA = A2 c1 + B2 s1 + C2 c3 + D2 s3 ,

(22.7)

with the substitution ci = cos (iγ τ ) and si = sin (iγ τ ). Inserting into Eq. 22.4, comparing sine and cosine coefficients according to the HBM, and ignoring all ci and si terms, i > 3 renders a system of eight nonlinear algebraic equations: (22.4), c1 :

  α3 A1 3 + A1 2 C1 + A1 B1 2 + 2A1 B1 D1 + 2A1 C1 2 + 2A1 D1 2 − B1 2 C1   + β3 ε A2 3 + A2 2 C2 + A2 B2 2 + 2A2 B2 D2 + 2A2 C2 2 + 2A2 D2 2 − B2 2 C2 

+ 1 − γ 2 A1 + 2ζS γ B1 + λ2 εA2 + 2ζA λεγ B 2 = 1,

22 Mitigation of Nonlinear Structural Vibrations by Duffing-Type Oscillators Using Real-Time Hybrid Simulation

157

  (22.5), c1 : α3 A1 3 + A1 2 C1 + A1 B1 2 + 2A1 B1 D1 + 2A1 C1 2 + 2A1 D1 2 − B1 2 C1   + β3 (ε + 1) A2 3 + A2 2 C2 + A2 B2 2 + 2A2 B2 D2 + 2A2 C2 2 + 2A2 D2 2 − B2 2 C2 

+ A1 + 2ζS γ B1 + λ2 (ε + 1) − γ 2 A2 + 2ζA λ (ε + 1) γ B 2 = 1,

(22.4), s1 :

  α3 A1 2 B1 + A1 2 D1 − 2A1 B1 C1 + B1 3 − B1 2 D1 + 2B1 C1 2 + 2B1 D1 2   + β3 ε A2 2 B2 + A2 2 D2 − 2A2 B2 C2 + B2 3 − B2 2 D2 + 2B2 C2 2 + 2B2 D2 2 

− 2ζS γ A1 + 1 − γ 2 B1 − 2ζA λεγ A2 + λ2 εB2 = 0,

  (22.5), s1 : α3 A1 2 B1 + A1 2 D1 − 2A1 B1 C1 + B1 3 − B1 2 D1 + B1 C1 2 + 2B1 D1 2   + β3 (ε + 1) A2 2 B2 + A2 2 D2 − 2A2 B2 C2 + B2 3 − B2 2 D2 + 2B2 C2 2 + 2B2 D2 2 

− 2ζS γ A1 + B1 − 2ζA λ (ε + 1) γ A2 + λ2 (ε + 1) − γ 2 B2 = 0,

(22.4), c3 :

 1  3 α3 A1 + 6A1 2 C1 − 3A1 B1 2 + 6B1 2 C1 + 3C1 3 + 3C1 D1 2 3   1 + εβ3 A2 3 + 6A2 2 C2 − 3A2 B2 2 + 6B2 2 C2 + 3C2 3 + 3C2 D2 2 3 − C1 9γ 2 + 2ζS D1 3γ + C1 + 2ζA λεD2 3γ + λ2 εC2 = 0,

(22.5), c3 :

 1  3 α3 A1 + 6A1 2 C1 − 3A1 B1 2 + 6B1 2 C1 + 3C1 3 + 3C1 D1 2 3   1 + (ε + 1) β3 A2 3 + 6A2 2 C2 − 3A2 B2 2 + 6B2 2 C2 + 3C2 3 + 3C2 D2 2 3 − C2 9γ 2 + 2ζS D1 3γ + C1 + 2ζA λ (ε + 1) D2 3γ + λ2 (ε + 1) C2 = 0,

(22.4), s3 :

 1  α3 3A1 2 B1 + 6A1 2 D1 − B1 3 + 6B1 2 D1 + 3C1 2 D1 + 3D1 3 3   1 + εβ3 3A2 2 B2 + 6A2 2 D2 − B2 3 + 6B2 2 D2 + 3C2 2 D2 + 3D2 3 3 − D1 9γ 2 − 2ζS C1 3γ + D1 − 2ζA λε C2 3γ + λ2 εD2 = 0,

(22.5), s3 :

 1  α3 3A1 2 B1 + 6A1 2 D1 − B1 3 + 6B1 2 D1 + 3C1 2 D1 + 3D1 3 3   1 + (ε + 1) β3 3A2 2 B2 + 6A2 2 D2 − B2 3 + 6B2 2 D2 + 3C2 2 D2 + 3D2 3 3 − D2 9γ 2 − 2ζS C1 3γ + D1 − 2ζA λ (ε + 1) C2 3γ + λ2 (ε + 1) D2 = 0.

(22.8)

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Fig. 22.3 Comparison of the FRF of the absorber relative displacement qA with different approaches. (a) Ansatz 1, qA = A2 % cos (γ τ ) + B2 sin (γ τ ) % 2 2 with qA,eff = A2 + B2 , and Ansatz 2, qA = A2 cos (γ τ ) + B2 sin (γ τ ) + C2 cos (3γ τ ) + D2 sin (3γ τ ) with qA,eff = A22 + B22 + C22 + D22 (b) especially at the high amplitudes, there is a visible difference

The description in the first column refers to the equation of motion and the sine/cosine expression used for comparison of coefficients. Although Eqs. 22.8 are given in analytical form, the evaluation for a specific case can only be done numerically. In Fig. 22.3, the FRF of the root mean square absorber displacement amplitude as well as their difference is shown for two different harmonic balance approximations. In the high-amplitude range, see Fig. 22.3b, minor differences between the absorber amplitudes can be detected.

22.4 Evaluation Approach and Results With the methods introduced, it is possible to select a set of design points and optimize the free parameters. The design points are dependent on the known parameter (mS , ωS , α 3 , ζ S ) of the host structure and a chosen mass ratio μ = mA /mS . The natural frequency ωA and the linear viscous damping ratio ζ A are determined using Den Hartog’s design rules [11] for linear systems, since the absorber must perform well in the low amplitude range: $ 1 , ωA = ωS 1+μ

ζA =

3μ . 8 (1 + μ)

(22.9)

Consequently, only the absorber’s cubic nonlinear stiffness β 3 remains unknown. It is determined according to Den Hartog equal-peak method: the FRF of qS is minimized which implies that the resonance peaks of the coupled system have approximately the same amplitude; see Fig. 22.4. If this numerical optimization procedure is repeated for all DOE points, the dependence of β 3 on the structural parmeter can be described using a DOE surface. In the present work, however, the computer simulation of the entire system is replaced by RTHS. Therefore, only the host structure is simulated in a computer model; the nonlinear dynamic absorber is designed, built, and tested in the laboratory. To obtain the characteristics of a Duffing oscillator, a couple of repelling magnets are added to the beam-type absorber: a large neodymium magnet is attached to the base to create a strong magnetic field, whereas a small magnet is fixed to the absorber mass. The beam stiffness and the interaction of the magnets create a complex force field, which can be approximated by a cubic polynomial. Furthermore, absorber damping is obtained from eddy current damping by inserting an aluminum plate in the air gap between the magnets. Since the damping force is proportional to the relative horizontal velocity of the magnet, almost linear viscous damping results [21]. In order to execute the applied RTHS optimization procedure efficiently, the absorber parameter has been identified for different geometric configurations. Figure 22.5 illustrates that both the linear and the nonlinear stiffness depend on the

22 Mitigation of Nonlinear Structural Vibrations by Duffing-Type Oscillators Using Real-Time Hybrid Simulation

Fig. 22.4 Generalized Den Hartog’s equal-peak method to determine β 3 , the effective structural displacement is qS,eff =

159

%

A21 + B12 + C12 + D12 ,

and the excitation amplitude is given by F = aF mS , — aF = 0.02m/s2 , — aF = 0.04m/s2 , — aF = 0.06m/s2 , —aF = 0.08m/s2 , — aF = 0.10m/s2

Fig. 22.5 Identification of linear and nonlinear stiffnesses of the absorber as a function of the magnet distance hM , (a) nonlinear stiffness kA3 , (b) linear stiffness kA

distance hM between the magnets. This has an impact on RTHS testing procedure because the stiffness coefficients cannot be chosen independently. However, in the absorber amplitude range considered, the coupling effect is rather small. Experimental measurements indicate that kA and kA3 linearly depend on hM : kA = −7.62 hM + 772.67,

kA3 = 179974 hM − 16594700.

(22.10)

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Fig. 22.6 Dependence of linear viscous damping ratio ζ A (a) on the distance to the permanent magnet on the absorber hP and (b) on the mass ratio with the required design points.

Fig. 22.7 Schematics of the RTHS experiments for the absorber parameter optimization for a given host structure with five different excitation levels

The viscous damping ζ A can be adjusted by changing the distance hP between the aluminum plate and the small magnet on the absorber [21]. The experimental identification results are given in Fig. 22.6a, where an analytical expression is obtained from curve fitting: ζA = 15.9 e−0.466 hP .

(22.11)

For the sake of completeness, the dependence of ζ A on the mass ratio μ is depicted in Fig. 22.5b, with the CCD and DM design points added. The results and derivations presented can now be combined to the RTHS-based absorber optimization process, whose schematics is illustrated in Fig. 22.7. Given the parameter of the host structure, the mass ratio μ, and the range of excitation levels, the numerical optimization renders β 3 which can be converted to the nonlinear absorber stiffness kA3 ; see Eq. 22.6. For the numerical optimization according to Den Hartog’s EPM, the analytical approximation given by [9]: β3 ∼ = 2μ2 α3 / (1 + 4μ)

(22.12)

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Fig. 22.8 Design point 2. (a) FRF of the absorber displacement, (b) FRF of the structure displacement, —.stable, − – unstable, × RTHS test decreasing excitation frequency, ¤ RTHS test increasing excitation frequency up, —aF = 0.02m/s2 , —aF = 0.04m/s2 , —aF = 0.06m/s2 , — aF = 0.08m/s2 , —aF = 0.10m/s2

is an excellent initial value. Subsequently, the corresponding experimental parameter hP , hM can be derived from Figs. 22.5 and 22.6a. After a physical adjustment of the absorber setup, the RTHS experiments are carried out for five different forcing levels. From the FRF measurements, the physical absorber parameter kA , kA3 , ζ A can be identified using the harmonic balance amplitude estimate of Eqs. 22.8. Thereby, a nonlinear gradient-based optimization method is applied. If the physical adjustment of hP and hM is not optimal, the FRF does not exhibit the equal-peak behavior, and fine-tuning of hP and hM is necessary in another optimization loop. Otherwise, the optimal absorber configuration is confirmed experimentally. The excellent agreement between RTHS experiment and simulation model is illustrated in Fig. 22.8, where the solid line represents the simulated FRF. The experimental results for increasing excitation frequencies are indicated by circles, those for decreasing excitation frequencies by crosses. The diagrams demonstrate the perfect agreement between theoretical predictions and RTHS experiments. Although the system is strongly nonlinear, the presented design methodology turns out to be fairly robust, and the dynamic vibration mitigation works excellently over a relatively wide amplitude and frequency range. Contrary to linear vibration absorber, whose relative vibration reduction is independent of the excitation amplitude, the desired behavior does no longer exist for Duffing-type systems at very high excitation levels. Following the procedure described, a single DOE point can be processed and verified using the RTHS method. If this approach is repeated for all DOE points, the desired absorber quantities can be presented in diagrams like Fig. 22.9, where both optimal absorber stiffness kA and nonlinear stiffness kA3 are given for different mass ratios. Similar diagrams are derived for the dependence on the other design input parameter kS and kS3 . Those diagram indicate that kA3 is almost independent from kS3 and μ. Having evaluated all DOE points, β 3 can be determined by surface fitting. In Fig. 22.10 the results of Eq. 22.12 and those obtained by the proposed DOE method are compared. Although the overall shape of the DOE β 3 surfaces appears fairly similar, there are distinct differences close to the boundaries.

22.5 Conclusion and Summary The vibration reduction of nonlinear host structures by dynamic absorbers of Duffing type is studied in this work. If both, structure and absorber, are modeled by SDOF oscillators, the method of harmonic balance renders an approximate analytical steady-state solution. To use the harmonic balance method reliably also for larger amplitudes, a multi-term analytical approximation is presented in addition to the usual single-term approach. For optimal absorber design, a generalization of Den Hartog’s equal-peak method defines the free tuning parameters. To verify the simulated results experimentally, a physical model of the absorber is tested in a real-time hybrid simulation setup. The nonlinear host structure is described

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Fig. 22.9 Optimal absorber parameter for all DOE points and varying mass ratio, (a) linear stiffness kA , (b) nonlinear stiffness kA3

Fig. 22.10 Nonlinear stiffness parameter β 3 , (a) surface obtained from Eq. 22.12, (b) surface rendered by proposed DOE method

by a virtual simulation model; the physical absorber is coupled to the simulation model using a transfer system. Since the nonlinear parameter of the physical absorber is difficult to adjust, the measured frequency response function is used for parameter identification with a nonlinear least squares method. In case of a suboptimal physical parameter set, the absorber is fine-tuned and the RTHS tests are repeated again. The proposed methodology allows reliable testing of physical absorbers without having to build the host structures. The presented hybrid simulation experiments perfectly agree with the theoretical predictions of this work and results published in literature. In accordance with theory, the experiments have shown that nonlinear vibration mitigation is strongly dependent on the applied forcing levels. Therefore, the nonlinear absorber parameter changes accordingly. To reduce the amount of experiments, DOE methods are used to investigate important design points and transfer the results to similar configurations. For the parameter range studied, optimal absorption is found if the nonlinearities of structure and absorber are of the same type. Furthermore, the coefficient of the absorber nonlinearity is only dependent on its corresponding host structure counterpart for a given mass ratio. The remaining absorber parameters are determined by the well-established Den Hartog equal-peak design criteria for linear systems, and, therefore, the absorber will also perform well for small vibration amplitudes.

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References 1. Oueini, S., Nayfeh, A.: Analysis and application of a nonlinear vibration absorber. J. Vib. Control. 6, 999–1016 (2000) 2. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley Interscience, New York (1979) 3. Viguié, R., Kerschen, G.: Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology. J. Sound Vib. 326, 780–793 (2009) 4. Agnes, G.S.: Performance of Nonlinear Mechanical Resonant-Shunted Piezoelectric, and Electronic Vibration Absorbers for Multi-Degree-ofFreedom Structures. Ph.D. dissertation, Virginia Polytechnic Institute and State University (1997) 5. Viguié, R., Kerschen, G.: On the functional form of a nonlinear vibration absorber. J. Sound Vib. 329, 5225–5232 (2010) 6. Denman, H.H.: Tautochronic bifilar pendulum torsion absorbers for reciprocating engines. J. Sound Vib. 159, 251–277 (1992) 7. Shaw, S.W., Schmitz, P.M., Haddow, A.G.: Dynamics of tautochronic pendulum vibration absorbers: theory and experiment. J. Comput. Nonlinear Dyn. 1, 283–293 (2006) 8. Gozen, S., Olson, B.J., Shaw, S.W.: Resonance suppression in multi-degree-of-freedom rotating flexible structures using order-tuned absorbers. J. Vib. Acoust. 134, 061016 (2012) 9. Habib, G., Detroux, T., Viguié, R., Kerschen, G.: Nonlinear generalization of den hartog’s equal-peak method. Mech. Syst. Signal Process. 52–53(1), 17–28 (2015). https://doi.org/10.1016/j.ymssp.2014.08.009. URL arXiv:1604.03868v1 10. Ormondroyd, J., Den Hartog, J.P.: The theory of the dynamic vibration absorber. Trans. ASME. 50, 9–22 (1928) 11. Den Hartog, J.P.: Mechanical Vibrations. McGraw-Hill, New York (1934) 12. Detroux, T., Habib, G., Masset, L., Kerschen, G.: Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber. Mech. Syst. Signal Process. 60, 799–809 (2015). https://doi.org/10.1016/j.ymssp.2015.01.035. URL arXiv:arXiv:1604.05524v1 13. Wagg, D., Neild, S.: Nonlinear Vibration with Control, 2nd edn. Springer (2015) 14. Kerschen, G.: Modal Analysis of Nonlinear Mechanical Systems CISM, vol. 555. Springer (2014) 15. Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems. Springer, Cham (2019) 16. Hochrainer, M., Puhwein, A.M.: Investigation of nonlinear dynamic phenomena applying real-time hybrid simulation. In: Kerschen, G. (ed.) Nonlinear Structures and Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series (2019). https:// doi.org/10.1007/978-3-030-12391-8_16 17. Witek-Krowiak, A., et al.: Application of response surface methodology and artificial neural network methods in modelling and optimization of biosorption process. Bioresour. Technol. 160, 150–160 (2014). https://doi.org/10.1016/j.biortech.2014.01.021 18. Box, G.E.P., Wilson, K.B.: On the experimental attainment of optimum conditions. J. R. Statist. Soc. Ser. B Methods. 13, 1–45 (1951) 19. Doehlert, D.H.: Uniform Shell design. J. R. Stat. Soc. C-App. 19, 231–239 (1970) 20. Brezani, I.: GUI for evaluation of a custom three variables multilevel DoE. Retrieved from https://www.mathworks.com/matlabcentral/mlcdownloads/downloads/submissions/45837/versions/1/previews/html/Manual.html. 16 Oct 2021 21. Hochrainer, M., Puhwein, A.: Design and characterization of a multi-purpose duffing oscillator with flexible parameter selection, in nonlinear structures and systems, volume 1. Conf. Proc. Soc. Exp. Mech. Series. (2020). https://doi.org/10.1007/978-3-030-47709-7_7

Chapter 23

Approximate Bayesian Inference for Piecewise-Linear Stiffness Systems Rajdip Nayek, Mohamed Anis Ben Abdessalem, Nikolaos Dervilis, Elizabeth J. Cross, and Keith Worden

Abstract This paper considers the problem of simultaneous model selection and parameter estimation for dynamical systems with piecewise-linear (PWL) stiffnesses. PWL models are a series of locally linear models that specify or approximate nonlinear systems over some defined operating range. They can be used to model hybrid phenomena common in practical situations, such as, systems with different modes of operation, or systems whose dynamics change because of physical limits or thresholds. Identifying PWL models can be a challenging problem when the number of operating regions and the boundaries of the regions are unknown. This study focusses on the joint problem of identifying the regions (their number and boundaries) as well as their associated parameters. PWL-stiffness models with up to four regimes are considered, and the identification problem is treated as a combined model selection and parameter estimation problem, addressed in a Bayesian framework. Because of the varying number of parameters across the PWL models, traditional Bayesian model selection would typically require reversible-jump Markov chain Monte Carlo (RJ-MCMC) for switching between model spaces. Here instead, a likelihood-free Approximate Bayesian Computation (ABC) scheme with nested sampling is followed, which simplifies the jump between model spaces. To illustrate its performance, the algorithm has been used to select models and identify parameters from four PWL-stiffness systems—linear, bilinear, trilinear, and quadlinear stiffnesses. The results demonstrate the flexibility of using ABC for identifying the correct model and parameters of PWL-stiffness systems, in addition to furnishing uncertainty estimates of the identified parameters. Keywords Piecewise-linear systems · Approximate Bayesian computation · Model selection · Parameter estimation

23.1 Introduction Confronted with the analysis of a nonlinear system, an age-old technique is to approximate the nonlinear system with one or more linear systems. Linear systems are easy to analyse and are solved more easily than nonlinear systems. If a single linear approximation around the operating point is accurate enough, the solution will be close to that of the nonlinear system. If not, then one can approximate the nonlinear system by a set of linear systems defined over different regions of operation; such a set of linear systems is called a piecewise-linear (PWL) system. PWL systems are quite an interesting class of models, since they possess universal function approximation properties [1], meaning that they can approximate any nonlinear function arbitrarily well, and they generalise the well-established theory of linear systems for analysing nonlinear systems. PWL systems (or more generally piecewise affine (PWA) systems) are very useful in modelling certain types of nonlinear phenomena occurring in many practical situations, such as systems with different modes of operation, or systems where the dynamics change because of thresholds, switches, or physical limits. One of the earliest practical implementations of PWL systems can be found in control engineering, where a piecewise-linear control formulation was introduced in anticipatory actions of a servomechanism [2]. In mechanical engineering, systems such as shock absorbers, vibration isolators, etc. involve different modes of operation, because of motion limiting constraints. Such systems can be modelled very well by PWL models. For example, shock absorbers possess different damping characteristics in compression and rebound which can be modelled using PWL damping [3]. Similarly, vibration isolators consisting of soft vibration-isolation springs and high stiffness and damping protection stops can be modelled with PWL stiffness and damping models [4].

R. Nayek () · M. A. B. Abdessalem · N. Dervilis · E. J. Cross · K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_23

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In the last few decades, there have been several studies on the closed-form response computation, stability analysis and control of PWL systems [5–7]. For such analyses, the PWL systems were assumed to specified, meaning that the parameters of each local linear system and the boundaries separating them were assumed to be known. However, in most practical cases, these parameters are unknown and have to be either arrived at based on appropriate application of first principles or they have to be identified from experimental data, the latter being the most commonly adopted approach. Identification of PWL/PWA models can be rather challenging, and a lot depends upon what information is assumed to be known. The full set of parameters of a PWA model includes (a) the number of operating regions, (b) the coefficients of the separation boundaries or partitions, and (c) the parameters of each local linear model. When the number of operating regions and their partitions are known, estimating the parameters of the local linear models is relatively easy and can be achieved with standard parameter estimation methods [8]. However, when coefficients of the partitions are unknown, the problem becomes considerably more challenging, and even more difficult when the number of operating regions are unknown as well. For a complete estimation of all parameters, one has to simultaneously classify the operating regions (i.e. identify the number of operating regions and estimate the coefficients of the partitions) and estimate the parameters of the linear submodels. In a sense, the estimation problem turns into a combined classification and regression problem. A variety of frequentist and Bayesian approaches for identifying PWL systems has been proposed in the literature; the approaches vary in the formulation and complexity of their identification procedure, see [9, 10] for a review. In most studies, identification of PWL systems have been proposed in the input–output space using piecewise AutoRegressive with eXogenous inputs (PWARX) models [11–13] and only a very few treated it in the state space using state-space models [8]. When the number of operating regions is treated as known, the coefficients of the partitions and parameters of linear submodels can be estimated via optimisation [14, 15]. However, schemes like gradient descent or Gauss–Newton search may get stuck in local optima and an evolutionary optimisation scheme (like a genetic algorithm) would be needed. As an extension to this problem with fixed number of operation regions, a few studies proposed to add new partitions (i.e., new sub-regions) progressively using several steps in identification [16–18]. Another group of methods [11, 12, 19–22] treat the number of operating regions as unknown and start by classifying the data points according to a certain criterion and then estimating the local linear/affine submodels simultaneously or iteratively. This study proposes a very straightforward and conceptually simple approach to identify PWL systems, using an approximate Bayesian computation-based procedure for model selection and parameter estimation [23, 24]. Model selection is used to determine the number of operating regions and Bayesian parameter estimation for inferring the posterior distribution over the parameters of partitions as well as those of the local linear maps. Unlike previous Bayesian approaches [11–13] that were developed as specific Bayesian computation algorithms tailored to the formulation of PWARX systems, the ABC procedure can work with any formulation as long as the model responses can be simulated given the parameters. The simplicity of the approach lies in the fact that one only needs to specify a set of PWL models with different model orders, i.e. different number of operating regions, and the ABC algorithm automatically determines the most suitable model, as well as estimate the posterior over its parameters. To demonstrate the working of this concept, the ABC-based combined Bayesian model selection and parameter estimation is applied to a single degree-of-freedom (SDOF) vibratory oscillator with PWL stiffness, i.e. one that has different linear stiffnesses defined over different regions of operation. The particular interest in PWL-stiffness dynamic models is because these models are quite useful in modelling vibration sinks [25], vibration isolators [4], and freeplay nonlinearities that occur because of worn-out hinges and loose rivets [26]. Moreover, PWL-stiffness nonlinearities arise in aircraft vibration tests from pylon–store–wing assemblies or pre-loading bearing locations [3]. The goal in this study is to use ABC to find the best model from a set of PWL-stiffness models (with different model orders) and to simultaneously estimate the posterior distribution of the selected PWL-stiffness model parameters along with the other parameters of the oscillator. The rest of the paper is organised as follows: Sect. 23.2 outlines the mathematical description of an SDOF vibratory oscillator with PWL stiffness. Section 23.3 details the procedure for model selection and parameter posterior estimation of the SDOF oscillator using ABC, followed by a numerical demonstration in Sect. 23.4. Finally, Sect. 23.5 provides some concluding remarks.

23.2 Description of SDOF Oscillator with PWL-Stiffness Consider a nonlinear SDOF mechanical oscillator with PWL stiffness subjected to a forcing function u, m¨z + c˙z + kPWL (z) = u

(23.1)

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Fig. 23.1 Mechanical models with bilinear (a), trilinear (b), and quadlinear (c) stiffnesses

where, z, z˙ , z¨ are displacement, velocity and acceleration responses, m is the mass, c is the damping coefficient, and kPWL denotes the PWL stiffness as a function of the displacement z. A dynamical system with PWL stiffness is obtained by partitioning the displacement space into a finite number of operating regions and by considering a linear map in each region. When there exists only a single region of operation, i.e., no partitions, the system is a linear system; however, when there are more than a single region of operation, one obtains a PWL system. Figure 23.1 shows three nonlinear mechanical models with bilinear, trilinear, and quadlinear stiffnesses and also depicts their corresponding stiffness-based-restoring-force versus displacement relations. Figure 23.1a illustrates a bilinear system that has two linear regions of operation separated by a breakpoint, each region characterised by its own linear-stiffness parameter, k0 and kR1 , respectively. The breakpoint between the two adjacent regions of operation represents a displacement partition and its location on the displacement axis is denoted by dR1 . For the purpose of convenience, the stiffness and partition parameters are denoted by subscripts L# or R# to refer to which side (left or right) of the mass they appear. If the partition for (say) the bilinear-stiffness system appears to the left of the mass (instead of right, as shown in Fig. 23.1a), the stiffness parameter would be denoted by kL1 and the partition parameter by dL1 . It must be emphasised that the adjacent stiffness and partition parameters for any PWL-stiffness system should be distinct from each another. For example, the two adjacent stiffnesses kR1 and kR2 of a QL stiffness system (see Fig. 23.1c) cannot be the same, else it would resemble a TL stiffness model. Such redundancies may be avoided by enforcing the adjacent stiffness and partition parameters to be different from each other. In this study, dynamic PWL-stiffness models exhibiting up to four regions of operations (i.e. quadlinear maps) are considered, since mechanical systems with more than four regions of operation are less frequent in practice. Mathematically, the piecewise-linear maps for bilinear (BL), trilinear (TL), and quadlinear (QL) stiffnesses used in this study are described by Eqs. (23.2), (23.3), and (23.4), respectively. Bilinear:  m¨z + c˙z + kBL (z) = u,

kBL (z) =

k0 z, k0 dR1 + kR1 (z − dR1 ) ,

for z ≤ dR1 for z > dR1

(23.2)

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Trilinear: m¨z + c˙z + kTL (z) = u,

⎧ ⎨ k0 dL1 + kL1 (z − dL1 ) , kTL (z) = k0 z, ⎩ k0 dR1 + kR1 (z − dR1 ) ,

for z ≤ dL1 for dL1 < z ≤ dR1 for z > dR1

(23.3)

Quadlinear:

m¨z + c˙z + kQL (z) = u,

⎧ k0 dL1 + kL1 (z − dL1 ) , ⎪ ⎪ ⎨ k0 z, kQL (z) = ⎪ k d + kR1 (z − dR1 ) , ⎪ ⎩ 0 R1 k0 dR1 + kR1 (dR2 − dR1 ) + kR2 (z − dR2 ) ,

for z ≤ dL1 for dL1 < z ≤ dR1 for dR1 < z ≤ dR2 for z > dR2

(23.4)

The parameters of the above SDOF models comprise the mass, the damping coefficient, and the stiffness and partition parameters of the PWL-stiffness map. Given these parameters, one can simulate forward responses from any of the above PWL models using a suitable numerical time-integration scheme, such as MATLAB’s ode45 [27]. In inverse identification, however, the goal is select a model with appropriate number of regions of operation and estimate its parameters based on some measured data. What follows next are the details of the ABC algorithm used for model selection and parameter estimation to help achieve this goal.

23.3 Bayesian Model Selection and Parameter Estimation via ABC " ! Given some observed dataset D = y 1 , . . . , y n , Bayesian parameter estimation framework seeks to estimate the posterior probability distribution of the parameters for a specified model. The Bayes’ rule for obtaining the posterior distribution of parameters θ i of model Mi is given by, p (θ i | D, Mi ) =

p (D | θ i , Mi ) p (θ i | Mi ) p (D | Mi )

(23.5)

where p (D | θ i , Mi ) is the likelihood of the dataset, p (θ i | Mi ) is the parameter prior distribution, and p (D | Mi ) is the marginal likelihood (also called the model evidence). The model evidence is a multi-dimensional integral over the modelparameter space, # p (D | Mi ) =

p (D | θ i , Mi ) p (θ i | Mi ) dθ i

(23.6)

and is typically analytically intractable, especially when the likelihood is non-Gaussian and/or the dimension of the parameter space θ i is high. Markov Chain Monte Carlo (MCMC) [28] techniques are often used to approximate the unnormalised parameter posterior distribution by bypassing the computation of model evidence. However, the computation of model evidence is important in Bayesian model selection, where the task is to determine the best model from a set of K models, M = {M1 , . . . , MK }; the best model is the one that has the highest relative model posterior probability among all models in the set. The expression for the model posterior for the ith model Mi is given by, p (Mi | D) =

p (D | Mi ) p (Mi ) ∝ p (D | Mi ) p (Mi ) p (D)

(23.7)

where p (Mi ) is the model prior probability and p (D) is a constant for all models in the model set. The computation of the model evidence p (D | Mi ) is not straightforward with standard MCMC algorithms. A specific MCMC algorithm that performs simultaneous model selection and parameter estimation is the Reversible-jump MCMC (RJMCMC) [29] and is commonly used in Bayesian model selection and parameter estimation. However, RJMCMC is difficult to implement and has its own drawbacks: Each time an RJMCMC jumps between models, the parameter values sampled in the previous model are lost and subsequent jumps into that model must start afresh, which can make the RJMCMC quite inefficient.

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Instead of adopting an MCMC algorithm, an alternative ABC-based approach [30, 31] is followed here, which is simple to implement and does not require jumping between model spaces like RJMCMC. Furthermore, ABC belongs to the class of simulation-based likelihood-free methods that can handle problems where the likelihood is analytically intractable or is difficult to compute. ABC methods operate by generating simulated 4 comparing them to the observed data. To 3 datasets and ∗ is compared to the observed data D, and the conduct posterior parameter estimation, a simulated dataset D∗ = y1∗ , . . . , yN t corresponding parameter sample is accepted if a suitable distance measure between them, ρ (D, D∗ ), is less than a specified threshold ε. By sampling several such parameter samples, the ABC algorithm forms an approximate parameter posterior of the form [31], #     pε (θ i | D, Mi ) ∝ p θ i , D∗ | ρ D, D∗ < ε, D, Mi dD∗ (23.8) where pε (θ i | D, Mi ) is the ABC-based approximate parameter posterior for model Mi , and it becomes increasingly close to the true parameter posterior p (θ i | D, Mi ) as ε gets smaller. The distance measure in this study has been chosen to be the Normalised Mean Squared Error (NMSE), between the simulated data D∗ and observed data D, Nt    ∗ 2 100  ρ D, D∗ = yi − yi 2 Nt σD i=1

(23.9)

2 is the variance of the observed dataset. where σD Alongside parameter estimation, ABC methods can perform model selection [23] quite conveniently. Model selection using ABC methods is based on relative frequencies of parameter samples for different models lying in the model set; the best model is selected as the one that has the highest proportion of parameter samples in a fixed population of samples [23]. This relatively simple framework of ABC is what makes it a desirable choice for model selection and parameter estimation over MCMC-based approaches in this study. There are several ABC algorithms that have been proposed in the last decade, see [30] for details. In this study, the ABC-NS [24] algorithm is applied for Bayesian model selection and parameter estimation, albeit with some modifications tailored for faster convergence. The ABC-NS method operates by generating a population of parameter samples, referred to as particles, from different models in the model set, and then iteratively moves the samples closer to the posterior parameter distribution. The ABC-NS algorithm implemented in this study consists of the following main steps:

1. Create an initial population of particles: This step generates an initial population of Ns particles from K models using model priors and parameter priors. First, the models are sampled from model priors, then model-specific parameters are sampled from parameter priors. For each sampled parameter θ ∗ belonging to model Mk , a dataset D∗ is simulated and compared with the observed dataset D, using NMSE given in Eq. (23.9). An Ns number of particles that have their NMSE values lower than an initial user-defined threshold ε1 are accepted and saved as the initial population of particles. See lines 2–12 in Algorithm 1. 2. Define the next threshold: The NMSE values of all particles of the initial population are sorted (or ranked) in a descending order, and the next threshold ε2 is defined as equal to (α0 Ns )th element of the ordered NMSE values. See lines 13–15 in Algorithm 1. 3. Assign weights to particles: The particles with NMSE values above threshold ε2 are dropped and the rest of “active” particles are assigned weights {wk }K k=1 based on their NMSE values; a particle with a lower NMSE value receives a higher weight. See lines 16–17 of Algorithm 1 and the weighting module in Algorithm 3. 4. Construct ellipsoids for sampling: The active particles, assembled in {Ak }K k=1 , are used to construct model-specific ! "K K ellipsoids {Ek }k=1 defined by their weighted means μk k=1 and weighted covariances { k }K k=1 respectively, =

μk

&1 wk,i i

k =

&1 wk,i i

&

wk,i θ k,i

(23.10a)

i

  T & wk,i θ k,i − μk θ k,i − μk

(23.10b)

i

Here θ k,i denotes the ith active parameter particle from model Mk and Wk,i is its corresponding scalar weight. Furthermore, the best parameter particles (with lowest NMSE values) from all models in the current population are saved "K ! as θ k,best k=1 . See lines 18–19 in Algorithm 1 and the ellipsoid construction module in Algorithm 4.

