Nonlinear Structures and Systems, Volume 1

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Structures and Systems, Volume 1 Gaetan Kerschen M. R. W. Brake Ludovic Renson Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 River Publishers

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

River Publishers Nonlinear Structures and Systems, Volume 1 Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 Gaetan Kerschen • M. R. W. Brake • Ludovic Renson Editors

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-982-5 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2020 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, or reproduction 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.

Preface Nonlinear Structures and Systems represents one of eight volumes of technical papers presented at the 37th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, on January 28–31, 2019. The full proceedings also include volumes on Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis & Testing. 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 the steps of the engineering design: 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 will be 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 are discussed. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Liége, Belgium G. Kerschen Houston, TX M. R. W. Brake Bristol, UK Ludovic Renson v

Contents 1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint ................................................................................................................ 1 Carlos Yoong and Mathias Legrand 2 A New Iwan/Palmov Implementation for Fast Simulation and System Identification........................... 11 Drithi Shetty and Matthew S. Allen 3 Analysis of Transient Vibrations for Estimating Bolted Joint Tightness.......................................... 21 M. Brøns, J. J. Thomsen, S. M. Sah, D. Tcherniak, and A. Fidlin 4 Spider Configurations for Models with Discrete Iwan Elements................................................... 25 Aabhas Singh, Mitchell Wall, Matthew S. Allen, and Robert J. Kuether 5 Predicting S4 Beam Joint Nonlinearity Using Quasi-Static Modal Analysis...................................... 39 Mitchell Wall, Matthew S. Allen, and Iman Zare 6 The Best Linear Approximation of MIMO Systems: First Results on Simplified Nonlinearity Assessment .. 53 Péter Zoltán Csurcsia, Bart Peeters, and Johan Schoukens 7 Forced Response of Nonlinear Systems Under Combined Harmonic and Random Excitation ................ 65 Alwin Förster, Lars Panning-von Scheidt, and Jörg Wallaschek 8 Gerrymandering for Interfaces: Modeling the Mechanics of Jointed Structures ................................ 81 T. Dreher, Nidish Narayanaa Balaji, J. Groß, Matthew R. W. Brake, and M. Krack 9 An Analysis of the Gimballed Horizontal Pendulum System for Use as a Rotary Vibrational Energy Harvester ................................................................................................................. 87 D. Sequeira, J. Little, and B. P. Mann 10 On the Dynamic Response of Flow-Induced Vibration of Nonlinear Structures ................................. 91 Banafsheh Seyed-Aghazadeh, Hamed Samandari, and Reza Abrisham Baf 11 Potential and Limitation of a Nonlinear Modal Testing Method for Friction-Damped Systems ............... 95 Maren Scheel, Tobias Schulz, and Malte Krack 12 Dynamics of a Magnetically Excited Rotational System............................................................ 99 Xue-She Wang and Brian P. Mann 13 Experimental Nonlinear Dynamics of a Post-buckled Composite Laminate Plate............................... 103 John I. Ferguson, Stephen M. Spottswood, David A. Ehrhardt, Ricardo A. Perez, Matthew P. Snyder, and Matthew B. Obenchain 14 Simulation of a Self-Resonant Beam-Slider-System Considering Geometric Nonlinearities ................... 115 Florian Müller and Malte Krack 15 Reinforcement Learning for Active Damping of Harmonically Excited Pendulum with Highly Nonlinear Actuator...................................................................................................... 119 James D. Turner, Levi H. Manring, and Brian P. Mann vii

viii Contents 16 Investigation of Nonlinear Dynamic Phenomena Applying Real-Time Hybrid Simulation..................... 125 Markus J. Hochrainer and Anton M. Puhwein 17 Experimental and Numerical Aeroelastic Analysis of Airfoil-Aileron System with Nonlinear Energy Sink .............................................................................................................. 133 Claudia Fernandez-Escudero, Miguel Gagnon, Eric Laurendeau, Sebastien Prothin, Annie Ross, and Guilhem Michon 18 On the Modal Surrogacy of Joint Parameter Estimates in Bolted Joints ......................................... 137 Nidish Narayanaa Balaji and Matthew R. W. Brake 19 Vehicle Escape Dynamics on an Arbitrarily Curved Surface....................................................... 141 Levi H. Manring and Brian P. Mann 20 Nonlinear Dynamical Analysis for Coupled Fluid-Structure Systems............................................. 151 Q. Akkaoui, E. Capiez-Lernout, C. Soize, and R. Ohayon 21 Experimental Nonlinear Vibration Analysis of a Shrouded Bladed Disk Model on a Rotating Test Rig...... 155 Ferhat Kaptan, Lars Panning-von Scheidt, and Jörg Wallaschek 22 The Measurement of Tangential Contact Stiffness for Nonlinear Dynamic Analysis............................ 165 C. W. Schwingshackl and D. Nowell 23 Investigating Nonlinearity in a Bolted Structure Using Force Appropriation Techniques ...................... 169 Benjamin R. Pacini, Daniel R. Roettgen, and Daniel P. Rohe 24 Techniques for Nonlinear Identification and Maximizing Modal Response ...................................... 173 D. Roettgen, B. R. Pacini, and R. Mayes 25 Influences of Modal Coupling on Experimentally Extracted Nonlinear Modal Models......................... 189 Benjamin J. Moldenhauer, Aabhas Singh, Phil Thoenen, Daniel R. Roettgen, Benjamin R. Pacini, Robert J. Kuether, and Matthew S. Allen 26 Dynamic Response of a Curved Plate Subjected to a Moving Local Heat Gradient ............................. 205 David A. Ehrhardt, B. T. Gockel, and T. J. Beberniss 27 A Test-Case on Continuation Methods for Bladed-Disk Vibration with Contact and Friction................. 209 Z. Saeed, G. Jenovencio, S. Arul, J. Blahoš, A. Sudhakar, L. Pesaresi, J. Yuan, F. El Haddad, H. Hetzler, and L. Salles 28 Dynamics of Geometrically-Nonlinear Beam Structures, Part 1: Numerical Modeling......................... 213 D. Anastasio, J. Dietrich, J. P. Noël, G. Kerschen, S. Marchesiello, J. Häfele, C. G. Gebhardt, and R. Rolfes 29 Dynamics of Geometrically-Nonlinear Beam Structures, Part 2: Experimental Analysis ...................... 217 D. Anastasio, J. Dietrich, J. P. Noël, G. Kerschen, S. Marchesiello, J. Häfele, C. G. Gebhardt, and R. Rolfes 30 Constructing Backbone Curves from Free-Decay Vibrations Data in Multi-Degrees of Freedom Oscillatory Systems...................................................................................................... 221 Mattia Cenedese and George Haller 31 Nonlinear 3D Modeling and Vibration Analysis of Horizontal Drum Type Washing Machines ............... 225 Cem Baykal, Ender Cigeroglu, and Yigit Yazicioglu 32 Comparison of Linear and Nonlinear Modal Reduction Approaches ............................................. 229 Erhan Ferhatoglu, Tobias Dreher, Ender Cigeroglu, Malte Krack, and H. Nevzat Özgüven 33 Reduced Order Modeling of Bolted Joints in Frequency Domain ................................................. 235 Gokhan Karapistik and Ender Cigeroglu 34 Comparison of ANM and Predictor-Corrector Method to Continue Solutions of Harmonic Balance Equations ................................................................................................................. 239 Lukas Woiwode, Nidish Narayanaa Balaji, Jonas Kappauf, Fabia Tubita, Louis Guillot, Christophe Vergez, Bruno Cochelin, Aurélien Grolet, and Malte Krack

