Nonlinear Structures & Systems, Volume 1

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Structures & Systems, Volume 1 Gaetan Kerschen Matthew R. W. Brake Ludovic Renson Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 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

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.

River Publishers Nonlinear Structures & Systems, Volume 1 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 Gaetan Kerschen • Matthew R. W. Brake • Ludovic Renson Editors

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-4380-011-8 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, 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 & Systems represents one of the nine volumes of technical papers presented at the 39th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held on February 8–11, 2021. The full proceedings also include volumes on 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, Energy Harvesting & 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 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 will be discussed. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Liège, Belgium Gaetan Kerschen Houston, TX, USA Matthew R. W. Brake London, UK Ludovic Renson v

Contents Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model......... 1 Eric Robbins, Trent Schreiber, Arun Malla, Benjamin R. Pacini, Robert J. Kuether, Simone Manzato, Daniel R. Roettgen, and Fernando Moreu Nonlinear Variability due to Mode Coupling in a Bolted Benchmark Structure ............................ 15 Mitchell P. J. Wall, Matthew S. Allen, and Robert J. Kuether Nonlinear Dynamic Analysis of a Shape Changing Fingerlike Mechanism for Morphing Wings ......... 19 Aabhas Singh, Kayla M. Wielgus, Ignazio Dimino, Robert J. Kuether, and Matthew S. Allen Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure ............................................................................................................. 33 Moheimin Khan, Patrick Hunter, Benjamin R. Pacini, Daniel R. Roettgen, and Tyler F. Schoenherr A Non-Masing Microslip Rough Contact Modeling Framework for Spatially and Cyclically Varying Normal Pressure .......................................................................................... 53 Justin H. Porter, Nidish Narayanaa Balaji, and Matthew R. W. Brake Finite Elements and Spectral Graphs: Applications to Modal Analysis and Identification ................ 61 Nidish Narayanaa Balaji and Matthew R. W. Brake Effects of the Geometry of Friction Interfaces on the Nonlinear Dynamics of Jointed Structure ......... 67 Jie Yuan, Loic Salles, and Christoph Schwingshackl Bifurcation Analysis of a Piecewise-Smooth Freeplay System................................................. 75 Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi Insights on the Bifurcation Behavior of a Freeplay System with Piecewise and Continuous Representations ..................................................................................................... 79 Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi Model Updating and Uncertainty Quantification of Geometrically Nonlinear Panel Subjected to Non-uniform Temperature Fields ................................................................................. 83 Kyusic Park and Matthew S. Allen On Affine Symbolic Regression Trees for the Solution of Functional Problems ............................. 95 M. D. Champneys, N. Dervilis, and K. Worden Comparative Analysis of Mechanical and Magnetic Amplitude Stoppers in an Energy Harvesting Absorber ............................................................................................................. 109 Tyler Alvis, Mikhail Mesh, and Abdessattar Abdelkefi NIXO-Based Identification of the Dominant Terms in a Nonlinear Equation of Motion ................... 113 Michael Kwarta and Matthew S. Allen vii