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Algorithm 1 Modified ABC-NS sampler 1: Input: Observed data D = {y1 , . . . , yn }, model set M = {M1 , . . . , MK }, total number of particles Ns , initial proportion of dropped particles α0 , initial threshold ε1 , importance-sampling probability pbest , shrink factor γ , stopping criterion 2: procedure GENERATE INITIAL POPULATION OF PARTICLES  For first population 3: p = 1, iter = 0 4: while iter < Ns do 5: Sample a model from model set: Mk ∼ p (M | M) 6: Sample a parameter from the model: θ ∗ ∼ p (θ | Mk ) 7: Simulate data: D∗ ∼ Mk (θ ∗ ) 8: Calculate NMSE value: ρ (D, D∗ ) 9: if ρ (D, D∗ ) < ε1 then 10: Save parameter sample θ ∗ in k,1 11: Save NMSE value ρ (D, D∗ ) in ek,1 12: iter = iter + 1  13: Concatenate the NMSE values for all models: eall,1 = e1,1 , . . . , eK,1 , eall,1 ∈ RNs 14: Sort eall,1 in descending order, and get an integer index ix = floor (α0 Ns ) 15: Define next threshold as ε2 = eall,1 (ix ) "K ! 16: procedure COMPUTE WEIGHTS OF PARTICLES( ek,1 k=1 , ε1 , ε2 ) 17: Output: {wk }K k=1 "K ! 18: procedure CONSTRUCT ELLIPSOIDS FOR SAMPLING( k,1 , ek,1 , wk k=1 ) ! "K 19: Output: μk ,  k , θ k,best , Ak , eA k k=1 , Na 20: ======================================================= 21: while stopping criterion is satisfied do 22: Increment population p = p + 1 ! "K 23: procedure SAMPLE FROM ELLIPSOIDS( μk ,  k , θ k,best k=1 , pis , γ , D, εp , Na , Ns ) " ! K 24: Outputs: S k , eS k k=1

32:

Define current population of particles: k,p = ! {Ak ,SS"k } for k = 1, . . . , K NMSE values for current population: ek,p = eA k , ek for k = 1, . . . , K "K ! procedure COMPUTE THE NEXT THRESHOLD( ek,p−1 , ek,p k=1 ) Output: εp+1 ! "K procedure COMPUTE WEIGHTS OF PARTICLES( ek,p k=1 , εp , εp+1 ) Output: {wk }K k=1 ! "K procedure CONSTRUCT ELLIPSOIDS FOR SAMPLING( k,p , ek,p , wk k=1 ) "K ! Output: μk ,  k , θ k,best , Ak , eA k k=1 , Na

33:

Check stopping criterion

25: 26: 27: 28: 29: 30: 31:

Algorithm 2 Automatic threshold selection module

! ! "K "K 1: Inputs: NMSE values of previous population ek,p−1 k=1 and current population ek,p k=1 2: Outputs: Threshold εp+1 3: procedure COMPUTE A NEW NMSE THRESHOLD 4: Set K equal to the total number of models in model set 5: for each model k = 1 : K do 6: if ek,p−1 and ek,p are then    non-empty   7: Estimate pdf, pˆ ek,p−1 , and find its maximum value: fk,p−1 = max pˆ ek,p−1      8: Estimate pdf, pˆ ek,p , and find its maximum value: fk,p = max pˆ ek,p   9: Store the ratio: rk = 1 − fk,p−1/fk,p

10: 11: 12: 13: 14:

 For the (p + 1)th population

Find the maximum ratio among all models: rmax = max {r1 , . . . , rK } Set quantile value α = max of {rmax , 0.1}  Concatenate NMSE values from all models: eall,p = e1,p , . . . , eK,p , eall,p ∈ RNs Sort eall,p in descending order, and get the integer index ix = floor (αNs ) Define temporary threshold as εp+1 = eall,p (ix )

5. Sample from the ellipsoids: Define the next population of Ns particles by adding particles to the existing set of active particles. These added particles are obtained by sampling from the ellipsoids {Ek }K k=1 . During this step, based on a ! "K certain prefixed probability pbest , either standard ellipsoids with mean μk k=1 and covariances { k }K k=1 are used for K sampling, or shrunk ellipsoids with reduced covariances {γ  k }k=1 (0 < γ < 1) and mean shifted at the location of best

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Algorithm 3 Weight assignment module

"K ! 1: Inputs: NMSE values ek,p k=1 , current threshold εp , next threshold εp+1 2: Outputs: Weights {wk }K k=1 3: procedure COMPUTE WEIGHTS FOR PARTICLES 4: for each model k = 1 : K do 5: if ek,p is non-empty then   6: Get the total number of particles: nk = length ek,p 7: Initialise weight vector to zeros: wk = zeros(nk , 1) 8: Find the set of indices, I , of elements of ek,p with NMSE values lower than εp+1 9: for each index i in I do   

10: 11: 12:

Set the corresponding weight, wk (Ii ) =

1 εp

1−

 Weights for each sample in each model  For the pth population

ek,p (Ii ) 2 εp

The weights for the rest of the elements remain zero Normalise the weights such that they add upto unity

Algorithm 4 Ellipsoid construction module

! ! "K "K 1: Inputs: Population of particles k,p k=1 , NMSE values ek,p k=1 , weights {wk }K ! "K ! "K k=1 K 2: Outputs: Mean vectors μk k=1 , covariances { k }K k=1 , best particles θ k,best k=1 , active particles {Ak }k=1 , NMSE values of active particles ! A "K ek k=1 , count of active particles Na 3: procedure CONSTRUCT ELLIPSOIDS FOR SAMPLING 4: Set the count of active particles to zeros: na = zeros(K, 1) 5: for each model k = 1 : K do 6: if k,p is non-empty then 7: Calculate the number of non-zero weights: na (k) = length (find(w k > 0)) 8: Save na (k) number of active particles from k,p in Ak 9: Select the corresponding NMSE values of the active particles from ek,p and save them in eA k 10: Save the parameter sample of Mk with the highest weight as θ k,best 11: Define ellipsoid Ek with weighted mean μk and weighted covariance  k using active particles in Ak 12: Enlarge the ellipsoid by a factor f0 & 13: Save the total count of active particles, Na = K k=1 na (k)

Algorithm 5 Sampling module

"K ! "K ! K 1: Inputs: Ellipsoids {Ek }K k=1 with means μk k=1 and covariances { k }k=1 , best particles θ k,best k=1 , importance-sampling probability pbest , shrink factor γ , observed data D, current threshold εp , count of active particles Na , total number of particles Ns ! S "K 2: Outputs: Sampled population {S k }K k=1 , NMSE values for sampled population ek k=1 3: Dropped number of particles: Nd = Ns − Na 4: procedure SAMPLE PARTICLES FROM ELLIPSOIDS 5: iter = 0, satisfiedConstraint = false 6: while iter < Nd do 7: Sample a model from model set; Mk ∼ p (M | M) 8: Sample a uniform random number: u ∼ U (0, 1)  U (0, 1) is a uniform distribution between (0,1) 9: if pis > u then  Condition for sampling around best particle 10: Sample a particle θ ∗∗ from ellipse Ek with center θ k,best and shrinked covariance γ  k 11: else 12: Sample a particle θ ∗∗ from ellipse Ek with center μk and covariance  k 13: Check if parameter sample satisfies constraints: satisfiedConstraint = true/false ? 14: if satisfiedConstraint = true then 15: Simulate data: D∗ ∼ Mk (θ ∗∗ ) 16: Calculate NMSE value: ρ (D, D∗ ) 17: if ρ (D, D∗ ) < εp then 18: Save parameter sample θ ∗∗ in Sk 19: Save NMSE value ρ (D, D∗ ) in eS k 20: iter = iter + 1

"K ! particles θ k,best k=1 are used. The shrunk ellipsoids serve the purpose of sampling particles that have greater chances of acceptance and are useful in scenarios where the acceptance rates of particles from standard ellipsoidal sampling become diminishingly small. However, sampling from shrunk ellipsoids is expected to occur only occasionally and is controlled by pbest , which is usually given a small value such as 0.05, meaning that shrunk ellipsoids may be used for sampling only

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5% of the time. Note that, the sampled particles are accepted only if they satisfy the constraint conditions (if any), and when their corresponding NMSE values fall below the previously generated threshold. 6. Computing the next threshold automatically: The next threshold is selected adaptively, based on NMSE values of particles of the current population as well as those of the previous population. The idea is compare the relative concentration of the distributions of model-specific NMSE values between two population of particles, and then gauge how much the tolerance should be decreased to (a) prevent diminishingly small acceptance rates (when the tolerance change is too big) and/or (b) prevent the algorithm getting stuck around the same tolerance (when the tolerance change is too little and the samples are dominated by a single model class). For comparing concentrations, a kernel density estimator is used to estimate the densities of two successive populations of particles for each model class, and the ratios of the maximum values of the densities for the two successive populations are computed. The ratio rk for a model class Mk is computed as, rk = 1 −

fk,p−1 fk,p

(23.11)

where fk,p−1 and fk,p are the maximum values of the densities of NMSE values for the successive populations p − 1 and p, respectively. A quantile α is assigned based on the maximum of the ratios {r1 , . . . , rK }. Once the value of α is determined, the NMSE values of all the particles of the current pth population are sorted in a descending order, and the next threshold εp+1 is defined equal to the (αNs )th element of the ordered NMSE values. See lines 27–28 in Algorithm 1 and the automatic threshold selection module in Algorithm 2. 7. Repeat iterations until stopping criterion is met: The steps of weight assignment (Step 3), ellipsoid construction (Step 4), sampling from ellipsoids (Step 5), and automatic calculation of next threshold (Step 6) are repeated until a stopping criterion is met. When the relative difference between NMSE thresholds between two successive populations is less than a preset tolerance tol , the ABC algorithm is deemed to have converged and the iterations are stopped. Post convergence of the ABC algorithm, the posterior model probability is approximated by the proportion of particles in each model class from the last population, p (Mk | D) ≈

Number of particles in Mk Total number of particles Ns

(23.12)

The model class that has the largest proportion of particles from the population of Ns particles is taken to be the “best” model, and the posterior distributions over its parameters are approximated by the particles belonging to the model.

23.4 PWL-Stiffness Model Identification Using ABC This section presents a numerical example to demonstrate the identification of PWL-stiffness systems using the aforementioned ABC framework. An SDOF spring–mass–damper model with trilinear stiffness is considered for identification, m¨z + c˙z + kTL (z) = u

(23.13)

with the true parameters of the model set to: m = 1 kg, c = 2 Ns/m, k0 = 103 N/m, kL1 = 3 × 103 N/m, kR1 = 7 × 103 N/m, dL1 = −1 mm, and dR1 = 0.5 mm. The trilinear-stiffness system is subjected to an input excitation u that is modelled as a zero-mean Gaussian bandlimited noise sequence, with a standard deviation of 2 N and passband of [0, 20] Hz. The system is simulated using ode45 function in MATLAB, with a sampling rate of 100 Hz for a time span of 10 s. It is assumed that only the acceleration response is available for measurement. The noiseless acceleration output from the “true” model, shown in Fig. 23.2, is corrupted with 5% zero-mean Gaussian white noise and then used as measured data for identification. As mentioned previously, identification of a PWL system involves determining the number of linear regions, the parameters associated with each linear map (stiffnesses and partitions), and the mass and damping parameters. A maximum of four linear regions are guessed for the system, which translates to performing model selection from a set of four models: a linear-stiffness model, a bilinear-stiffness model, a trilinear-stiffness model, and a quadlinear-stiffness model. Note that the number of parameters to be identified varies with each model class: three parameters for a linear model, five for a bilinear model, seven for a trilinear model, and nine for a quadlinear model. It must be remarked at this point, that the stiffness and partition parameters of two neighbouring linear regions must differ from each other to avoid redundancies. Hence, two constraint conditions are imposed while sampling the parameter particles for a given model: (a) the partition parameters

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Fig. 23.2 Noiseless acceleration from “true” SDOF model with trilinear stiffness Table 23.1 Uniform parameter priors for linear (L), bilinear (BL), trilinear (TL), and quadlinear (QL) stiffness models

Model L, BL, TL, QL L, BL, TL, QL L, BL, TL, QL BL, TL, QL BL, TL, QL

Parameter m c k0 kL# /kR# dL# /dR#

Lower bound 0.5 kg 0.1 Ns/m 0.5 × 103 N/m 0.5 × 103 N/m −10 mm

Upper bound 1.5 kg 4 Ns/m 1.5 × 103 N/m 10 × 103 N/m 10 mm

Fig. 23.3 Plot of variation of acceptance rate and threshold with ABC populations. Small windows on the top of the plot depict model posterior probabilities for a selected number of populations

dL# /dR# of a particle must be unique, and (b) the adjacent local stiffness parameters of a particle must differ by a minimum of 5% from each other. All particles that are accepted in a population must satisfy these two constraints. Bayesian inference requires specification of prior beliefs over parameters as well as over models. Here, a uniform discrete prior is assumed over the models, that is, p (M1 ) = p (M2 ) = p (M3 ) = p (M4 ) = 14 . The priors over all parameters of the models in the model set are set using uniform distributions, with their corresponding ranges given in Table 23.1. In this study, the ABC hyperparameters are set to ε1 = 200, α0 = 0.4, pbest = 0.05, γ = 0.1, and tol = 0.005. The ABC algorithm takes 60 iterations (or populations) to converge, which takes around 3.5 hours. The variation of acceptance rates and decrement of thresholds over several populations of ABC are shown in Fig. 23.3. Alongside, the model posterior probabilities for a selected number of populations are also depicted in Fig. 23.3. It can be seen that the acceptance rate decreases with populations, as the simpler models exit and the parameter search-space becomes larger for trilinear and quadlinear models, while the thresholds becomes more tight. After population 10, the linear and bilinear models exit, as they are not able to produce datasets that are closer to the observed data than the NMSE thresholds of

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Fig. 23.4 ABC-based marginal posterior distributions over parameters of the trilinear-stiffness model; the true values of parameters are shown in triangle markers

the next population. With the remaining trilinear and quadlinear models, the parameter search spaces become relatively more complex, making it difficult to sample good particles—particles with NMSE lower than thresholds—and hence a decline in the acceptance rates is observed post population 10. Nonetheless, the ABC algorithm is able to clearly select the trilinearstiffness model over the quadlinear-stiffness model at the end of convergence, demonstrating the principle of parsimony in Bayesian model selection. The marginal parameter posterior distributions for the trilinear system are shown in Fig. 23.4, in the form of histograms. The true values of the parameters are observed to be lie around the peaks of the histograms.

23.5 Conclusion In this study, the problem of identifying dynamical systems with piecewise-linear stiffness is attempted, as PWL-stiffness systems are useful in modelling a variety of engineering systems. Identification of PWL-stiffness systems involves estimating the number of linear regions of operation and the parameters associated with each linear region. In this study, a maximum of four linear regions of stiffnesses are considered, and the estimation of the number of regions in a PWL-stiffness system is treated as a problem of selecting the correct model from a set of models with different numbers of piecewise-linear regions. As such, the identification problem turns into a straightforward task of Bayesian model selection and parameter estimation. Instead of using a standard MCMC-based technique for model selection, an ABC-based approach is followed which is more intuitive and simple to implement. To demonstrate the performance of the proposed methodology, an ABC-NS algorithm has been used to identify an SDOF oscillator with trilinear stiffness, from among four models with linear, bilinear, trilinear, and quadlinear stiffnesses. The results show that the ABC algorithm is able to select the correct model as well as identify the partition coefficients, regional stiffness parameters and other oscillator parameters like mass and damping. The results show the simplicity of the proposed approach and demonstrate its potential for identifying PWL systems in a Bayesian framework.

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Acknowledgments The authors gratefully acknowledge the support of the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant references EP/R006768/1, EP/N018427/1, EP/J016942/1, and EP/S001565/1.

References 1. Lin, J.-N., Unbehauen, R.: Canonical piecewise-linear approximations. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 39(8), 697–699 (1992) 2. Schwartz, W.J.: Piecewise linear servomechanisms. Trans. Amer. Inst. Electr. Eng. Part II Appl. Ind. 71(6), 401–405 (1953) 3. Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. CRC Press, Boca Raton (2019) 4. Wang, X., Zheng, G.: Two-step transfer function calculation method and asymmetrical piecewise-linear vibration isolator under gravity. J. Vibr. Control 22(13), 2973–2991 (2016) 5. Chicurel-Uziel, E.: Exact, single equation, closed-form solution of vibrating systems with piecewise linear springs. J. Sound Vibr. 245(2), 285–301 (2001) 6. Natsiavas, S.: Periodic response and stability of oscillators with symmetric trilinear restoring force. J. Sound Vibr. 134(2), 315–331 (1989) 7. Ji, J.-C., Leung, A.Y.T.: Periodic and chaotic motions of a harmonically forced piecewise linear system. Int. J. Mech. Sci. 46(12), 1807–1825 (2004) 8. Verdult, V., Verhaegen, M.: Subspace identification of piecewise linear systems. In: 2004 43rd IEEE Conference on Decision and Control (CDC), vol. 4, pp. 3838–3843. IEEE, Piscataway (2004) 9. Paoletti, S., Juloski, A.L., Ferrari-Trecate, G., Vidal, R.: Identification of hybrid systems a tutorial. Eur. J. Control 13(2–3), 242–260 (2007) 10. Garulli, A., Paoletti, S., Vicino, A.: A survey on switched and piecewise affine system identification. IFAC Proc. Vol. 45(16), 344–355 (2012) 11. Juloski, A.L., Weiland, S., Heemels, W.P.M.H.: A Bayesian approach to identification of hybrid systems. IEEE Trans. Autom. Control 50(10), 1520–1533 (2005) 12. Wågberg, J., Lindsten, F., Schön, T.B.: Bayesian nonparametric identification of piecewise affine ARX systems. IFAC-PapersOnLine 48(28), 709–714 (2015) 13. Piga, D., Bemporad, A., Benavoli, A.: Rao-Blackwellized sampling for batch and recursive Bayesian inference of piecewise affine models. Automatica 117, 109002 (2020) 14. Chan, K.S., Tong, H.: On estimating thresholds in autoregressive models. J. Time Ser. Analy. 7(3), 179–190 (1986) 15. Julián, P., Jordán, M., Desages, A.: Canonical piecewise-linear approximation of smooth functions. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 45(5), 567–571 (1998) 16. Breiman, L.: Hinging hyperplanes for regression, classification, and function approximation. IEEE Trans. Inf. Theory 39(3), 999–1013 (1993) 17. Johansen, T.A., Foss, B.A.: Identification of non-linear system structure and parameters using regime decomposition. Automatica 31(2), 321–326 (1995) 18. Heredia, E.A., Arce, G.R.: Piecewise linear system modeling based on a continuous threshold decomposition. IEEE Trans. Signal Process. 44(6), 1440–1453 (1996) 19. Bemporad, A., Garulli, A., Paoletti, S., Vicino, A.: A bounded-error approach to piecewise affine system identification. IEEE Trans. Autom. Control 50(10), 1567–1580 (2005) 20. Ferrari-Trecate, G., Muselli, M., Liberati, D., Morari, M.: A clustering technique for the identification of piecewise affine systems. Automatica 39(2), 205–217 (2003) 21. Nakada, H., Takaba, K., Katayama, T.: Identification of piecewise affine systems based on statistical clustering technique. Automatica 41(5), 905–913 (2005) 22. Breschi, V., Piga, D., Bemporad, A.: Piecewise affine regression via recursive multiple least squares and multicategory discrimination. Automatica 73, 155–162 (2016) 23. Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.P.H.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009) 24. Abdessalem, A.B., Dervilis, N., Wagg, D., Worden, K.: Model selection and parameter estimation of dynamical systems using a novel variant of approximate Bayesian computation. Mech. Syst. Signal Process. 122, 364–386 (2019) 25. Gendelman, O.V.: Targeted energy transfer in systems with non-polynomial nonlinearity. J. Sound Vibr. 315(3), 732–745 (2008) 26. Abdelkefi, A., Vasconcellos, R., Marques, F.D., Hajj, M.R.: Modeling and identification of freeplay nonlinearity. J. Sound Vibr. 331(8), 1898–1907 (2012) 27. MATLAB version 9.9.0.1570001 (R2020b). The Mathworks, Inc., Natick, Massachusetts (2020) 28. Brooks, S., Gelman, A., Jones, G., Meng, X.-L.: Handbook of Markov Chain Monte Carlo. CRC Press, Boca Raton (2011) 29. Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711–732 (1995) 30. Sisson, S.A., Fan, Y., Beaumont, M.: Handbook of Approximate Bayesian Computation. CRC Press, Boca Raton (2018) 31. Marin, J.-M., Pudlo, P., Robert, C.P., Ryder, R.J.: Approximate Bayesian computational methods. Statist. Comput. 22(6), 1167–1180 (2012)

Chapter 24

Experimental Model Update for Single Lap Joints Simone Gallas, Hendrik Devriendt, Jan Croes, Frank Naets, and Wim Desmet

Abstract With the increasing trend toward lightweight assemblies, the designer has to face new NVH challenges. Particularly, various structural joints have a different impact on the vibration response of these assemblies, which should be numerically modelled since the early stage of the design. The goal of this work is to experimentally assess the accuracy of simplified finite element models for joints and to evaluate how model updating techniques can improve the model accuracy. Four different lap joint samples are considered, each of them constituted by two aluminum plates connected with a different technology: one epoxy adhesive, one toughened acrylic adhesive, one acrylic foam tape, and one bolted joint, which is assessed for multiple preload configurations. For each of these cases, an experimental modal analysis is performed, a finite element model with input parameters from the datasheet of the bonding method is set up, and finally these joint parameters are updated with respect to the experimental reference. We apply the same procedure on an industrial case: the adhesive connection between the tube and the bracket of an automotive shock absorber. It is shown that the initial finite element models generally show a good agreement with the experiments for stiff adhesives. However, for the more flexible adhesives, the model update offers a significant accuracy improvement which can be important to account for during the design. Keywords Adhesive bonding · Bolted joints · Model update · Modal analysis · Finite element model

24.1 Introduction One of the major sources of uncertainty and inaccuracy for the modelling of mechanical assemblies is the presence of structural joints. In fact, the physical mechanism of energy dissipation and transmission at the joints’ interfaces is still the object of open research questions [1]. For traditional material combinations, such as metal substrates connected by bolts, the scientific literature is quite wide and structured. Depending on the application and on the requirements regarding accuracy and computational power, finite element models with different levels of complexity can be used for the connections between metal substrates [2, 3]. Our group has recently experimentally validated linear joint models, predefined within a commercial finite element software, by using vibration measurements [4]. Four different joining technologies were modelled with an accuracy of 12% maximum relative error for the first 6 modal frequencies. However, how the accuracy can be improved by model updating still remains an open question. Thus, this new experimental study tries to answer this question applied to bolted joints with different preloads as well as to adhesive joints.

24.2 Materials and Methods The experimental dynamic behavior of plate-to-plate connections is captured in terms of modal frequencies, modal shapes, and modal damping values. This data is obtained via a roving hammer test, in which 2 accelerometers are placed on the plate-to-plate system that is freely suspended and excited with an automatic hammer over a predefined grid of 36 points [5].

S. Gallas () · H. Devriendt · J. Croes · F. Naets · W. Desmet KU Leuven, Mechanical Engineering Department, Leuven, Belgium Flanders Makes, DMMS CoreLab, Lommel, Belgium e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_24

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Fig. 24.1 Plate-to-plate experimental samples

The choice of the automatic hammer improves the repeatability of the hits, allows to excite the frequency range of interest (0–500 Hz), and limits the influence of possible excitation amplitude nonlinearities. Adhesively bonded and bolted joints are the two connecting technologies chosen for our tests. As it can be seen in Fig. 24.1, four different lap joint samples are considered, each of them constituting of two 2.55-mm-thick A4 aluminum plates. For the adhesive bonding, a 19 mm overlap is realized on the short edges of the two plates that are connected with either one epoxy adhesive (Sadechaf 1671), one toughened acrylic adhesive (3M 8405), or one acrylic foam tape (3M VHB 5952). For the bolted joint, the two plates are connected with an array of four M4-bolted connections, which is assessed for five preload configurations. The preload configurations are labelled with the preload levels at each of the four M4 bolts used, ordered from one side of the overlap to the other. The thickness of the adhesive layer is 1 mm for all the types of glue. On the finite element modelling side, the plate-to-plate system is modelled using Siemens Simcenter 3D. Both the plate and the adhesive models consist of a mesh of 3D solid hex elements with three elements over the thickness. For the bolted connection, a spider model is used, where a beam element represents the bolt shank. For both joining technologies, the model update is set as an optimization problem, where the input material parameters, namely, the adhesive Young’s modulus and the bolt shank Young’s modulus, are tuned in order to minimize the error between the simulated and experimental modal parameters.

24.3 Results and Discussion The following observations can be made considering the experimental modal frequencies and damping values reported for all the considered joint types in Table 24.1: (i) Frequency differences between stiff adhesives with datasheet Young’s modulus higher than 1GPa are negligible (lower than 0.5% relative difference). (ii) Frequency differences between all bolts tightened to 1.5 Nm and 3.0 Nm are negligible (lower than 0.5% relative difference). (iii) Frequency decrease due to a loose bolt is larger when the loose bolt is at the edge of the array of bolts (i.e., compare case [0, 3, 3, 3] to case [3, 0, 3, 3]). (iv) Frequencies for the bolt cases are lower than for the adhesive cases due to the mass addition and the noncontinuous nature of the bolted connection. (v) The influence of the joints on the modal frequencies is larger for modes “Bending 1” and “Bending 3.” (vi) The influence of the joints on the modal damping values is larger for modes “Torsional 1” and “Torsional 2.”

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Adhesive

Bolts

Table 24.1 Experimental modal frequencies for the adhesive and bolted lap joints

Joint type [3, 3, 3, 3] Nm [1.5,1.5,1.5,1.5] Nm [0, 3, 3, 3] Nm [0, 0, 3, 3] Nm [3, 0, 3, 3] Nm 3M VHB 5952 3M 8405 Sadechaf 1671

Experimental modal frequencies [Hz] and modal damping values [%] Bending 1 Torsion 1 Bending 2 Torsion 2 Bending 3 Torsion 3 38.9 0.09 66.0 0.11 110.8 0.05 131.2 0.02 208.2 0.09 232.8 0.03 39.0 0.10 66.7 0.37 110.8 0.03 131.3 0.02 208.5 0.06 232.8 0.04 37.3 0.12 65.4 0.33 110.5 0.06 128.8 0.04 199.7 0.16 230.6 0.05 35.3 0.39 65.1 0.48 110.4 0.08 128.1 0.29 192.8 0.64 229.5 0.16 37.8 0.06 66.5 0.24 110.7 0.05 131.0 0.02 203.5 0.12 232.1 0.04 39.9 0.57 66.9 0.06 113.5 0.04 135.2 0.13 215.9 0.42 237.1 0.08 41.8 0.08 72.4 0.23 114.1 0.09 136.6 0.02 221.1 0.06 249.0 0.22 41.6 0.14 72.0 0.05 113.3 0.07 134.9 0.02 220.1 0.06 248.4 0.09

Adhesive

Bolts

Table 24.2 Model errors on modal frequencies, before and after the joint model update

Joint type [3, 3, 3, 3] Nm [1.5,1.5,1.5,1.5] Nm [0, 3, 3, 3] Nm [0, 0, 3, 3] Nm [3, 0, 3, 3] Nm 3M VHB 5952 3M 8405 Sadechaf 1671 Average

Relative modal frequency error [%] - Before and after the update Bending 1 Torsion 1 Bending 2 Torsion 2 Bending 3 Torsion 3 -3.1 -1.1 2.7 2.9 0.7 0.7 3.1 3.5 -2.4 -0.8 1.5 1.7 -3.5 -1.5 1.6 1.8 0.7 0.7 3.1 3.4 -2.6 -0.9 1.5 1.7 -3.9 -3.7 -2.4 1.9 -1.1 0.3 3.4 3.4 -2.3 -2.2 -5.6 0.3 -10.9 -1.4 -14.8 3.9 -7.6 1.0 2.4 4.1 -9.1 0.1 -20.8 2.9 -4.5 -0.2 1.9 2.0 0.7 0.8 3.1 3.3 -3.0 -0.1 1.8 1.9 -4.7 -0.3 1.3 1.5 -1.2 -1.1 0.9 1.8 -4.2 -0.5 1.2 0.2 -2.0 -1.7 -1.0 0.4 -1.4 -1.4 1.0 1.0 -1.5 -1.1 -0.8 0.3 -0.9 -1.4 2.5 0.4 -0.7 -0.8 2.4 2.3 -0.4 -0.8 1.9 0.1 4.2 1.4 3.5 1.9 1.8 0.9 2.4 2.9 3.2 0.8 4.4 1.1

Average 2.3 1.8 2.2 1.7 3.1 2.0 10.9 2.3 2.5 1.4 2.3 0.9 1.3 1.0 1.5 1.0

In Table 24.2 it is shown that the model update allows a substantial improvement for those modes that are most influenced by the joint: for the adhesive 3 M VHB 5952, the average error on the modes diminishes from 2.3% to 0.9%; for all the types of joint, the average error on “Bending 1” diminishes from 4.2% to 1.4%. In general, all the considered cases on average show an improvement after the model is updated. Therefore, the same parametric bolted model can be used to represent both fully tightened connection and loose fasteners. The same conclusion can be drawn for the adhesive case where the same parametric model can be used to represent several adhesives types. The same steps are repeated for the case of the adhesive connection between a tube and a bracket. It is important to notice that the geometry of the joint has an impact on the model update results: the updated parameter should not be intended as a purely “physical” property but rather a property of the whole modelled object. In fact, if the adhesive Youngs’ modulus of the adhesive 3 M VHB 5952 would be estimated on the plate-to-plate connection and then used on the tube to bracket, the error would already be reduced from 25% (input from datasheet) to 5% (input from plate-to-plate model update). If the model update would be performed directly on the tube-to-bracket connection, the error would be further decreased below 4%.

24.4 Conclusions Finite element models for bolted and adhesive joints are validated and updated with reference to the experimental modal parameters. The accuracy improvement due to the model update is quantified in terms of modal frequency error. On the one hand, for the case of flexible adhesives, the modal based model update allows a substantial accuracy improvement. On the other hand, stiff adhesives have lower influence on the system’s modal frequencies and datasheet values can be used as model input to obtain already quite accurate results. The same parametric adhesive model was updated and validated for three adhesive types. The same procedure was successfully applied to the parametric bolted model to represent five bolt preload configurations. Finally, it is shown that the geometry of the joint has a relevant influence on the updated joint model parameters. For the considered tube-to-bracket case, the model update on the simplified geometry already allows an accuracy improvement with respect to the case in which datasheet parameters are used. The accuracy could be even further improved when the model update is directly performed on the actual geometry. Acknowledgments This research was partially supported by Flanders Make, the strategic research center for the manufacturing industry.

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References 1. Brake, M.R.W.: The Mechanics of Jointed Structures (2018) 2. Kim, J., Yoon, J.C., Kang, B.S.: Finite element analysis and modeling of structure with bolted joints. Appl. Math. Model. 31(5), 895–911 (2007). https://doi.org/10.1016/j.apm.2006.03.020 3. Porter, J.H., Balaji, N.N., Little, C.R., Brake, M.R.W.: A quantitative assessment of the model form error of friction models across different interface representations for jointed structures. Mech. Syst. Signal Process. 163(September 2020), 108163 (2022). https://doi.org/10.1016/ j.ymssp.2021.108163 4. Van Belle, L., et al.: Experimental validation of numerical structural dynamic models for metal plate joining techniques. J. Vib. Control. 24(15), 3348–3369 (2018). https://doi.org/10.1177/1077546317704794 5. Heylen, W., Lammens, S., Sas, P.: Modal Analysis Theory and Testing. KUL/Faculty of engineering/Department of mechanical engineering/Division of production engineering, machine design and automation, Heverlee (1997)

Chapter 25

Data-Driven Identification of Multiple Local Nonlinear Attachments Installed on a Single Primary Structure Aryan Singh and Keegan J. Moore

Abstract The goal of this study is to find mathematical models for the dynamics of multiple local nonlinear attachments on a single primary structure. We focus on the application of the characteristic nonlinear system identification (CNSI) method to multiple attachments and attachments tuned higher than the primary structure’s first linear mode. The characteristic nonlinear system identification method is a data-driven method for building mathematical models of nonlinear attachments. To produce representative mathematical models, the method only requires the transient experimental response measurements, the general frequency content, and the mass of the attachments. A two-story tower with a nonlinear attachment installed on each floor in two configurations is used to demonstrate the applicability of the CNSI method for identifying multiple nonlinear attachments. The attachments are tuned to interact with the tower’s first mode in the first configuration. The linear stiffness of the attachment on the first floor is increased in the second configuration so that it interacts with the second mode rather than the first mode. We numerically integrate the models and compare the resulting displacements with experimental measurements to validate the CNSI method’s success and strength. Keywords Nonlinear dynamics · Data-driven modeling · Data-driven identification · System identification · Nonlinear energy sink

25.1 Introduction A recently developed approach called the characteristic nonlinear system identification was proposed for identifying mathematical models describing the dynamics of local nonlinear attachments installed on linear structures [1–3]. The approach is based on the partitioning of the dynamics of local attachment from the dynamics of the parent structure. This partitioning allows the attachment to be treated like a single-degree-of-freedom (SDOF) oscillator with the exception that the dissipative and elastic restoring force be dependent on the relative displacement between the attachment and its installation points. This method has previously identified the dynamics of a local nonlinear attachment attached to a linear oscillator with a smooth nonlinearity [1] and the dynamics of a non-smooth clearance-type nonlinear attachment installed on a model wing [2]. This work aims to assess the applicability of the CNSI method when multiple attachments are attached to a parent structure. We also determine the applicability of the method when the nonlinear attachments are tuned to have modes higher than the first linear mode of the primary structure. Thus, in this research, we consider the dynamics of two different configurations. In the first configuration, both the attachments are tuned to interact with the first linear mode of the parent structure. While in the second configuration, we tune the attachment on the first floor by increasing the linear stiffness such that it only interacts with the second mode of the parent structure.

A. Singh () · K. J. Moore Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_25

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Fig. 25.1 Two-story tower: (a) CAD and (b) experimental setup with instrumentation

25.2 Methods The experimental setup consists of a two-story structure with two nonlinear attachments known as nonlinear energy sinks (NESs) [1, 2] installed on each floor. Figure 25.1 presents the CAD schematic and the actual assembled structure with the instrumentation for acceleration measurements. We term the attachments installed on the first and second floors as NES1 and NES2, respectively. The first and second floors are linear oscillators and are termed floor 1 and floor 2, respectively. The experimental setup thus has four DOFs in total. All the experimental measurements consist of the transient response of the tower when excited using a PCB Piezotronics impact hammer. We measured the response using four PCB Piezotronics accelerometers directly screwed into each component.

25.3 Results In Fig. 25.2, we present the displacement and the wavelet spectra of the model identified using the CNSI method for the first configuration. Both NESs are tuned to have linear frequencies below that of the first linear mode of the tower. We also include the experimental measurement and its wavelet spectra for a direct comparison between the identified model and the measurements. It can be observed from the time series that the model accurately predicts the amplitude decay for each of the four DOFs. Interestingly, the model also accurately captures the strongly nonlinear beats that arise in the displacement of NES1. These beats are strongly dependent upon the nonlinear interactions that occur between the NES and the first floor and thus are difficult to capture by a model and can be easily missed if the nonlinearity and damping are not represented accurately in the model. Using the wavelet spectra, we also find that the model reproduces the dominant frequency transitions during the nonlinear portion of the response and the lack of time dependence in the linear parts.

25 Data-Driven Identification of Multiple Local Nonlinear Attachments Installed on a Single Primary Structure

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Fig. 25.2 Comparison of the model with the measured responses and corresponding wavelet transform spectra for the measurement case (impact of 509 N) used in the identification

References 1. Moore, K.J.: Characteristic nonlinear system identification: A data-driven approach for local nonlinear attachments. Mech. Syst. Signal Process. 131, 335–347 (September 2019). https://doi.org/10.1016/j.ymssp.2019.05.066 2. Singh, A., Moore, K.J.: Characteristic nonlinear system identification of local attachments with clearance nonlinearities. Nonlinear Dyn. 102(3), 1667–1684 (November 2020). https://doi.org/10.1007/s11071-020-06004-8 3. Singh, A., Moore, K.J.: Identification of multiple local nonlinear attachments using a single measurement case. J. Sound Vib. 513, 116410 (November 2021). https://doi.org/10.1016/j.jsv.2021.116410

Chapter 26

Supervised Learning for Abrupt Change Detection in a Driven Eccentric Wheel Samuel A. Moore, Dean Culver, and Brian P. Mann

Abstract Event detection is often a predominant challenge in processing non-stationary signals. In engineering mechanics, events may result from non-smoothness in the form of loss of contact, impact, or the onset of sliding-friction. An interesting example of such a mechanical system is a wheel whose center of mass does not coincide with its geometric center. An eccentric wheel may evolve in three distinct phases: roll without slip, roll with slip, and hop. Therefore, this paper seeks to explore and compare supervised learning methods for phase identification (i.e., roll, slip, and hop) in simulated data from a driven eccentric wheel. The mechanics of a torque driven wheel on a flat surface are derived through an augmented Lagrangian formulation and Coulomb friction is adopted to model transverse contact forces. To accommodate for nonsmoothness, the system is broken down in complementary sub-problems and the simulation is conducted using event-based methods. The simulated data is then used to train a Naive Bayes classifier, a Support Vector Machine (SVM), and an Extreme Gradient Boosting (XGBoost) classifier. Lastly, the methods as well as their performance, merits, and drawbacks are discussed in detail. Keywords Non-smooth · Event detection · Machine learning

26.1 Introduction Abrupt changes in mechanical systems due to parameter shifts, or the dropping and adding of entire terms can make data from these systems difficult to analyze. Further complicating the analysis is the fact that experimental data is typically collected continuously. In other words, data from each phase of motion cannot easily be collected in isolation, resulting in an unsegregated series of observations. For instance, data can be challenging to quantitatively analyze from motion tracking of a system with stick-slip dynamics [1, 2], or data from a machining process where tool contact is intermittent [3, 4]. Moreover, one may need to quickly and accurately determine the phase of motion in real time in systems where abrupt changes can be important, costly, or indicate damage. Therefore, this investigation seeks to explore the use of three supervised learning methods (e.g., Naive Bayes, Support Vector Machines, and Extreme Gradient Boosting) for phase of motion identification in non-smooth dynamical systems. An analogous problem to phase identification in mechanical systems is known in data science and signal processing more generally as change point detection. A change point is a transition point from one state to another in a system that generates time-series data [5]. In the context of this work, a change point is the transition between the distinct phases of motion. Supervised learning methods that have been successful at identifying change points include Naive Bayes [6], Support Vector Machines [7], and Boosting [8]. Some examples of previously explored change point detection problems are: using mobile phone data to identify transportation modes [6], analyzing heart beat variability to detect different stages of sleep [9], and audio signal segmentation for speech recognition [10]. Thus, this work will test and extend the use of the aforementioned supervised learning tools to identify change points in nonlinear mechanical systems.