Contents ix 35 A Priori Methods to Assess the Strength of Nonlinearities for Design Applications ............................. 243 E. Rojas, S. Punla-Green, C. Broadman, Matthew R. W. Brake, B. R. Pacini, R. C. Flicek, D. D. Quinn, C. W. Schwingshackl, and E. Dodgen 36 Predictive Modeling of Bolted Assemblies with Surface Irregularities............................................ 247 Matthew Fronk, Gabriela Guerra, Matthew Southwick, Robert J. Kuether, Adam Brink, Paolo Tiso, and Dane Quinn 37 A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures .............. 259 Hamed Samandari and Ender Cigeroglu 38 Experimental-Numerical Comparison of Contact Nonlinear Dynamics Through Multi-level Linear Mode Shapes ............................................................................................................. 263 Elvio Bonisoli, Domenico Lisitano, and Christian Conigliaro 39 Dynamic Behavior and Output Charge Analysis of a Bistable Clamped-Ends Energy Harvester............. 273 Masoud Derakhshani and Thomas A. Berfield

Chapter 1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint Carlos Yoong and Mathias Legrand Abstract The present contribution describes a numerical technique devoted to the nonsmooth modal analysis (natural frequencies and mode shapes) of a non-internally resonant elastic bar of length Lsubject to a Robin condition at x = 0 and a frictionless unilateral contact condition at x = L. When contact is ignored, the system of interest exhibits noncommensurate linear natural frequencies, which is a critical feature in this study. The nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the boundary conditions. In order to find a few of the above families, the unknown displacement is first expressed using the well-known d’Alembert’s solution incorporating the Robin boundary condition at x =0. The unilateral contact constraint at x =Lis reduced to a conditional switch between Neumann (open gap) and Dirichlet (closed gap) boundary conditions. Finally, T-periodicity is enforced. It is also assumed that only one contact switch occurs every period. The above system of equations is numerically solved for through a simultaneous discretization of the space and time domains, which yields a set of equations and inequations in terms of discrete displacements and velocities. The proposed approach is non-dispersive, nondissipative and accurately captures the propagation of waves with discontinuous fronts, which is essential for the computation of periodic motions in this study. Results indicate that in contrast to its linear counterpart (bar without contact constraints) where modal motions are sinusoidal functions “uncoupled” in space and time, the system of interest features nonsmooth periodic displacements that are intricate piecewise sinusoidal functions in space and time. Moreover, the corresponding frequency-energy “nonlinear” spectrum shows backbone curves of the hardening type. It is also shown that nonsmooth modal analysis is capable of efficiently predicting vibratory resonances when the system is periodically forced. The pre-stressed and initially grazing bar configurations are also briefly discussed. Keywords Nonsmooth systems · Modal analysis · Internal resonance · Unilateral contact constraints · Wave equation 1.1 Introduction The concept of linear modes (natural frequencies and mode shapes) is a widely studied subject in the field of structural dynamics [7]. A possible extension of this notion to nonlinear conservative systems sees a mode of vibration as a oneparameter continuous family of periodic orbits displaying similar qualitative features [5]. In the phase space, nonlinear modes emerge as invariant surfaces of periodic trajectories, referred to as invariant manifolds [10], where invariant implies that the motion initiated on the manifold stays on it as time unfolds. To some extent, nonlinear modal analysis can be employed for predicting vibratory resonances, computing the nonlinear spectra of vibration or performing model-order reduction. Techniques traditionally employed for nonlinear modal analysis require a certain degree of smoothness in the nonlinearities [11] and thus fail for systems withnonsmooth nonlinearities such as unilateral contact constraints. Certainly, an accurate characterization of the vibratory response of these systems is essential to achieving enhanced and safer engineering applications [12]. Modal analysis of nonsmooth mechanical systems, also called nonsmooth modal analysis, has been recently proposed for a finite elastic bar of lengthLsubject to a Dirichlet boundary condition at x =0 and a unilateral contact constraint at x =L[13]. This system satisfies a complete internal resonance condition, i.e. all linear natural frequencies are commensurate with the first one, which has drastic consequences on the nonlinear modal response. Despite the simplicity of the system, the computed nonsmooth modes (NSMs) indicate highly intricate vibratory behaviour. Corresponding periodic displacements were observed to be unseparated piecewise linear functions of space and time, as opposed to their linear counterparts which are sinusoidal functions separated in space and time. Moreover, for certain NSMs such internal resonance C. Yoong ( ) · M. Legrand Department of Mechanical Engineering, McGill University, Montréal, QC, Canada e-mail: carlos.yoong@mail.mcgill.ca © Society for Experimental Mechanics, Inc. 2020 G. Kerschen et al. (eds.), Nonlinear Structures and Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12391-8_1 1

2 C. Yoong and M. Legrand generates a discontinuity between the linear and nonlinear portions of the invariant manifold. To further explore the nonlinear dynamics of this one-dimensional contact problem, a non-internally resonant configuration is investigated in the present work. The complete internal resonance condition is annihilated by changing the boundary condition (BC) at x = 0 from Dirichlet type u(0, t) = 0 to a Robin type ∂xu(0, t) −αu(0, t) = 0 which reflects that the elastic bar is now connected to a rigid support through a simple linear spring.1 Analytical derivations are first proposed to facilitate the construction of the sought NSMs. Then, a numerical scheme based on the simultaneous discretization of the space and time domains is employed and the nonsmooth modes of vibration are constructed. 1.2 Non-internally Resonant Elastic Bar The system of interest is a homogeneous elastic bar of length L > 0 and constant cross-sectional area S > 0 subject to a conservative unilateral constraint at its right end. Its left end is connected to a rigid support through a spring of stiffness κ >0, as depicted in Fig. 1.1. The displacement, velocity, strain and stress fields are denoted by u(x, t), v(x, t), (x, t) and σ(x, t) respectively where x is the coordinate of a point of the bar in the initial configuration and t denotes time. Young’s modulus is denoted by E >0 and ρ >0 stands for the mass per unit volume, which are both, by assumption, space and time independent. Hence, the propagation speed of any longitudinal wave is √E/ρ. In the framework of linear elasticity, the stresses read σ = E , where = ∂xu should be infinitesimal and satisfy |∂xu| < 1 [1, p. 5]. The unilateral contact force r(t) is related to the stresses by σ(L, t) =E∂xu(L, t) =r(t)/S. Further, the signed distance between the right extremity of the bar and the rigid obstacle, termed gap function, is defined as g(u(L, t)) =g0 −u(L, t), where g0 is the signed distance between the unrestricted resting configuration and the obstacle: it is strictly negative in the pre-stressed configuration, for instance. Unless stated otherwise, there is no external excitation on the system. The full formulation reads: Local equation ∂ 2 tt u(x, t) −c 2 ∂ 2 xxu(x, t) =0, ∀x ∈]0;L[, ∀t >0, (1.1) RobinBC ∂xu(0, t) −αu(0, t) =0, ∀t >0, (1.2) Signorini BC g(u(L, t)) ≥0, r(t) ≤0, r(t)g(u(L, t)) =0, ∀t >0, (1.3) Initial conditions u(x, 0) =u0(x), v(x, 0) =v0(x), ∀x ∈]0;L[. (1.4) where α =κ/(ES). This formulation possesses a unique solution which conserves the total energy [9]. It is worth noting that the local equation (1.1) is the well-known wave equation (a hyperbolic partial differential equation) defined on a one-dimensional finite domain. The natural frequencies ωk and k of the underlying linear system are solutions to the transcendental equations: spring−freeBCs ωk −αccot(ωkL/c) =0, k ∈ N>0, (1.5) spring−fixedBCs k +αctan( kL/c) =0, k ∈ N>0. (1.6) The corresponding natural periods are Tk =2π/ωk and Pk =2π/ k. In both configurations, the natural frequencies are incommensurate, in the sense that ωk and k for k =2, 3, . . . , ∞are not multiples of ω1 nor 1, respectively [8, p. 245]. Accordingly, the complete internal resonance condition emerging when the bar is clamped at x =0 no longer holds [13]. Non-trivial solutions of the unilateral contact problem described by Eqs. (1.1)–(1.4) are successions of free phases (open gap) and contact phases (closed gap) [2]. Hence, these solutions can be perceived as the combination of motions satisfying the wave equation together with a switching boundary condition at x =Lbetween∂xu(L, ·) =0 when the gap is open, referred x u(x, t) L g(u(L, t)) Fig. 1.1 One-dimensional finite elastic bar attached to a spring at its left extremity and subject to unilateral contact constraint on its right tip 1In this document, operators ∂ξ(•) and ∂ 2 ξξ(•) stand for the first and second derivatives of (•) with respect to the argument ξ.