S. A. Moore () · B. P. Mann Department of Mechanical Engineering and Materials Science, Pratt School of Engineering, Duke University, Durham, NC, USA e-mail: [email protected]; [email protected] D. Culver U.S. Army Research Laboratory, Interdisciplinary Mechanics Group, Aberdeen, MD, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_26

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The physical system to be examined in this work is a rolling wheel whose center of mass does not coincide with its geometric center, otherwise known as an eccentric wheel. This system exhibits particularly intriguing behavior as the center of mass rises and falls with rotation resulting in a non-constant normal force. As a consequence, the wheel may evolve in three distinct phases of motion: roll without slip, roll with slip, and hop. Many studies have examined the behavior of the interesting eccentric wheel both theoretically and experimentally [1, 11–14]. This peculiar behavior is not unique to eccentric disks also having been shown in elliptic disks and cylinders [2, 15, 16]. An eccentric wheel serves as a great example system to test the abilities of different supervised learning methods for phase identification because the dynamics are highly nonlinear and the differences between phases are nuanced. The rest of this paper is organized as follows. The next section outlines the physical system of interest and derives the math model used to generate the time-series data. After that, the numerical simulation procedure and data preparation steps are examined. Finally, the learning methods and their performance are discussed at length.

26.2 System Description and Model Derivation This section describes the physical system of interest and derives the math model used to generate the training and validation data for machine learning. Figure 26.1 shows a uniform circular lamina of radius R and mass M—which represents a wheel—driven by a torque τ undergoing general planar motion. Attached to the wheel is a point mass m at distance d from the geometric center. The added mass shifts the center of mass of the entire system c a distance e from the geometric center. To begin, a position vector that tracks c in a fixed reference frame is defined such that rc = (x + e sin θ )i + (y − e cos θ )j .

(26.1)

Differentiating rc with respect to time the velocity of the mass center is given by r˙ c = (x˙ + eθ˙ cos θ )i + (y˙ + eθ˙ sin θ )j .

(26.2)

Note that the convention of using an over dot is adopted to indicate the derivative with respect to time. Next, the kinetic energy of the system is given by

Fig. 26.1 Schematic of a torque driven eccentric undergoing general planar motion. The wheel’s mass center is not aligned with its geometric center and thus has a variable normal force and may spontaneously hop

26 Supervised Learning for Abrupt Change Detection in a Driven Eccentric Wheel

T =

1 1 (m + M)˙rc · r˙ c + Ic θ˙ 2 , 2 2

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(26.3)

where Ic is the moment of inertia about the mass center. Following from Eq. 26.1, the potential energy is U = (m + M)g(−y + e cos θ ) .

(26.4)

Now, the holonomic constraints f1 and f2 are imposed that prevent translation in the j direction and maintain roll without slip respectively. These constraints are given by f1 = y = 0 , f2 = x − Rθ = 0 .

(26.5) (26.6)

Then, the Lagrangian L = T − U for the system is given by L=

1 1 (m + M)[x˙ 2 + y˙ 2 + 2eθ˙ (x˙ cos θ + y˙ sin θ )] + Io θ˙ 2 − (m + M)g[−y + e cos θ ] + λ1 y + λ2 (x − Rθ ) , (26.7) 2 2

where Io = Ic + (m + M)e2 is the moment inertia about the geometric center of the disk and λi is the Lagrange multiplier associated with constraint fi . This Lagrangian results in the three following equations of motion after including the force from sliding friction Ff (m + M)x¨ + (m + M)[eθ¨ cos θ − eθ˙ 2 sin θ ] = H2 λ2 + (1 − H2 )Ff ,

(26.8)

(m + M)y¨ + (m + M)[eθ¨ sin θ + eθ˙ 2 cos θ ] = H1 λ1 + (m + M)g ,

(26.9)

Io θ¨ + (m + M)[ex¨ cos θ + ey¨ sin θ ] = −eg(m + M) sin θ + τ − H2 λ2 R − (1 − H2 )Ff R .

(26.10)

In the equations above, λ1 and λ2 may be interpreted as the force preventing translation in the j direction and the force maintaining roll without slip respectively. But, since λ1 is only valid when the constraint y = 0 is upheld, the heaviside step function H1 is included. Likewise, the step function H2 is included because λ2 and Ff are not simultaneously valid and are dependent upon if x − Rθ = 0. Furthermore, when y = 0 the Lagrange multiplier λ1 is equivalent to the normal force FN = −(m + M)[eθ¨ sin θ + eθ˙ 2 cos θ − g] .

(26.11)

Now, we may define H1 such that  H1 =

0 FN  0 , 1 FN < 0 ,

y=  0 y = 0.

(26.12)

Note that in reality FN cannot be positive but may momentarily be positive as a consequence of numerical integration. Similarly, we define H2 in terms of λ2 and the maximum force from static friction such that  H2 =

0 |λ2 |  μs |FN | , 1 |λ2 | < μs |FN | ,

x − Rθ =  0 x − Rθ = 0 ,

(26.13)

where μs is coefficient of static friction. Lastly, we define the sliding frictional force Ff as Ff = μd FN sgn(x˙ − R θ˙ ) ,

(26.14)

where μd is the coefficient of dynamic friction and the direction of Ff is determined by the relative velocity of the contact point on the wheel and the surface. For the purposes of numerical simulation, it will be useful to isolate the highest order derivatives. As a result, Eqs. 26.8–26.10 become

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⎡ ⎤ x¨ ⎣y¨ ⎦ = θ¨

⎡ 1 Io −(m+M)e2

⎤ Io + e2 (m + M)[1 + cos2 θ ] e2 (m + M) sin θ cos θ −e cos θ ⎣ e2 (m + M) sin θ cos θ Io + e2 (m + M)[1 + sin2 θ ] −e sin θ ⎦ −e(m + M) cos θ −e(m + M) sin θ 1 ⎡ ⎤ H2 1−H2 λ2 + m+M Ff eθ˙ 2 sin θ + m+M ⎢ ⎥ H1 ×⎣ −eθ˙ 2 cos θ + g + m+M λ1 ⎦. τ − eg(m + M) sin θ − H2 Rλ2 − (1 − H2 )RFf

(26.15)

If the three degree-of-freedom system is examined closely, it becomes evident that the system evolves in three distinct regimes determined by H1 and H2 . For the first case, roll without slip, H1 = H2 = 1, x = Rθ , and y = 0. Therefore, equation (15) reduces to the following single degree-of-freedom system after substituting for λ2 and replacing x and its derivatives with Rθ and its derivatives θ¨ =

eR θ˙ 2 sin θ + τ/(m + M) − ge sin θ . Io /(m + M) + R 2 + 2eR cos θ

(26.16)

Moreover, λ2 must be found in terms of the parameters and state variables to determine the onset of slip. So, by equating x¨ and R θ¨ the following expression for λ2 was found λ2 = −

(R + e cos θ )[(m + M)ge sin θ − τ ] + [(m + M)eR cos θ + Io ]eθ˙ 2 sin θ . Io /(m + M) + e2 cos2 θ + R 2

(26.17)

If λ2 exceeds the threshold for slip, then H2 = 0 and x = Rθ but H1 = 1 and y = 0. This is roll with slip, and the dynamics are determined the following two degree-of-freedom system 

 1 1 Ff eθ˙ 2 sin θ + m+M −e cos θ Io x¨ = . θ¨ 1 Io − e2 cos2 θ −(m + M)e cos θ τ − eg(m + M) sin θ − RFf

(26.18)

Finally, if FN becomes zero, this indicates hop, which results in H2 = H1 = 0, x = Rθ , and y = 0. The dynamics during hop are determined by the three following differential equations ⎡ ⎤ ⎡ ⎤ x¨ Io + e2 (m + M)[1 + cos2 θ ] e2 (m + M) sin θ cos θ −e cos θ 1 ⎣y¨ ⎦ = ⎣ e2 (m + M) sin θ cos θ Io + e2 (m + M)[1 + sin2 θ ] −e sin θ ⎦ 2 I − (m + M)e o θ¨ −e(m + M) cos θ −e(m + M) sin θ 1 ⎡ ⎤ eθ˙ 2 sin θ × ⎣ −eθ˙ 2 cos θ + g ⎦ . τ − eg(m + M) sin θ

(26.19)

26.3 Numerical Simulation and Data Preparation This section describes the event-driven numerical integration of the equations derived in the previous section and the data preparation steps for machine learning. The equations were simulated with a variety of constant torques using the fourth order Runge–Kutta method with event detection. If the conditions for roll without slip are satisfied (i.e., H1 = H2 = 1), then Eq. 26.16 determines the evolution of x and θ . However, the values for FN and λ2 must also be determined to detect the onset of slip. If slip is detected, the integration proceeds as the two degree-of-freedom system given in Eq. 26.18. While slip is occurring, two events may occur: return to roll without slip and hop. Entering the roll without slip phase from the slip phase is dictated by the relative velocity of the contact point of the wheel and the surface. If this relative velocity is sufficiently close to zero then the system has entered roll without slip. In other words, if |x˙ − R θ˙ | < , where  is machine error, the integration proceeds as the single degree of freedom system outlined previously. This event-driven integration scheme allows for the system to alternate between roll with and without slip as required. Furthermore, while slip is occurring it must be

26 Supervised Learning for Abrupt Change Detection in a Driven Eccentric Wheel Table 26.1 Parameters for the eccentric disk used in numerical simulation

189 Parameter M m R d e Io μf μd

Value 1.0 0.15 0.30 0.30 0.0391 0.0468 0.30 0.18

Units kg kg m m m kg · m2

a)

b)

Fig. 26.2 Sample numerical simulation for a torque driven eccentric wheel including: (a) the time-series for position and (b) the time-series for velocity

checked if hop has occurred. Which, again, is settled by the sign of the normal force. If hop is detected, then the integration proceeds as the three degree-of-freedom system given in Eq. 26.19. Finally, the only event that may occur during hopping is landing, which is designated by the sign of y. In the event of landing, the integration is terminated. Table 26.1 shows the parameters for the wheel used in numerical simulation to generate data for the learning process. Figure 26.2 shows a sample time-series generated from the process given above. In total, twenty time-series simulations of various duration were generated with a wide range of constant torques in MATLAB [17]. During numerical simulation, each time step in the series was also tagged with a class identifier (i.e., 0, 1, or 2) corresponding to roll, slip, or hop. Then, the states, inputs, and their associated classes for each series were agglomerated into single matrix resulting in a total of over 35,000 samples in time. This matrix was then randomly divided into an 80–20 training and validation split.

26.4 Learning Methods and Performance This section describes the learning methods used for phase identification and their performance. The Naive Bayes, SVM, and XGBoost algorithms were chosen for their ability to handle overlap among clusters, success with similar problems, and

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their ease of accessibility in common programming languages. In this work, the learning algorithms were implemented in Python using the Scikit-learn and XGBoost libraries [18, 19]. The hyperparameters for both the SVM and XGBoost models were tuned with cross validation on the training data via the Scikit-learn function GridSearchCV(). The performance metrics included for model evaluation are precision or positive predictive value and recall or sensitivity. Precision is a metric specific to each class and is defined as the ratio of the number of true positives over the total number of true and false positives. The number of true positives or TP is considered to be the count of hits (e.g., the count of predictions that match ground truth). On the other hand, the number of false positives or FP is considered the number of false alarms. Thus, precision is defined as P recision =

TP . T P + FP

Recall is also specific to each class but is defined as the ratio of the number of true positives to the number true positives and false negatives. Here, the number of false negatives or FN may thought of as the number of misses (e.g., the number of times a class is failed to be predicted). So, recall is given by Recall =

TP . T P + FN

These metrics allow for a more in-depth performance evaluation than a miss-classification rate alone; providing information on the predictability of each phase of motion. Furthermore, a low precision rate for a class is interpreted as predicting that class too often when ground truth suggests otherwise. For instance, if a classifier has a low precision rate for slip that means that slip is predicted when in reality the wheel is rolling or hopping. Additionally, a low recall rate for a class is interpreted as predicting other classes over the true class. For example, if a classifier has a low recall rate for slip, this indicates that when the wheel is slipping the classifier predicts rolling or hopping. A well-rounded classifier will have both high precision and recall rates for all classes. The first, and simplest, learning method employed was a Naive Bayes classifier. This method gets its name from the fact that it assumes conditional independence. In other words, given a class (i.e., roll, slip, or hop) the features (i.e., position, velocity, torque, etc.) are independent. Thus, the probability of a sample (i.e., all the states and inputs at a time-step) given a class is easily calculated with the probabilities of each observation as P (S|C) =

k 5

p(si |C) ,

(26.20)

i=1

where S is the k dimensional vector of feature observations s1 , . . . , sk at each time-step and C is the class. This assumption allows for quick and easy estimations of probabilities but when violated, as it is most applications, the estimates will be biased. However, in practice, this assumption may be violated and the method will still successfully identify class membership. In general though, the more the valid the assumption of conditional independence, the better Naive Bayes will perform. Using Bayes Theorem and the conditional independence assumption given in Eq. 26.20, estimated class membership is determined by Cˆ = argmaxc P (C)P (S|C) = argmaxc P (C)

k 5

p(si |C) .

(26.21)

i=1

More information on the methodology and uses of the Naive Bayes classifier can be found in works [20, 21]. The precision rates for the Naive Bayes Classifier on validation data are shown in the first column of Table 26.2. Noteworthy here is the relatively low precision rate of 85.6% for the slip phase of motion compared to the roll and hop phases. Like previously mentioned, this indicates that the Naive Bayes Classifier has a non-negligible chance of predicting slip when the true class is either roll or hop. That being said, the precision rates for roll and hop are fairly high at 97.8% and 99.1% respectively; suggesting that when these classes are predicted, the predictions are true. Next, the recall rates for the Naive Bayes Classifier are given in the first column of Table 26.3. The recall rates across the classes are fairly high with the lowest occurring for the roll phase. This recall rate of 93.4% indicates that when the wheel is actually rolling, the predictions estimate that the wheel is slipping or hopping approximately 7% of the time.

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Table 26.2 Precision rates on validation data for the three supervised learning methods employed

Precision rates Learning Naive method Bayes Roll 97.8% Slip 85.6% Hop 99.1%

Support machine 95.9% 99.1% 94.5%

Extreme gradient boosting 99.9% 100% 100%

Table 26.3 Recall rates on validation data for the three supervised learning methods employed

Recall rates Learning Naive method Bayes Roll 93.4% Slip 94.9% Hop 96.1%

Support machine 100% 88.9% 96.2%

Extreme gradient boosting 100% 99.9% 100%

The next learning algorithm employed was a Support Vector Machine. SVMs transform the k-dimensional feature data into an n-dimensional space where n  k. Then, an optimal (n − 1) dimensional hyperplane is selected to separate the data in that higher dimensional space. The transformation of the data is performed by a kernel function which may in general be nonlinear. For the purposes of this investigation a polynomial kernel was chosen. Furthermore, while the hyperplane which separates the data is clearly linear in n-dimensional space, its projection may be nonlinear in the original k-dimensional feature space. However, a hyperplane can only perform binary separation, therefore multiple classifiers must be fit and compiled to constitute a multi-class model. For this work, the “one-vs-one” approach was used, which constructs a classifier to distinguish between every pair of classes. Additional information on the methodology and applications of SVMs can be found in references [21, 22] Column two of Table 26.2 shows the precision rates for the SVM model on validation data. Overall, the SVM model has high precision compared to the Naive Bayes Classifier described previously. Notably is the precision rate for Slip at over 99%, which suggests that when slip is predicted this classification is incorrect less than one percent of the time. The recall rates for the SVM model vary widely with the lowest rate for slip at 88.9% and the highest for roll at 100%. This indicates that when the wheel is rolling the SVM model will predict roll. However, when the wheel is slipping the model will predict roll or hop approximately 11% of the time. Overall, the SVM classifier performs similarly to the Naive Bayes classifier with comparable values for precision and recall. The final classification model was fit using Extreme Gradient Boosting. Boosting, in general, is method that agglomerates the predictions from many classifiers whose estimations in isolation are poor to form an overall strong consensus. Through iteration, the base classifiers are trained on many versions of the training data that are successively modified. The training data is modified in the sense that the observations which were miss-classified in the previous iteration are given more weight or priority in the following iteration. The base learners fit in each iteration of XGBoost are trees; which divide the feature space into a set of rectangles and assign a class to each one based on a majority vote. XGboost also employs regularization and shrinkage to prevent overfitting and is optimized for speed. Once the group of classifiers are trained and assembled, the class with the most votes across all models is chosen. For more information on the applications and specifics of XGBoost and Boosting in general see references [18, 21]. The precision and recall rates for the model trained with XGBoost are shown in the third column of Tables 26.2 and 26.3. It is clear that the XGBoost algorithm outperforms the models fit using the Naive Bayes and SVM algorithms with nearly 100% precision and recall rates across classes.

26.5 Conclusion This paper tested the efficacy of using the Naive Bayes, Support Vector Machine, and Extreme Gradient Boosting algorithms for identifying roll, slip, and hop in a torque driven eccentric wheel. An event-driven integrator was adopted to simulate the system whose math model was derived using an augmented Lagrangian formulation. The state and input data from many series were combined and randomly split into training and validation sets. Lastly, the learning algorithms, fitting procedures, and each classifier’s performance on validation data were examined at length. In general, the results in the previous section suggest that the Naive Bayes, SVM, and XGBoost algorithms are all capable of change point detection in an eccentric wheel. However, while the performance of Naive Bayes and SVM are similar, the recall and precision rates given Tables 26.2 and 26.3 signal that XGBoost performs the best. These results also demonstrate

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that if a model for a non-smooth system is known, supervised learning is an effective approach to identify abrupt changes to a system and different phases of motion. That being said, one may wish to have a model free approach in cases where the dynamics of a system are unknown. Therefore, future works should focus on unsupervised methods to cluster distinct phases of motion in mechanical systems. Acknowledgments Partial support from ARO awards W911NF2120117 and W911NF12R001204 is gratefully acknowledged.

References 1. Moore, S., Culver, D., Mann, B.P.: The eccentric disk and its eccentric behaviour. Eur. J. Phys. 42(6), 065012 (2021) 2. Lindén, J., Källman, K.-M., Lindberg, M.: The rolling elliptical cylinder. Amer. J. Phys. 89(4), 358–364 (2021) 3. Khasawneh, F. A., Mann, B. P., Insperger, T., and Stépán, G. (August 24, 2009). Increased stability of low-speed turning through a distributed force and continuous delay model. ASME. J. Comput. Nonlinear Dynam. 4(4), 041003–1 (October 2009). 4. Patel, B.R., Mann, B.P., Young, K.A.: Uncharted islands of chatter instability in milling. Int. J. Mach. Tools Manuf. 48(1), 124–134 (2008) 5. Aminikhanghahi, S., Cook, D.J.: A survey of methods for time series change point detection. Knowl. Inf. Syst. 51(2), 339–367 (2017) 6. Reddy, S., Mun, M., Burke, J., Estrin, D., Hansen, M., Srivastava, M.: Using mobile phones to determine transportation modes. ACM Trans. Sensor Netw. 6(2), 1–27 (2010) 7. Zheng, Y., Liu, L., Wang, L., Xie, X.: Learning transportation mode from raw GPS data for geographic applications on the web. In: Proceedings of the 17th International Conference on World Wide Web, pp. 247–256 (2008) 8. Aminikhanghahi, S., Cook, D.J.: Using change point detection to automate daily activity segmentation. In: 2017 IEEE International Conference on Pervasive Computing and Communications Workshops (PerCom Workshops), pp. 262–267. IEEE, Piscataway (2017) 9. Staudacher, M., Telser, S., Amann, A., Hinterhuber, H., Ritsch-Marte, M.: A new method for change-point detection developed for on-line analysis of the heart beat variability during sleep. Phys. A Statist. Mech. Appl. 349(3–4), 582–596 (2005) 10. Rybach, D., Gollan, C., Schluter, R., Ney, H.: Audio segmentation for speech recognition using segment features. In: 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 4197–4200. IEEE, Piscataway (2009) 11. Cross, R.: Pendulum motion of a biased cylindrical tube. Eur. J. Phys. 41(1), 015006 (2019) 12. Yanzhu, L., Yun, X.: Qualitative analysis of a rolling hoop with mass unbalance. Acta Mech. Sinica 20(6), 672–675 (2004) 13. Maritz, M.F., Theron, W.F.D.: Experimental verification of the motion of a loaded hoop. Amer. J. Phys. 80(7), 594–598 (2012) 14. Theron, W.F.D.: The rolling motion of an eccentrically loaded wheel. Amer. J. Phys. 68(9), 812–820 (2000) 15. Cross, R.: Dynamics of a rolling egg. Eur. J. Phys. 42(5), 055015 (2021) 16. Heppler, G.R., D’Eleuterio, G.M.T.: Rock and roll of an ellipse. Amer. J. Phys. 89(7), 666–676 (2021) 17. MATLAB. Version 9.8.0.1417392 (R2020a). The MathWorks Inc., Natick, Massachusetts (2020) 18. Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 785–794 New York, NY. ACM, New York (2016) 19. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011) 20. Rish, I. et al.: An empirical study of the Naive Bayes classifier. In: IJCAI 2001 Workshop on Empirical Methods in Artificial Intelligence, vol. 3, pp. 41–46 (2001) 21. Friedman, J., Hastie, T., Tibshirani, R., et al.: The Elements of Statistical Learning, vol. 1. Springer Series in Statistics. Springer, New York (2001) 22. Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)

Chapter 27

Bolt-Jointed Structural Modelling by Including Uncertainty in Contact Interface Parameters Nidhal Jamia, Hassan Jalali, Michael I. Friswell, Hamed Haddad Khodaparast, and Javad Taghipour

Abstract In jointed structures, both stiffness and damping properties are affected by the way that the contact interfaces behave. The overall behaviour of the contact interface is the result of different parameters; among them are the contact surface quality and interface pressure distribution. Small changes in these parameters result in a considerable change in the physics of the contact interface and hence introduce variability in structural dynamics properties. The variability in the contact interface can happen among nominally identical structures. These features raise the need for robust models of the joint contact interface to predict their performance in the structure. In this paper, experimental modal testing of a set of nominally identical Brake-Reuss beam structures is employed to investigate the effect of preload on the variability in the natural frequencies and damping ratios for different modes of the structure. The contact interface is modelled using a representative stochastic model to account for the variabilities in the contact interface parameters. Finally, the distribution of the joint model stiffness parameters is identified using the variability in dynamic properties. Keywords Contact interface · Brake-Reuss · Stiffness parameters

27.1 Introduction Bolted joints are widely used in mechanical structures to assemble the various substructures due to their simplicity and ability to transfer forces and moments between the substructures. Safety and durability are the main advantages of bolted joints. However, the inherent dynamics of this type of joint is very complex, which highlights the significant complexity of the modelling and analysis of bolted joint structures. Under a dynamic load, bolted joints suffer from a reduced repeatable performance which is due mainly to the loosening phenomenon, which can be avoided using locking mechanisms. This approach will add an extra layer of uncertainty to the contact interface and will increase the complexity of modelling the structure accurately. Modelling the contact interface of a bolted joint presents one of the most challenging steps in structural engineering design. This is due to the high levels of uncertainty and lack of information about the dynamic properties at the contact interface. The deformation of asperities in the joint is one of the main sources of uncertainty that can lead to a relaxation of the joint [1]. The second source of uncertainty in the joint is due to the assembly process and the manufacturing engineering tolerances. Brake et al. [2] studied the variability and repeatability in the dynamic response of the assembled structure and showed that the randomness in the preload and contact surface characteristics, such as the roughness quality, can cause significant variability in the contact interface parameters. Guo and Zhang [3] simulated the joint parameters as probabilistic variables and employed a probabilistic approach to identify the joint parameters. Castelluccio and Brake [4] investigated the model input and uncertainties in threaded fasteners by performing finite element simulations with various input parameters and model simplifications. Brake et al. [2] investigated the effect of residual stress on the variability and repeatability of measurements of natural frequency and energy dissipation of a contact interface of a Brake-Reuss beam.

N. Jamia () · M. I. Friswell · H. Haddad Khodaparast · J. Taghipour Faculty of Science and Engineering, Swansea University, Swansea, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] H. Jalali Department of Mechanical & Construction Engineering, Northumbria University, Newcastle upon Tyne, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_27

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Two wide approaches are used to model the joint properties. The first approach is non-parametric models, where no assumptions about the properties of the joint are required. Ren and Beards [5] coupled a stiff joint using generalized coupling techniques and identified the properties of the joint using an identification technique. Ma et al. [6] presented a method that relies on comparing the overall dynamics of the bolted structure to that of a similar but unbolted one. Parametric models of joints can be categorized into two theories: the zero-thickness element and the thin layer element theories. Frictional sliders are a well-known example of the zero-thickness theory. A Valanis model was employed by Gaul and Lenze [7] to simulate the stick-slip in the joint interface of a joint structure. Iwan [8] introduced one of the earliest models of mechanical joint interfaces consisting of a series of Jenkin’s elements in parallel or series combinations. He presented a model to investigate the yielding behaviour of continuous and composite materials and structures by complementing and extending some of the earliest works [9–13]. This model has been the basis of many other joint models developed thenceforth [14–16]. Segalman [17] showed that the Iwan model successfully represents the micro\macro-slip at joint interfaces. A one-dimensional physically motivated micro-slip model was developed by Menq et al. [18]. This model allows partial slip at the friction interface. Regarding the thin layer element theory, the joint is represented as an element with physical dimensions and a well-defined force-displacement relationship. Ahmadian et al. [19–21] showed the potential of the linear thin layer element in modelling the linear behaviour of different types of joints. Jalali et al. [22, 23] used the generic joint model [24] to model the contact interfaces. They identified the joint model parameters using a Bayesian identification approach. In this paper, a method based on the element thin layer approach is employed. A generic model capable of generating an acceptable model of the joint interface is constructed. Extensive modal testing on a set of nominally identical Brake-Reuss beams is performed. The effect of the variation of the preload on the variability in the natural frequencies and damping ratios is investigated. After developing a generic model with unknown parameters of the generic joint interface element, an identification approach is employed to verify the ability of the constructed model to predict the measured dynamic properties.

27.2 Experimental Modal Testing In this section, a study of the effects of preload on the variability of the dynamic characteristics at the contact interface is performed. The well-known Brake-Reuss [25] beam was used as a test structure in this study. The structure consists of a pair of stepped beams bolted together through a three-bolt lap joint. The design and dimensions of the joint structure are shown in Fig. 27.1.

27.2.1 Experimental Set-Up A set of 10 beams was used to obtain 25 nominally identical joint structures, as shown in Fig. 27.2. Identical material properties and geometric dimensions are considered for the different beams. Random variability is present in the different joint structures due to the manufacturing process. One of the main sources of this variability is the variation of the contact interface profile of each beam. The beams are made of structural steel with the following material properties, ρ = 7850 kg/m3 , E = 210 GPa and v = 0.3, where ρ is the material density, E is Young’s modulus and v is Poisson’s ratio. In order to minimize the variability in the joint system, a set of identical M8 bolts, washers and nuts was used. In addition, a unique assembly procedure was followed during the assembly of each joint sample (Fig. 27.3a). Linear modal testing was employed using hammer testing to obtain the linear modal parameters of the different joint samples. Free-free boundary

Fig. 27.1 BRB joint – geometry details (Balaji and Brake [26])

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Fig. 27.2 Set of 25 BRB joint samples

Fig. 27.3 BRB joint – experimental setup

conditions were used for the joint structure by suspending it using flexible strings, as shown in Fig. 27.3b. Three PCB accelerometers bonded to the middle and the tip of the beams were used to measure the response of the structure. A KTC digital torque wrench was used to tighten the bolts to a specific level. In this study, three levels of torque are considered, 5 Nm, 10 Nm and 15 Nm. For each torque level, tests at three force levels were performed, 250 N, 550 N and 850 N. Therefore a total of 9 * 25 tests were performed.

27.2.2 Variability of Modal Properties The modal properties were extracted using the peak-picking modal approach for the first six modes of the different structures. The experimental measurements presented in Figs. 27.4 and 27.5 show the variability in the natural frequencies and damping ratios for different modes due to the bolt torque variation. An important level of variability of the natural frequencies is observed in the different modes. However, this variability is lower for the damping ratios. As expected, the natural frequencies increase when the bolt torque increases for the lower and higher modes. The statistical properties of the first four natural frequencies corresponding to three different levels of bolt-torque are listed in Table 27.1. It’s observed that by increasing the bolt-torque, the coefficient of variation decreases for all modes. This is due to the increase of stiffness in the contact interface which reduces the effect of the variability. Moreover, the damping ratio variation decreases with increasing torque level. This behaviour is expected at high levels of torque since the energy dissipated in the contact interface would be low due to the high stiffness of the joint. Therefore,

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Fig. 27.4 Natural frequency distribution at F = 250 N for three levels of bolt torque

Fig. 27.5 shows that the variability of the damping ratios decreases significantly at high levels of torque. In addition, a high correlation of the modal properties is observed for the lower modes, as shown by the reduced area of the ellipses area compared to the higher modes for the three different levels of torque, as shown in Figs. 27.4 and 27.5. The same observation can be seen in the statistical properties of the damping ratios for different level of bolt-torque listed in Table 27.2. By increasing the preload in the bolts from 5 Nm to 15 Nm, the mean and the standard deviation decrease. In addition, an important variability is observed from the high values of the coefficient of variation compared to the variability of the stiffness shown in Table 27.1.

27.3 Finite Element Modelling of the BRB In this section, a finite element model is constructed to predict the dynamic properties of the BRB joint. The model is utilized to represent the variability in the dynamic properties in the contact interface observed in the previous section. A bolted structure model is considered as shown in Fig. 27.6a. The joint structure is divided to three sections, namely two symmetric beam sections and one joint section. A finite element model of the joint structure is developed by modelling each beam portion, as shown in Fig. 27.6b. The beam section is modelled using Euler-Bernoulli beam elements [26]. The number of elements shown in Fig. 27.6b is only for schematic purposes. By considering only the transverse and rotational degrees of

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Fig. 27.5 Damping ratio distribution at F = 250 N for three levels of bolt torque Table 27.1 Statistical properties of the first four natural frequencies corresponding to three levels of bolt-torque Modes (Hz) Bolt-torque (NM) Mean (Hz) Standard deviation (Hz) Coefficient of variation (%)

f1 5 147.4 13.13 8.91

10 153.3 12.08 7.88

15 155.7 11.47 7.37

f2 5 566.1 17.93 3.17

10 572.26 17.67 3.09

15 574.75 17.33 3.02

f3 5 1154 25.84 2.24

10 1170.8 20.07 1.71

15 1177.3 17.733 1.51

f4 5 2892.8 74.89 2.59

10 2950.5 52.43 1.78

15 2975.3 43.28 1.45

freedom in the beam elements, the stiffness and mass matrices of the Euler-Bernoulli beam elements are obtained as follows: ⎡

12 EI ⎢ 6l e [Kb] = 3 ⎢ le ⎣ − 12 6le

6le 4le2 −6le 2le2

−12 6le 12 −6le

⎡ ⎤ 6le 156 ⎢ 22le ρAl 2le2 ⎥ e ⎢ ⎥ and [Mb] = −6le ⎦ 420 ⎣ 54 4le2 − 13le

22le 54 4le2 13le 13le 156 −3le2 −22le

⎤ −13le −3le2 ⎥ ⎥ −22le ⎦

(27.1)

4le2

where le is the element length, A is the cross-section area and I is the area moment of inertia. Generic elements [27, 28] are implemented using an FE approach to model the joint section. The generic element stiffness matrix is defined using only two independent parameters, kw and kt . These two parameters represent, respectively, the lateral

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Table 27.2 Statistical properties of the first four damping ratios corresponding to three levels of bolt-torque Modes (Hz) Bolt-torque (NM) Mean (%) Standard deviation (%) Coefficient of variation (%)

ξ1 5 0.296 0.295 99.66

10 0.151 0.126 83.33

15 0.136 0.137 101.29

ξ2 5 0.074 0.066 89.07

10 0.053 0.037 69.37

15 0.052 0.046 87.98

ξ3 5 0.088 0.072 81.72

10 0.051 0.024 47.47

15 0.042 0.031 73.64

ξ4 5 0.155 0.068 44.42

10 0.082 0.025 30.81

15 0.059 0.019 32.37

Fig. 27.6 (a) Schematic (b) Corresponding FE model of the BRB

and the flexural stiffness of the joint element. The lateral and the flexural stiffnesses are equivalent to the stiffnesses of the slap and slip mechanisms in the contact interface, respectively. Therefore, the stiffness matrix of the generic joint element is considered as follows: ⎡

⎤ 2kw le −4kw 2kw le 4kw 1 ⎢ 2kw le (kw + kt ) le2 −2kw le (kw − kt ) le2 ⎥ ⎥ [Kj ] = 2 ⎢ −6le 4kw −2kw le ⎦ le ⎣ − 4kw 2kw le (kw − kt ) le2 −2kw le (kw + kt ) le2

(27.2)

where kw and kt are stiffness parameters of the joint element. In this study, it is assumed that the damping in the system is embedded in the imaginary part of the stiffness parameters. The mass matrix corresponding to the joint section element is equal to the Euler-Bernoulli beam element mass matrix [Mb ] given in Eq. (27.1). After assembling the mass and stiffness matrices of the beam and joint elements, the dynamic model governing the free vibration of the assembled structure given in Fig. 27.6a can be solved to relate the natural frequencies and damping ratios to the unknown joint element stiffness parameters.

27.4 Deterministic Identification of the Joint Model Parameters The constitutive relations in the generic joint stiffness model are defined using two unknown parameters kw and kt. These parameters are identified using deterministic model updating. By minimizing the difference between the model predictions and the corresponding experimental measurement, the stiffness parameters may be identified. The first two natural frequencies are used to identify the unknown parameters of the joint. The identification is performed using a minimization function based on the simplex search method of Lagarias et al. [29]. For every joint structure sample, the first measured natural frequencies are used to identify the unknown stiffness parameters kw and kt . Overall, 20 beam elements and 10 joint elements are used in the model. Therefore, 25 values of these two parameters are identified using the 25 sets of measured frequencies. In Table 27.2, the measured NFs, predicted NFs, the error rate and the identified stiffness parameters for four different BRB structures are listed. The results presented in Table 27.3 and Fig. 27.7 show that the modelling approach and the identification procedure presented in this study can effectively predict the experimental results for the first two natural frequencies which were used in the identification. For higher frequencies, this approach shows a limitation of the ability to predict the experimental results for modes higher than the first and second mode. An example of this limitation can be seen in the results shown in Fig. 27.8

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Table 27.3 The identified deterministic joint element stiffness parameters Joint sample # Measured NF (Hz) Predicted NF (Hz) Error (%) Identified stiffness parameters (Nm)

Mode 1 Mode 2 Mode 1 Mode 2 Mode 1 Mode 2 kw kt

3 156.27 576.239 156.27 576.239 2.71e-07 5.05e-06 1.107 4410.6

5 145.02 567.18 145.02 567.18 5.23e-07 7.3e-08 1.0924 3795.4

6 129.35 533.91 129.35 533.91 1.1e-07 9.3e-08 0.788 3015.2

7 140.03 555.08 140.03 555.08 5.65e-07 4.66e-07 0.9815 3537.2

Fig. 27.7 Experimental vs. identified natural frequencies using first two natural frequencies

where the prediction accuracy decreased significantly for the correlation of third mode with mode 1 and 2, comparing to the correlation of mode 1 and 2 shown in Fig. 27.7. In addition, increasing the number of natural frequencies that contribute to the identification process will result in an overdetermined optimization. However, the local deformation of the joint will be different for the higher modes than for the lower modes, and this presents a limitation of joint modelling used. Improved modelling which will account for the damping in the stiffness parameter expression and an optimization approach based on a convenient likelihood function that guarantees the convergence of the identification will be investigated in future work.

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Fig. 27.8 Experimental vs. identified natural frequencies using first two natural frequencies

27.5 Conclusion In this paper, the effect of the variation of the bolt preload on the variability of the dynamic modal properties was investigated using experimental measurements on a nominally identical set of BRBs. It was shown that the variability is significant in the natural frequencies and damping ratios. An FE model of the joint structure was constructed using generic joint elements. Using deterministic modal updating, the stiffness properties of the contact interface were identified, and the results showed a good agreement with experimental results using the lower modes in the identification. Acknowledgements The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council through the award of the Programme Grant “Digital Twins for Improved Dynamic Design”, grant number EP/R006768/1.

References 1. Ibrahim, R.A., Pettit, C.L.: Uncertainties and dynamic problems of bolted joints and other fasteners. J. Sound Vib. 279(3–5), 857–936 (2005) 2. Brake, M.R., Reuss, P., Segalman, D.J., Gaul, L.: Variability and repeatability of jointed structures with frictional interfaces. In: Allen, M., Mayes, R., Rixen, D. (eds.) Dynamics of Coupled Structures. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 1. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04501-6_23

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3. Guo, Q.T., Zhang, L.M.: Identification of the mechanical joint parameters with model uncertainty. Chin. J. Aeronaut. 18(1), 47–52 (2005). https://doi.org/10.1016/S1000-9361(11)60281-1 4. Castelluccio, G.M., Brake, M.R.W.: On the origin of computational model sensitivity, error, and uncertainty in threaded fasteners. Comput. Struct. 186, 1–10 (2017). https://doi.org/10.1016/j.compstruc.2017.03.004 5. Ren, Y., Beards, C.F.: Identification of ‘effective’ linear joints using coupling and joint identification techniques. Am. Soc. Mech. Eng. J. Vib. Acoust. 120(2), 331–338 (1998). https://doi.org/10.1115/1.2893835 6. Ma, X., Bergman, L., Vakakis, A.F.: Identification of bolted joints through laser vibrometry. J. Sound Vib. 246(3), 441–460 (2001). https:// doi.org/10.1006/jsvi.2001.3573 7. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125, 169–181 (1997). https://doi.org/10.1007/ BF01177306 8. Iwan, W.D.: On a class of models for the yielding behavior of continuous composite systems. J. Appl. Mech. 34(3), 612–617 (1967). https:// doi.org/10.1115/1.3607751 9. Massing, G.: Eigenspannungen und Verfestigung beim Messing. In: Proceedings of the Second International Congress of Applied Mechanics, pp. 332–335 (1926) 10. Duwez, P.: On the plasticity of crystals. Phys. Rev. 47(6), 494–501 (1935). https://doi.org/10.1103/PhysRev.47.494 11. Drucker, D.C.: On the continuum as an assemblage of homogeneous elements or states. In: Parkus, H., Sedov, L.I. (eds.) Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. IUTAM Symposia (International Union of Theoretical and Applied Mechanics). Springer, Vienna. https://doi.org/10.1007/978-3-7091-5581-3_4 12. Ivlev, D.E.: The theory of complex media. Soviet Physics—Doklady. 8(1), 28–30 (1963) 13. Prager, W.: Models of Plastic Behavior, Proceedings of the Fifth U. S. National Congress of Applied Mechanics, pp. 447–448. ASME (1966) 14. Argatov, I.I., Butcher, E.A.: On the Iwan models for lap-type bolted joints. Int. J. Non-Linear Mech. 46(2), 347–356 (2011). https://doi.org/ 10.1016/j.ijnonlinmec.2010.09.018 15. Li, Y., Hao, Z.: A six-parameter Iwan model and its application. Mech. Syst. Signal Process. 68-69, 354–365 (2016). https://doi.org/10.1016/ j.ymssp.2015.07.009 16. Brake, M.R.W.: A reduced Iwan model that includes pinning for bolted joint mechanics. Nonlinear Dyn. 87, 1335–1349 (2017). https://doi.org/ 10.1007/s11071-016-3117-2 17. Segalman, D.J.: An Initial Overview of Iwan Modeling for Mechanical Joints, Report SAND 2001–0811. Sandia National Laboratories, Albuquerque (2001) 18. Menq, C.-H., Griffin, J.H., Bielak, J.: The influence of microslip on vibratory response. Part II: A comparison with experimental results. J. Sound Vib. 107(2), 295–307 (1986). https://doi.org/10.1016/0022-460X(86)90239-7 19. Ahmadian, H., Ebrahimi, M., Mottershead, J.E., Friswell, M.I.: Identification of Bolted Joint Interface Models, pp. 1741–1747. ISMA 27, Leuven (2002) 20. Ahmadian, H., Jalali, H., Mottershead, J.E., Friswell, M.I.: Dynamic Modeling of Spot Welds Using Thin Layer Interface Theory, pp. 7–10. Tenth International Congress on Sound and Vibration, Stockholm (2003) 21. Ahmadian, H., Mottershead, J.E., James, S., Friswell, M.I., Reece, C.A.: Modeling and updating of large surface-to-surface joints in the AWE-MACE structure. Mech. Syst. Signal Process. 20(4), 868–880 (2006). https://doi.org/10.1016/j.ymssp.2005.05.005 22. Jalali, H., Khodaparast, H.H., Friswell, M.I.: The effect of preload and surface roughness quality on linear joint model parameters. J. Sound Vib. 447, 186–204 (2019). https://doi.org/10.1016/j.jsv.2019.01.050 23. Jalali, H., Haddad Khodaparast, H., Madinei, H., Friswell, M.I.: Stochastic modelling and updating of a joint contact interface. Mech. Syst. Signal Process. 129, 645–658 (2019). https://doi.org/10.1016/j.ymssp.2019.04.003 24. Ahmadian, H., Jalali, H.: Generic element formulation for modeling bolted lap joints. Mech. Syst. Signal Process. 21, 2318–2334 (2007). https://doi.org/10.1016/j.ymssp.2006.10.006 25. Balaji, N.N., Brake, M.R.W.: The surrogate system hypothesis for joint mechanics. Mech. Syst. Signal Process. 126, 42–64 (2019). https:// doi.org/10.1016/j.ymssp.2019.02.013 26. Han, S.M., Benaroya, H., Wei, T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 255(5), 935–988 (1999). https://doi.org/10.1006/jsvi.1999.2257 27. Gladwell, G.M.L., Ahmadian, H.: Generic element matrices suitable for finite element model updating. Mech. Syst. Signal Process. 9(6), 601–614 (1995). https://doi.org/10.1006/mssp.1995.0045 28. Ahmadian, H., Mottershead, J.E., Friswell, M.I.: Physical realization of generic parameters in updating. J. Vib. Acoust. 124(4), 628–633 (2002). https://doi.org/10.1115/1.1505028 29. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9(1), 112–147 (2006). https://doi.org/10.1137/S1052623496303470

Chapter 28

Parameter Estimation of Jointed Structures Using Alternating Frequency-Time Harmonic Balance Javad Taghipour, Nidhal Jamia, Michael I. Friswell, Hamed Haddad Khodaparast, and Hassan Jalali

Abstract Despite all the efforts made to investigate the dynamics of jointed structures, they still remain as one of the uncertain dynamical systems in structural dynamics. Modelling the dynamic behaviour of bolted joint contact interfaces is one of the most challenging tasks in structural dynamics. The multi-frequency response of nonlinear systems, particularly bolted joint structures, is an important characteristic for the model identification of such structures. The alternating frequencytime approach using a harmonic balance (AFTHB) is used in this study to identify the nonlinear model of the bolted joint of the Brake-Reuß beam. Vibration tests with harmonic excitation were performed over a range of frequencies to measure the dynamic response of the structure. The two beams of the structure are modelled using Timoshenko beam theory. A model selection procedure using the measured data is applied to a range of candidate models to model the dynamics of the joint contact interface. The AFTHB approach is then applied to estimate the unknown parameters of the assumed nonlinear model of the structure utilizing the experimentally measured data. The estimated parameters are used to reconstruct the measured dynamic response of the structure. Keywords Jointed structures · Nonlinear model identification · Parameter estimation · Multi-harmonic response · Brake-Reuß beam

28.1 Introduction Bolted joints play important roles in mechanical assembled structures. They have been widely used in many engineering applications, yet their dynamic behaviour is still mysterious in spite of a vast number of research works dedicated to the dynamic modelling of bolted joint structures. The behaviour of jointed structures is mainly dependent on the damping of the structure which is dominated by the damping of the joint interface. Damping of the joint interface is related to various factors, including the roughness of the contact interface, lubrication condition of the joint, the geometry of joints, residual stress, bolt preload, and also the amplitude of excitation. In order to accurately estimate the dynamic behaviour of jointed structures, an accurate model of the assembled structure, including an appropriate model of the joint contact interface, is crucial. However, it is very difficult to consider the effect of all the above-mentioned factors on the assumed model of the joint contact interface. There are different nonlinear model identification approaches in the literature [1–3] with advantages and disadvantages. In addition to an efficient model identification method, an appropriate model of the joint section or a model selection procedure for the dynamics of the joint interface is a key step in modelling bolt-jointed assembled structures. Relying on the early efforts of joint modelling [4–8] and benefiting from simple but efficient joint models such as the Jenkins element and the Iwan model of the joint interface [9, 10], the investigation of the dynamic behaviour of jointed structures has improved during recent decades. Gaul and Lenze [11] showed the application of a Valanis model in predicting the stickslip mechanism in the dynamic response of a jointed structure. Gaul and Nitsche [12] reviewed methods to investigate the nonlinear dynamic behaviour of frictional contacts in assembled structures. The article described various theoretical and

J. Taghipour () · N. Jamia · M. I. Friswell · H. Haddad Khodaparast Faculty of Science and Engineering, Swansea University, Swansea, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] H. Jalali Department of Mechanical and Construction Engineering, Northumbria University, Newcastle-upon-Tyne, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_28

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practical constitutive and phenomenological friction models introduced to estimate the nonlinear force in contact interfaces. Segalman [13] provided a review study on the formulation of Iwan joint modelling. Argatov and Butcher [14], Li and Hao [15], and Brake [16] studied the application of different versions of the Iwan model to predict the dynamics of various jointed assemblies. Ahmadian and Jalali [17] introduced a generic element formulation to model the nonlinear dynamic behaviour of bolted lap joints. They provided a nonlinear formulation for parametric modelling of a joint interface using admissible nonlinear models for joints. The Brake-Reuß beam (BRB) is a benchmark introduced by the Nonlinear Mechanics and Dynamics (NOMAD) Research Institute [18] to study the variability and uncertainty of the dynamic behaviour of jointed structures with frictional interfaces. This benchmark was described by Brake and Reuß [19]. Lacayo et al. [20] presented a numerical benchmark to investigate the performance of two joint modelling approaches in predicting the dynamic behaviour of a Brake-Reuß beam: a time-domain whole-joint approach and a frequency-domain node-to-node approach. Balaji et al. [21] introduced a new modelling framework for bolt-jointed assembled structures based on multiscale traction-based contact constitutive laws. In this framework, the nonlinear internal forces of contact interfaces are modelled by implementing the multiscale traction-based laws via zero-thickness elements (ZTE). They employed the presented model to predict the behaviour of a BRB benchmark. Considering all the aspects of the joint modelling approaches, two main issues remain complicated in analysing the dynamics of jointed structures: dealing with the multi-harmonic response and force signals which is common in the nonlinear behaviour of jointed structures and the identification of jointed structures utilizing spatially incomplete measurements. The aim of this study is to identify the nonlinear mathematical model of a bolt-jointed Brake-Reuß beam (BRB) structure utilizing spatially incomplete experimental measurements. To this end, the BRB structure is divided into three sections: two beam substructures and a joint section. The two beam substructures are modelled using Euler-Bernoulli beam theory with two-node elements. The degrees of freedom of the beam elements are the in-plane transverse vibration and the rotational deflection of the beam. The joint section is modelled using generic joint elements. The alternating frequency-time approach using a harmonic balance (AFTHB) [22] is used as the identification method to estimate the unknown parameters of the nonlinear model of the jointed structure in the presence of multi-harmonic responses.

28.2 Experimental Set-Up The experimental data used in this study is provided by Balaji et al. [23]. The experimental set-up is composed of a stainless steel Brake-Reuß beam (BRB) structure. The dimensions of the BRB structure are given in Fig. 28.1. The two beam substructures of the BRB are attached together using three bolts at joint section A, as shown in Fig. 28.1a. The bolts of the joint section are tightened with 12 kN bolt preload. The BRB structure is attached to a shaker at around 24.023 cm from one end (near end) of the BRB via a stinger. The excitation force and the response at the location of excitation are measured using an impedance head installed between the stinger and the beam. Also, the response at both ends of the BRB structure is measured using two triaxial accelerometers. The experimental data used in this study are measured from the very early tests carried out on the fresh BRB structure without any accumulated damage at the joint interface.

28.3 Theory/Methodology To model the jointed structure of Fig. 28.1a, three substructures are considered as shown in Fig. 28.1b: two beam substructures and a joint section between the two beam substructures. Each beam substructure is modelled using EulerBernoulli beam theory with 30 two-node elements. In this section, the equation of motion of the BRB system is first obtained using the assumed model for the joint section of the beam. The model of the joint section is generated using 12 two-node generic elements [17]. The model of the joint section is then introduced and the equation of motion of the system is provided. Afterwards, the alternating frequency-time approach using harmonic balance (AFTHB) is described.

28 Parameter Estimation of Jointed Structures Using Alternating Frequency-Time Harmonic Balance

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Fig. 28.1 (a) Bolt-jointed BRB structure; (b) substructures of the BRB structure; (c) generic joint element and its degrees of freedom

28.3.1 Joint Section The generic joint model assumed for the structure of Fig. 28.1a is composed of 12 two-node generic elements with four degrees of freedom, as shown in Fig. 28.1c. The degrees of freedom at each node of the joint element are assumed to be similar to the beam substructures, {wi , θ i }. wi denotes the transverse displacement and θ i the rotation of the beam at node i. The linear stiffness matrix of the joint element is written as follows: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ Kj = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

kwL

kwL

Le 2

L2e kθL +

kwL 4



⎤

−kwL

⎥ kwL



Le 2 ⎥ ⎥ kwL ⎥ Le − 2 L2e −kθL + kwL ⎥ 4

 ⎥, ⎥ Le − kwL 2

 ⎥ ⎥ kwL ⎥ L2e kθL + kwL ⎦ 4

(28.1)

Sym. where kwL and kθL are linear stiffness coefficients and Le is the length of element. There are different classical and practical methods in the literature to model the nonlinearity of the joint contacts in assembled structures [17]. The damping matrix of the generic element is defined as follows: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ Cj = ⎢ ⎢ ⎢ ⎢ ⎣

cwL

L cwL  e 2 cwL  2 Le cθL + 4

−cwL − Le cwL 2 cwL

⎤ ⎥ ⎥ Le cwL ⎥ 2  cwL  ⎥ 2 Le −cθL + 4 ⎥ ⎥, ⎥ − Le cwL 2   cwL ⎥ 2 ⎥ Le cθL + 4 ⎦

(28.2)

Sym. where cwL and cθL are the coefficients of linear damping of the joint element. The mass matrix of the generic element is identical to the mass matrix of the Euler-Bernoulli beam element given by:

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⎡ ⎢ ⎢ ⎢ ⎢ ρAL ⎢ ⎢ Mj = ⎢ 420 ⎢ ⎢ ⎢ ⎢ ⎣

156 22Le 4L2e

54

−13Le

13Le

− 3L2e

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, − 22Le ⎥ ⎥ ⎥ ⎥ ⎦

156

(28.3)

4L2e

Sym.

where ρ and A are, respectively, the density and cross-section area of the joint section. The nonlinearity of the joint element is assumed to be due to a connected (ungrounded) cubic stiffness between two nodes of each generic joint element. The force-displacement relation of the nonlinear force of the joint contact interface is defined as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−

⎧ ⎫ Vi ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ Mi = ⎪ ⎪ Vi+1 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎪ ⎪ ⎪ Mi+1 ⎪ ⎪ ⎩

⎫ 3 ⎪ ⎪ (θi + θi+1 ) ⎪ ⎪ ⎪

3 ⎪ kwN Le 3⎪ ⎬ w L − w + + θ − k L − θ (θ ) (θ ) e i i+1 i i+1 θN e i i+1 2 2 , 3

⎪ ⎪ kwN wi − wi+1 + L2e (θi + θi+1 ) ⎪ ⎪ ⎪

3 ⎪ kwN Le 3⎪ − 2 Le wi − wi+1 + 2 (θi + θi+1 ) + kθN Le (θi − θi+1 ) ⎭

−kwN wi − wi+1 +

Le 2

(28.4)

Accordingly, the nonlinear force is defined in the following form:

f NG =

⎫ ⎧ −kwN h1 ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎪ ⎬ ⎨ Le − kwN h1 − kθN h2 ⎪ 2

⎪ kwN h1 ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎭ ⎩ L − kwN h + k h ⎪ e 1 θN 2 2

,

(28.5)

where kwN and kθN are the coefficients of the nonlinear stiffness of the joint element and h1 = ( w)3 ,

w = wi − wi+1 +

h2 = ( θ )3 ,

Le (θi + θi+1 ) , 2

θ = (θi − θi+1 ) .

(28.6a)

28.3.2 Equation of Motion Considering the mass and stiffness matrices of the beam elements obtained using Euler-Bernoulli beam theory, the damping of the beam substructures is assumed to be proportional, given in the form: Ce = αMe + βKe ,

(28.6b)

where Me , Ce , and Ke denote the mass, damping, and stiffness matrices of a Euler-Bernoulli beam element with proportional damping. The shaker-structure interaction is modelled as a linear damper-spring system with linear damping csh and linear stiffness ksh . The linear damping csh and stiffness ksh are included in the global matrices of the equation of motion of the system at the location of excitation force. The governing equations of motion of the jointed BRB structure are obtained by assembling the element matrices of all three substructures and considering the nonlinear force of the joint section. M¨z + C˙z + Kz + f N G (z, z˙ ) = f ex (t).

(28.7)

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M, C, and K denote the mass, damping, and stiffness global matrices of the structure; z, z˙ ,and z¨ are the vectors of displacement and its derivatives: "T ! z = w1 , θ1 , w2 , θ2 , . . . , wi , θi , . . . , wNe , θNe ,

(28.8)

where Ne is the number of elements considered for the structure, including the joint elements. f N G (z, z˙ ) is the unknown nonlinear internal force, and fex (t) is the external force. It is assumed that the mass matrix of the joint element is known in this model, having the accurate dimensions and density of the beam substructures. Although all the parameters of the underlying linear system of the structure can be obtained using well-known linear model updating methods utilizing the measured modal parameters, this study focused on applying the AFTHB approach to identify the unknown parameters of the assumed nonlinearity of the jointed structure. To identify the mathematical model of the restoring force in Eq. (28.7), the unknown parameters of the system should be identified. To this end, the AFTHB method is used in this study which is introduced in Sect. 28.3.3.

28.3.3 AFTHB Identification Approach The alternative frequency-time harmonic balance-based approach (AFTHB) [22] is an identification method introduced based on the harmonic balance method to deal with multi-harmonic response and force signals. In this study, the AFTHB approach is applied to the equation of motion of the BRB structure given by Eq. (28.7). There are two options for applying the AFTHB method: the first is the case with all coordinates measured, and the second is the case with incomplete measurements. (a) In the first case, all unknowns of the assumed model, including the proportional damping of the beam substructures, the linear stiffness and damping of the joint section, and the parameters of the nonlinear model of the contact interface can be estimated using the AFTHB method. Therefore, the equation of motion of the system is rewritten in the following form so that all unknowns of the assumed model are included in the nonlinear internal force. Thus:

M¨z + Kz + Ga = f ex (t),

(28.9)

All unknown internal forces of the system are included in the term Ga. This includes the nonlinear function fNG of Eq. (28.7). As the two coefficients of the proportional damping are unknown, the linear damping term of Eq. (28.7) is also moved into the term Ga. G and a denote, respectively, the matrix of nonlinear functions and the vector of unknown parameters. ⎡

g1,1 ⎢g ⎢ 2,1 ⎢ . ⎢ . ⎢ . Ga = ⎢ ⎢ gi,1 ⎢ ⎢ .. ⎣ . gN,1

g1,2 · · · g1,Np g2,2 · · · g2,Np .. .. .. . . . gi,2 · · · gi,Np .. .. .. . . . gN,2 · · · gN,Np

⎤ ⎧ ⎫ a1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ a2 ⎪ ⎬ . .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ aNp N

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

(28.10)

p ×1

N ×Np

where G denotes the matrix of assumed nonlinear functions and gi, j is the nonlinear function corresponding the ith degree of freedom and jth parameter. a is the vector of Np unknown parameters of the assumed model. The unknown parameters of the assumed model of the BRB structure in this study are α, β, csh , ksh cwL , cθL , kwL , kθL , kwN , kθN . Therefore, all terms, including the unknown parameters, are moved to the term Ga, and the matrix and the vector are constructed as follows:

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⎡

g1,1 ⎢g ⎢ 2,1 ⎢ . ⎢ . ⎢ . Ga = ⎢ ⎢ gi,1 ⎢ ⎢ .. ⎣ . gN,1

g1,2 · · · g1,Np g2,2 · · · g2,Np .. .. .. . . . gi,2 · · · gi,Np .. .. .. . . . gN,2 · · · gN,Np

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ N ×Np

⎫ α ⎪ ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ ⎪ csh ⎪ ⎪ ⎪ ⎪ ksh ⎪ ⎪ ⎪ ⎬ cwL × ⎪ cθL ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k wL ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ θL ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k wN ⎪ ⎪ ⎪ ⎩ ⎭ kθN N

,

(28.11)

p ×1

The first two unknown parameters of a in Eq. (28.11) are the coefficients of proportional damping. The functions corresponding to these parameters in the matrix G are given as follows: gi,1 = M (i, :) z˙ ,

gi,2 = K (i, :) z˙ ,

(28.12)

The functions related to the unknown damping and stiffness of the shaker stinger are the transverse oscillation and velocity of the BRB structure at the location of excitation force. The nonlinearity of the BRB structure is assumed to be located at the joint section. Using the vector of nonlinear force given in Eq. (28.4) and the stiffness and damping matrices of the joint section, the part of G corresponding to the joint section is given as follows: ⎤ h5 −h1 0 0 0 h3  ⎢ h3 /2 h4 h5 /2 h6 − h1 /2 − h2 ⎥  ⎥, G Nj : Nj + 4, 5 : 10 = ⎢ ⎣ − h3 0 0 ⎦ − h5 0 h1 h3 /2 − h4 h5 /2 − h6 − h1 /2 h2 ⎡

(28.13)

where: h3 = w, ˙

˙ h4 = θ,

h5 = w, h6 = θ,

(28.14)

(b) In the second case, where the responses at all coordinates are not measured, the AFTHB approach cannot be used directly. Therefore, the identification process is divided into two steps: the model updating of the underlying linear system using well-known conventional methods and the estimation of the parameters of the nonlinear internal force. First, the underlying linear model is updated using the linear response of the structure. Then, the measured responses of the structure are expanded utilizing the updated underlying linear system to estimate the response at unmeasured coordinates. The nonlinear model of the system is then identified using the AFTHB method. Note that using the updated linear model to predict the response at unmeasured coordinates of a nonlinear system can lead to erroneous estimation of the response, particularly for structures with strong nonlinearities. After separating the unknown nonlinearity of the structure from its known linear part and formulating the nonlinear force in the form of Eq. (28.10), the AFTHB method is applied to the assumed model. The system is assumed to be excited by a harmonic force in the form of a truncated Fourier series: f ex (t) = F 0 +

Hf 

 I F n ej ωt ,

j=

√ −1,

(28.15)

n=1

where I represents the imaginary part; F0 and Fn denote, respectively, the static value and the complex amplitude of the nth harmonic of force; ω is the excitation frequency; and Hf is the number of harmonics contributing to the force signal. The response to the harmonic excitation can also be assumed as a harmonic response:

28 Parameter Estimation of Jointed Structures Using Alternating Frequency-Time Harmonic Balance

z(t) = Z 0 +

Hr 

 I Z n ej ωt .

209

(28.16)

n=1

Z0 and Zn denote the vectors of the static value and the complex amplitude of the nth harmonic of response, respectively. Hr is the total number of harmonics considered in the response, which can be different from Hf . The unknown internal force G (z, z˙ ) is a function of system response, and it is defined by a Fourier series: Hg 

 G (z, z˙ ) = U0 + I Un ej ωt ,

(28.17)

n=1

where U0 and Un denote the static value and the dynamic amplitude of the nth harmonic of the nonlinear force G (z, z˙ ). Hg is the number of harmonics contributing to the nonlinear force, which can be different from both Hf and Hr . Substituting Eqs. (28.15), (28.16) and (28.17) into Eq. (28.9) and averaging over each harmonic (i.e. balancing the coefficients of each harmonic), the equation of motion of the system at one frequency point ω is obtained: Un a = En (ω) ,

n = 1, . . . , NH ,

(28.18)

where NH is the total number of harmonics considered in the identification process and

 En = Fn − −n2 ω2 M + K Zn .

(28.19)

Solving Eq. (28.18) will result in the estimation of the unknown parameters of the assumed model at a specific excitation frequency. That is, for an individual unknown parameter, different estimates are obtained at different excitation frequencies. For an accurate assumed model, the variation of the estimated parameter should be minimum if the parameter is not dependent on the excitation frequency. Combining the equations for all harmonics at different frequency points gives the following: ⎫ ⎧ ⎤ U (ω1 ) E (ω1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ U (ω2 ) ⎥ ⎬ ⎨ E (ω2 ) ⎪ ⎢ ⎥ , a = ⎢ ⎥ .. .. ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ . . ⎪ ⎪   ⎭ ⎩  U ωNω E ωNω ⎡

(28.20)

where: ! "T U = U0 U1 · · ·UNH −1 UNH , U ∈ C

"T ! E = E0 E1 · · ·ENH −1 ENH , E ∈ C

(28.21)

and C denotes complex numbers. To apply the AFTHB approach in an experimental case, the response and force signals are measured in the time domain. Using the fast Fourier transform (FFT), the amplitude Zn and Fn of the nth harmonic of the response and force signals are obtained. Also, the matrix G (z, z˙ ) of the unknown internal force is reconstructed using the measured response z, z˙ . Then, the amplitude of each harmonic of G is obtained by taking FFT of the reconstructed signal of G (z, z˙ ).

28.4 Results and Discussion The experimental data obtained from sine tests carried out on the test rig of Fig. 28.1 is used to identify the nonlinear model of the BRB structure. The vibration tests with sine excitations were performed within the frequency range f = 150~190 Hz with a frequency step of f = 0.5 Hz. At each frequency, a sampling frequency of fs = 8.533 × 103 Hz (dt = 0.1172 ms)

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Fig. 28.2 The experimental time history of the force signal at the location of the force application and the response of the BRB structure at one end of the beam with respect to the location of the force application

was used to capture the time history of the excitation force and the response of the structure. The response of the system was measured at three different locations: at the location of the excitation force and at the two ends of the beam. An impedance head was used to measure the force signal and response at the location of excitation. Two triaxial accelerometers were used to measure the response of the BRB structure at the two ends of the beam. Figure 28.2 shows the time history of the excitation force and the measured acceleration of the BRB structure at the excitation frequency ω = 170 Hz. The measured data illustrates the significant contribution of the higher harmonics to the dynamics of the structure. The frequency content of the measured data is obtained using the fast Fourier transform (FFT). The frequency response function (FRF) of the structure is obtained using the frequency content of the force signal and the measured response. Figure 28.3 shows the frequency response of the BRB structure calculated using the first harmonic of the force and response signals. The results of the figure show a softening nonlinearity in the dynamic behaviour of the BRB structure. This nonlinear softening behaviour comes from the nonlinear stiffness of the joint contact interface. The measured data obtained from the vibration test at a force level of 0.05 V is used to update the underlying linear model of the BRB structure. As a result of this process, the proportional damping coefficients of the beam substructures, the stiffness and damping of the shaker attachment, and the linear damping coefficients and linear stiffnesses of the joint model are obtained. The updated values of the parameters are given in Table 28.1. Figure 28.4 shows the comparison between the measured data and the numerically regenerated response using the measured force signal. Figure 28.5 shows the first six mode shapes of the updated underlying linear model of the BRB structure. These mode shapes are used to expand the measured response of the structure and estimate the response at unmeasured coordinates. The estimated responses at unmeasured coordinates and the measured responses are used in the process of AFTHB identification to estimate the parameters of the nonlinear internal force given by Eq. (28.4). Table 28.2 gives the estimated values of kwN and kθN . The estimated values of parameters are then used to regenerate the experimentally measured response of the BRB structure. The comparison between the measured and the numerical response is given in Fig. 28.6. The discrepancy between the measured and numerical response is likely due to usage of linear expansion methods to estimate the nonlinear response at unmeasured coordinates of the system. Although updating the underlying linear system using the results of low-amplitude excitation is commonly used, this may also lead to some level of error in the updated model.

28 Parameter Estimation of Jointed Structures Using Alternating Frequency-Time Harmonic Balance

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Fig. 28.3 The experimental frequency response of the BRB structure at one end with respect to the location of the force. The response of the structure was obtained for four different levels of excitation Table 28.1 Updated parameters of the underlying linear system of the BRB structure Parameter  (unit)

N.s α m.kg

 csh N.s

m  cwL N.s

m N kwL m

Initial value

Updated value

Parameter (unit)

Initial value

Updated value

2

4.65

β (s)

 N ksh m

 cθ L N.s

rad  N kθ L rad

1 × 10−5

1 × 10−8

1×

1.25 × 104

0.5

2

5 × 103

1.16 × 103

5×

7.84 ×

1010

1010

105

5 × 102

97

4.5 ×

1.93 × 109

109

28.5 Conclusions This study investigated the nonlinear identification of a Brake-Reuß beam structure assembled using a three-bolt joint. The aim of this study was to identify the nonlinear model of a jointed structure with spatially incomplete measurement in the presence of multi-harmonic response and force signals. The BRB structure was modelled using two beam substructures and a joint section. The beam substructures were modelled using Euler-Bernoulli beam theory, and the joint section of the BRB was modelled using two-node generic elements. The nonlinearity of the joint contact interface was considered in the joint element. The AFTHB identification approach was used to estimate the unknown parameters of the assumed nonlinear model of the joint section. Although the results of linear model updating showed a good compatibility between the experimental and numerical results, the identified nonlinear model was not capable of predicting the nonlinear response. This may be due in part to the problem in linear model updating, as only the results of one mode have been used to update the underlying linear model of the structure. In addition, more nonlinear stiffness should be considered as the candidate nonlinearity of the joint contact interface. These problems will be addressed in the future to improve the quality of the identified model of the jointed structure.

212

Fig. 28.4 Comparison between the experimental and numerical linear response obtained from low-amplitude excitation

Fig. 28.5 The first six linear mode shapes of the BRB structure obtained using the updated linear model

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Table 28.2 Estimated values of the parameters of the nonlinear internal force of the joint interface Parameter

(unit) kwN mN3

Estimated value 5.4 × 1017

Parameter

(unit) N kθ N 3 rad

Estimated value 6.1 × 1013

Fig. 28.6 Comparison between the nonlinear experimental response and the numerically regenerated response using the identified model

Acknowledgement The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council through the award of the Programme Grant “Digital Twins for Improved Dynamic Design”, grant number EP/R006768/1.

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8. Prager, W.: Models of Plastic Behavior. Proceedings of the Fifth U. S. A. National Congress of Applied Mechanics, pp. 447–448. ASME (1966) 9. Iwan, W.D.: A distributed-element model for hysteresis and its steady-state dynamic response. ASME J. Appl. Mech. 33, 893–900 (1966) 10. Iwan, W.D.: On a class of models for the yielding behavior of continuous composite systems. J. Appl. Mech. 89, 612–617 (1967) 11. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125, 169–181 (1997) 12. Gaul, L., Nitsche, R.: The role of friction in mechanical joints. Appl. Mech. Rev. 54, 93–106 (2001) 13. Segalman, D.J.: An Initial Overview of Iwan Modeling for Mechanical Joints. No. SAND2001–0811. Sandia National Labs, Albuquerque/Livermore (2001) 14. Argatov, I.I., Butcher, E.A.: On the Iwan models for lap-type bolted joints. Int. J. Non-Linear Mech. 46, 347–356 (2011) 15. Li, Y., Hao, Z.: A six-parameter Iwan model and its application. Mech. Syst. Signal Process. 68-69, 354–365 (2016) 16. Brake, M.R.W.: A reduced Iwan model that includes pinning for bolted joint mechanics. Nonlinear Dyn. 87, 1335–1349 (2017) 17. Ahmadian, H., Jalali, H.: Generic element formulation for modelling bolted lap joints. Mech. Syst. Signal Process. 21(5), 2318–2334 (2007) 18. Brake, M.R.W.: The Mechanics of Jointed Structures. Springer, Cham (2016) 19. Brake, M.R.W., et al.: The 2014 Sandia Nonlinear Mechanics and Dynamics Summer Research Institute. SAND2015–1876. Sandia National Laboratories, Albuquerque (2015) 20. Lacayo, R., Pesaresi, L., Groß, J., Fochler, D., Armand, J., Salles, L., Schwingshackl, C., Allen, M., Brake, M.: Nonlinear modeling of structures with bolted joints: a comparison of two approaches based on a time-domain and frequency-domain solver. Mech. Syst. Signal Process. 114, 413–438 (2019) 21. Balaji, N.N., Chen, W., Brake, M.R.W.: Traction-based multi-scale nonlinear dynamic modeling of bolted joints: formulation, application, and trends in micro-scale interface evolution. Mech. Syst. Signal Process. 139, 106615 (2020) 22. Taghipour, J., Khodaparast, H.H., Friswell, M.I., Shaw, A.D., Jalali, H., Jamia, N.: Harmonic-balance-based parameter estimation of nonlinear structures in the presence of multi-harmonic response and force. Mech. Syst. Signal Process. 162, 108057 (2022) 23. Balaji, N.N., Smith, S.A., Brake, M.R.W.: Evolution of the Dynamics of Jointed Structures over Pro-longed Testing. Proceedings of the IMAC-XL, Orlando (2021)

Chapter 29

A Novel Test Rig for the Validation of Non-linear Friction Contact Parameters of Turbine Blade Root Joints Daniel J. Alarcón Cabana, Jie Yuan, and Christoph W. Schwingshackl

Abstract The assembly of components into a large-scale engineering system naturally leads to the presence of joints with frictional interfaces. The degree of agreement between numerical models and their experimental counterparts decreases when assemblies based in this kind of interfaces are studied due to the non-linear dynamic behaviour that joints introduce. This is, for example, the case in turbine blade root joints. The main cause for these deviations is the friction-related non-linear damping and stiffness effects influencing the dynamic behaviour of the assembly. The experimental measurement of these damping effects poses a challenge due to the presence of the excitation rig itself, which can introduce significant parasitic damping in the system. A free decay measurement is consequently the ideal way to extract the non-linear behaviour; however, the exciter must be initially in physical contact with the test fixture in order to reach the high excitation amplitudes that lead to macroslip friction in the fixture joints. The test setup proposed in this paper is developed for a beam on which two blade root designs have been machined at both ends (dog bone). This beam is fitted between two clamps equipped with dovetail roots and pulled into tension to simulate rotational centrifugal loading, thus creating a blade root contact joint at either end of the beam. The novel excitation method excites the beam harmonically with a rigidly connected shaker to macroslip deflection amplitudes before decoupling from the beam to release it into free decay. This test procedure allows the contactless measurement of the variation in vibrational decay in the beam and the subsequent extraction of the resulting non-linear frictional behaviour associated with the joints. Keywords Blade root · Non-linear damping · Friction interface · Test rig · Macroslip

29.1 Project Motivation Traditional model updating techniques based in modal testing, in which the linearity and time invariance of the system are mathematical assumptions [1], have reached high levels of accuracy and efficiency in the last years. As a consequence, the discrepancies observed between computational models and their experimental counterparts can now be mostly attributed to effects that escape the assumptions on which linear modal analysis is founded. Non-linearities significantly complicate modelling and make the classical CAD/CAE-based high-fidelity approach prohibitive from a computational cost perspective [2]. The assembly of engineering structures and components into large-scale systems naturally leads to the presence of joints based on friction interfaces. These joints induce a non-linear behaviour in the system dynamics at large vibration amplitudes and consequently have a significant influence in the system vibration response. Complex assemblies such as aircraft engines and other turbomachines present a wide range of different types of damping joints such as underplatform dampers, shrouds, blade roots, etc. [3]. These joints are often designed to decrease the amplitude response of their associated components by providing the appropriate damping at large excitation amplitudes, with the aim of improving efficiency and reducing the risk of high cycle fatigue [3]. Turbomachine blade roots are of great importance in aircraft engine design, as they have significant influences on the overall system vibration response. Turbine blades are inserted into their disc via a root-slot system, which, besides making the assembly possible, allows the slippage of the blades within the fixture. This kind of contact interface creates a hysteretic type of damping with microslip (when some parts of the contact are slipping and some parts are stuck) and macroslip (when

D. J. Alarcón Cabana () · J. Yuan · C. W. Schwingshackl Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_29

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Fig. 29.1 Logarithmic decay method to calculate the logarithmic decrement δ [3]

the entire contact surface slips) regimes when the turbine is in service [4]. This behaviour can be studied experimentally in a rotating rig, where the load in the root is introduced by centrifugal forces. A rotating rig is however very complex and expensive to build, operate and instrument. Exciting the blade and its associated root to cause macroslip friction in the joint is particularly challenging since large forces are needed to excite the system to operating conditions. Solutions such as excitation by means of air jets [5] or alternating magnets [6] have been implemented by researchers in this field, but generally very low amplitudes are achieved on the tested blades. Alternatively static rigs have been used, where the centrifugal forces are replaced by pulling forces [4, 7]. The advantages related to the use of a static rig for this study are manifold. The rig excitation can be controlled more accurately at a fraction of the design and operating costs of a rotating rig, a single trained operator can be in charge of the entire rig, and changes in the design and construction of the rig can be implemented in a straightforward manner. On the other hand, rig designs tend to become overcomplicated by trying to isolate the specimen from rigid body rig resonances and parasitic damping. Operational challenges related to loss of load in the specimen over time, material creep/plastic deformation, and the inability of the rig to hold shear forces related to the dynamic excitation of the specimen may arise as well. Nevertheless, the main challenge posed by static rigs lies in the excitation system. In order to cause macroslip friction in the joints, large deflection amplitudes must be excited. This can only be effectively achieved by using a traditional electrodynamic shaker. Schwingshackl et al. [3] found that attaching the shaker permanently to the specimen would yield the required excitation levels, but the excitation fixture itself added parasitic damping to the rig. Instead, detaching the excitation fixture from the bone after a running-in phase would yield satisfactory results with very little added parasitic damping, but the chosen mechanism would not reach the required amplitudes. The authors propose in this work a novel excitation fixture for the Dogbone test rig, capable of both evenly pulling and pushing the bone with a stable shaker connection and then cleanly detaching the excitation system once the required amplitudes are reached. Detaching the excitation fixture, namely, the shaker and its pushrod, allows the free vibrational decay measurement of the bone without the impact of the shaker and in consequence an extraction of amplitude-dependent damping via the piecewise logarithmic decay equation (Fig. 29.1) [3].