1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint 3 to as “spring–free BCs” (or equivalently Robin–Neumann BCs), and prescribed displacement u(L, ·) =g0 which implies v(L, ·) = 0 when the gap is closed, named “spring–fixed BCs” (or equivalently Robin–Dirichlet BCs). The nonlinearity in the formulation arises in the dependence of the solution to the unknown switching time. To further elaborate on this statement, consider the general solution to the local equation (1.1) comprising the superposition of forward hand backward f travelling waves defined on the real line R[1, p. 91] u(x, t) =f(ct +x) +h(ct −x), (1.7) At x =0, the effect of the attached spring on the reflection of an incident backward wave f is computed by inserting the general solution (1.7) in (1.2), yielding (∂ξf(ξ) −∂ξh(ξ)) =α(f(ξ) +h(ξ)), ∀ξ ∈ R, which in turns leads to the identity h(ξ) =f(ξ) +e−αξ β −2α ξ 0 eαsf(s)ds , ∀ξ ∈R (1.8) Without going into the details, it is straightforward to check that the BC at x = Limply that β = 0. Incorporating (1.8) in (1.7) leads to an integral expression of the displacement in terms of f only u(x, t) =f(ct +x) +f(ct −x) −2αeα(x−ct) ct−x 0 eαsf(s)ds, (1.9) which incorporates the reflection mechanism of the travelling waves at x =0.At x =L, the boundary condition successively switches from Neumann to Dirichlet and vice-versa. The corresponding reflections require additional considerations elaborated later in the text. Let us first denote by f|[a;b] the set of all ordered pairs (ξ,f(ξ)), ∀ξ ∈ [a;b], which is also referred to as the graph of function f over the interval [a;b]. Provided that f|[0;L] is known, the boundary condition at x =Lis utilized to specify f(ξ) over the remaining portions of the real axis R, that is for ξ ∈] −∞;0[ and ξ ∈]L;∞[. During free phases (open gap), a homogeneous Neumann BC is active at x = L, that is ∂xu(L, ·) = 0. Inserting Eq. (1.7) into the latter yields ∂ξf(ξ +2L) = ∂ξh(ξ), which in turn results in f(ξ +2L) −h(ξ) = f(2L) −h(0). Incorporating the influence of the attached spring via Eq. (1.8), the latter is re-formulated in terms of the backward wave f only, that is f(ξ +2L) −f(ξ) +2αe−αξ ξ 0 eαsf(s)ds =f(2L) −f(0). (1.10) During contact phases (closed gap), the right end of the bar satisfies a non-homogeneous Dirichlet BC, corresponding to a prescribed displacement at x = L, which reads u(L, ·) = g0. Again, combining this boundary condition and the general solution (1.7), the interaction of forward and backward travelling waves shall satisfyf(ξ +2L)+h(ξ) =g0, which can also be expressed in terms of f only as follows: f(ξ +2L) +f(ξ) −2αe−αξ ξ 0 eαsf(s)ds =g0, (1.11) Accordingly, if f is known in any region of length 2L, it can be expanded over the real axis, via (1.10) for a free phase (open gap) and (1.11) for a contact phase (closed gap). It is then employed in (1.9) to calculate the corresponding displacement. During each of the above phases, f|[0;2L] is obtained from the associated initial and boundary conditions [4, p. 80]. For the free phase, knowing that the displacement and velocity waves reflect at a free BC without changing signs, f|[0;2L] satisfies f|[0;2L]= ⎧ ⎪⎪⎨ ⎪⎪⎩ u0(ξ) 2 + 1 2c ξ 0 v0(s)ds ξ ∈ [0;L], u0(2L−ξ) 2 + 1 2c L 0 v0(s)ds + 1 2c ξ L v0(2L−s)ds ξ ∈]L;2L]. (1.12) However, for the contact phase, knowing that u0(L) =g0 and that the displacement and velocity waves reflect with opposite sign, f|[0;2L] shall satisfy

4 C. Yoong and M. Legrand f|[0;2L]= ⎧ ⎪⎪⎨ ⎪⎪⎩ u0(ξ) 2 + 1 2c ξ 0 v0(s)ds ξ ∈ [0;L], g0 − u0(2L−ξ) 2 + 1 2c L 0 v0(s)ds − 1 2c ξ L v0(2L−s)ds ξ ∈]L;2L], (1.13) Consequently, f can be completely defined over Rfor every phase. To summarize, the displacement of the elastic bar with Robin BC on the left end is obtained through Eq. (1.9) provided that f is defined everywhere on the real axis. The successive switches in boundary conditions at x =L, reflecting the unilateral contact constraint, are incorporated through appropriate extensions: Eq. (1.10) for the free phase or Eq. (1.11) for contact phase. The following section is concerned with analytical derivations for the computation of periodic solutions by employing the expressions (1.9)–(1.11). 1.3 Periodic Solutions and Nonsmooth Modal Analysis Nonsmooth modes of vibration of the spring–bar system depicted in Fig. 1.1 are defined as continuous families of periodic solutions satisfying the formulation (1.1)–(1.4) together with periodicity conditions in displacement and velocity: ∃T >0 such that u(x, t +T) =u(x, t) and v(x, t +T) =v(x, t), ∀x ∈ [0;L] and ∀t >0. Finding such solutions translates into finding corresponding initial conditions u0 andv0 andperiodT which generate periodic motions. Without loss of generality, it is assumed that within one period, over the interval t ∈ [0;T], the initial time segment is a free phase that initiates at t =0+ and the final time segment is a contact phase that ends at t =T− and switches back to the initial free phase state at t =T+. In general, various successions of free and contact phases might arise within one period. Knowing that the motion of the bar can be uniquely defined by a single function f, the targeted trajectory is then an unknown periodic sequence of functions uin the form (1.9) where f switches between (1.10) and (1.11). Let us consider the simplest combination of one free phase and one contact phase of duration tf and tc = T −tf, respectively. When the gap is open, displacement and velocity satisfy the following equalities ∀x ∈ [0;L] and ∀t ∈ [0;tf]: u1(x, t) =f0(ct +x) +f0(ct −x) −2αe α(x−ct) ct−x 0 eαsf0(s)ds, (1.14a) 1 c v1(x, t) =∂t f0(ct +x) +∂t f0(ct −x) −2αf0(ct −x) +2α 2eα(x−ct) ct−x 0 eαsf0(s)ds, (1.14b) where f0|[0;2L], calculated via Eq. (1.12) with the initial conditions u0 and v0, is then expanded throughout the real axis R via (1.10). During gap closure, the motion is described by the composite function f1(f0(·)) corresponding to a boundary condition switch. This function arises by defining the graphf1|[0;2L] withu1(·, tf) andv1(·, tf) as the “initial conditions” in Eq. (1.13) and then expanding it onRvia Eq. (1.11). During a contact phase, the displacement and velocity read∀x ∈ [0;L] and ∀t ∈]tf ;T] u2(x, t) =f1(c(t −tf) +x) +f1(c(t −tf) −x) −2αe α(x−c(t−tf)) c(t−tf)−x 0 eαsf1(s)ds, (1.15a) 1 c v2(x, t) =∂t f1(c(t −tf)+x) +∂t f1(c(t −tf)−x)−2αf1(c(t −tf)−x)+2α 2eα(x−c(t−tf)) c(t−tf)−x 0 eαsf1(s)ds. (1.15b) Accordingly, admissibleT-periodic motions involving one lasting contact phase per period are described by functions f1◦f0 and f0 that satisfy the following periodicity condition, arising fromu0(x) =u2(x,T) and v0(x) =v2(x,T), ∀x ∈ [0;L] u0(x) =f1(ctc +x) +f1(ctc −x) −2αe α(x−ctc) ctc−x 0 eαsf1(s)ds, (1.16a) 1 c v0(x) =∂t f1(ctc +x) +∂t f1(ctc −x) −2αf1(ctc −x) +2α 2eα(x−ctc) ctc−x 0 eαsf1(s)ds. (1.16b) together with the admissibility conditions reflecting Signorini’s conditions at x =Lin Eq. (1.3):