δ=

1 xn ln m xn+m

ζ =6 1+

1

2π δ

2

In these equations and in Fig. 29.1, m is the number of peaks for which the logarithmic decrement is calculated for. xn refers to the point in the x axis where the calculation starts, namely, the highest peak; and xn + m is the last point for which the logarithmic decay is calculated. In the results and discussion chapter of this paper, m is always 1, as the logarithmic decay

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Fig. 29.2 Schematics of a dog bone setup for the study of blade root contacts

is calculated in a peak-to-peak basis. ζ is the damping ratio, a non-dimensional characterization of the decay rate relative to the frequency.

29.2 The Dogbone Test Rig The test setup presented in this paper is an evolution of a concept first introduced by Allara et al. [8]. This is known as a “dog bone” setup (Fig. 29.2), which has a beam with a rectangular cross section as its central component. Two blade roots have been machined at either end of this beam, which enables its attachment to two compatible roots machined into the edge of a solid disc. These discs are pulled apart to simulate a centrifugal load, while a harmonic excitation load is applied to excite a particular mode shape in the bone. Both joints behave in consequence as a blade root joint. This concept was implemented by Marquina et al. [7] and by Schwingshackl et al. [3] in the past. The centre of these discs has a hardened bore with rounded edges, which allows sliding a U-shaped shackle through the bore as Fig. 29.3a shows. Each shackle is trapped in the grips of an electromechanical tensile/compressive test machine (Instron 5980, Fig. 29.3a), capable of loads of up to 250 kN with a gripping pressure of 6 bar. The entire dog bone assembly is loaded by moving the upper crossbar a few tenths of a millimetre to a nominal loading of 2.2 kN. This creates two blade root friction contacts at either end of the beam as shown in Fig. 29.3b. This arrangement allows, as proven in [3], to minimize the parasitic damping introduced by the supporting structure. The main difference between the current dog bone setup and the one in [3, 7] is that the current bone is shorter and thinner, measuring 315 mm in length and with a rectangular cross section of 18 × 3 mm, made of AISI 316 stainless steel. The shorter length allows to mount the dog bone assembly in the more compact tensile/compressive machine, thereby increasing the versatility of this rig.

29.3 A Novel Excitation Methodology The main idea of this excitation mechanism consists of a power-to-release electromagnet that couples the bone rigidly to the shaker during the excitation. The magnet can be released at any point during a vibration cycle, decoupling this way the shaker from the bone (Fig. 29.4). An offset introduced to the shaker, and triggered at the same time as the magnet release, ensures that the magnet is pulled away from the bone. A power-to-hold electromagnet behind the shaker is used to hold the shaker back from its excitation position during the release action. This magnet ensures that the pushrod assembly does not impact the bone after a release. A schematic representation of technical implementation of the excitation fixture on the Dogbone test rig is shown in Fig. 29.5. The shaker assembly is represented attached to the bone for illustrative purposes. The excitation is provided by an electrodynamic shaker (Data Physics GW-V20) connected to a PA300E amplifier that can deliver up to 100 N of excitation force. The shaker hangs off a 40 × 40 mm aluminium struts frame equipped with stiff springs and turnbuckles to easily regulate the shaker height according to the experimental requirements. The frame is clamped to a laboratory table by means of four C-clamps.

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Fig. 29.3 (a) Dog bone assembly under load in the Instron 250 kN tensile/compressive machine. (b) Detail view of the upper root contact. The lower contact is analogous

Fig. 29.4 Graphical representation of the working principle of this excitation fixture

The shaker is equipped with a pushrod assembly (Fig. 29.6) with an array of components screwed to one another. Component number 1 in the figure is a 40 × 30 mm power-to-release electromagnet (Bunting XH4030) with a load capacity of 160 N. Two M4 drillings in the electromagnet’s casing allow screwing it to an interface part (part 2) 3D printed of Formlabs Tough 2000 resin. The other side of this interfacing part is equipped with an M6 hole to attach a quartz piezoelectric force sensor (PCB 208C02 (3)) by means of an adapting stud. The force sensor is directly screwed to a custom-built pushrod (4) with a 10-32 UNF mounting stud. The opposite end of the pushrod presents a rounded contact for an easy alignment and mounting of the shaker and is finished on a female brass connector (5). This connector is compatible with a male brass

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Fig. 29.5 Scheme of the Dogbone test rig with its excitation fixture, measurement and data acquisition system as of October 2021

Fig. 29.6 Pushrod assembly (ruler for scale) with numbered main components

connector attached to another 3D-printed part (6) made of blue Formlabs Tough V5 resin. This printed part allows screwing the pushrod assembly to all seven mounting holes on the shaker armature and to minimize operational strain on the central armature screw. Current is supplied to both electromagnets by a dual DC power source, which simultaneously supplies 24 VDC, 1.33 A to the power-to-release magnet and 12 VDC, 0.18 A to the power-to-hold electromagnet. Two solid-state relays (Schneider Electric 70S2(S)) close the power circuit that supplies the magnets when they detect a DC voltage higher than 3 V. This trigger signal is supplied by the user-defined signal generator built in the Polytec PSV-500 data acquisition card (DAQ) together with the excitation signal for the shaker. The signal generator is connected to both relays and to the shaker amplifier; this solution controls the circuit action and the shaker with one fully customizable single input signal. The excitation and relay trigger signal (Fig. 29.7), based in the work of Mace et al. [10], has an absolute amplitude of 10 VDC, which is the maximum voltage supplied by the PSV-500 DAQ, and has three stages: (I) a sinusoidal waveform of amplitude ±1 VDC is supplied at the frequency of interest to excite the bone for 10 s; (II) a steep ramp is then added to the

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Fig. 29.7 Sample of a signal fed to the electrodynamic shaker and the relays with three numbered stages

excitation signal to pull the shaker back from the bone. This ramp is used as well to trigger the relays to close the power circuit and thus feeding the magnets. The power-to-release electromagnet demagnetizes approximately at the same time as the shaker starts pulling back leading to a release of the bone and the removal of the excitation system. (III) A sinusoidal waveform to keep some vibration in the shaker during the free decay of the bone for another 5 s. The shaker and pushrod assembly stay retracted for about 5 s, which is close to the maximum feeding time the power-to-release electromagnet can safely hold. After this, the power is cut from the electromagnets, and the shaker attaches back to the bone, returning the setup to its original state. The vibrational response from the bone is measured with a Polytec scanning laser Doppler vibrometer (LDV) head connected to the PSV-500 DAQ. This upgraded model of scanning head does not require any prior surface treatment or the application of retroreflective adhesives on the bone due to its use of an infrared laser. Data is sampled with 2.5 k Samples for a bandwidth of 1 kHz discretized with 12800 fast Fourier transformation (FFT) lines. This results in a measured time block of 12.8 s and a resolution of 0.1 Hz.

29.4 Results and Discussion The current Dogbone test rig concept, where the pushrod and the shaker retract from a specimen due to the action of two electromagnets, was investigated on a proof-of-concept rig before its full-scale implementation (Fig. 29.8). Two heavy steel L-profiles were clamped to a laboratory bench, and a steel bar measuring 300 × 50 × 5 mm was rigidly clamped to each of these profiles. This bar was first swept with a stepped sine sweep between 50 and 500 Hz to find the first natural frequency of bar and shaker at 138 Hz. A test campaign was carried out to study the influence of different variables in the release, for example, the phase angle at which the offset ramp starts, the shaker boundary conditions, etc. The influence of the shaker and magnet trigger signal was studied, and the signal was made more adaptable by adding two phase parameters. A phase multiplier parameter makes possible starting the offset ramp at angles 0, π/2, π or 3π/2 rad (Fig. 29.9). A phase-tuning parameter, taking in values of 0, π/8, π/4 or 3π/8 rad, contributes to a further fine-tuning of the phase angle at which the offset ramp starts. The shaker release was studied under three different boundary conditions: (1) with the shaker mounted on its base and simply resting on a laboratory bench, (2) with the shaker mounted on its base and resting on two steel cylinders (which

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Fig. 29.8 Proof-of-concept implementation of the Dogbone test rig on a double-clamped stiff test specimen

allowed the movement of the shaker in the axial direction) and (3) with the shaker hanging from two pillars as Fig. 29.8 shows. It was found that hanging the shaker yielded notably cleaner releases on the proof-of-concept rig (Fig. 29.10). The hanging setup was still stiff, but nevertheless it successfully decoupled the shaker from the assembly. The measured response signals required some post-processing. Firstly, given that the Dogbone test rig pursues the measurement of damping as a function of vibrational amplitude, the velocity signal measured by the LDV must be integrated to have a displacement signal. This tends to amplify the signals’ low-frequency components. The measured response was windowed with a tapered Hanning window (also known as a Tukey window or cosine-tapered window) with an alpha parameter of 0.1. In essence, this window attempts to smoothly set the data to zero at the boundaries of the signal while not significantly reducing the processing gain of the windowed transform [10]. A bandpass filter with order 5 between 80 and 200 Hz is also applied as Fig. 29.11 shows. This is necessary to remove the very low-frequency vibration in the proof-ofconcept rig caused by the laboratory bench, which is not rigidly attached to the floor, and to remove higher-order harmonics in the specimen. A peak-to-peak envelope of this decaying signal is calculated (Fig. 29.12a) using a free-domain Python implementation by Bergman [11] of the MATLAB peakdet code created by Billauer [12]. This code requires controlling for the so-called look ahead value and the delta value. The look ahead value is the distance to look ahead from a peak candidate to determine if it is the actual peak and is set to 1 after some trial and error (a value of 1.25 is suggested in [12]), and the delta is kept at a default value of 0. Each pair of consecutive envelope points is used in the calculation of the logarithmic decay with the equations in the project motivation section. Figure 29.12b shows the calculated damping ratio ζ for each point plotted against the vibration amplitude. Little variation of the damping ratio against the vibration amplitude was expected as the specimen used for this proof-of-concept rig is made of structural steel and is rigidly double clamped to the test fixture. However, the data points below 1 in Fig. 29.12b are related to a sudden change in the decrement of the envelope function observable in Fig. 29.12a at a displacement amplitude of about 0.12–0.14 mm. It was felt at this stage that the approach proven in the proof-of-concept rig could be transitioned to the full-scale Dogbone test rig. Figure 29.13 shows an initial implementation of the excitation fixture. An extra fixture to keep the pushrod from dropping after releasing was installed (Fig. 29.14) after a few initial trials. Figure 29.14 shows as well a small structural steel interfacing part between bone and power-to-release electromagnet. The construction material of the bone (AISI 316 stainless steel) is non-magnetic, and therefore, the attachment of this interfacing part is necessary to make the setup work. The interface part is glued to the bone with Loctite 495 adhesive. Screwing both parts would have required drilling the bone, which would create a stress concentration point, decreasing thus the safety factor of the design. A bone built of magnetic AISI 430 stainless steel is being currently manufactured, which will render this interfacing part unnecessary in the future.

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Fig. 29.9 Effect of different phase multiplier values in the offset ramp start

A resonance test was firstly conducted by means of a handheld modal hammer (PCB 086C02) and the PSV-500 on the bone and its root discs to rule out any influences of the rig in the resonance frequencies of the bone. Figure 29.15 shows an overlay of all frequency response functions (FRFs) measured in the test. The low-frequency peaks of these FRFs are related to out-of-plane rigid body motion of the bone and rotation motion in the root discs, while the peaks around 192 Hz and beyond are flexible bone modes. The FRFs that do not present peaks beyond the 30 Hz range are FRFs measured on the root discs. Consequently, there seems to be no significant flexible deflection at the discs, highlighting the decoupling of rig from the bone. The release mechanism proven in the proof-of-concept rig was then employed after loading the bone to 2.2 kN and exciting it with an excitation force of 40 N, but unfortunately the first tests on the full-scale Dogbone test rig did not yield the same quality of release results as the proof-of-concept rig did. At the moment of writing this paper, all datasets present an extremely high-amplitude low-frequency vibration of about 9 Hz after release due to the rigid body rotation of the two discs, which makes a decent extraction of the free decay damping rather challenging (Fig. 29.12, plotted in blue). A high-pass filter set at 30 Hz would remove this undesired frequency (Fig. 29.16, plotted in orange), but this is, however, not ideal since the target is to work on an unfiltered dataset. Even though the low-frequency component could be filtered out, most of the vibration in the root contact will still be dissipated by friction damping related to this low-frequency vibration, not by the first flexible mode of the bone.

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Fig. 29.10 Top − velocity signal measured after releasing the double-clamped beam from the shaker mounted on its base and rolling on cylinders. Bottom – velocity signal measured after releasing the double-clamped beam from the hanging shaker

Fig. 29.11 Freely decaying signal measured after detaching the shaker from the test specimen on the proof-of-concept rig. In blue, the unfiltered displacement signal. In orange, the filtered displacement signal

Fig. 29.12 (a) Envelope of the decaying signal in orange, decaying signal plotted in blue. (b) Amplitude (displacement) plotted versus the damping ratio ζ

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Fig. 29.13 Full-scale Dogbone test rig with the bone assembly mounted with a load of 2.2 kN on the Instron 250 kN tensile/compressive test stand, shaker amplifier, dual DC power source and the PSV-500 SLDV head

Fig. 29.14 Extra fixture installed added to the frame to hold the pushrod from falling after a release

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Fig. 29.15 Overlay of all frequency response functions (FRFs) measured in the resonance test

Fig. 29.16 Overlay of the response in the full-scale Dogbone test rig after an unsuccessful release (blue) and the same signal with a high-pass filter at 30 Hz (orange)

Fig. 29.17 Plot of the signal fed to the electrodynamic shaker and the relays with timestamps at the 3 VDC threshold and the relays’ specified maximum turn-on time

29.5 Further Work The authors are currently striving to solve this issue in order to obtain clean releases from the full-scale rig. It is hypothesized that the retracting action from the shaker and the disengagement of the power-to-release magnet do not happen in a synchronous manner, but there is a small delay in the magnet action. While this issue seemed to be overcome by the larger stiffness of the system in the proof-of-concept test rig, the Dogbone test rig is too flexible to operate in this way, and the bone experiences a strong pull as the shaker is pulled backwards, while the power-to-release magnet is still engaged during this time lapse. The relays have, according to their datasheet [13], a control voltage range of 3–15 VDC. Therefore, they will close the circuit when triggered with 3 VDC or higher. With the current ramp gradient shown in Fig. 29.17, this happens at t = 0.0024 s after an offset ramp starts. A lapse of approximately 0.002 can be as well observed in the time signals measured in unsuccessful releases. The relay datasheet also specifies a maximum relay turn-on time of 75 ms [13], which is represented by the area shaded in red in Fig. 29.14. This time lapse would encompass the entire offset ramp, and the release would take case, in a worst-case scenario, when the shaker has already pulled back from the bone. This accumulated delay does not take into account the demagnetizing action delay of the electromagnet, which cannot be found specified in any datasheet.

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To address this potential release delays, it is necessary to make changes in the way the shaker pullback and the relays are triggered. Shaker and relays must receive signals independent from each other. This cannot be achieved with the DAQ built in the PSV-500 to the authors’ knowledge, as only a single user-defined signal generator is available in the software in its current version. Currently a new setup is being implemented with an external National Instruments cDAQ-9188 equipped with a NI 9263 analogue output module. In this way the excitation and offset signal and the relay trigger signal can be sent independently, with the shaker ramp starting at a time t, while an independent voltage step signal is sent to the relays earlier to close the power circuit at a time t, which is now an adjustable interval. Finding the appropriate gap between relay activation and offset ramp start will be a matter of trial and error if the maximum relay turn-on time and the unspecified magnet demagnetization time are taken into account.

29.6 Conclusions The Dogbone test rig presented in this work is a solution to study the amplitude-dependent, frictional behaviour associated to turbine blade root joints. It tries to overcome the shortcomings that similar setups presented in past investigations, particularly the excitation of the root to representative amplitudes. The innovation in the experimental setup lies in the use of power-torelease and classical electromagnets to detach the electrodynamic shaker and pushrod from the bone after reaching steadystate high-frequency vibration and to allow the test specimen to freely vibrate. This work has described the complexity of such a rig, which is composed of different components working together simultaneously. Initial tests showed a clean release with the proposed system, but currently more work is underway to adapt the setup for the actual full-scale test rig. Acknowledgements The work in this paper is part of a work package in the project Systems Science-based Design and Manufacturing of Dynamic Materials and Structures (SYSDYMATS), funded by the Engineering and Physical Sciences Research Council with grant reference EP/R032793/1.

References 1. Ewins, D.J.: Modal Testing: Theory and Practice, 5th edn, p. 95. Wiley, Chichester 2. Yuan, J., Schwingshackl, C.W., Wong, C., Salles, L.: On an improved adaptive reduced-order model for the computation of steady-state vibrations in large-scale non-conservative systems with friction joints. Nonlinear Dyn. 103, 3283–2200 (2021) 3. Schwingshackl, C.W., Joannin, C., Pesaresi, L., Green, J.S., Hoffmann, N.: Test method development for nonlinear damping extraction of dovetail joints. In: Dynamics of Coupled Structures, vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series, pp. 229–237 (2014) 4. Schwingshackl, C.W., Zolfi, F., Ewins, D.J., Coro, A., Alonso, R.: Nonlinear friction damping measurements over a wide range of amplitudes. In: Proceedings of the IMAC-XXVII, Orlando, FL, 9–12 February 2009 5. Kurstak, E., D’Souza, K.: An experimental and computational investigation of a pulsed air-jet excitation system on a rotating bladed disk. In: Proceedings of the ASME Turbo Expo 2020, 21–25 September 2020 6. Kruse, M.J., Pierre, C.: An experimental investigation of vibration localization in bladed disks. Part I: Free response. In: Proceedings of the International Gas Turbine & Aeroengine Congress & Exhibition, Orlando, FL, 2–5 June 1997 7. Marquina, F.J., Coro, A., Gutiérrez, A., Alonso, R.: Friction damping modelization in high stress contact areas using microslip friction model. In: Proceedings of GT2008 ASME Turbo Expo 2008: Power for Land, Sea and Air, Berlin, Germany, 9–13 June 2008 8. Allara, M., Filippi, S., Gola, M.M.: An experimental method for the measurement of blade-root damping. In: Proceedings of the ASME Turbo Expo, Barcelona, Spain, May 2006 9. Mace, T., Taylor, J., Schwingshackl, C.W.: A novel technique to extract the modal damping properties of a thin blade. In: Topics in Modal Analysis, vol. 8. Conference Proceedings of the Society for Experimental Mecanics Series, pp. 247–250 (2019) 10. Harris, F.J.: On the use of windows for harmonic analysis with the discrete Fourier transformation. Proc IEEE. 66(1) (1978) 11. Bergman, S.: Peakdetect [Online]. Available: https://pypi.org/project/peakdetect/1.1/. Retrieved October 2021 12. Billauer, E.: Peakdet: Peak detection using MATLAB [Online]. Available: http://billauer.co.il/peakdet.html. Retrieved October 2021 13. Schneider Electric USA, Inc.: Legacy Schneider Electric Solid-State Relays – Catalog 2017, p. 20. Version 06/2017

Chapter 30

A Study on Data-Driven Identification and Representation of Nonlinear Dynamical Systems with a Physics-Integrated Deep Learning Approach: Koopman Operators and Nonlinear Normal Modes Abdolvahhab Rostamijavanani, Shanwu Li, and Yongchao Yang

Abstract In this study, we investigate the performance of data-driven Koopman operator and nonlinear normal mode (NNM) on predictive modeling of nonlinear dynamical systems using a physics-constrained deep learning approach. Two physicsconstrained deep autoencoders are proposed: one to identify eigenfunction of Koopman operator and the other to identify nonlinear modal transformation function of NNMs, respectively, from the response data only. Koopman operator aims to linearize nonlinear dynamics at the cost of infinite dimensions, while NNM aims to capture invariance properties of dynamics with the same dimension as original system. We conduct numerical study on nonlinear systems with various levels of nonlinearity and observe that NNM representation has higher accuracy than Koopman autoencoder with same dimension of feature coordinates. Keywords Nonlinear normal modes · Koopman operators · Data-driven system identification · Deep learning · Modal analysis

30.1 Introduction Identifying proper coordinates or modes of dynamical systems is essential to understand and characterize the underlying dynamics for system identifications, controls, and reduced-order modeling for dynamical systems [1]. Although a linear system can be universally represented as a superposition of linear normal or eigen modes (LNMs) using modal transformation, there exists no such a general mathematical framework for representing nonlinear dynamical systems. Seeking some nonlinear generalization of the modal superposition is thus critically needed for more accurate representation and characterization of nonlinear dynamical systems. Koopman operators and nonlinear normal modes (NNMs) are two leading frameworks of such intrinsic coordinates or modes to represent nonlinear dynamics. The challenging part of the Koopman method is obtaining the observable function that transforms nonlinear dynamics to a new state space where the underlying nonlinear dynamics can be approximately represented linearly. Theoretically, it needs infinite dimension to linearize the nonlinear dynamics. NNMs, first introduced by Rosenberg and studied by many following research, on the other hand, is a natural extension of linear normal modes (LNMs). In this study, we investigate and compare the effectiveness of Koopman operators and NNMs in representing the nonlinear dynamics with a physics-integrated deep learning-based data-driven approach. For each representation method, we devise a physics-constrained DNN autoencoder which transforms original coordinates (system response measurements) to identify the intrinsic coordinates (Koopman modes or NNMs) and evaluate their representation effectiveness of nonlinear dynamics by reconstruction and prediction accuracy of the nonlinear system response using different dimension numbers of the identified Koopman modes and NNMs, respectively. We conduct numerical study on nonlinear systems with various levels of nonlinearity and observe that NNM representation has higher accuracy than Koopman autoencoder with same dimension of feature coordinates.

A. Rostamijavanani · S. Li · Y. Yang () Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_30

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Fig. 30.1 Left: Architecture of our physics-constrained deep autoencoder. Original coordinates are transformed to latent coordinates through encoder block, and then they evolve in time with dynamic block, and finally they turn back to original coordinate system by decoder block. Right: The NNMs’ invariant manifolds of a conservative 2-DOF Duffing system identified from response data only using the physics-integrated deep learning approach. In-phase and out-of-phase mode shapes and manifolds. Energy level of different case studies denoted by different colors

30.2 The Effectiveness of Physics Constraints in the Deep Learning Framework We devise our DNN architecture with modal transformation and integrate the physics constraints of NNMs and Koopman into the loss functions for learning our DNNs, respectively, which are physically interpretable. For instance, we have enforced the identified modal coordinates to be independent or decoupled, i.e., they should have a single distinct frequency. In original coordinates, if the system is not initiated with one of the mode shape ratio, then they would have a mixture of modal coordinates. By using correlation loss function, we obtain decoupled modal coordinates for Koopman modes and NNMs.

30.2.1 Identified NNM Invariant Manifolds The identified NNMs of an undamped 2-DOF Duffing system for different energy levels using the NNMs’ DNNs are illustrated in Fig. 30.1. Each energy level corresponds to a specific initial condition, and its NNM manifold is identified by the corresponding NNMs’ DNNs. We represent these identified in-phase and out-of-phase manifolds separately, by using the corresponding pair of latent coordinates which are assumed as modal coordinates for a single-mode reconstruction. When we increase system energy, we can observe higher nonlinearity as the in-phase and out-of-phase mode-shapes change from flat manifolds (linear) to curved manifolds which is consistent with the analytical results.

30.3 Conclusion In this study, we investigate the abilities of the Koopman operators and nonlinear normal modes (NNMs) on representing nonlinear dynamical systems in a data-driven framework without knowing the closed-form models or equations of dynamical systems. For each representation method, we devise a physics-constrained DNN autoencoder which transforms original coordinates to identify the intrinsic coordinates (Koopman modes or NNMs) and observe that NNM representation has higher accuracy than Koopman autoencoder with same dimension of feature coordinates.

Reference 1. Lusch, B., Kutz, J.N., Brunton, S.L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9(1), 1–10 (2018)

Chapter 31

Data-Driven Nonlinear Modal Analysis: A Deep Learning Approach Shanwu Li and Yongchao Yang

Abstract We present a data-driven method based on deep learning for identifying nonlinear normal modes of unknown nonlinear dynamical systems using response data only. We leverage the modeling capacity of deep neural networks to identify the forward and inverse nonlinear modal transformations and the associated modal dynamics evolution. We test the method on Duffing systems with cubic nonlinearity and observe that the identified NNMs with invariant manifolds from response data agree with those analytical or numerical ones using closed-form equations. Keywords Nonlinear normal modes · Invariant manifolds · Nonlinear system identification · Data-driven · Deep learning

31.1 Introduction Nonlinear systems typically exhibit complex dynamical behaviors, characterizing which has remained a long-standing challenge across science and engineering fields. While a linear system can be universally represented as a superposition of normal or eigenmodes using modal transformation that exactly characterizes the underlying dynamical characteristics and enables reduced-order modeling for a wide range of systems, there exists no such a general mathematical framework for nonlinear dynamical systems. Seminal work by Shaw and Pierre extended the definition of normal modes for nonlinear systems with a generalized nonlinear transformation. To date, however, the majority research on identifying the defined NNMs and nonlinear modal transformation has focused on theoretical derivation or numerical computation of the closedform equation of the system or input-output experiment, which, however, is seldom available in practice; usually only the observed or measured system response data are available. Data-driven identification of characteristic modes of nonlinear systems from response data only has thus been widely studied, including those by utilizing advanced signal processing techniques. However, these identified modes typically lack modal invariance property of NNMs. This work focuses on developing a data-driven framework based on physics-integrated deep learning for identifying both nonlinear normal modal transformation functions with nonlinear invariant manifolds and corresponding modal dynamics simultaneously from the system response data only. We conduct a series of study to validate on the conservative and nonconservative Duffing systems with cubic nonlinearity, subject to different levels of excitation energy. It’s observed that the proposed data-driven framework can identify NNMs with invariant manifolds, energy-dependent nonlinear modal spectrum, and future-state predictive model for unknown nonlinear dynamical systems from response data only.

31.2 Physics-Integrated Deep Learning for NNM Identification from Response Data Only We take the NNM definition which was proposed by Rosenberger and extended by Shaw and Pierre [1] as the prior knowledge to devise our deep learning framework for the NNM identification from response data only. The deep learning framework includes a physics-constrained autoencoder and a dynamics coder, as shown in Fig. 31.1a. The encoder and decoder identify the forward modal transformation function and inverse transformation function, respectively. We enforce the transformed

S. Li · Y. Yang () Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_31

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Fig. 31.1 Illustration on a 2DOF Duffing system: deep learning framework for NNM identification from system response only. (a) shows the deep neural network framework consisting of a physics-constrained autoencoder and modal dynamics coder. (b) shows the identified energy-dependent modal frequencies and modal curves. (c) shows the identified invariant manifold of the out-of-phase NNM

coordinates to be NNM coordinates by defining loss functions based on NNM properties such as independence between NNMs to optimize the autoencoder. Meanwhile, we devise a dynamics coder to identify the modal dynamics (i.e., temporal evolution function of identified NNM coordinates) to ensure that the identified NNM space by the autoencoder retains the dynamics of the nonlinear system. By incorporating the autoencoder and dynamics coder, we require the overall network framework to perform future-state predictions.

31.3 Numerical Experiments We validate the proposed deep learning framework for data-driven identification of NNM on 2DOF Duffing system numerical simulation. We use data sets of free response excited by random initial displacements with various amplitudes (energy levels) to train the deep neural networks. The trained encoder is used to transform measured system response (typically consisting of multiple modal components) to modal response. The trained decoder is used to transform each single pair of modal coordinates to the original space separately to achieve the mode separation. With the obtained NNM response in the original space, we can easily identify the corresponding NNM modal curve. Given various energy response data, we can identify the energy-dependent plot (modal spectrum) as shown in Fig. 31.1b. Also, we can identify the invariant manifold by plotting the obtained NNM response in the subspace as shown in Fig. 31.1c.

31.4 Conclusion In this study, we present a data-driven framework based on physics-integrated deep learning to identify the underlying nonlinear normal modes (NNMs) with invariant manifolds and the energy-dependent modal spectrum of unknown nonlinear dynamic systems from response data only. Leveraging the universal modeling capacity and learning flexibility of deep neural networks, the proposed framework first represents and identifies the forward and inverse nonlinear modal transformations through the physically interpretable encoder-decoder structure, generalizing the modal superposition to nonlinear systems.

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Furthermore, to guarantee the correctness of the identified modal transformations, we also incorporate physical constraints to guide the network training, including modal dynamics evolution, generalized NNM properties, and future-state prediction. The proposed deep learning framework is validated on the numerical experiments 2DOF Duffing systems. Acknowledgments This research was partially funded by the Physics of Artificial Intelligence Program of US Defense Advanced Research Projects Agency (DARPA) and the Michigan Technological University faculty startup fund.

Reference 1. Shaw, S., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150(1) (1991)

Chapter 32

Higher-Order Invariant Manifold Parametrisation of Geometrically Nonlinear Structures Modelled with Large Finite Element Models Alessandra Vizzaccaro, Andrea Opreni, Loic Salles, Attilio Frangi, and Cyril Touzé

Abstract In this contribution we present a method to directly compute asymptotic expansion of invariant manifolds of large finite element models from physical coordinates and their reduced-order dynamics on the manifold. We show the accuracy of the reduction on selected models, exhibiting large rotations and internal resonances. The results obtained with the reduction are compared to full-order harmonic balance simulations obtained by continuation of the forced response. We also illustrate the low computational cost of the present implementation for increasing order of the asymptotic expansion and increasing number of degrees of freedom in the structure. The results presented show that the proposed methodology can reproduce extremely accurately the dynamics of the systems with a very low computational cost. Keywords Reduced-order modelling · Invariant manifold · Large-scale FE models · Direct parametrisation

32.1 Introduction Slender structures in large amplitude vibration exhibit geometric nonlinear effects that modify their dynamics. To accurately model structures with complex geometries, large finite element models with a high number of degrees of freedom are often required. Numerically, the geometric nonlinear forces affect all the elements in the structure, thus making the simulation of such problems very demanding in terms of computational cost. Reduced-order models of geometrically nonlinear structures are then an attractive solution to drastically reduce the size of the problem, whilst maintaining the accuracy high.

32.2 Background Direct model order reduction methods, such as implicit condensation and quadratic manifold method, give explicit expressions directly in physical coordinates of the reduced models of geometrically nonlinear structures. Conversely, exact reduction methods are not directly applicable to large models because they work in modal coordinates, which are not available for large models. Recent developments by the authors [1–3] have extended the applicability of exact reduction methods to A. Vizzaccaro () Imperial College London, London, UK University of Bristol, Bristol, UK e-mail: [email protected] A. Opreni · A. Frangi Politecnico di Milano, Milano, Italy e-mail: [email protected]; [email protected] L. Salles Imperial College London, London, UK Skolkovo Institute of Science and Technology, Moscow, Russia e-mail: [email protected] C. Touzé ENSTA Paris – CNRS – EDF – CEA – Institut Polytechnique de Paris, Palaiseau, France e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_32

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large FE structures, by deriving explicit expressions directly in physical coordinates of the reduced models. A similar work is done in [4], where spectral submanifolds are computed directly in physical coordinates.

32.3 Method Mechanical structures in large deformations display geometric nonlinearities in the form of quadratic and cubic restoring forces. When discretised with the finite elements method, the dynamics of such structures can be written as a system of N ordinary differential equations in time, with N number of degrees of freedom: M U¨ + C U˙ + K U + G (U , U ) + H (U , U , U ) = F EXT The idea of using invariant manifolds for model order reduction has been proposed by several authors [5–7] in the context of mechanical systems. The modal coordinates of the system are expressed as a polynomial function of the master coordinates z, which are tangent to the linear modal coordinates of the nonlinear mode of interest. The reduced dynamical model is also built as a polynomial function of the master coordinates in first-order form: z˙ = f (z). The coefficients of the change of coordinates and reduced dynamics are then found by solving the so-called homological equation for each order of the polynomial expansion. Here, the displacement in physical coordinates is directly expressed as a polynomial function of the master coordinates, as U = (z), and the vectors I multiplying each monomial zI = zi1 zi2 ..zip are found by solving a N × N linear system of equations.

32.4 Results In Fig. 32.1, the forced response and the manifold geometry of a large FE model of a MEMS micromirror are captured by the reduced model up to high rotation angles (12◦ ). The dynamics of this structure is particularly challenging due to the nonlinear coupling of multiple modes and due to strong inertia nonlinearities. The expansion up to order 3 is not accurate enough and a higher order is necessary. In Fig. 32.2, the 1:3 internal resonance between the second and fourth mode of a microbeam resonator is perfectly captured by the reduced model consisting in two coupled oscillators. In this case the invariant manifold is 4D.

Fig. 32.1 Forced response of a MEMS micromirror in large rotations in the vicinity of the resonance of its third linear mode. (a) FE model of the mirror and mode shape. (b) Invariant manifold obtained from the parametrisation and orbits of the full model solution. (c) FRFs obtained with full and reduced model for different values of the excitation force

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Fig. 32.2 Forced response of a microbeam in the vicinity of the internal resonance between its first and third mode. (a) Second mode shape. (b) Fourth mode shape. (c) FRFs obtained with full and reduced model at order 9 for different values of the excitation force

Fig. 32.3 Computational time to build a single oscillator ROM from a 3483 DoF FE model for increasing order of expansion (left). Computational time to build a single oscillator ROM of order 3 for increasing size of the FE model (right)

32.5 Computational Cost The cost of model order reduction is composed of an offline phase to compute the invariant manifold and the reduced dynamics and an online phase to solve the dynamics of the reduced model itself. The latter is negligible because, regardless of the size of the original system, the reduced model is either a single oscillator or two coupled oscillators in case an internal resonance occurs. In Fig. 32.3, the offline computational cost to build the model is reported as a function of the order of the expansion and of the degrees of freedom of the original FE model. It is shown that higher orders of expansion are attainable at a very low computational cost, which scales quadratically with the order of expansion p. Moreover, the computational cost of large-scale FE models’ reduction is also extremely low, not only as compared to full-order HB simulations but also as compared to other model reduction techniques.

32.6 Conclusion Nonlinear modes have become an essential tool for engineers in that they provide crucial insights on the dynamics of structures. The computation of nonlinear modes and their associated invariant manifolds for model order reduction is of utmost importance for modern industry. In this respect, the extension of nonlinear mode calculations to large-scale finite element models is a critical issue that will allow the use of this tool in industrial-scale problems. The method proposed here is based on a classical asymptotic expansion of invariant manifolds, but its computation is adapted to the case of FE models

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with a high number of degrees of freedom. The method is a cutting-edge development that will enable a better understanding of the nonlinear dynamics of real-world structures exhibiting large displacements and rotations. The rigourous mathematical foundation of the method coupled with a thoughtful implementation makes it extremely accurate and fast, thus making it the ideal candidate for reduced-order modelling of geometrically nonlinear structures in an industrial framework.

References 1. Vizzaccaro, A., Shen, Y., Salles, L., Blahoš, J., Touzé, C.: Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures. Comput. Methods Appl. Mech. Eng., 113957 (2021) 2. Opreni, A., Vizzaccaro, A., Frangi, A., Touzé, C.: Model order reduction based on direct normal form: application to large FE MEMS structures featuring internal resonance. Nonlinear Dyn. 105, 1237–1272 (2021) 3. Vizzaccaro, A., Opreni, A., Salles, L., Frangi, A., Touzé, C.: High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point. Submitted to Nonlinear Dyn. (2021) 4. Jain, S., Haller, G.: How to compute invariant manifolds and their reduced dynamics in high-dimensional finite-element models? Nonlinear Dyn., 1573–269X (2021) 5. Shaw, S.W., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150(1), 170–173 (1991) 6. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273(1–2), 77–101 (2004) 7. Ponsioen, S., Haller, G.: Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. J. Sound Vib. 420, 269–295 (2017)

Chapter 33

Application of Black-Box NIXO to Experimental Measurements Michael Kwarta and Matthew S. Allen

Abstract Nonlinear identification methods seek to create a mathematical representation of a mechanical system, which can then be used to: (1) predict the structure’s motion or (2) design, redesign or optimize the structure. In a prior work the Nonlinear Identification through eXtended Outputs (NIXO) algorithm was found to work well if the model form is known a priori. Moreover, the black-box NIXO-based algorithm was successful for the data generated numerically. However, when it comes to actual experimental measurements, the black-box identification procedure has proven more challenging. This work builds on the previous efforts seeking to create a black-box NIXO and to demonstrate it on experimental measurements. The identification attempt is performed on a 3D-printed flat beam, and the results are validated against experimental measurements collected during sweep sine vibration testing. Keywords Nonlinear system identification · Nonlinear parameter estimation · Black-box methods · Nonlinear normal modes · NIXO methods

33.1 Introduction The NIXO-based black-box system identification procedure was first introduced in [1, 2]. The authors showed there a successful application of the algorithm to the case studies with signals generated numerically. The objective of this publication is to test this black-box technique with the experimental measurements. Before we share the details from the experimental case studies performed, we would like to provide a brief overview of the black-box NIXO procedure. It consists of three steps: 1. Assuming the (most) general form of the nonlinear equation of motion (EOM) describing the mechanical system (e.g. EOM consisting of every quadratic and cubic term, see Eq. (33.1)). k 2 k k k q¨k + 2ζk ωk q˙k + ω2 qk + α11 q1 + α12 q1 q2 + · · · + β111 q13 + β112 q12 q2 + · · · = Tk f(t)

(33.1)

2. Providing the measured input and output signals to the D1 - and/or D2 -NIXO algorithms with the nonlinear EOM assumed in the previous step. The algorithms return estimates of the frequency response of the underlying linear system as well as the parameters describing the mechanical system’s nonlinearities, as illustrated in Fig. 33.1. 3. Grouping the nonlinear parameters into the dominant and irrelevant sets, where the division is based on the values of two indicators: ∗ and ∗∗ . The inequalities that should be satisfied (simultaneously) by these two parameters are presented in Eq. (33.2). For the detailed description of the -indicators please refer to [1, 2]. 7

∗ < 5% ∗∗ > 95%

(33.2)

M. Kwarta () University of Wisconsin–Madison, Mechanical Engineering Dept., Madison, WI, USA e-mail: [email protected] M. S. Allen Brigham Young University, Mechanical Engineering Dept., Provo, UT, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_33

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Fig. 33.1 System identification process with NIXO. The algorithms use the input and output time series to estimate the linear and nonlinear parts of the equation of motion

Fig. 33.2 (a) Photograph of the experimental set-up. (b) Cross-section of the SolidWorks model of the flat beam; small rectangles illustrate locations of the accelerometers attached to the beam

33.2 Experimental Set-Up and Signals Measured The experimental set-up considered here is shown in Fig. 33.2a. The structure is excited using a Modal Exciter 100 lbf Model 2100E11 powered by a 2050E05 Linear Power Amplifier. The shaker is connected with the backing structure of the beam using a metal stinger. The experimental data is collected with the Polytec software, where the oscillations of: – The point at the beam’s center is measured with the PSV-400 Scanning Vibrometer. – The two points distant by 37.08 mm from each of the beam’s ends (see Fig. 33.2b) are measured with the PCB352C23 accelerometers. The input signals used to excite the structure are multiple 204.8-second-long linear sweep sines of various amplitudes and frequencies increasing from 50 to 150 Hz. The time series of these signals are presented in Fig. 33.3.

33.3 System Identification Results The motion of the beam is modeled with the first two symmetric modes, i.e. modes 1 and 3. Moreover, the authors decided to represent the nonlinear part of EOM with the most general form of the polynomial consisting of the quadratic and cubic terms. Hence, the resulting number of the nonlinear terms that occur in each modal equation is 7, see Eq. (33.3). 1 2 k 1 2 k k k k q¨k + 2ζk ωk q˙k + ω2 qk + α11 q1 + α12 q1 q2 + α22 q2 + β111 q13 + β112 q12 q2 + β122 q1 q22 + β222 q23 = Tk f(t),

(33.3)

where: ωk ζk , k are, respectively, the linear natural frequency, damping ratio, and mode shape of the k-th mode; the quantities qk (t) and f(t) are the time representations of the k-th modal coordinate and the force distribution; α’s and β’s are,

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Fig. 33.3 Time response of the beam measured at its center: (a) displacement (obtained by integrating velocity) and (b) velocity. The signals measured with the two accelerometers are analogous. The NNM curves computed for the equation of motion with the nonlinear part consisting of only the β111 q13 term are overlayed on the time signals (β111 ∈ (1.1, 1.6) × 1013 kg m12 s2 )

Fig. 33.4 Underlying linear system estimated by the NIXO algorithm in the (left) black-box and (right) white-box identification attempts. The FRFs labeled H1 , H2 , and Hv were found by applying the named estimator to low-amplitude, white noise excited measurements Table 33.1 Estimated values of the nonlinear coefficients that satisfy the accuracy criteria specified in Eq. (33.2). The parameters that do not meet the criteria are not shown in this table   β111 estimate kg m12 s2 Min Max Average st. dev. [%] Black-box White-box

D1 NIXO D2 NIXO D1 NIXO D2 NIXO

2.44e13 2.10e13 1.14e13 1.07e13

3.19e13 2.47e13 1.66e13 1.27e13

2.81e13 2.36e13 1.37e13 1.15e13

9.16 5.07 13.58 5.62

respectively, the quadratic and cubic coefficients, and k ∈ {1, 2}. In this work we focus on identification of the nonlinear mode 1 only. Different pairs of signals were provided to the NIXO algorithms in order to identify the structure; this approach worked well for the data generated numerically. The estimates of the underlying linear as well as the nonlinear parts of the system are presented in, respectively, Fig. 33.4 and Table 33.1.

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The NIXO-based black-box algorithm identified β111 q13 as the only dominant term in the nonlinear equation of motion. However, the values of β111 that were identified areapproximately 2–3 times larger than expected. Table 33.1 shows that the  1 13 , while the accurate value of the coefficient most likely belong estimates of β111 belong to set (2.10, 3.19) × 10 kg m2 s2   to (1.10, 1.60) × 1013 kg m12 s2 (see Fig. 33.3). On the other hand, the linear frequency response function is found fairly accurately by both D1 - and D2 -NIXO based techniques. However, in the prior black-box identification we sought to identify all of the system parameters and their values in a single step. If we instead repeat the NIXO identification, while only seeking to identify those terms that black-box algorithm found to be important, we obtain the results labeled “white-box” in Table 33.1  and Fig.  33.4. Using this approach, the estimated 1 13 . However, the resultant linear FRF is less values of β111 are satisfactory since they belong to (1.07, 1.66) × 10 kg m2 s2 accurate than the one obtained in the black-box identification.

33.4 Conclusion and Future Work This brief publication presents the application of the black-box NIXO-based methods to the experimental data. The results presented here are preliminary, yet still show that the algorithms can identify the underlying linear system as well as point out which nonlinear terms should be kept in the equation of motion. However, the estimated values of the nonlinear coefficients are frequently found to be inaccurate when using this approach. Nonetheless, if we extend the 3-step-long procedure (described in the beginning of this work) by adding a fourth step in which NIXO only seeks to identify the values of the dominant parameters, we obtain accurate estimates of the nonlinear parameters. In future work, the authors will apply these algorithms to other structures to see if the proposed approach also works well for different nonlinear systems.

References 1. Kwarta, M., Allen, M.S.: NIXO-based identification of the dominant terms in a nonlinear equation of motion. In: Proceedings of the 39th International Modal Analysis Conference (IMAC), Houston, TX (2021) 2. Kwarta, M., Allen, M.S.: NIXO-Based Identification of the Dominant Terms in a Nonlinear EOM (2021). https://www.youtube.com/watch?v= soGiG_MlTYg. Online video. YouTube. Version 2021. Accessed 17 Oct 2021

Chapter 34

Reliable Damage Tracking in Nonlinear Systems via Phase Space Warping: A Case Study He-Wen-Xuan Li and David Chelidze

Keywords Fatigue damage · Structural health monitoring · Nonlinear vibrations · Phase space warping · Moon beam

34.1 Introduction Phase space warping (PSW) was proposed to resolve hidden slow-time deterministic processes from observable fast-time system responses in a hierarchical dynamical system (HDS). In an HDS, the two categories of dynamics with disparate time scales are assumed to be coupled through slowly varying system parameters. PSW was proposed as a practical algorithm to process sampled field measurement data, which has a lower dimensionality than the total number of system states, and resolve the hidden dynamics through nonlinear time series analysis. Since its introduction, PSW has been used successfully in many engineering applications. These applications range from estimating of slow-time multivariate parameter drifts in electromechanical systems, identifying physiological fatigue, to tracking and characterizing damage evolution in mechanical and biomechanical systems [1, 2]. PSW tries to resolve the underlying slow-time dynamics by investigating how the original high-dimensional fast-time nonlinear dynamical system’s manifold changes over a longer period of time over which the slow subsystem’s effects kick in. Despite its success in the applications mentioned above, there is no systematic discussion regarding selecting the parameters to ensure a reasonably good estimation of the underlying slow-time dynamics. Furthermore, there were six variables involved in the original PSW algorithm, randomly selecting these parameters takes one almost infinite amount of time to figure out a somewhat optimal set of parameters P to make the tracking work. The PSW algorithm has a solid theoretical background but may seem like a gray-box to a black-box data-driven method if one uses it for the first time. Practitioners definitely should not anticipate a random set of parameters to work out, and they may get lost in searching for such a set of parameters. As a primer study, this extended abstract tries to provide a rough strategy to systematically select a set of parameters that yields reasonably good or even optimal tracking results. First, a brief graphical illustration of how the PSW algorithm works is discussed. Then, simulations with various types of fast-time and slow-time dynamics will be conducted to guide parameter selection. These systems are well-known nonlinear oscillators derived from mechanical and electrical systems to make the illustrations approachable. Simulations with known damage dynamics are conducted to further illustrate the power of PSW algorithm when an optimal parameter set is used.

34.2 The Phase Space Warping Algorithm PSW algorithm is well documented in the previous studies. The authors assume that one has obtained the field measurement data from an experimental setup and is ready to use the PSW algorithm to track the slowly evolving damage dynamics. The PSW-based damage tracking is mainly composed of four parts, namely, (1) Data pre-processing, (2) constructing the reference model, (3) estimating the PSW function, and (4) extracting smooth damage trends using smooth orthogonal decomposition. The four steps are visually illustrated in Fig. 34.1.

H.-W.-X. Li () · D. Chelidze Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_34

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Fig. 34.1 PSW algorithm schematic. Data pre-processing: The data is thoroughly visualized, truncated, and filtered, if necessary. Constructing the reference model: Based on delay coordinate embedding (a1), the phase space of the reference system (a2) (where damage is at its initial value φ0 ) is constructed using a set of parameters {τ, d}, delay time, and embedding dimension pair. After embedding the observed data into the reconstructed phase space, the phase space is partitioned into nb number of hypercuboids, each of which can be treated as a local Euclidean space with its coordinate system (local observer). Estimating the PSW function: For a windowed section of the damage-affected data (window is dictated by the rl and nr, the record length and the number of records, respectively), statistical inference using the local linear regression based on the reference model is used to resolve how the current phase flow would evolve as if the system is healthy, see the comparison between the yellow vector e(t) ˆ and the red vector e(t) in (b). For every time step, a fast-time PSWF e(t) ˆ is estimated and then averaged in each of the hypercuboids (spatially on the sectioned attractor) and averaged over the windowed data section. Therefore, there are nb spatially averaged PSWF vectors for each of the windowed data section, each PSWF vector has d independent coordinates, resulting in nb × d coordinates for multivariate damage tracking; and there are nr such windowed sections, representing nr slow-time PSWF vectors in each of the hypercuboid (the slow-time evolution is described by the magenta vectors in the subplot (g)). Finally, a PSWF matrix can be formed by concatenating the nr slow-time snapshots, and it becomes the error matrix E ∈ Rnr×(nb×d) ; a smooth orthogonal decomposition can, then, be applied to the PSWF matrix, which produces the smooth damage evolution time histories. The readers shall refer to the text shown in the subplots as descriptive captions for subplot (a1)–(f)

34.3 A Case Study: Numerical Example of a Stiffened Moon Beam The simulated system carries the idea from the double-well magnetoelastic oscillator, which provides the structural response of an elastic beam that ranges from harmonic motion to chaotic ones [3]. This system has been used for studying damage tracking of structures that undergo irregular responses. As it is shown in Fig. 34.2, the only slow-time variable is assumed to be the degrading stiffness of the stiffened beam. One can express this system in the following form of a hierarchical dynamical system: ⎧ ⎪ ⎪ ⎪x˙ ⎪ ⎨y˙ ⎪θ˙ ⎪ ⎪ ⎪ ⎩˙ φ

=y = −cy − (κ(φ) + α)x − βx 3 − Aω2 cos(θ t) =1 = f (φ, t)

,

(34.1)

34 Reliable Damage Tracking in Nonlinear Systems via Phase Space Warping: A Case Study

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Fig. 34.2 Simulated system for damage tracking and its dynamic behavior. Left: a stiffened Moon’s beam setup; right: a bifurcation diagram that illustrates the range of dynamics of the simulated responses. A damage range of 0–0.5 is used to ensure quasi-chaotic response dominates

where μ = {κ(φ), c, α, β} are treated as system parameters, among which the slow variable is the degrading stiffness κ(φ); here, the damage function is a linear exponential function of time f (φ, t) =. The response x is used to track the hidden dynamics φ(t), and the fast subsystem response is set to chaotic-dominant according to the damage-response seconditerative bifurcation diagram shown in Fig. 34.2. In this particular case study, the parameters are set to κ(0) = 0.5, c = 0.3, α = −1.5, β = 1.5, A = 0.32, θ = 1.3. The simulation takes up to T = 1e4 seconds, where the stationary or quasistationary part of the response is used for bifurcation analysis. Then, time series are obtained and ready for parameter space exploration. To save the computation effort without losing the generality in exploring the parameter space  = {|Pop ∀ practical combinations}, a Sobel set is used to randomly sample the designated uniformly distributed parameters. MonteCarlo simulations are conducted to obtain the PSWF estimates when at distinct subsets of the sampled parameter space.

34.4 Preliminary Results: PSW-Based Damage Tracking For the considered set of system parameters, the optimal set of PSW algorithm parameters ensures a reasonably close tracking to the ground truth, which can be seen in the left subplot of Fig. 34.3. The variation in the tracking quality is induced by distinct record lengths and overlapping ratios. If a non-optimal averaging parameter set is used, the tracking is subjected to large error if a range normalization is considered. The error as a function of the record length and the overlapping ratio is given in the right subplot of Fig. 34.3, indicating shorter record length and about 50% overlapping is preferred for PSW tracking in this particular case. The parameter exploration results indicates the optimal set of parameters are nm = 212 , nb = 25 , nn = 25 , step = 1, rl = 432, and olr = 50%.

34.5 Conclusions This presented work numerically illustrates the power of phase space warping to track the underlying hidden dynamics in a hierarchical dynamical system. When an optimal set of parameters is selected for the PSW algorithm, the tracking is reliable and can be used for damage diagnosis/prognosis purposes. A series of numerical studies may be conducted to provide a more comprehensive strategy in parameter selection for damage tracking purposes.

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Fig. 34.3 Preliminary results of the conducted Monte-Carlo simulation. Left: representative damage-normalized time plot indicating an optimal set of parameters makes PSW accurately tracking the underlying damage variable over the slow time; right: the tracking error as a function of rl and overlapping ratio olr (number of record, nr), showing an optimal averaging strategy prefers shorter record length with a moderate amount of overlapping

Acknowledgement National Science Foundation Grant No. 1561960.

References 1. Chelidze, D., Liu, M.: Multidimensional damage identification based on phase space warping: an experimental study. Nonlinear Dyn. 46(1–2), 61–72 (2006) 2. Chelidze, D., Liu, M.: Reconstructing slow-time dynamics from fast-time measurements. Phil. Trans. R. Soc. A Math. Phys. Eng. Sci. 366(1866), 729–745 (2008) 3. Moon, F., Holmes, P.J.: A magnetoelastic strange attractor. J. Sound Vibr. 65(2), 275–296 (1979)

Chapter 35

Nonlinear Modes of Cantilever Beams at Extreme Amplitudes: Numerical Computation and Experiments Marielle Debeurre, Aurélien Grolet, Pierre-Olivier Mattei, Bruno Cochelin, and Olivier Thomas

Abstract A novel method for the numerical computation of the nonlinear normal modes (NNMs) of a highly flexible cantilever beam is presented. The flexible cantilever is modeled using a 2D finite element discretization of the geometrically exact beam model, wherein geometric nonlinearities relating to the rotation are kept entirely intact. The model is then solved using the proposed solution method, which is fully frequency domain-based and involves a novel combination of a harmonic balance (HBM) Fourier expansion with asymptotic numerical (ANM) continuation for periodic solutions. The NNMs are also calculated experimentally using a flexible cantilever specimen mounted to a shaker table. The experimental NNMs can be compared to their numerical counterparts in order to validate the frequency domain numerical technique. Keywords Nonlinear modes · Geometric nonlinearity · Geometrically exact beam · Finite element analysis · Continuation methods

35.1 Introduction The scientific literature on the subject of cantilever beams is very dense across a variety of different studies. Modern studies on highly flexible beam structures often use the geometrically exact beam model, wherein any geometric nonlinearities related to the rotation of the structure are kept exact. Based on the work of Reissner [1] and Simo [2], existing methods for the simulation of flexible beams often apply a finite element discretization and are typically resolved using time domain numerical schemes [3–5]. In this paper, a novel method is introduced for the numerical simulation of highly flexible beam structures. This method proposes a finite element discretization of the geometrically exact beam model to be solved using a frequency domain numerical scheme as opposed to time domain formulations. The method of solving, rooted entirely in the frequency domain, illustrates the novelty of this approach. The discretized equations of motion are solved through a unique combination of the harmonic balance method (HBM) and asymptotic numerical method (ANM), the latter of which represents a continuation method based on pseudo-arclength parameterization and first mentioned in the work of Damil and Potier-Ferry [6]. This unique scheme for the continuation of periodic solutions has several consequential advantages. Most importantly in the present context, this method allows direct and efficient computation of the nonlinear normal modes (NNMs) of the system. The NNMs in this context are defined as the periodic solutions of the free and undamped dynamical system [7] and represent a visual characterization of the nonlinear mode shape dependence on the amplitude of vibration, also called the backbone curve. They are equivalent to invariant manifolds of the phase space [8, 9]. Several methods exist for their analytical calculation (e.g., see Ref. [10]); their numerical and experimental calculations are described here.

M. Debeurre () · A. Grolet · O. Thomas Arts et Métiers Institute of Technology, LISPEN, HESAM Université, Lille, France e-mail: [email protected]; [email protected]; [email protected] P.-O. Mattei · B. Cochelin Laboratoire de Mécanique et d’Acoustique, Ecole Centrale Marseille, Marseille, France e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_35

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35.2 Analytical Background The analytical background, including the derivation of the geometrically exact model and the subsequent finite element discretization, is outlined in detail in previous work [11] and is therefore largely omitted for brevity. The 2D finite element model, based on a total Lagrangian nonlinear formulation and using Timoshenko beam elements, is written, for all time t: M q¨ + C q˙ + f int (q) = f ext

(35.1)

where M and C represent the mass and damping matrices, respectively, of size 3n × 3n for n × n degrees of freedom; q the vector of the discretized degrees of freedom, of size 3n; fint (q) the nonlinear internal force vector, which implicitly contains the stiffness matrix K = ∂fint /∂q|q = 0 , of size 3n; and fext the external force vector, also of size 3n. The nonlinear modes, however, are defined as the free solutions of the underlying conservative system. In this case, Eq. (35.1) contains no damping C q˙ or forcing fext terms. It is this autonomous system that is solved using the numerical scheme outlined in the next section in order to compute the nonlinear modes of the system.

35.3 Numerical Analysis The numerical procedure for solving the finite element model described above is rooted in continuation of periodic solutions. A two-step, frequency domain formulation for continuation of periodic solutions has been implemented into a custom solver coded in MATLAB. The custom solver, called MANLAB, is capable of tracing the solution(s) of periodic systems by combining the aforementioned two-step procedure: a harmonic balance method (HBM) expansion followed by asymptotic numerical method (ANM) continuation. In the first step, the system unknowns u(t) (the displacement of the cantilever beam, in the present case) are expanded as a Fourier series truncated to a number of harmonics H: u(t) = u0 +

H   c uk cos (kt) + usk sin (kt)

(35.2)

k=1

where u0 , uck , and usk represent the Fourier coefficients of the unknowns u(t). It is known that solving via HBM proves difficult when H is large or if the system contains complex (i.e., non-polynomial) nonlinearities, as is the case with geometric nonlinearities. This difficulty is easily resolved, however, given that the ANM procedure necessitates a unique procedure called the quadratic recast, whereby non-polynomial nonlinearities are “recast” as polynomial nonlinearities of quadratic order or less through the introduction of so-called auxiliary variables. For example, cos(θ ) and sin(θ ) may be redefined as c = cos (θ ) and s = sin (θ ) where c and s represent solutions of linear differential equations [14], yielding the term u1 cos (θ ) as u1 c instead. The HBM is then easily applied to the larger system of simplified equations [12]. Finally, ANM continuation solves the system for u(t) as a function of a varying parameter, called λ. The ANM implements a pseudo-arclength parametrization to compute the branches of solution u(t), a method which is derived in detail in [13, 14]. An appropriate phase condition is applied to the system in order to initialize the solution on the backbone, leading to calculation of the NNMs as λ increases.

35.4 Experiments An experimental setup capable of reproducing the behavior of the flexible cantilever in extreme amplitude vibration is designed using a vibration table. A slender, highly flexible cantilever beam specimen (29.1 cm × 1.3 cm × 0.4 mm, stainless steel) is mounted to a shaker table forced with a periodic signal F(t) = F0 cos (t): ¨ + C u(t) ˙ + Ku(t) + f nl [u(t)] = F 0 cos (t) M u(t) corresponding to the same parameters as defined above (Fig. 35.1).

(35.3)

35 Nonlinear Modes of Cantilever Beams at Extreme Amplitudes: Numerical Computation and Experiments

247

1.2 1 0.8 0.6 [m] 0.4 0.2 0 −0.2 −1

−0.8 −0.6 −0.4 −0.2

0 0.2 [m]

0.4

0.6

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1

Fig. 35.1 (Left) Views of the vibration table used in experiments involving transverse vibration of slender, flexible cantilevers and (right) a numerical simulation of extreme amplitude vibration of a flexible cantilever 1 Amplitude [mm/s]

0.8 0.7 0.6

F0 = 0 01 10

F0 = 0 03 F = 0 05

5

F0 = 0 07

0

F0 = 0 1

0.5

0 3.8

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0.9

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4

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1 0 −1 −2 3.8

Fig. 35.2 and 35.3 Amplitude of the first harmonic of the transverse displacement and NNMs at the tip of a cantilever beam, numerical simulations and experiments. (Left) Numerical forced response and NNMs of a dimensionless cantilever beam. (Right) Experimental forced response and NNMs of the cantilever beam system

Velocity measurements (captured via laser vibrometers) and the forcing of the table are routed through a dSpace MicroLabBox, which integrates a phase-lock loop (PLL) control schematic [15]. The PLL controller governs the behavior of the system, permitting experimental calculation of the nonlinear normal modes (NNMs) based on the work of Peeters, Kerschen et al. [16]. It is hypothesized that the NNMs being defined as free solutions of the underlying conservative system un (t): M u¨ n (t) + Kun (t) + f nl [un (t)] = 0

(35.4)

can be estimated experimentally by controlling the system such that a phase resonance is attained [15, 16]. The phase resonance occurs at a phase difference of π /2 between the relative position un (t) and the forcing F(t), [15] or π if measuring the relative velocity u˙ n (t), a value that the PLL controller “locks” onto before increasing the forcing amplitude in order to trace the backbone curve of the system, equivalent to the NNMs (Figs. 35.2 and 35.3).

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35.5 Conclusion The modeling of highly flexible structures is a current and competitive subject of research, with new methods extending the capabilities of numerical modeling with each passing year. A novel method for the simulation of highly flexible cantilevers in the frequency domain along with an experimental comparison with physical reality is described. The proposed numerical scheme, rooted fully in the frequency domain, offers several advantages when compared with time domain formulations: most importantly, the capability to calculate directly the nonlinear normal modes (NNMs) of the system. Based on a finite element discretization of the geometrically exact beam model, this method is able to simulate the deformation of a flexible cantilever even at extreme amplitudes of vibration. Finally, the numerical results are compared to experiments based on flexible cantilever specimens in order to validate the accuracy of the proposed frequency domain numerical scheme. Acknowledgments This project is part of the THREAD European Training Network “Joint Training on Numerical Modeling of Highly Flexible Structures for Industrial Applications.” This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 860124.

References 1. Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. J. Appl. Math. Phys. 23, 795–804 (1972) 2. Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985) 3. Cardona, A., Géradin, M.: A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Eng. 26, 2403–2438 (1988) 4. Jeleni´c, G., Crisfield, M.A.: Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Computat. Methods Appl. Mech. Eng. 171, 141–171 (1999) 5. Zupan, E., Saje, M., Zupan, D.: The quaternion-based three-dimensional beam theory. Comput. Methods Appl. Mech. Eng. 198, 3944–3956 (2009) 6. Damil, N., Potier-Ferry, M.: A new method to compute perturbed bifurcation: application to the buckling of imperfect elastic structures. Int. J. Eng. Sci. 26, 943–957 (1990) 7. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes. Part I: A useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23, 170–194 (2009) 8. Shaw, S.W., Pierre, C.: Normal modes for nonlinear vibratory systems. J. Sound Vib. 164(1), 85–124 (1993) 9. Touzé, C., Vizzaccaro, A., Thomas, O.: Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dyn. 105, 1141–1190 (2021) 10. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273, 77–101 (2004) 11. Thomas, O., Sénéchal, A., Deü, J.-F.: Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams. Nonlinear Dyn. 86, 1293–1318 (2016) 12. Guillot, L., Cochelin, B., Vergez, C.: A generic and efficient Taylor series-based continuation method using quadratic recast of smooth nonlinear systems. Int. J. Numer. Methods Eng. 119(4), 261–280 (2019) 13. Cochelin, B., Damil, N., Potier-Ferry, M.: Asymptotic-numerical method for Padé approximations for non-linear elastic structures. Int. J. Numer. Methods Eng. 37, 1187–1213 (1994) 14. Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vib. 324(1–2), 243–262 (2009) 15. Denis, V., Jossic, M., Giraud-Audine, C., Chomette, B., Renault, A., Thomas, O.: Identification of nonlinear modes using phase-locked-loop experimental continuation and normal forms. Mech. Syst. Signal Process. 106, 430–452 (2018) 16. Peeters, M., Kerschen, G., Golinval, J.C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330, 486–509 (2011)

Chapter 36

Stability and Convergence Analysis of the Harmonic Balance Method for a Duffing Oscillator with Free Play Nonlinearity Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

Abstract In this work, we determine the quality of the harmonic balance method (HBM) using a single degree-of-freedom forced Duffing oscillator with free play. HBM results are compared to results obtained using direct time integration with an event location procedure to properly capture contact behavior and identify nonperiodic motions. The comparison facilitates an evaluation of the accuracy of nonlinear, periodic responses computed with HBM, specifically by comparing super- and subharmonic resonances, regions of periodic and nonperiodic (i.e., quasiperiodic or chaotic) responses, and discontinuityinduced bifurcations, such as grazing bifurcations. Convergence analysis of HBM determines the appropriate number of harmonics required to capture nonlinear contact behavior, while satisfying the governing equations. Hill’s method and Floquet theory are used to compute the stability of periodic solutions and identify the types of bifurcations in the system. Extensions to multi- degree-of-freedom oscillators will be discussed as well. Keywords Harmonic balance · Piecewise-smooth · Bifurcations · Floquet stability · Convergence analysis

36.1 Introduction The harmonic balance method (HBM) is a well-known and popular method for obtaining periodic solutions of dynamical systems, whether linear or nonlinear, with few or many degrees of freedom (DOF). Its largest advantages over time integration numerical methods include much lower computational costs and the ability to capture unstable solution responses otherwise undetectable even in experiments. Other valuable uses of HBM include computing the nonlinear normal modes (NNMs) of a system [1, 2], bifurcation tracking [3–5], and stability analysis [3, 6]. In the past decade, HBM has seen increased application to problems involving contact, impact, or friction between parts. These kinds of phenomena are non-smooth and can lead to very complex behavior even in simple systems. Further, a larger number of harmonics is often required to obtain converged results than is required for smooth systems. Alcorta et al. [4], for example, used 15 modes/harmonics when analyzing a single DOF system with both cubic and contact nonlinearities, while Detroux et al. [3] analyzed a 37-DOF system and used 5 harmonics in their calculations. Residual error analysis can be used to determine the approximate number of harmonics required to get results accurate to a given error tolerance and to ensure that sub- and superharmonic resonances are not being missed. In this work, we determine the quality of the harmonic balance method for a single degree-of-freedom forced Duffing oscillator with free play [7]. HBM results are compared to time integration results to facilitate an evaluation of the accuracy of nonlinear periodic responses computed with HBM. Super- and subharmonic resonances, regions of periodic and nonperiodic (i.e., quasiperiodic or chaotic) responses, and discontinuity-induced bifurcations, such as grazing bifurcations, are all analyzed to study HBM solution quality. Convergence analysis is performed to find the appropriate number of harmonics

B. E. Saunders () · A. Abdelkefi Mechanical & Aerospace Engineering Department, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected] R. M. G. Vasconcellos Campus of São João da Boa Vista, São Paulo State University, São João da Boa Vista, Brazil e-mail: [email protected] R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_36

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required to capture nonlinear contact behavior for various nonlinearity strengths. Hill’s method and Floquet theory are used to compute the stability of periodic solutions and identify bifurcations. In the final conference presentation, extensions to multi-degree-of-freedom oscillators will also be discussed.

36.2 System Modeling and Numerical Methods The equations of motion for the Duffing free play system as used by deLangre et al. [7] are given by: ⎧ x < −j1 ⎨ Kc (x + j1 ) , α F p c x¨ + 2ωn ζ x˙ + ωn2 x + x 3 + = cos (ωt) , Fc = − j1 < x < j2 0, ⎩ m m m x < j2 Kc (x − j2 ) ,

(36.1)

®

where α denotes the cubic stiffness, Kc is the contact stiffness, and j1 , j2 represent the free play boundaries. MATLAB ode45 with event location was used to provide time integration solutions used as reference data [8] to compare the HBM solution. The harmonic balance works by assuming the solution, nonlinear forcing terms, and external forcing term(s) can all be expressed as a Fourier series with the following general form: h  x cx M¨x + C˙x + Kx + fnl (x, x˙ ) = fext (t), x(t) = √0 + sk sin (kωt) + cxk cos (kωt) 2 k=1

N

(36.2)

The mathematical details are omitted here but are given by Detroux et al. [3]. The result is a system of nonlinear algebraic equations. The piecewise-smooth contact force is regularized with a hyperbolic tangent approximating function [9]. This system is solved iteratively using the Newton-Raphson method. The step size is adaptively controlled within the continuation algorithm using the strategy of Colaïtis et al. [10] and also controlled by the curvature at each solution point.

36.3 Convergence Analysis Figure 36.1a shows the frequency response curves of the system for a moderately hard contact stiffness and a small symmetric free play gap. The colored dots correspond to time integration results for various initial conditions (IC), one color for each IC, and the solid lines correspond to HBM results for 12, 24, and 36 harmonics along with the zero-frequency/DC component. Overall, all three HBM cases capture the majority of the behavior with good agreement including the isolated subharmonic resonance between 15 and 28 Hz. The agreement breaks down at the lower frequencies, though, as more harmonics are required to capture the various superharmonic resonance responses present between 0 and 4 Hz. There also appear to be multiple isolated responses not captured, in addition to chaotic responses which will naturally also not be captured. Figure 36.1b shows the Floquet stability of the previous HBM result curves excluding the subharmonic resonance. Dots indicate unstable regions. Nearly all the unstable regions start and end at vertical tangents which indicate saddle-node bifurcations have occurred. The relative residual error of the three HBM results in the superharmonic resonance region is inset in the upper right and indicates the error is highest in the frequency ranges where amplitude is large enough for contact to occur. Error reduces by a factor of three between each curve (NH = 12 → 24 → 36).

36.4 Conclusions In this work, we presented convergence and stability analyses of a single degree-of-freedom forced Duffing oscillator with free play using the harmonic balance method. We showed that, for this combination of contact stiffness and free play gap size, the majority of the frequency response can be captured with 12 harmonics including the behavior of the isolated subharmonic resonance branch. However, the superharmonic resonances can require many more harmonics in order to be captured. Floquet analysis showed that the system experiences a large number of saddle-node bifurcations at low frequency. The continuation procedure was subject to some persistent difficulties near turning points; the procedure was robust against turning back on

36 Stability and Convergence Analysis of the Harmonic Balance Method for a Duffing Oscillator with Free Play Nonlinearity

251

Fig. 36.1 (a) Frequency response curves of the system comparing HBM with different numbers of harmonics to time integration and (b) frequency response divided into stable and unstable regions with relative residual error also inset

itself, but it was prone to “jumping” to a previously solved portion of a branch and continuing in reverse. A more robust step size control or error control may mitigate this issue. Acknowledgments The authors B. Saunders and A. Abdelkefi gratefully acknowledge the support from Sandia National Laboratories. This study describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the US Department of Energy or the US government. This study is supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, 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 US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2021-12822 C. R. Vasconcellos acknowledges the financial support of the Brazilian agency CAPES (grant 88881.302889/2018-01).