1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint 5 • the elastic bar shall not contact the obstacle during a free phase except at gap closure when t =tf; grazing is permitted during free phase g0 −f0(ct +L) −f0(ct −L) +2αe α(L−ct) ct−L 0 eαsf0(s)ds ≥0, ∀t ∈ [0;tf], (1.17a) g0 −f0(ctf +L) −f0(ctf −L) +2αe α(L−ctf) ctf−L 0 eαsf0(s)ds =0, (1.17b) • the contact force should be non-positive during gap closure until contact separation at t =tf +tc =T ∂xf1(ct +L) −∂xf1(ct −L) +2αf1(ct −L) −2α 2eα(L−ct) ct−L 0 eαsf1(s)ds ≤0, ∀t ∈ [0;tc], (1.18a) ∂xf1(ctc +L) −∂xf1(ctc −L) +2αf1(ctc −L) −2α 2eα(L−ctc) ctc−L 0 eαsf1(s)ds =0. (1.18b) From Eq. (1.16), the equality c∂xu0(x) +v0(x) =c∂xu2(x,T) +v2(x,T) provides an additional relationship between f0 and f1 for potential periodic motions to exist: f1(ctc +x) −f1(ctc) =f0(x) −f0(0), ∀x ∈ [0;L]. Finding solutions to Eqs. (1.16)–(1.18) is a noticeably challenging task. The complexity for solving such problems lies in the difficulty of defining a simple relationship between functions f0 and f1 incorporating the BC switching mechanism. Additionally, the effect of the spring, arising as a delay integral term in the displacement (1.9) further complicates the identification of admissible periodic motions. 1.4 Numerical Scheme The main objective of the proposed numerical scheme is to simultaneously discretize the space and time domains of the above formulation in order to accurately mimic the propagation of discontinuous waves along the characteristics lines: x ±ct = constant. The main limitation of the proposed technique is the fact that the travelling-wave solution shall be partially known (in the sense that identities such as Eqs. (1.16)–(1.18) can be derived) before discretization. To compute families of periodic orbits, the space and time domains of the integral equations (1.16) are simultaneously discretized in order to approximate the initial conditions that generate a periodic motion. Space is divided into N intervals of identical length x =xi+1 −xi =L/Nwithi =0, 1, . . . ,N. Since travelling waves are required to propagate along the characteristic lines, the time-step shall satisfy t =tn+1 −tn = x/c for n =0, 1, . . . ,nT, where nT satisfies nT t =T. To approximatef1 in Eq. (1.16), the discretization of the real axis emerges from the discretized space-time coupling argument x ±ct of the travelling-wave solution, hence the discretized f1 function is denoted as f1j ≈f1(ξj) such that ξj =xi ±ctn where j =i ±n for i =0, 1, . . . ,N and n =0, 1, . . . ,nT. Similar notations also apply to f0j. The discretization of the displacement and velocity during free phase produces u (n) 1i ≈u1(xi, tn) and v (n) 1i ≈v1(xi, tn) respectively, and in a similar fashion during contact phase for u2 and v2. The discretized initial conditions read u0i ≈u0(xi) and v0i ≈v0(xi). Since the duration of the free phase is tf =nf t and the duration of the contact phase is tc =nc t, the approximations of f0|[0;2L] andf1|[0;2L] needed to approximately solve (1.16) are computed via a trapezoidal rule to compute the integrals: • for a free phase via discretization of Eq. (1.12) with discrete initial conditions u0i ≈u0(xi) and v0i ≈v0(xi) f0j = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ u0j 2 + 1 4c j i=0 x(v0i +v0(i+1)) j =0, 1, . . . ,N, a + u0(2N−j) 2 + 1 4c j i=N+1 x(v0(2N−i) +v0(2N−i+1)) j =N+2,N+3, . . . , 2N, (1.19)

6 C. Yoong and M. Legrand • for a contact phase via discretization of Eq. (1.13) with discrete “initial conditions”u (nf) 1i ≈u1(xi, tf) andv (nf) 1i ≈v1(xi, tf) f1j = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ u (nf) 1j 2 + 1 4c j i=0 x v (nf) 1i +v (nf) 1(i+1) j =0, 1, . . . ,N, b− u (nf) 1(2N−j) 2 − 1 4c j i=N+1 x v (nf) 1(2N−i) +v (nf) 1(2N−i+1) j =N+2,N+3, . . . , 2N, (1.20) where4ac = N i=0 x(v0i +v0(i+1)) and4bc = N i=0 x v (nf) 1i +v (nf) 1(i+1) +4g0c. The above expressions (1.19) and (1.20) known for j =0, 1, . . . , 2Nare then expanded on the real axis: f0(j+2N) −f0(j) +sgn(j)αe−αj x j k=0 (eαξkf0(k) +e αξk+1f0(k+1)) =f0(2N) −f0(0), (1.21) f1(j+2N) +f1(j) +sgn(j)αe−αj x j k=0 (eαξkf1(k) +e αξk+1f1(k+1)) =g0, (1.22) where sgn(•) is the sign function. Since the period T =tf +tc =nT t is known, where nT =nf +nc, the approximated displacement can be calculated for each phase as follows: • free phase, via discretization of Eq. (1.14), for i =0, 1, . . . ,Nand n =0, 1, 2, . . . ,nf u (n) 1i =f0(n+i) +f0(n−i) −sgn(n−i)αe α(xi−ctn) n−i k=0 x(eαξkf0(k) +e αξk+1f0(k+1)), (1.23) • contact phase, via discretization of Eq. (1.15), for i =0, 1, . . . ,Nand n =nf +1,nf +2, . . . ,nT u (n) 2i =f1(n−nf+i) +f1(n−nf−i) −sgn(n−nf −i)αe α(xi−c(tn−tf)) n−nf−i k=0 x(eαξkf1(k) +e αξk+1f1(k+1)). (1.24) The corresponding velocity is calculated through a numerical time-differentiation scheme, such as the forward Euler method v (n) i =(u (n+1) i −u (n) i )/ t. It is worth remarking that because f1j in Eq. (1.24) is compounded with f0j, and the latter is computed from discrete initial conditions, then all displacements points u (n) 1i and u (n) 2i can be expressed in terms of u0i and v0i. The discretization of the periodicity condition (1.16) together with the unilateral contact condition yields a constrained system of linear equations written in terms of the discrete initial conditions u0i andv0i, free-phase time-steps nf and contactphase time-steps nc: periodicity: u0i −u (nf+nc) 2i = 0 and v0i −v (nf+nc) 2i = 0, i =0, 1, . . . ,N (1.25) impenetrability: g0 −u (n) 1N ≥ 0, n =0, 2, . . . ,nf (1.26) compressive contact: E (n) 2N ≤ 0, n =nf +1,nf +2, . . . ,nf +nc (1.27) where (n) 2N is the strain at x = L at time tn calculated from the numerical space-differentiation of discrete displacements u (n) 2i ; for instance, the implementation of the forward Euler scheme yields discrete strains in the form of (n) 2i =(u (n) 2i −u (n) 2(i−1) )/ x. The discretized formulation given by Eqs. (1.25)–(1.27) is solved through the following steps: 1. Set number of time-steps nf and nc. 2. Calculate potential initial conditions u0i and v0i that generate a periodic motion satisfying Eq. (1.25) only. 3. Verify that the generated periodic motion satisfies the conditions of impenetrability (1.26) and compressive contact (1.27): the corresponding initial conditions are admissible.

1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint 7 4. If the initial conditions are admissible, compute and store the period of vibration and total energy. 5. Change values of nf and nc. For the identification of families of periodic orbits, every feasible pair (nf,nc) ∈ N2 >0 is examined. Nevertheless, verifying among all potential combinations is computationally very inefficient. For that reason, the iteration intervals are bounded for each family of admissible periodic motions by analyzing every possible combination of nf and nc on coarse meshes. 1.5 Spectrum of Nonsmooth Vibration Three initial configurations of the bar are explored: unstressed (g0 >0), prestressed (g0 <0) and initially grazing (g0 =0). Without loss of generality, the mechanical parameters E, ρ, S andLare arbitrarily chosen to be unity and units are discarded. The autonomous dynamics of the bar is investigated for two spring-bar stiffness ratios α. The linear natural frequencies ωk(α) and k(α) respectively calculated using Eqs. (1.5) and (1.6), now only depend on the parameter α. The proposed discretization technique with t = x/c = 10−3 is implemented. Modal responses were obtained with g0 = ±10−3 in both unstressed and prestressed configurations. The response of the autonomous elastic bar is depicted in frequency-energy plots (FEPs) where a backbone curve (also known as branch) represents a NSM. Each point pertaining to a backbone curve represents a modal motion whose frequency is indicated on the horizontal axis and constant total energy along the vertical axis. The frequencies in the FEPs are not normalized while the energy is normalized with respect to the energy of the first linear mode grazing orbit. NSMs computed for α = 1 and α = 1/2 are now investigated. In contrast to the results exposed in [13], the intricate dynamics caused by the spring complicates the identification of NSM branches and a highly fine discretization is required to discern how admissible periodic solutions organize to form a continuum. The backbone curves, emerging in the frequency range [ω1 ; 1] are depicted in Fig. 1.2 as sets of scattered points supposedly belonging to NSMs. Contrary to main NSMs of the internally resonant bar studied in [13] where the energy continuously depends on the frequency, the depicted scattered points indicate more complicated backbone curves. Figure 1.2 suggests that a “nicely” connected continuum representing a possible “main NSM” does not exist. However, several independent branches emerge around subharmonics frequencies lying in the range for the unstressed configuration. This agrees with NSMs in discrete systems without linear internal resonance [6, 12]. An equivalent phenomenon occurs for the prestressed configuration where independent branches align in a seemingly softening branch. For the initially grazing configuration, the NSMs emerge as vertical lines embedding periodic solutions with identical frequency and increasing energy. It also appears that every given collection of scattered points represents a purely independent subharmonic vibration, as opposed to the internally resonant bar which features subharmonic as well as w1 w2 2 w3 5 w3 7 w4 8 W1 w5 11 w5 13 w3 6 100 101 102 103 1:27w1 c b a = 1 Normalized energy w1 1 w2 2 w2 3 w2 4 w3 7 w3 9 100 102 104 1:48w1 1:79w1 c b a = 1/2 Frequency of vibration w Normalized energy Fig. 1.2 Backbone curves in the frequency interval [ω1 ; 1] for α = 1 and α = 1/2 with bar configurations: unstressed [blue solid line], prestressed [grey solid line ] and initially grazing [brown solid line]. Subharmonics frequencies [dashed line]