References 1. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes. Part I: A useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23(1), 170–194 (2009). https://doi.org/10.1016/j.ymssp.2008.04.002 2. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes. Part II: Toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009). https://doi.org/10.1016/j.ymssp.2008.04.003 3. Detroux, T., Renson, L., Masset, L., Kerschen, G.: The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Comput. Methods Appl. Mech. Eng. 296, 18–38 (2015). https://doi.org/10.1016/j.cma.2015.07.017 4. Alcorta, R., Baguet, S., Prabel, B., Piteau, P., Jacquet-Richardet, G.: Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances. Nonlinear Dyn. 98, 2939–2960 (2019). https://doi.org/10.1007/s11071-019-05245-6 5. Xie, L., Baguet, S., Prabel, B., Dufour, R.: Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems. Mech. Syst. Signal Process. 88(1), 445–461 (2017). https://doi.org/10.1016/j.ymssp.2016.09.037 6. Lazarus, A., Thomas, O.: A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus Mécanique. 338(9), 510–517 (2010). https://doi.org/10.1016/j.crme.2010.07.020 7. De Langre, E., Lebreton, G.: An experimental and numerical analysis of chaotic motion in vibration with impact. In: ASME 8th International Conference on Pressure Vessel Technology, Montreal, QC, Canada (1996) 8. Saunders, B.E., Vasconcellos, R., Kuether, R.J., Abdelkefi, A.: Relationship between the contact force strength and numerical inaccuracies in piecewise-smooth systems. Int. J. Mech. Sci. 210, 106729 (2021). https://doi.org/10.1016/j.ijmecsci.2021.106729 9. Saunders, B.E., Vasconcellos, R., Kuether, R.J., Abdelkefi, A.: Insights on the continuous representation of piecewise-smooth nonlinear systems: limits of applicability and effectiveness. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06436-w 10. Colaïtis, Y., Batailly, A.: The harmonic balance method with arc-length continuation in blade-tip/casing contact problems. J. Sound Vib. 502, 116070 (2021). https://doi.org/10.1016/j.jsv.2021.116070

Chapter 37

A Physics-Based Modeling Approach for the Dynamics of Bolted Joints: Deterministic and Stochastic Perspectives Nidish Narayanaa Balaji and Matthew R. W. Brake

Abstract Tribomechadynamics is an emerging field seeking to advance the theoretical understanding of the dynamics of jointed structures by considering the influences of physical phenomena at different length scales (micro-level tribology, mesolevel contact, and macro-level geometrical features). The current work presents a fully physics-based modeling approach attempting to make blind predictions of the near-resonant dynamics of a bolted joint through non-intrusive scans of the contacting interfaces aloe (from an interferometer). Drawing inspiration from earlier work, the study presents a novel rough contact modeling approach that considers micro-scale asperity distributions along with meso-scale interface geometry (surface topology) for the contact-constitutive modeling. The developed framework is validated against hammer impact tests conducted on a three-bolted lap joint benchmark structure known as the Brake-Reuß Beam (BRB). The results demonstrate that the current framework can predict the linearized (low amplitude) natural frequencies of the first three modes with an accuracy of less than 0.25%. The model, however, seems to yield relatively inferior predictions when it comes to the nonlinear trends, characterized here as amplitude-dependent resonant frequency and damping. The Rayleigh-Quotient based Nonlinear Modal Analysis (RQNMA) technique is employed for all the simulations. In order to explore the sensitivity of the model to errors/deviations in the parameters assumed (or measured), Polynomial Chaos (PC) based uncertainty propagation studies are undertaken. The factors studied in this stochastic manner are: the micro-level contact model parameters, bolt prestress levels, and deviations in the meso-scale topologies of the contact interfaces. Initial results indicate that the response is most sensitive to the micro-level contact model properties. Finally, the blind predictions of the model are also assessed for the case of a damaged beam, in which case the model performs relatively poorly in the quantitative sense, although capturing the qualitative trends satisfactorily well. It is inferred, from these observations, that further improvements in the micro-scale contact model are necessary in order to develop a modeling approach that is capable of predicting the dynamics of structures with considerably worn/damaged contact interfaces. Keywords Joint dynamics · Rough contact modeling · Uncertainty propagation · Polynomial chaos · Nonlinear modal analysis

37.1 Introduction Dealing with the uncertainty in the frictional properties of a jointed structure has long been a challenging task in the nonlinear structural dynamics community. Although significant advances have been made in the computational modeling of hysteretic models appropriate for contact modeling [1], fixing parameters for these to best represent the dynamics of a real-world jointed structure has proven to be challenging (see, for instance [2] and [3] for a predictive and a calibration approach respectively). Following previous studies [2, 4], the present study seeks to develop a hysteretic description of the contact interactions in a jointed structure through a fully physics-based formulation. Specifically, the approach entails the employment of asperitylevel normal and tangential contact models applied over a rough surface modeled as a large number of asperities with heights distributed statistically. In [2], such an approach was employed to model the dynamics of the Brake-Reuß Beam by first estimating the asperity distribution empirically and employing curve-fits to obtain the surface-level contact properties. Further, the approach also presented a convenient way of distinguish micro- and macro-scale features and their influence on the contact model. Since the developed model is informed by properties coming from such different length scales, this multi-

N. N. Balaji () · M. R. W. Brake Department of Mechanical Engineering, William Marsh Rice University, Houston, TX, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_37

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scale approach is referred to as a Tribomechadynamics Approach. There have been other studies taking a similar approach too, and the interested reader is referred to [4–7]. In the present work, a rough contact modeling approach as described above is adopted, combined with a modified elasticdry friction model with a Hertzian normal contact model applied at the asperity level. In order to understand how the predictions compare with reality, Rayleigh Quotient-based Nonlinear Modes (RQNMs) [8, 9] are computed and compared with nonlinear modal backbones estimated from hammer impact-based ringdown data using the Peak-Finding and Fitting (PFF) method [10]. The Brake-Reuß Beam (BRB), a three-bolted lap-jointed beam structure [11], is used as the benchmark for all the studies in this paper. In addition to the assessment of the nonlinear modal properties, sensitivity of the predictions to a selected number of experimental parameters are studied through uncertainty propagation. For this, Polynomial Chaos Expansions (PCE) of the nonlinear characteristics are estimated using generalized PCE (gPCE) [12]. Sobol indices are calculated from the PCE coefficients to investigate and rank the sensitivities of the factors contributing to the dynamics of the response. They are also used to develop percentile bounds on the modal characteristics to better understand the relative spread of the uncertainties.

37.2 Modeling Approach A finite-element model of the Brake-Reuß Beam [11] (BRB) is developed in ABAQUS. The mesh as well as the detail on the interface are shown in Fig. 37.1. HCB CMS (Hurty/Craig-Bampton Component Mode Synthesis) is next employed to extract the linear mass and stiffness matrices of this model, with the interface-nodal Degrees-of-Freedom (DOFs) kept as retained DOFs. The following subsections describe the contact modeling and uncertainty propagation methodology adopted in the paper. The interfacial mesh was refined until the predicted natural frequency did not vary more than 1–2% for the first mode (around 1.5 Hz for the current model). It is noted that the present methodology allows one to employ relatively coarse interfacial meshes (in comparison to studies such as [13]) due to the statistical nature in which the interfacial properties are employed. The amplitude natural frequency and damping factors are estimated using the Rayleigh Quotient-based Nonlinear Modal Analysis (RQNMA) procedure (see [8, 9] for details). The nonlinear system of algebraic equations that are solved for this are, Ku + fnl (u; θ ) − fs − ωn2 M(u − us ) = 0 ¯ ¯¯ ¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ 1 T 2 (u − us ) M(u − us ) − q = 0, 2 ¯ ¯ ¯¯ ¯ ¯

(37.1a) (37.1b)

where K, M are the linear stiffness and mass matrices; fnl is the nonlinear function parameterized by internal variables theta; f¯¯s , u¯¯s are the static force and displacement vector ¯(from bolt prestress); q is the modal amplitude specification; and ¯ ¯ (u, ωn2 )¯is the nonlinear eigen-pair (eigen-vector and square of natural frequency respectively). This is repeated for a set of ¯

Fig. 37.1 The finite-element model of the Brake-Reuß Beam Benchmark, and the interfacial mesh used for the study

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Fig. 37.2 Overview of the twelve segments with the identified features and the finite-element mesh imposed over it

q values, followed by the cyclic averaging procedure (outlined in [9]) to obtain the dynamic natural frequency and damping factor for each cyclic amplitude level.

37.2.1 Contact Modeling and Surface Feature Processing A Keyence VR-5100 white light interferometer (with a measurement accuracy of about 4 μm1 ) is employed to conduct detailed scans of the interface to develop the contact model. For the present work, the planar resolution of the scans was set to 23.6 μm. Since the work area of the interferometer is not large enough to accommodate the whole BRB interface, the scans were done in twelve separate segment and then stitched together. Using edge and hole identification, the finite-element mesh of the interface is super-imposed on the scan data in a “best-fit” fashion (see chapter 7 of [14] for details). Figure 37.2 shows sample data processing results for one interface. Following this (and some cross-segment adjustments), the data in each element is fit to a bi-linear plane in a Total Least-Squares (TLS) sense. This plane is interpreted as the meso-scale topology of the interface (Fig. 37.3a) and the residuals/deviations from this fitted plane are interpreted as micro-scale features of the asperities. The watershed algorithm is employed (see [15]) to detect the asperities from this data (in each element) and each is fitted in a TLS sense to a 3D ellipsoid. The estimated asperity heights are then fitted to a two-parameter exponential distribution (as in [16]) parameterized by the mean asperity height exponent λ and the reference height z0 . The height exponent is then changed to ensure that exactly a single asperity is in contact (given the number of asperities detected) when the statistical expectation for the same is one (a similar idea was employed in [17]). Figure 37.4 shows the gap function (meso-surface heights), asperity height exponents and mean asperity radii for one of the interfaces under consideration, over each element. Note that no rigid body rotations are applied on the data other than those necessary for stitching the raw data. This is the reason for the slope observed in the gap function. Errors in stage orientation are accounted for in the next section (Sect. 37.2.2). The parameters are estimated over two runs for each interface and, as can be observed, the parameters are strongly correlated. This is also used as an additional outlier rejection strategy to reject outliers (with their data replaced by an average of their surrounding elements’ counterparts). The normal contact is modeled using a Hertzian circular contact model at the asperity level. Conducting the statistical expectation of this model leads to an exponential model (f ∼ eλ(un −gn ) ), with un and gn being the normal interference and static gap function respectively). For the tangential direction, the Cattaneo–Mindlin model [18] is linearized at the asperity level and then the statistical expectation is taken on top of that, leading to an exponential model for the tangential model in terms of normal interference (kt ∼ eλ(un −gn ) ). The tangential force is saturated by setting the slip limit equal to the normal force scaled by a coefficient of friction μ. Note that this is a simplification of the Cattaneo–Mindlin model that is not necessarily physical since linearizing at the asperity level and applying the slip saturation at the rough surface level requires all the asperities to slip simultaneously. This is mainly carried out here as a computational simplification since retaining the original Cattaneo–Mindlin model is unwieldy when it comes to taking the statistical expectation.

1 See

www.keyence.com/products/microscope/macroscope/vr-3000/models/vr-5100/ for detailed specifications.

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Fig. 37.3 (a) Identified Meso-Scale topology of the dataset and (b) a sample fitted rough surface

Fig. 37.4 Contact model parameters estimated from the scan data: (a) Gap function; (b) Asperity Height exponent λ; and (c) the mean asperity curvature radius. Estimates from two repeats of the scans are shown to demonstrate the variability of the parameters

37.2.2 Uncertainty Modeling with Polynomial Chaos Expansion Seven parameters are identified for the uncertainty propagation of the model and are summarized in Table 37.1 below. The parameters related to the contact model above are the coefficient of friction μ, asperity height exponent λ, gap function gap, and mean asperity curvature radius R. Except μ, all these parameters are estimated from TLS fits. Therefore, the mean and variance from these fits are used and they are treated as Gaussian-distributed random variables. Other than this, stage rotation errors θX,Y are modeled as Gaussian-distributed random variables with an arbitrarily large standard deviation of 15◦ . The prestress force applied at the bolt is also taken to have a Gaussian-distribution with the mean coming from strain gauges

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Table 37.1 Summary of uncertain parameters, distributions, and the corresponding quadrature points (exp. refers to experimental; s.d. refers to standard deviation; and b.s. refers to boot-strapped S.No. 1. 2. 3. 4. 5. 6. 7.

Description Coefficient of friction Asperity height exponent Bolt prestress force Stage rotation X Stage rotation Y Gap function Mean curvature radius

Symbol μ λ P θX θY gap R

Distribution Exponential (mean ≈ 0.1183) Normal (fit parameters) Normal (exp. mean, 2100 N s.d.) Normal (0 mean, 15◦ s.d.) Normal (0 mean, 15◦ s.d.) Normal (fit parameters) Normal (b.s. parameters)

Quadrature Gauss-Laguerre Gauss-Hermite Gauss-Hermite Gauss-Hermite Gauss-Hermite Gauss-Hermite Gauss-Hermite

instrumented to the bolts (identical experimental setup as in [2]), which came to around 12,002 N, and a worst-case standard deviation of 2100 N. Fixing a density distribution for the coefficient of friction is generally a challenging problem even for metal-on-metal contacts. In the present work, the Constant Friction Coefficient model (see [4] for discussions) is employed to set the mean value for μ as the ratio of the shear strength and hardness of the material (μ = s/H ), which comes to around 0.1183. An exponential probability distribution is chosen with the above value as the mean. This is considered to be reasonable since most experimental studies yield a friction coefficient between 0–1.00, and the CDF of the distribution evaluates to 99.97% at μ = 1.00. It must be noted that this might be excessively skewed toward μ = 0 in comparison with experimental studies ([19], for instance, reports μ ∼ 0.85), but the spirit of the present investigation is to develop a model that can make blind predictions, so no parameter tuning is performed here. Given the above parameters (referred to as factors henceforth), a functional expansion for the response characteristics, say ωn , is sought, of the form, ωn (q; {μ, λ, P , θX , θY , gap, R}), i.e., parameterized by the factors while retaining the amplitude-dependence on q as a deterministic dependence (not considered for the uncertainty propagation). The generalized Polynomial Chaos Expansion (gPCE) [12] approach allows to write this output as a sum of weighted orthogonal polynomials (orthogonal in the factor-space). The choice of these polynomials may be made based on the distributions of each parameter (see chapter 3 in [14] for a detailed presentation). Several techniques exist for the estimation of the gPCE coefficients, with one class of techniques employing the orthogonality of the polynomials and evaluating expectation integrals numerically, and another class of techniques treating this as a simple weighted regression problem (mathematically however, they can be shown to be similar, see [20], for instance). Here, numerical quadrature is employed with rules chosen based on the distributions of each factor (see Table 37.1) on a rectangular grid. Since the influence of the stage rotations are mathematically identical, only one is retained for the full-factor analysis, which leads to a six-factor PCE study. Five quadrature points are employed for each dimension of the six-dimensional factorspace, leading to a total of 15,625 quadrature points, upon which the full model is evaluated. The results are multiplied by each of the polynomial bases and summed using the appropriate weights to obtain the gPCE coefficients. As already mentioned, the main purpose of developing the gPCEs is to rank the factors based on sensitivity, and Sobol indices [21] are estimated for this purpose (see [22] for another recent study conducting uncertainty propagation using PCE in joint dynamics). One convenience here is since the Sobol indices are merely decompositions of the variance of the outputs (second moments of the gPCEs), they can be evaluated analytically from the gPCE coefficients directly. For constructing uncertainty bounds (say, percentile bounds) on the modal characteristics on the other hand, such an analytical formulation is a non-trivial problem, so the gPCE model is sampled to numerically estimate the bounds.

37.3 Results Applying the developed model on the BRB model shown in Fig. 37.1 allows one to make blind predictions on the statics as well as dynamics of the beam since none of the parameters have been estimated from testing. Figure 37.5 shows the interfacial tractions at the end of a nonlinear static simulation (with bolt prestress being the only input force). The x and y directions are tangential to the plane of the interface and the z direction is normal to it. Interesting contact traction distributions are observed (completely different from flat-interface simulations), which seem to show a lot of traction concentration around machining lay patterns on the interface. All the parameters here are fixed to be the mean values (see Table 37.1).

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Fig. 37.5 (a) Static Tractions: 1, 2, and 3 denote x, y, and z coordinates, and (b) an image of the worn surface after several rounds of testing Table 37.2 Comparison of linearized natural frequency

S.No. 1 2 3

Experimental (Hz) 179.56 594.71 1199.8

Mean Model (Hz) 179.41 594.72 1197.1

Error (%) 0.0845 0.0016 0.2209

Fig. 37.6 (a) First, (b) Second, and (c) Third transverse bending mode shapes linearized about a friction-less “hard”-contact prestress simulation on ABAQUS

Also shown in the figure above is a figure of the interface after several cycles of testing, showing damage in areas that are reminiscent of the regions of contact pressure concentration. The mesh employed for the present study is relatively too coarse for conducting any quantitative investigations into the traction distribution, but this indicates that studies involving finer meshes with a similar formulation could potentially be interesting. Linearizing the system around this state allows one to compute low-amplitude natural frequency predictions with experimental data. Table 37.2 summarizes the results from this and the match between the experimental data and the mean model predictions can be seen to be less than 0.25% for the modes under consideration (see Fig. 37.6). Previous studies [2] showed a deviation as high as 10 Hz in the low-amplitude regime but the results of the current study seem to suggest a significant improvement. Figure 37.6 shows the three bending mode shapes (depicted around a linearized analysis of a rigid-contact analysis on ABAQUS) of interest for the current investigation. Figure 37.7 shows the predictions of the nonlinear backbones of the mean model against the experimental data for the three modes. Looking at the natural frequency trends, it is clear that although the low-amplitude natural frequency has good agreement, the nonlinear trend has not been captured well. Typically the amplitude level at which the drop in natural frequency starts is related to the coefficient and this is thought to be the main issue here. For the damping factor, the linear viscous damping factor (estimated from tests conducted on the parts of the unassembled beams) is added in addition to the contributions from the nonlinear model. Since the friction model here is just a modification of the elastic-dry friction model, the friction model does not dissipate at all until it starts slipping, leading

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Fig. 37.7 Comparison of the modal backbone predictions from the mean model against experimental data for (a) mode 1, (b) mode 2, and (c) mode 3

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Fig. 37.8 Comparison of the modal backbone predictions from the mean model against experimental data for (a) mode 1, (b) 2, and (c) 3, for the damaged beam (only frequency backbones shown)

to an abrupt initiation of dissipation (relative to the experimental data) uncharacteristic of the data. Therefore, the match in damping factor seems to be largely unsatisfactory. As previously mentioned, the low-amplitude match in natural frequency is very surprising. In order to investigate this further, the BRB was subjected to around 12 hrs of continuous steady state testing around the first mode frequency and the beam was disassembled after this after another round of impact testing. The beam’s interface showed clear signs of damage (see Fig. 37.5) and the new scan data was used to build another set of parameters for the mean model simulations (Fig. 37.8). Comparing the predictions from these simulations with the experimental data from the last round of hammer testing showed a mismatch of more than 5 Hz for each mode. Note that the beams used for the first round of testing were freshly manufactured for this study, i.e., that was the first time they were assembled. This seems to indicate that the accuracy of the present modeling framework drops as the joint accumulates damage, showing that the present modeling approach misses some physics at the epistemic level. This is a consistent observation since the modeling approach does not incorporate plasticity or fretting wear, two phenomena commonly attributed to interfacial damage.

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Fig. 37.9 The propagated uncertainty in the modal backbones (left axis) along with the total sensitivity indices (right axis, square-root scale)

37.3.1 Uncertainty Propagation Results Figure 37.9 shows the results from the gPCE uncertainty propagation. The nonlinear modal backbones are plotted along with percentile limits, to indicate the spread of the backbone around the statistical mode of the distribution. Note that this is not related to the mean model (where the parameters values were fixed as their mean values). The percentile limits show maximal spread at larger amplitudes (regimes of stronger nonlinearity). Also plotted in the same plot are the total Sobol indices (in square-root scale). It can be observed that the results are most sensitive to the coefficient of friction, while the second highest sensitivity is shown by the bolt prestress. The other parameters seem to be much less sensitive in comparison. The sensitivity on the coefficient of friction indicates that the asperity-level simplification of the Cattaneo–Mindlin model is perhaps not justified and a more detailed model is necessary for the proper interpretation of the coefficient of friction. What can be said however, though, is that the present model is most sensitive to the tangential model and this show the importance of further research in this area. The sensitivity to bolt prestress, on the other hand, can be interpreted from a purely physical standpoint. Uncertainty in bolt pre-load leads to uncertainties in the loading state of the interface and its large sensitivity is justified. This indicates, also, that care has to be exercised in bolt tightening in predictive studies such as the present one. While the developed model may be very good, if the bolt pretension is not monitored appropriately, the results will not be satisfactory at all.

37.4 Conclusions A novel, predictive tribomechadynamics modeling approach is presented for modeling the dynamics of bolted joints. The present work indicates significant improvements over earlier studies in the same area, improving the prediction of lowamplitude natural frequencies from more than 10 Hz to less than 0.10 Hz for the Brake-Reuß Beam. The predictive capability of the modeling approach is compared for the jointed system before and after it accumulates interfacial damage through persistent testing. The drop in accuracy with damage accumulation demonstrates the importance of damage/wear mechanics in predictive joint dynamics in a quantifiable fashion. Uncertainty propagation studies are conducted using generalized Polynomial Chaos Expansion (gPCE). The study indicates that more research on predictive rough surface tangential contact models will improve the predictability in this area considerably. Further, the importance of bolt prestress monitoring is brought out by the large sensitivity of the modal characteristics to variances in the bolt prestress.

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References 1. Mathis, A.T. et al.: A review of damping models for structures with mechanical joints. Appl. Mech. Rev. 72(4), 040802 (2020). American Society of Mechanical Engineers Digital Collection. ISSN: 0003-6900. https://doi.org/10.1115/1.4047707 2. Balaji, N.N., Chen, W., Brake, M.R.W.: Traction-based multi-scale nonlinear dynamic modeling of bolted joints: formulation, application, and trends in micro-scale interface evolution. Mech. Syst. Signal Process. 139, 106615 (2020). ISSN: 0888-3270. https://doi.org/10.1016/j.ymssp. 2020.106615 3. Balaji, N.N., Brake, M.R.W.: The surrogate system hypothesis for joint mechanics. Mech. Syst. Signal Process. 126, 42–64 (2019). ISSN: 0888-3270. https://doi.org/10.1016/j.ymssp.2019.02.013 4. Eriten, M., Polycarpou, A.A., Bergman, L.A.: Physics-based modeling for partial slip behavior of spherical contacts. Int. J. Solids Struct. 47(18), 2554–2567 (2010). ISSN: 0020-7683. https://doi.org/10.1016/j.ijsolstr.2010.05.017 5. Armand, J. et al.: A multiscale approach for nonlinear dynamic response predictions with fretting wear. J. Eng. Gas Turbines Power 139(2) (2017). ISSN: 0742-4795. https://doi.org/10.1115/1.4034344 6. Pesaresi, L. et al.: An advanced underplatform damper modelling approach based on a microslip contact model. J. Sound Vibr. 436, 327–340 (2018). ISSN: 0022-460X. https://doi.org/10.1016/j.jsv.2018.08.014 7. Porter, J.H., Balaji, N.N., Brake, M.R.W.: A rough contact modeling framework for arbitrarily varying normal pressure. In: Tribomechadynamics Conference. Houston, TX (2021) 8. Balaji, N.N., Brake, M.R.: A quasi-static non-linear modal analysis procedure extending Rayleigh quotient stationarity for non-conservative dynamical systems. Comput. Struct. 230, 106184 (2020). ISSN: 00457949. https://doi.org/10.1016/j.compstruc.2019.106184 9. Balaji, N.N., Brake, M.R.W.: Nonlinear modal analysis through the generalization of the eigenvalue problem: applications for dissipative dynamics. In: Nodycon 2021 (2021) 10. Jin, M. et al.: Identification of instantaneous frequency and damping from transient decay data. J. Vibr. Acoust. 142(5), 051111 (2020). ISSN: 1048-9002. https://doi.org/10.1115/1.4047416 11. Brake, M.R.W., Reuß, P.: The Brake-Reuß beams: a system designed for the measurements and modeling of variability and repeatability of jointed structures with frictional interfaces. In: Brake, M.R. (ed.) The Mechanics of Jointed Structures: Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics, pp. 99–107. Springer International Publishing, Cham (2018). ISBN: 978-3-31956818-8. https://doi.org/10.1007/978-3-319-56818-8_9 12. Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991). ISBN: 978-1-4612-7795-8. https:// doi.org/10.1007/978-1-4612-3094-6 13. Wall, M., Allen, M.S., Zare, I.: Predicting S4 beam joint nonlinearity using quasi-static modal analysis. In: Kerschen, G., Brake, M.R.W., Renson, L. (ed.) Nonlinear Structures and Systems, vol. 1, pp. 39–51. Springer International Publishing, Cham (2020). ISBN: 978-3-03012390-1 978-3-030-12391-8. https://doi.org/10.1007/978-3-030-12391-8_5 14. Balaji, N.: Dissipative Dynamics of Bolted Joints. PhD Thesis. Rice University (2021) 15. Wen, Y. et al.: A reconstruction and contact analysis method of three-dimensional rough surface based on ellipsoidal asperity. J. Tribol. 142, 041502 (2020). ISSN: 0742-4787. https://doi.org/10.1115/1.4045633 16. Polycarpou, A.A., Etsion, I.: Analytical approximations in modeling contacting rough surfaces. J. Tribol. 121(2), 234–239 (1999). ISSN: 0742-4787, 1528-8897. https://doi.org/10.1115/1.2833926 17. Lu, P., O’Shea, S.J.: Mechanical contact between rough surfaces at low load. J. Phys. D Appl. Phys. 45(47), 475303 (2012). ISSN: 0022-3727, 1361-6463. https://doi.org/10.1088/0022-3727/45/47/475303 18. Mindlin, R.D.: Compliance of elastic bodies in contact. J. Appl. Mech. 71, 259–268 (1949) 19. Fantetti, A. et al.: The impact of fretting wear on structural dynamics: experiment and Simulation. Tribol. Int. 138, 111–124 (2019). ISSN: 0301-679X. https://doi.org/10.1016/j.triboint.2019.05.023 20. Sudret, B.: Global sensitivity analysis using polynomial chaos expansions.Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008). Bayesian Networks in Dependability. ISSN: 0951-8320. https://doi.org/10.1016/j.ress.2007.04.002 21. Sobol , I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55(1), 271–280 (2001). The Second IMACS Seminar on Monte Carlo Methods. ISSN: 0378-4754. https://doi.org/10.1016/S0378-4754(00)00270-6 22. Yuan, J. et al.: Propagation of friction parameter uncertainties in the nonlinear dynamic response of turbine blades with underplatform dampers. Mech. Syst. Signal Process. 156, 107673 (2021). ISSN: 0888-3270. https://doi.org/10.1016/j.ymssp.2021.107673

Chapter 38

A Review of Critical Parameters Required for Accurate Model Updating of Geometrically Nonlinear Dynamic Systems Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

Abstract Model updating is an essential part of any dynamic system study. In an ideal situation, an experimental analysis and a corresponding modeling analysis must establish equivalent characteristics for the same dynamic system. This is a tall task, particularly for nonlinear systems where known parameters are surpassed by many unknown parameters crucial for establishing the right model. In contrast, many robust techniques and processes are available for model updating of linear vibration systems. However, the linear updated models typically provide erroneous characteristics when directly employed for nonlinear studies. In this paper, a test structure exhibiting geometric nonlinearity is experimentally studied, and the same is modeled using finite element method (FEM). A study of various parameters involved during the modeling processes such as any assumed parameters and boundary conditions is thoroughly reviewed. The deficiencies of linear model updating processes are highlighted, and suitable workarounds that provide more meaningful and well correlating models for nonlinear dynamic systems are discussed in this paper. Keywords Geometric nonlinearity · Boundary models and update

38.1 Introduction Experimental analysis and modeling the underlying physics are two important methods of analyzing dynamic systems. For linear dynamic systems where the vibratory motion of the system is generally assumed very small, an experimental modal analysis (EMA) [1–6] is performed to establish its modal characteristics such as modal frequency (λ), the modeshape (ψ), and other relevant information. A corresponding model of the same linear vibratory system is used developed using and analytical of finite element methods (FEM), and an eigenanalysis of the system provides its modal characteristics. In an ideal world, the experimentally obtained modal parameters (λx , ψ x ) and the results from a corresponding model (λa , ψ a ) should yield the same characteristics. This however is seldom the case, and to overcome this issue, several model updating techniques may be employed to update one or more parameters that are involved during the modeling process, viz., material properties, boundary properties, geometry, connectivity definition, etc. which are generally assumed a nominal value before any model updating is performed. For nonlinear dynamic systems, a nonlinear modal testing using sine-sweep techniques or closed-loop feedback techniques may be employed to determine the nonlinear modal characteristics [7–12]. Unlike linear systems where it is assumed that modal characteristics are invariant, the modal characteristics of nonlinear systems show large variations with increased vibratory response amplitudes when the external forcing levels are increased. Traditionally, a backbone curve (BBC) is derived from a large family of sine-sweep profiles, where the mode is estimated as that frequency where the response and excitation are in phase-lag quadrature. Conversely, a closed-loop nonlinear modal testing system utilizes the phase-lag quadrature definition of a mode to directly estimate the BBC. Accurately modeling nonlinear systems is a major challenge. In general, a lumped system with limited degrees of freedom may be considered, and a well-established technique such as the harmonic balance method [10, 13] is applied to obtain the nonlinear characteristics of the system. For complex system, a nonlinear reduced-order modeling (NLROM) is adopted where a large model is reduced two a system with limited degrees of freedom, and then a harmonic balance method is adopted to obtain the nonlinear characteristics [14–18]. For a thorough and comprehensive analysis, the full model without

M. Nagesh () · R. J. Allemang · A. W. Phillips Department of Mechanical Engineering, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_38

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any reduction techniques is required to understand the nonlinear characteristics. However, unlike a linear system where an eigenanalysis provides a direct estimation of the modal characteristics, a nonlinear system requires sophisticated techniques and formulation in order to obtain the backbone curve and other relevant information. A general framework for updating linear dynamic systems is described in [10, 19]. Here, the inverse eigensensitivity method (IESM) [20–22] is used to update a linear FE model using a two-step approach. First, the individual components of the system are analyzed in free-free condition (experimental and modeling), and Young’s modulus is updated. The second step involves formulating the clamped boundary region as a combination of translational and rotational springs of large stiffness values. The IESM is employed to update these high stiffness values and hence a FE model of the full system whose characteristics agree well with experimentally estimated data [10, 19]. The IESM method uses the experimental modal characteristics as a reference which is in turn estimated by a mathematical formulation of the frequency response function (FRF). FRFs are force normalized and hence the exact force provided to the system is unavailable for any updating purposes. Moreover, the linear behavior assumption of the system does not demand knowing exact forcing values for broadband multimode excitation during EMA. As a consequence, the initial and updated boundary stiffness estimated using this method is highly prone to errors and may fail to capture the actual stiffness of the high stiffness springs that are required to model the true nature of the nonlinearities in the dynamic systems. This inability to accurately estimate the boundary stiffness values majorly impacts the modeling and analysis of nonlinear dynamic systems. In this paper, the FE modeling of a dynamic system exhibiting geometric nonlinearity is studied with particular emphasis on the various clamped boundary models along with other assumptions and updates performed for material properties. A comparison is provided between idealized clamping boundary assumptions, updated boundary stiffness using the IESM technique, and any further updating and corrections issues required to model the nonlinear dynamic system accurately.

38.2 Experimental Analysis and Test System Description A T-structure shown in Fig. 38.1 is considered for experimental analysis in this paper. The structure comprises of a thick cantilevered steel beam (ν = 0.3) with its free end attached to two thin double-clamped steel beams (ν = 0.3) on either sides with specifications described in Table 38.1. The entire test structure is mounted on a frame described in [10, 23]. A multiple reference impact test (MRIT) using miniature impact hammer is performed on the individual beams and the assembled T-structure for EMA. The first combined bending mode of the full structure, which is a combination of the first bending mode of the cantilevered beam and the double-clamped beams, is chosen for detailed study. The mode shape of this mode at 44.301 Hz is shown in Fig. 38.2. Fig. 38.1 T-structure exhibiting geometric nonlinearity

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Table 38.1 Individual beam specifications and properties

Dimensions | | ( | ) |

Beam Type

Cantilevered Beam

.

|

.

Estimated Density ( )

Beam Mass ( )

Nominal Young’s ) Modulus (

Updated Young’s Modulus ( )

| .

7813

300.49

200

210.44

DoubleClamped Beam

.

|

.

| .

7830

71.2

200

198.78

DoubleClamped Beam

.

|

.

| .

7911

71.9

200

200.57

Fig. 38.2 First bending mode of T-structure

For the nonlinear modal testing, a modal shaker of adequate stroke length is attached to location at the junction of the beams. This shaker attachment along with a force sensor slightly decreases the first mode frequency to 42.45 Hz. A software phase-locked loop (sPLL) previously described in [11, 12] is used to obtain the BBC for the first bending mode. Along with the BBC, frequency response curves (FRC), which are not force normalized but rather the direct response values for sinesweep across the considered mode, are also presented as shown in Fig. 38.3. The FRC information is useful for estimation and assessment of damping models for the test structure system.

38.3 Finite Element Modeling: Assumptions and Limitations An FE model of the structure is developed using 3D beam elements as shown in Fig. 38.4 using both MATLAB [24] and ANSYS Mechanical [25]. The linear model and model updating is performed using MATLAB, and the nonlinear modeling is performed using ANSYS Mechanical. A nominal material model of values presented in Table 38.1 is initially assumed

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Fig. 38.3 Experimentally determined BBC and FRC: Raw acceleration data at location 10

Fig. 38.4 FE Model of T-structure with individual beams and boundary descriptions

for the individual constitutive beams, which is updated using the IESM method described in the framework provided in [19] for free-free analysis. A 6-DOF spring system in each of [x y z θ x θ y θ z ]T directions is attached to nodes in the clamping region, and the stiffness values of these springs are updated using the linear model updating process. The modal parameters estimated from the updated linear FE model correlates well with the experimentally determined modal parameters. For damping modeling, a viscous damper is attached to the junction of beams for both linear and nonlinear analysis. While the linear (infinitesimal) strain theory holds valid for linear analysis, a finite strain theory (Green-Lagrange/PiolaKirchhoff) model is used for nonlinear analysis. For nonlinear modeling, a direct method to estimate the nonlinear modal frequency or BBC does not exist. Instead, a sine-sweep (linear chirp) analysis across the mode is considered for modeling at varying forcing and damping values. The sine-sweep forcing is applied to the node at the junction of beams, corresponding to the actual forcing applied to the system. By the normal mode (nonlinear) definition, a modal frequency is defined by the

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Table 38.2 FE modeling case descriptions Case name NOM NLUP BC 1

Parameters used Young’s modulus Nominal Experimentally updated

BC 2 BC 3

Clamping region boundary conditions Ideal clamping boundary on all beams with [x y z θ x θ y θ z ]T = [0 0 0 0 0 0]T Ideal clamping on cantilevered beam 6-DOF spring boundary on double-clamped beam 6-DOF spring boundary on cantilevered beam Ideal clamping on double-clamped beam 6-DOF spring boundary on all beams

Fig. 38.5 BBCs obtained from FE model – clamped boundary modeled as ideal at location 10

phase-lag quadrature between response and forcing. For the various forcing and damping values used in the sine-sweep, the locus of all such quadrature is utilized to obtain characteristics of the underlying conservative system, and hence the BBC is obtained. For the nonlinear analysis using sine-sweep forcing, several cases listed in Table 38.2 are considered, where each case is a unique combination of several assumptions in material characteristics and boundary properties and in some instances includes results from the linear model updating performed on the entire rig using the framework mentioned in [19]. The results are also compared against a Duffing equivalent model of the T-structure available in [11]. While the Duffing equivalent modeling shows a good correlation as shown in [11], analyzing the entire model highlights several issues with assumptions made during the modeling process and the linear model updating. Figure 38.5 shows a comparison between the experimentally determined BBC and those obtained from assuming an ideal clamped boundary condition for the cantilevered thick beam and both thin double-clamped beams, at the junction of the beams (location 10). The underlying FE model in both cases indicates that an ideally clamped boundary provides a model with greater overall stiffness characteristics than the experimental test structure. The comparison also highlights the importance of accurately estimating Young’s modulus values for the individual constituent beams and its importance FE models which correlate better with experimental data, particularly at smaller response amplitudes. Having established the importance of updating material characteristics, Fig. 38.6 shows a comparison of a combination of boundary modeling using the 6-DOF spring at clamping nodes with values updated using IESM. The comparison in Fig. 38.6 shows a severe underestimating of the boundary springs’ stiffness values, while highlighting the importance of using the 6-DOF spring clamping region description for the beam clamping regions, particularly for the springs attached to the double-clamped thin beams since the geometric nonlinearity characteristics of the structure are determined by these beams. To obtain a FE model that correlates well with the nonlinear BBC, a trial-and-error approach is adopted to obtain a suitable value of a multiple M that is multiplied with the spring stiffness values of all the 6-DOF springs. It is found that M = 51 gives the best FE

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Fig. 38.6 BBCs obtained from FE model – clamped boundary modeled as 6-DOF spring at location 10

Fig. 38.7 Fully updated FE model with BBC and FRC correlation at location 10

model that has an underlying conservative system characteristic that correlates well with the BBC as seen in Fig. 38.7 [10]. provides a detailed analysis of the spatial response correlation criterion used to assess the quality of the nonlinear FE model at all spatial response locations located along the test structure for a sine-sweep-type analysis. Figure 38.7 also shows that although a viscous damping model is suitable to obtain the BBC using the sine-sweep analysis, it fails to capture the responses accurately at the higher forcing levels.