8 C. Yoong and M. Legrand Space x Time t L 0 0 Tb g0 Space x Time t Displacement u(x, t) Displacement u(x, t) 0 0 c g0 L T Fig. 1.3 Periodic displacement fields for α =1 corresponding to points b (left) and c (right) in Fig. 1.2 Displacement u(x, t) Displacement u(x, t) Space x Time t L 0 0 Tb g0 Space x Time t 0 0 c g0 L T Fig. 1.4 Periodic displacement fields for α =1/2 corresponding to points b (left) and c (right) in Fig. 1.2 internally resonant NSMs [13]. This observation coincides with the fact that the system of interest does not satisfy the full internal resonance condition. Figures 1.3 and 1.4 depict NSM periodic displacement fields, for α = 1 and α = 1/2 respectively, corresponding to points b and c in Fig. 1.2. Both points b represent periodic motions belonging to backbone curves that emerge in the vicinity of the respective ω1. On the other hand, points c live in apparent NSM branches that arise in the neighborhood of linear subharmonic frequencies. The complicated pattern of each solution is caused by an intricate interplay between various travelling waves embedding the Robin and Signorini boundary conditions. In contrast to linear modes that are purely harmonic functions, the nonsmooth modes of the non-internally resonant system are nonsmooth piecewise-sinusoidal functions. The velocity fields shall present several jump discontinuities per period, which would not be accurately described by traditional semi-discretization strategies [3]. From the reported motions, it also seems that the displacement field for t ∈ [0;tf] presents an axis of symmetry in time located at the middle of the free phase: t = tf/2. This observation could facilitate the derivation of closed-form expressions; this is left to future investigations. 1.6 Response to Periodic External Forcing This section numerically investigates the relationships between NSMs and the system response under periodic excitation. A slight amount of structural viscous damping is introduced in the governing equation (1.1) and a weak external damper is attached at x =0 so that the system can possibly reach a periodic steady-state with bounded energy. The system is forced via a harmonically moving rigid wall that periodically compresses the bar, as seen in Fig. 1.5. The formulation for the forced spring–bar system, to be compared to Eqs. (1.1)–(1.4), reads ∂ 2 tt u(x, t) +ζ1∂t u(x, t) −c 2 ∂ 2 xxu(x, t) =0, ∀x ∈]0;L[, ∀t >0, (1.28) ∂xu(0, t) −αu(0, t) −ζ2∂t u(0, t) =0, ∀t >0, (1.29) g(u(L, t), w(t)) ≥0, r(t) ≤0, r(t)g(u(L, t), w(t)) =0, ∀t >0, (1.30) u(x, 0) =u0(x), v(x, 0) =v0(x), ∀x ∈]0;L[, (1.31) where ζ1 and ζ2 are respectively the structural and external damping coefficients. The moving wall excitation corresponds to w(t) = w0 sinωt. For an elastic bar initially at rest, the response of the periodically-forced system is constructed by

1 Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint 9 2 x u(x, t) L g(u(L, T), (t)) (t) Fig. 1.5 Spring–bar system excited by a moving rigid wall w1 1 w1 1 w1 1 w1 1 w1 1 w1 1 100 101 102 103 d Normalized energy 100 102 104 Normalized energy Frequency of vibration w Frequency of vibration w Frequency of vibration w Fig. 1.6 Response to periodic forcing over [ω1 ; 1] of the spring–bar system under various damping coefficients: α = 1 (top) and α = 1/2 (bottom). Bar configuration at rest: unstressed g0 >0 (left), grazing g0 =0 (center) and prestressed g0 <0 (right). Bar damping coefficient ζ1: low [red solid line] to high [solid line]. Grayed regions where NSMs were not found and forced-responses are not periodic Fig. 1.7 Displacement over one steady-state period due to external excitation via moving wall [pink dashed line]. Result computed with (ω,α) =(1.25, 1) and g0 >0. It corresponds to point d in Fig. 1.6 in the vicinity of NSM motion b in Fig. 1.2 top and Fig. 1.3 left Space x Time t Displacement u(x , t) L 0 0 T g0 computing, when possible, the steady-state solution for each frequency of excitation. In this study, such solutions are obtained via the Wave Finite Element Method [13]. The total energy of the steady-state solution, averaged over one forcing period, for increasing frequencies of excitation and various damping coefficients is illustrated in Fig. 1.6 in the interval [ω1 ; 1]. In the frequency intervals where NSMs possibly do not exist,2 highlighted with grayed areas in Fig. 1.6, the forced system does not seem to exhibit periodic steady-states. When non-periodic forced responses are detected, corresponding portions of the response curves in the frequency diagram consider, for each frequency of excitation, the total energy of the forced motion averaged over ten periods of the external forcing. It is observed that forced responses with high-energy periodic steady-state align well with the NSM backbone curves. Also, Fig. 1.7 illustrates a steady-state displacement of the slightly-damped bar under periodic forcing computed in the vicinity of an NSM. It clearly resembles the corresponding NSM motion, see Fig. 1.3. Although this forced solution seems to be identical to the autonomous oscillation, its free phase is actually not symmetric with respect to a time axis located in the middle of the free phase. 2The number of computable NSMs depends on the mesh size, and finer meshes should provide a more detailed spectrum.

10 C. Yoong and M. Legrand 1.7 Conclusions This contribution targeted the nonsmooth modal analysis of a non-internally resonant bar through a numerical strategy based on simultaneous space-time discretization of the travelling-wave solution. The system of interest consisted of a finite elastic bar of length L subject to a Robin boundary condition at x = 0 and a unilateral contact constraint at x = L. Such configuration annihilates the full internal resonance condition featured by its internally resonant counterpart [13]. In addition, the Robin BC, physically corresponding to a simple spring attachment, causes a distortion in the waveform that complicates the numerical construction of nonsmooth modes. A semi-analytical approach is derived from the exact solution to the autonomous wave equation together with a switch in the boundary condition where unilateral contact arises. As nonsmooth periodic solutions in closed form are inaccessible, a discretization strategy is proposed to find families of periodic motions. The periodic nonsmooth motions are piecewise-sinusoidal functions, as opposed to the internally resonant counterparts where modal displacements are piecewise-linear functions. The forced response plots illustrated the capability of the Nonsmooth Modal Analysis to predict the vibratory resonances of the one-dimensional periodically-forced elastic bar. Even though the vibratory characterization is more challenging than for the internally resonant counterpart [13], the computed NSMs forecast most intervals involving nonlinear resonances. References 1. Achenbach, J.: Wave Propagation in Elastic Solids. Elsevier, Amsterdam (1973). https://doi.org/10.1016/C2009-0-08707-8 2. Amerio, L.: Continuous solutions of the problem of a string vibrating against an obstacle. Rendiconti del Seminario Matematico della Università di Padova 59, 67–96 (1978). EUDML: 107691 3. Doyen, D., Ern, A., Piperno, S.: Time-integration schemes for the finite element dynamic Signorini problem. SIAM J. Sci. Comput. 33(1), 223–249 (2011). OAI: hal-00440128 4. Graff, K.: Wave Motion in Elastic Solids. Dover, New York (1975) 5. 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). OAI: hal-01863233 6. Legrand, M., Junca, S., Heng, S.: Nonsmooth modal analysis of a n-degree-of-freedom system undergoing a purely elastic impact law. Commun. Nonlinear Sci. Numer. Simul. 45, 190–219 (2017). OAI: hal-01185980 7. Meirovitch, L.: Analytical Methods in Vibrations. Macmillan, New York (1967) 8. Rao, S.: Vibration of Continuous Systems. Wiley, London (2007). DOI: 10.1002/9780470117866 9. Schatzman, M., Bercovier, M.: Numerical approximation of a wave equation with unilateral constraints. Math. Comput. 53(187), 55–79 (1989). OAI: hal-01295436 10. Shaw, S., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150(1), 170–173 (1991). OAI: hal-01310674 11. Thorin, A., Legrand, M.: Nonsmooth modal analysis: from the discrete to the continuous settings. In: Leine, R., Acary, V., Brüls, O. (eds.) Advanced Topics in Nonsmooth Dynamics. Transactions of the European Network for Nonsmooth Dynamics, chapter 5, pp. 191–234. Springer, Berlin (2018). OAI: hal-01771849 12. Thorin, A., Delezoide, P., Legrand, M.: Non-smooth modal analysis of piecewise-linear impact oscillators. SIAM J. Appl. Dyn. Syst. 16(3), 1710–1747 (2017). OAI: hal-01298983 13. Yoong, C., Thorin, A., Legrand, M.: Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint. Nonlinear Dyn. 91(4), 2453–2476 (2018). OAI: hal-01471341

Chapter 2 A New Iwan/Palmov Implementation for Fast Simulation and System Identification Drithi Shetty and Matthew S. Allen Abstract While Iwan elements have been shown to be an effective model for the stiffness and energy dissipation in bolted joints, they are presently somewhat expensive to integrate. Currently, the Newmark-beta algorithm is used to integrate the equations of motion when a structure contains Iwan elements, and a small time step is needed to maintain accuracy. This paper presents a new way of simulating Iwan elements that speeds up the simulations dramatically by using closed form expressions for the micro-slip regime and using an averaging method for regions of time in which no external force is applied. With this method the response can be computed in about a hundredth of the time. The proposed algorithm is demonstrated on a single degree-of-freedom (SDOF) system to understand the range over which it retains accuracy. Although current implementation is applicable to SDOF systems, it can simulate the response of each mode in a structure that is modeled using the modal Iwan approach (i.e. assuming uncoupled, weakly-nonlinear modes). Keywords Non-linear damping · Iwan model · method of averaging · Newmark-beta integration · Runge-Kutta 2.1 Introduction Built-up structures are typically modeled using linear solvers with springs approximating the joints and using modal or proportional damping to account for energy dissipation. However, mechanical joints are a major contributor to the overall damping of structures [1]. Their behavior is predominantly non-linear and the development of a predictive model for the same is a challenge [2]. At lower amplitudes, only the edges of the joint surfaces slide relative to each other while a majority of the joint remains intact, known as micro-slip. In the micro-slip regime, the stiffness of the joint decreases only slightly, but there is significant energy loss [3]. Hence, the response is nearly linear although the damping is observed to change significantly with the amplitude of vibration [4]. As the amplitude increases, the slip region gradually expands until macro-slip occurs. In this case, relative motion occurs between the surfaces and the stiffness of the joint is significantly affected. While the physics just described could be captured by modeling each joint in detail with a suitable friction law between parts, to compute the dynamic response with such an approach would take months or years on current computers. As a result, it is more common to use a reduced model with nonlinear elements between each component to account for microand macro-slip in the joints. The Iwan element, initially introduced for metal elasto-plasticity [5], has been shown to be effective in capturing joint behavior [4]. It is a lumped, hysteretic model consisting of a parallel system of spring-slider units known as Jenkins elements. The most common Iwan model is Segalman’s four parameter model [6], with the four parameters accounting for the joint stiffness, the force at which the joint slips completely and the power law energy dissipation that many joints have been found to exhibit in microslip. While this is far less computationally expensive than modeling the contact in detail, the computational burden is significant when many joints are present or when performing parameter studies. As an alternative, Segalman proposed a modal approach [7], which exploits the fact that the modes of a structure tend to be mostly uncoupled when the joints remain in the micro-slip regime [8] and hence, each mode can be modeled as a single degree of freedom system but with an Iwan element to account for the nonlinearity. In either case, the response of a structure that contains Iwan elements is typically found using the average acceleration Newmark-beta integration method [9], with a Newton-Raphson iteration loop for the nonlinear force in the Iwan model. This method is effective and quite reliable, but requires a small time step, making it computationally expensive. This paper presents an alternative that is much less expensive. One study that is relevant to the current work is that on the RIPP joint, D. Shetty ( ) · M. S. Allen University of Wisconsin, Madison, WI, USA e-mail: ddshetty@wisc.edu; msallen@engr.wisc.edu © Society for Experimental Mechanics, Inc. 2020 G. Kerschen et al. (eds.), Nonlinear Structures and Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12391-8_2 11

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