38.4 Conclusions and Future Work Through this paper, a procedure for updating a FE model of a structure exhibiting geometric nonlinearity is studied in detail. The procedure involves an initial linear model updating step using a well-established EMA procedure, and the IESM capable of updating specific parameters that are critical to the modeling process. The paper also establishes the

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importance of updating some parameters such as material characteristics, etc. which are of particular importance to the nonlinear modeling of the structure. It is also established that ideal description of boundaries such a clamped boundary is inaccurate and do not capture the conservative characteristic of the dynamic system accurately. A 6-DOF spring per node attached to all nodes describing a clamped region with adequate updating through IESM and further updating through a trialand-error process is required to obtain a BBC that correlates well with experimentally determined characteristics. Through this paper, the potential advantages and disadvantages of a viscous damping model are understood, and the need for a more detailed description of damping for accurately obtaining FRCs from the FE model that correlate well with the experimentally determined characteristics is also highlighted.

References 1. Phillips, A.W., Zucker, A.T., Allemang, R.J.: A comparison of MIMO-FRF excitation/averaging techniques on heavily and lightly damped structures. In: 17th International Modal Analysis Conference; Kissimmee, FL, USA, pp. 1395–1404 (1999) 2. Phillips, A.W., Allemang, R.J.: An overview of MIMO-FRF excitation/averaging/processing techniques. J. Sound Vib. 262(3), 651–675 (2003) 3. Allemang, R.J., Phillips, A.W.: The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proceedings of the 2004 International Conference on Noise and Vibration Engineering, pp. 2373–2401. ISMA (2004) 4. Allemang, R.J.: The modal assurance criterion – twenty years of use and abuse. Sound Vib. 37(8), 14–21 (2003) 5. Allemang, R.J.: UC SDRL-CN-20-263-662 Vibrations: Analytical and Experimental Modal Analysis. University of Cincinnati (2013) 6. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. Wiley (2006) 7. Müller, F., et al.: Comparison between control-based continuation and phase-locked loop methods for the identification of backbone curves and nonlinear frequency responses. Nonlinear Struct. Syst. 1, 75–78 (2021) 8. Renson, L., Gonzalez-buelga, A., Barton, D.A.W., Neild, S.A.: Robust identification of backbone curves using control-based continuation. J. Sound Vib. 367, 145–158 (2016) 9. Renson, L., Kerschen, G., Cochelin, B.: Numerical computation of nonlinear normal modes in mechanical engineering. J. Sound Vib. 364, 177–206 (2016) 10. Nagesh, M.: Nonlinear modal testing and system modeling techniques. PhD dissertation, University of Cincinnati, Cincinnati (2021) 11. Nagesh, M., Allemang, R.J., Phillips, A.W.: Characterization of nonlinearities in a structure using nonlinear modal testing methods. In: Nonlinear Structures & Systems, vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series 12. Nagesh, M., Allemang, R.J., Phillips, A.W.: Challenges of characterizing geometric nonlinearity of a double-clamped thin beam using nonlinear modal testing methods. In: Nonlinear Structures & Systems, vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series 13. Bernárdez, A.S.: Nonlinear Normal Modes in Mechanical Systems: Concept and Computation with Numerical Continuation. Universidad Politecnica de Madrid (2016) 14. Van Damme, C.I.: Model correlation and updating of geometrically nonlinear structural models using nonlinear normal modes and the multiharmonic balance method by. PhD dissertation, University of Wisconsin-Madison (2019) 15. Kwarta, M., Allen, M.S.: Nonlinear normal mode estimation with near-resonant steady state inputs. Mech. Syst. Signal Process. 162(2021), 85–88 (2021) 16. Van Damme, C.I., Allen, M.S., Hollkamp, J.J.: Evaluating reduced order models of curved beams for random response prediction using static equilibrium paths. J. Sound Vib. 468, 115018 (2020) 17. Kuether, R.J., Deaner, B.J., Hollkamp, J.J., Allen, M.S.: Evaluation of geometrically nonlinear reduced-order models with nonlinear normal modes. AIAA J. 53(11), 3273–3285 (2015) 18. Spottswood, S.M.: University of Cincinnati. University of Cincinnati (2006) 19. Nagesh, M., Allemang, R.J., Phillips, A.W.: Finite element (FE) model updating techniques for structural dynamics problems involving nonideal boundary conditions. In: Proceedings of ISMA 2020 – International Conference on Noise and Vibration Engineering and USD2020 – International Conference on Uncertainty in Structural Dynamics, pp. 1937–1949 (2020) 20. Lin, R.M., Lim, M.K., Du, H.: Improved inverse eigensensitivity method for structural analytical model updating. J. Vib. Acoust. Trans. ASME. 117(2), 192–198 (1995) 21. Lin, R.M., Lim, M.K., Du, H.: A new complex inverse eigensensitivity method for structural damping model identification. Comput. Struct. 52(5), 905–915 (1994) 22. Canbalo˘glu, G., Özgüven, H.N.: Model updating of nonlinear structures from measured FRFs. Mech. Syst. Signal Process. 80, 282–301 (2016) 23. Pandiya, N.: Design and validation of a MIMO nonlinear vibration test rig with hardening stiffness characteristics in multiple degrees of freedom. MS thesis, University of Cincinnati, p. 133 (2017) 24. MATLAB: 9.7.0.1190202 (R2019b). The MathWorks Inc., Natick (2018) 25. Ansys® Academic Research Mechanical, Release 2021 R1 (2021)

Chapter 39

Magnetic Excitation System for Experimental Nonlinear Vibration Analysis Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

Abstract Traditional modal shakers are generally unsuitable for direct modal testing of extremely lightweight and thin structures. For small structures, a base excitation technique may be applied but the technique has many disadvantages. In this paper, a novel magnetic excitation system is explored for obtaining linear and nonlinear modal characteristics of an extremely thin double-clamped beam. The linear modal testing uses established experimental modal analysis techniques, and a software phase-locked loop (sPLL) system is employed for nonlinear modal testing of the double-clamped beam. Comprehensive results and challenges associated with the magnetic excitation system are discussed in this paper. A comparison is also provided to highlight the advantages of a magnetic excitation over traditional modal testing based on an earlier similar work. Keywords Nonlinear modal testing · sPLL · Magnetic excitation

39.1 Introduction Experimentally estimating dynamic characteristics is an important step in determining functionality and optimum design of mechanical systems. For general linear vibratory systems, an experimental modal analysis (EMA) is performed on the system to estimate its modal parameters, viz., modal frequencies (λ) and eigenvectors (ψ) along with other relevant information. EMA is generally classified into phase-resonance techniques and phase-separation techniques. The former has historical significance where modal shaker(s) are used for exciting one mode at any given time using a sinusoidal signal in order to study and estimate modal parameters. In contrary, most modern linear EMA techniques are formulated based on a phaseseparation approach where a broadband simultaneous multimode excitation using a multiple-input multiple-output (MIMO) configuration is possible. Many subsequent mathematical formulations that are used to estimate the modal parameters in each case are briefly described in [1–4]. Hardware used in linear EMA typically consists of impact hammers or modal shakers that are used for excitation of dynamic systems (external forcing), and response is usually measured using accelerometers, lasers, strain gages, etc. to estimate the frequency response function (FRF). Performing multiple reference impact test (MRIT) using impact hammers is a quick technique and has minimum leakage issues associated with it, but the input force cannot be controlled. This is disadvantageous particularly when nonlinearities are present in the dynamic system. In addition, impact hammers are generally disadvantageous when systems have significant damping, since the short decay time provides poor frequency resolution. On the other hand, modal shakers are used to excite structures using a wide range of excitation signals that may be random, periodic, single frequency, or any combination that can provide a good estimate of the modal parameters. These shakers however require direct attachment to the test structure along with force sensors which can cause significant changes in its dynamic characteristics, particularly for extremely thin and lightweight structures. Modal shakers of adequate stroke length are also increasingly employed in nonlinear modal testing where a closed-loop control system is generally employed to obtain nonlinear characteristics such as backbone curves (BBC), amplitude-frequency dependence, etc. of a nonlinear dynamic system. While the use of modal shakers of adequate stroke and power rating for nonlinear modal testing has seen popular application, the system cannot be directly employed for those structures which are extremely thin and lightweight as shown in [5]. To overcome this limitation, during nonlinear modal testing of smaller beams or similar structures, a rigid heavy base,

M. Nagesh () · R. J. Allemang · A. W. Phillips Department of Mechanical Engineering, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_39

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which is in turn connected to a modal shaker in a manner not to alter dynamic characteristics of the test system, is popularly studied. This base excitation generally provides a uniform loading to the test structure, and further analysis can hence be performed as seen in [6–12]. Larger structures however cannot be studied experimentally using this base excitation because it is impossible to develop a very large base of adequate stiffness and modal characteristics that does not alter the underlying dynamics of the test structure. In addition, many nonlinear modal testing applications may demand a point load being applied to the test structure instead of a uniform loading which cannot be accommodated in the base excitation case. In order to overcome some of the inherent limitations of modal shakers and their applications to nonlinear modal testing, it is implicit that a noncontact excitation technique capable of addressing both linear and nonlinear EMA and testing requirement is needed to understand dynamics of lightweight and thin structures. A magnetic excitation technique, previously described for linear EMA in [13, 14], is detailed in this paper as a technique suited for both linear and nonlinear modal testing. The advantages of the technique over traditional EMA hardware, along with some challenges that require adequate understanding and addressing, are also discussed.

39.2 Magnetic Excitation System: Working Details Force exerted by an electromagnet on a magnetic body placed in its vicinity is directly proportional to the square of the electrical current passing through it and inversely proportional to the square of distance between the electromagnet and the body [15]. This property of electromagnets can be used for excitation of thin and lightweight dynamic systems. If the test system is magnetic by nature, two electromagnets of opposite polarities can be used in tandem to obtain a constant force field region for excitation of the system as shown in Fig. 39.1. For excitation of nonmagnetic test structures, extremely small neodymium magnets can be installed on the test structure along with electromagnets of similar polarities to obtain a similar constant force field region. The combined pair of two of such electromagnets provides a point loading on the test structure which can be used for linear and nonlinear modal testing purposes. This technique is hence termed as magnetic excitation technique. The magnetic excitation used in this paper comprises of a Honeywell Industrial VRS magnetic speed sensor (model 3010S20) as the electromagnet. A voltage source similar to those used for modal shakers is used for magnetic excitation, along with an IRD MB-100 audiophile monoblock amplifier to provide a high current to the electromagnets and hence generate adequate forcing. Instead of direct force measurements, the force induced by the electromagnet is measured in terms of a voltage drop measured across a resistor connected in series with the electromagnet. This voltage measured across the resistors provides phase information that is consistent with the excitation signal. The electromagnets can be suitably positioned as shown in Fig. 39.2, and the initial distance between the test structure and the electromagnets may be adequately adjusted to provide the required forcing. From Fig. 39.2, it is observed that the magnetic excitation system along with its

Fig. 39.1 Magnetic excitation system for magnetic (steel) double-clamped beam

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Fig. 39.2 Thin double-clamped beam and magnetic excitation system at UC-SDRL

mounting and positioning structures does not alter any dynamics of the test specimen considered unlike traditional modal shakers or base excitation techniques. The excitation voltage and other relevant information for linear and nonlinear modal testing are detailed in the later sections.

39.3 Test Structure and Modal Testing Procedure A double-clamped steel beam of geometry shown in Fig. 39.3 is considered for both linear and nonlinear modal testing using magnetic excitation systems. The double-clamped thin beam readily exhibits geometric nonlinearity, and the variation of its modal frequency with response amplitudes is investigated under nonlinear modal testing. The beam is fixed to a frame structure detailed in [1, 16], whose first natural frequency is around 400Hz. For the experimental analysis in this paper, only those modes occurring well below this frequency are only considered. EMA is performed for this beam under extremely small forcing levels where the response of the beam is considered to be generally linear. All acceleration data is recorded using PCB Piezotronics model 352C23 miniature wheat grain accelerometers. Seven accelerometers are spread across the length of the beam as shown in Fig. 39.3. The modal parameters for the linear experimental modal analysis are obtained using the RFP-Z algorithm available in UC-SDRL’s X-Modal software program. The magnetic excitation uses a pure random signal for linear EMA where the response amplitudes are extremely small. Both single-input multiple-output (SIMO) and MIMO analysis are performed as part of the linear tests with the varying distances between the magnets and the beam structure as shown in Table 39.1. For the same beam structure, a MRIT is performed using PCB Piezotronics model 086D80 miniature impact hammer with impacts at location 2 through location 18 as shown in Fig. 39.3. A comparison is made between the FRFs obtained from the MRIT and the electromagnetic

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Fig. 39.3 Double-clamped beam geometry and details Table 39.1 Magnetic exciter configuration details Configuration type Config 1 Config 2

Distance of magnetic exciter(s) from double-clamped beam (mm) 2.38 3.175

Input voltage (V) 0.5 0.5

Test type SIMO, MIMO SIMO, MIMO

  where excitation. It must be noted that the MRIT using the miniature impact hammer provides FRFs of units response N   m   m s2 or displacement X the response is generally acceleration FA = N F = N . In case of the magnetic excitation, there is no   direct measurement of force and hence the FRF has units response . V For the nonlinear modal testing, a software phase-locked loop (sPLL) previously discussed in [1, 5, 17] is used to estimate the backbone curve (BBC) for both the first and second lateral bending modes. In addition, a frequency response curve (FRC), where the response is not normalized with force, is also generated at various excitation voltage levels. A sine-sweep is performed across the modes being considered, and the recorded FRC characteristics are presented in the subsequent sections. It must be noted once again that since a direct measurement of force is not available in magnetic excitation systems; all data is presented in terms of the input voltage of the sinusoidal signal provided to the magnetic excitation system. As mentioned earlier, the force generated by an electromagnet is inversely proportional to the square of distance between the electromagnet and the magnetic material upon which the force is generated. In case of linear EMA, since the amplitudes are extremely small, it can be assumed that the distance between the electromagnet and the test beam structure generally does not show significant variations that could alter the force generated on it by the electromagnets. This assumption is invalid in nonlinear modal testing since, by definition, larger amplitudes of vibration are under investigation. Although the dualmagnet configuration supports a constant force field between the two magnets, the varying distance between the magnets and the beam structure causes significant variations in the force generated on the beam structure. Therefore, the FRC generated during sine-sweep across the modal frequency under investigation provides profiles whose input voltage only is known unlike traditional FRCs generated using sine-sweep where each FRC represents data at a constant forcing level [8–10, 12, 17].

39.4 Results: Linear Experimental Modal Analysis Several distinct FRFs are considered (Hinput, output ) for comparison of linear EMA results from both the MRIT and magnetic excitation methods. SIMO and MIMO FRFs for the two configurations of distance between the beam and magnets are considered as shown in Table 39.1. For the experimental analysis of predominantly the first bending mode, the MIMO configuration consists of two pairs of magnetic exciters located at location 9 and location 11, whereas the SIMO case has a pair of magnetic excitation system at location 11 only. In case of experimental analysis of the second bending mode, a config 2 placement of the magnetic excitation system at location 13 is considered. The units for FRFs obtained from the MRIT and magnetic excitation have units described in the previous section. Table 39.2 shows the results of the first and second natural frequencies of the double-clamped beam in lateral bending estimated from both the MRIT and magnetic excitation systems. It is evident from the FRFs shown in Figs. 39.4, 39.5 and 39.6 that the natural frequencies from the FRFs obtained from MRIT and the magnetic excitation system align well. A similar comparison can be made for excitation of predominantly the second bending mode as shown in Fig. 39.7. The FRFs obtained from the magnetic excitation are of magnitudes several

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Table 39.2 First two lateral bending modes of double-clamped beam – experimental modal analysis results Later bending mode type First bending mode – Symmetric Second bending mode – Antisymmetric

Natural frequency (Hz) 45.2 107

Fig. 39.4 H11, 10 : magnetic excitation vs. MRIT comparison

Fig. 39.5 H11, 10 : magnetic excitation vs. MRIT mode 1 only comparison

orders less than the magnitude obtained from the MRIT process. This is attributed to FRFs obtained from the MRIT being normalized by the actual force, whereas the magnetic excitation responses are only normalized with the input voltage to the system.

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Fig. 39.6 H9, 10 : magnetic excitation vs. MRIT comparison

Fig. 39.7 H13, 12 : magnetic excitation vs. MRIT mode 2 comparison

39.5 Results: Nonlinear Modal Testing Using sPLL System The sPLL system described in [1, 5, 17] is used for nonlinear modal testing of the double-clamped beam using magnetic excitation. It must be noted that for nonlinear modal testing, only a SIMO-type testing is performed. Since the magnetic excitation is a point load, a BBC is shown for the magnetic excitation system located at two distinct locations in order to excite locations closer to the peak response amplitude of any considered mode. The first two lateral bending modes are considered for study, and the respective backbone curves obtained for the two modes are shown in Figs. 39.8, 39.9, 39.10, 39.11 and 39.12 for the excitation at the corresponding locations described in the previous section. In addition to the backbone curves, FRCs for four different excitation voltages are provided for the first bending mode at the center of the beam (location 10) as seen in Figs. 39.8, 39.9, 39.10 and 39.11. The FRCs and the BBC obtained for the first bending mode clearly depict the nonlinear response of the double-clamped beam structure for varying forcing and response amplitude levels. Also shown in Figs. 39.13 and 39.14 are the responses from two similar locations along the length of the beam describing the symmetric first bending mode characteristic and the antisymmetric second bending mode characteristics.

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Fig. 39.8 FRC for location 10: 0.1V source voltage

Fig. 39.9 FRC for location 10: 0.2V source voltage

39.6 Conclusions and Future Work A novel magnetic excitation system for both linear and nonlinear experimental modal testing and analysis is successfully demonstrated using a thin double-clamped beam. For the linear EMA performed using the magnetic excitation, the results obtained from a MRIT performed on the same beam is used to compare the quality of data and the natural frequencies obtained. The magnetic excitation system shows good correlation for both a SIMO and MIMO test setup for two varying configurations of physically setting up the exciter system. Similarly, backbone curves are obtained for both the first and second lateral bending modes of the double-clamped beam using the software phase-locked loop (sPLL) system described in [1, 5, 17]. The BBCs clearly show a generally hardeningtype nonlinearity for both the modes which is similar to trends obtained in [18, 19]. A slight initial softening is observed for both BBCs which is primarily attributed to a slight asymmetry caused by the miniature accelerometers being attached on only one side of the double-clamped beam for measurement. This is further confirmed using the time responses in Figs. 39.13 and 39.14. A noncontact measurement is strongly recommended for noncontact excitation techniques as seen in [13,

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Fig. 39.10 FRC for location 10: 0.5V source voltage

Fig. 39.11 FRC for location 10: 1.0V source voltage

14]. In addition to the BBC, FRCs are obtained for the first bending mode only for varying source voltage levels. The trends depicted by these FRCs in both sweep-up and sweep-down profiles are consistent with the characteristics obtained from the BBC. While the necessity and advantages of a Non-contact excitation technique for extremely lightweight and thin structures is evident from the current work in comparison with [5], the magnetic excitation technique requires further investigation and fine-tuning. A key concern in this excitation technique is the nonavailability of forces that are generated by the magnetic excitation system unlike traditional EMA where a force sensor is available (attached to modal shakers or impact hammers) for direct measurement of forces. An indirect method for force estimation and calibration is hence recommended by using the MRIT results as a baseline case and performing a voltage-distance-force calibration per magnetic excitation system. This can be elaborated for a pair of magnetic excitation systems, therefore providing a profile for various physical configurations of magnetic excitation setup and the force developed for any given voltage source. Another alternate calibration technique is using a high-sensitivity force similar to PCB Piezotronics model 209C11 that potentially has a nonmagnetic material as the outer casing placed in a nonmagnetic environment. The magnetic excitation can then be used to exert a force on a small

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Fig. 39.12 Backbone curve for mode 2 – second bending mode of double-clamped beam

Fig. 39.13 Symmetric mode 1 response at similar locations along length of double-clamped beam

specimen attached to such a sensor at various source voltage and distances to obtain the voltage-distance-force calibration characteristics. The availability of such a sensor with nonmagnetic casing is a major hurdle for this method. This voltage-distance-force calibration is important particularly for obtaining FRC which is typically represented as curves at a constant force for showing variation of modal frequency with varying response levels. This is crucial since any modeling and analytical analysis of the system requires an understanding of the forced response at various forcing levels. In addition, any analytical or modeling of damping present in these structures also requires knowing the actual force values instead of voltage values alone. Subject to a successful calibration providing the voltage-distance-force profile, the sPLL system requires modifications to provide a constant force excitation that in turn depends on the large amplitude displacements of the double-clamped beam along with the calibration profile for successful estimation of FRC at constant forcing levels. In addition, developing a MIMO nonlinear modal testing procedure for larger continuous structures using magnetic excitation technique requires further investigation.

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Fig. 39.14 Antisymmetric mode 2 response at similar locations along length of double-clamped beam

References 1. Nagesh, M.: Nonlinear modal testing and system modeling techniques. PhD dissertation, University of Cincinnati, Cincinnati (2021) 2. Allemang, R.J., Phillips, A.W.: The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proceedings of the 2004 International Conference on Noise and Vibration Engineering, pp. 2373–2401. ISMA (2004) 3. Phillips, A.W., Allemang, R.J.: An overview of MIMO-FRF excitation/averaging/processing techniques. J. Sound Vib. 262(3), 651–675 (2003) 4. Allemang, R.J.: UC SDRL CN-20-263-663/664 Vibrations: Experimental Modal Analysis. University of Cincinnati (2013) 5. Nagesh, M., Allemang, R.J., Phillips, A.W.: Challenges of characterizing geometric nonlinearity of a double-clamped thin beam using nonlinear modal testing methods. In: Nonlinear Structures & Systems, vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series 6. Van Damme, C.I.: Model correlation and updating of geometrically nonlinear structural models using nonlinear normal modes and the multiharmonic balance method by. PhD dissertation, University of Wisconsin-Madison (2019) 7. Boivin, N., Pierre, C., Shaw, S.W.: Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems. Nonlinear Dyn. 8(3), 315–346 (1995) 8. Kwarta, M., Allen, M.S.: Nonlinear normal mode estimation with near-resonant steady state inputs. Mech. Syst. Signal Process. 162(2021), 85–88 (2021) 9. Florian, M., et al.: Comparison Between Control-Based Continuation and Phase-Locked Loop Methods for the Identification of Backbone Curves & Nonlinear Frequency Responses Department of Mechanical and Process Engineering, Characterization Methods (1827) 10. Renson, L., Gonzalez-buelga, A., Barton, D.A.W., Neild, S.A.: Robust identification of backbone curves using control-based continuation. J. Sound Vib. 367, 145–158 (2016) 11. Van Damme, C.I., Allen, M.S., Hollkamp, J.J.: Evaluating reduced order models of curved beams for random response prediction using static equilibrium paths. J. Sound Vib. 468, 115018 (2020) 12. Müller, F., et al.: Comparison between control-based continuation and phase-locked loop methods for the identification of backbone curves and nonlinear frequency responses. Nonlinear Struct. Syst. 1, 75–78 (2021) 13. Baver, B.C., Phillips, A.W., Allemang, R.J., Kim, J.: Magnetic excitation and the effects on modal frequency and damping. Top. Modal Anal. Test. 10, 347–354 (2016) 14. Baver, B.C.: Property Identification of Viscoelastic Coatings Through Non-contact Experimental Modal Analysis. University of Cincinnati (2016) 15. Walker, J., Halliday, D., Resnick, R.: Fundamentals of Physics, 8th edn. Wiley, Hoboken (2008) 16. Pandiya, N.: Design and validation of a MIMO nonlinear vibration test rig with hardening stiffness characteristics in multiple degrees of freedom. MS thesis, University of Cincinnati, p. 133 (2017) 17. Nagesh, M., Allemang, R.J., Phillips, A.W.: Characterization of nonlinearities in a structure using nonlinear modal testing methods. In: Nonlinear Structures & Systems, vol. 1. Conference Proceedings of the Society for Experimental Mechanics Series 18. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley (2008) 19. Sathyamoorthy, M.: Nonlinear Analysis of Structures. Taylor & Francis (1997)

Chapter 40

Predicting Nonlinearity in the TMD Benchmark Structure Using QSMA and SICE Drithi Shetty, Kyusic Park, Courtney Payne, and Matthew S. Allen

Abstract Built-up structures exhibit nonlinear dynamic behavior due to friction between interfaces that are fastened together. On the other hand, aircraft, spacecraft, and even automotive structures consist of thin panels to reduce their weight, which can exhibit geometric nonlinearity for displacements on the order of the thickness. As part of the Tribomechadynamics Research Challenge (TRC), this work seeks to predict these effects a priori, whereas most prior works have focused on tuning a model to experimental measurements. While methods are beginning to mature that can predict micro-slip nonlinearity of structures, and methods are well established for reduced-order modeling of geometrically nonlinear structures, these have not been combined previously. This paper presents a simulation approach used to predict the nonlinear response of a benchmark structure proposed in the TRC, which exhibits both geometric nonlinearity and micro-slip due to friction in the bolted connections. A two-dimensional model of the structure is created to enable a wide range of simulations to be performed with minimal computational cost, including some dynamic simulations where both friction and geometric nonlinearity are considered. The nonlinear modal behavior is predicted using quasi-static modal analysis (QSMA) and a recent extension called single-degree-of-freedom implicit condensation and expansion (SICE). Static load-displacement data is also used to define a non-parametric Iwan element that reproduces the modal behavior with high fidelity and yet with minimal computational cost. Additionally, limited simulations are performed on three-dimensional models, which are much more expensive but should be predictive, at least so long as Coulomb Friction is appropriate to model the interactions at the interfaces. Keywords Iwan element · Friction · Geometric nonlinearity · Reduced-order modeling

40.1 Introduction The aircraft, spacecraft, and automotive industry are increasingly making use of thin, curved panels in their structures for higher strength-to-weight ratios and lower fuel consumption. Additionally, these structures contain mechanical fasteners used to assemble the various parts together, making future maintenance convenient. Curved panels can exhibit highly nonlinear responses due to geometric nonlinearity [1, 2]. Mechanical fasteners, too, contribute to the nonlinear dynamic behavior due to frictional energy dissipation at the contact surfaces [3, 4]. In order to cut testing and manufacturing costs, it is essential to predict the resulting changes in stiffness and damping of the structure. This, however, poses some challenges, since the observed nonlinear behavior cannot be easily explained using linear vibration theory, and is an area of active research. Several reduced-order modeling (ROM) techniques have been developed for geometrically nonlinear structures. The Enforced Displacements [5] and Implicit Condensation and Expansion [6] methods have matured considerably and are popular in the research community, even if they have yet to gain significant traction in industry. Another similar technique is the single degree-of-freedom implicit condensation and expansion ROM, or the SICE-ROM, proposed by Park and Allen [7]. In this method, an SDOF ROM is generated to represent the nonlinear normal mode of interest. The corresponding nonlinear stiffness terms are identified by fitting a polynomial to the force–displacement backbone curve of the nonlinear mode of

D. Shetty () · K. Park UW-Madison - Department of Mechanical Engineering, Madison, WI, USA e-mail: [email protected]; [email protected] C. Payne · M. S. Allen Brigham Young University, Department of Mechanical Engineering, Provo, UT, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2023 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-04086-3_40

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interest. On the other hand, lumped hysteretic models are commonly used to represent the nonlinearity observed due to bolted joints [8, 9]. Shetty and Allen [10] presented a non-parametric Iwan model that can be derived from force–displacement data. Dynamic simulations of the resultant reduced-order model can then be performed to estimate the amplitude-dependent frequency and damping of the structure. Both the SICE-ROM for geometric nonlinearity and the non-parametric Iwan model for friction nonlinearity are derived from a force–displacement curve in modal coordinates. For uncoupled modes, this curve can be obtained using quasi-static modal analysis (QSMA) [11, 12]. In QSMA, a distributed load in the shape of the mode of interest is applied and the quasi-static response of the system is solved for. Since the load is applied quasi-statically, QSMA is much more computationally efficient than performing dynamic simulations. The tribo-mecha-dynamics research challenge presents a benchmark structure designed to test the predictive capabilities of the latest ROM techniques developed by the community. The structure consists of a thin, curved panel that is clamped at the ends with the help of bolts, thus potentially consisting of both geometric and frictional nonlinearity. The aim is to predict the first five linear frequencies of the benchmark structure and the amplitude-dependent frequency and damping of the first mode. This work aims to tackle the presented challenge by using QSMA to estimate the backbone curve of the first nonlinear mode. The SICE-ROM and the non-parametric Iwan modeling techniques can then be used to estimate the amplitude-dependent nonlinear behavior of the system. The following section summarizes the FE modeling and nonlinear analysis approach, which is followed by initial results and a summary of conclusions.

40.2 FE Modeling The benchmark structure consists of a 1.5 mm thick panel mounted on a rectangular support with the help of two rectangular blocks, known as blades. Two rows of three bolts each are used to clamp the panel at the ends. The surface of the support that is in contact with the panel has an inclination of about 1.1◦ . Thus, the panel has some curvature when assembled. The support will be mounted to a shaker through a fixture along the vertical face. Three-dimensional (3D) FE models of gradually increasing complexity were created. First, only the panel was modeled with fixed boundary conditions. Ten-node tetrahedral elements were used and three different global mesh sizes were considered. This model was used to quickly assess the effect of mesh density on the natural frequencies of the modes of interest. Next, the support and blades were added and the nodes in contact were tied together, without friction being considered. This model was used to determine the effect of the boundary conditions on the natural frequencies of the system, and hence to quantify the level of approximation in the 2D Model, which neglects much of the flexibility of the support structure. Finally, the bolts were modeled and the effect of friction was simulated using a Coulomb friction model. The resultant FE model is shown in Fig. 40.1b. It can be seen that a much finer mesh was implemented near the interfaces to

Fig. 40.1 (a) Left-hand side of the complete 2D FE model, and (b) full 3D FE model. Note that the 2D model extends to the right, including the full panel length and a second set of blade and support pieces

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better capture slip behavior. This model simulates the boundary conditions of the panel more accurately than the partial assembly, thus resulting in better predictions of the linear frequencies. Note that in all 3D models, the curvature of the panel was added when defining the geometry itself. In reality, however, the panel would be manufactured nominally flat and the curvature would be added by clamping the ends on the inclined surface of the support. 3D finite element analysis of models containing contact can be highly computationally expensive. For example, Wall et al. [13] observed that a 3D model of two beams with two lap joints required a minimum of 6 hours to perform a quasi-static analysis, with the solve time going up to 48 hours or more if the contact surfaces were not initially flat. Therefore, the 3D geometry was represented with a 2D model in this project so many more iterations on the model could be performed, in order to understand the potential for slip. Certain approximations need to be made to model the 3D structure in 2D. First, the bolt preload was simulated by applying a pressure load along a line of length equal to the washer diameter. The pressure was calculated as an average of the total pressure acting along the width of the support, i.e. an average of the preload force acting on the area covered by the three washers and zero force acting on the rest of the area. Second, it was estimated that the stiffness arising due to the cantilever nature of the support-fixture arrangement is large enough for the support to be assumed to behave as a rigid structure. Therefore, fixed translational and rotational boundary conditions were applied at all the nodes along the bottom edge of the support. Changing the boundary conditions in the 3D model to replicate this resulted in negligible change in linear frequencies, thus verifying the assumption made. A Coulomb friction model, with a friction coefficient μ = 0.6, was implemented to model the interface interactions. Three-node linear (CPE3 in Abaqus) plane strain elements were used to mesh all the parts. Initially, a mesh size of 1 mm was used near all the contact regions. After running a modal and quasi-static nonlinear analysis, this was refined further to better capture slip at the interfaces, reducing the size to 0.2–0.5 mm. Five elements were used across the thickness of the panel. The mesh was gradually made coarser further away from the contact regions, with a maximum mesh size of 3 mm. Figure 40.1a shows a magnified version of the resulting 2D FE model obtained, zoomed in to show the support, blade, and panel assembly at the left-hand side. As seen in the figure, the panel in this case was modeled flat. The application of line pressure in the preload step resulted in curvature and pre-stress in the panel, making it closer to what will be observed experimentally.

40.3 Nonlinear Analysis To simulate the nonlinear behavior of the first mode, the method of QSMA can be applied on the 2D model. The structure consists of geometric and friction nonlinearity. This paper proposes characterizing this combined nonlinearity using a parallel arrangement of an SDOF SICE-ROM and a non-parametric Iwan element. Consider the equation of motion for a single mode of the system, given by Eq. 40.1. q¨r + 2ξ ωr q˙r + ωr2 rr + fN L (qr , t, φ) = f (t)

(40.1)

The nonlinear force, fN L (qr , t, φ), can then be written as a sum of the nonlinear restoring force of a SICE-ROM and that of an Iwan element, as shown in Eq. 40.2, fN L (qr , t, φ) =

l  i=2

# ki qri +

∞

ρ(φ)[qr (t) − qr,s (t, φ)]dφ

(40.2)

0

where l is the number of polynomials in the SICE-ROM, ki is the polynomial coefficient, qr,s (t) is a continuous variable representing the location at which each slider is stuck; this is typically represented with a discrete set of the Jenkins elements, and φ is the displacement at which ρ(φ) number of sliders slip. While the method of QSMA can be used to obtain the polynomial coefficients forming the SICE-ROM and the distribution function that defines the Iwan model, the two sources of nonlinearity need to first be isolated. This can be done by performing two static analyses—one with the coefficient of friction μ = ∞, that simulates just the geometric nonlinearity, and the other with μ = 0.6 that includes both geometric and joint nonlinearity. Note that the use of the “Rough contact” interface property in Abaqus enforces no slip, i.e. μ = ∞. Least-squares can then be used to fit a SICE-ROM to the backbone curve obtained from the μ = ∞ analysis. The difference between the backbone curves from the μ = ∞ analysis and the μ = 0.6 analysis, on the other hand, gives the nonlinear restoring force due to joints. A non-parametric Iwan model can be fit to this curve, using a process similar to the one outlined in [10].

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Once a nonlinear model has been fit, established approaches can be used to estimate the amplitude-dependent frequency and damping. The increase in frequency due to geometric nonlinearity can be computed using the shooting and pseudoarclength continuation method [14]. The amplitude-dependent softening and nonlinear damping behavior due to friction nonlinearity, on the other hand, can be calculated by simulating its response to an impulse and then using the Hilbert transform [15].

40.4 Initial Results Table 40.1 shows the linear natural frequencies of the first five modes obtained from the 3D model of the complete assembly, as well as the 2D model. The results from the 3D model are reported as the initial prediction, and the uncertainty equals the difference in the predictions of the 2D and 3D models. The uncertainty due to mesh convergence was separately quantified. The 3D model of the panel alone, created using shell elements, was considered, and the mesh size was varied until the element size did not significantly affect the linear natural frequencies. It was found that with solid tetrahedral elements, reducing the global mesh size from 5 to 2 mm changed the frequency of mode 1 by 0.08%. Therefore, a mesh size of 5 mm was considered sufficient. The uncertainty in the linear frequency of the first mode was therefore estimated to be