viii Contents Nonlinear Dynamics and Characterization of Beam-Based Systems with Contact/Impact Boundaries .. 119 M. Trujillo, M. Curtin, M. Ley, B. E. Saunders, G. Throneberry, and A. Abdelkefi Experimental Modal Analysis of Geometrically Nonlinear Structures by Using Response-Controlled Stepped-Sine Testing ...................................................................... 123 Taylan Karaag˘açlı and H. Nevzat Özgüven On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness ........... 135 T. Gowdridge, N. Dervilis, and K. Worden A Hybrid Static and Dynamic Model Updating Technique for Structures Exhibiting Geometric Nonlinearity.......................................................................................................... 151 Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips Insights on the Dynamical Responses of Additively Manufactured Systems ................................. 163 M. Curtin, M. Ley, M. Trujillo, B. E. Saunders, G. Throneberry, and A. Abdelkefi Characterization of Nonlinearities in a Structure Using Nonlinear Modal Testing Methods .............. 167 Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips Challenges of Characterizing Geometric Nonlinearity of a Double-Clamped Thin Beam Using Nonlinear Modal Testing Methods ................................................................................ 179 Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips Establishing the Exact Relation Between Conservative Backbone Curves and Frequency Responses via Energy Balance..................................................................................... 189 Mattia Cenedese and George Haller Joint Interface Contact Area Predictions Using Surface Strain Measurements ............................. 193 Aryan Singh and Keegan J. Moore Towards Compact Structural Bases for Coupled Structural-Thermal Nonlinear Reduced Order Modeling ............................................................................................................. 197 X. Q. Wang and Marc P. Mignolet Ensemble of Multi-time Resolution Recurrent Neural Networks for Enhanced Feature Extraction in High-Rate Time Series........................................................................................... 207 Vahid Barzegar, Simon Laflamme, Chao Hu, and Jacob Dodson Modelling the Effect of Preload in a Lap-Joint by Altering Thin-Layer Material Properties.............. 211 Nidhal Jamia, Hassan Jalali, Michael I. Friswell, Hamed Haddad Khodaparast, and Javad Taghipour

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model Eric Robbins, Trent Schreiber, Arun Malla, Benjamin R. Pacini, Robert J. Kuether, Simone Manzato, Daniel R. Roettgen, and Fernando Moreu Abstract A proper understanding of the complex physics associated with nonlinear dynamics can improve the accuracy of predictive engineering models and provide a foundation for understanding nonlinear response during environmental testing. Several researchers and studies have previously shown how localized nonlinearities can influence the global vibration modes of a system. This current work builds upon the study of a demonstration aluminum aircraft with a mock pylon with an intentionally designed, localized nonlinearity. In an effort to simplify the identification of the localized nonlinearity, previous work has developed a simplified experimental setup to collect experimental data for the isolated pylon mounted to a stiff fixture. This study builds on these test results by correlating a multi-degree-of-freedom model of the pylon to identify the appropriate model form and parameters of the nonlinear element. The experimentally measured backbone curves are correlated with a nonlinear Hurty/Craig-Bampton (HCB) reduced order model (ROM) using the calculated nonlinear normal modes (NNMs). Following the calibration, the nonlinear HCB ROM of the pylon is attached to a linear HCB ROM of the wing to predict the NNMs of the next-level wing-pylon assembly as a pre-test analysis to better understand the significance of the localized nonlinearity on the global modes of the wing structure. Keywords Nonlinear dynamics · Nonlinear normal modes · Backbone curves · Craig-Bampton reduction · Multi-harmonic balance 1 Introduction Large deformations, materials, and displacement-dependent boundary conditions are all potential sources of nonlinearity in engineering applications. Effects of nonlinearity on structural dynamic response include internal resonances, amplitudedependent modal characteristics, self-excited oscillation, and non-repeatability, to name a few. These physics have been studied by numerous researchers for several decades, resulting in major developments toward modeling, analysis, and experimental techniques [1]. While linear models can yield adequate results for predicting and characterizing structural dynamic response, nonlinear effects can influence the accuracy of these models and introduce behavior not supported by linear theory. Including nonlinear physics in engineering models can often improve the model’s predictive capability and E. Robbins · F. Moreu University of New Mexico, Albuquerque, NM, USA e-mail: erobbins@unm.edu; fmoreu@unm.edu T. Schreiber Georgia Institute of Technology, Atlanta, GA, USA e-mail: trent.schreiber@gatech.edu A.Malla Virginia Polytechnic Institute and State University, Blacksburg, VA, USA e-mail: arun.malla@vt.edu B. R. Pacini ( ) · R. J. Kuether · D. R. Roettgen Sandia National Laboratories, Albuquerque, NM, USA e-mail: brpacin@sandia.gov; rjkueth@sandia.gov; drroett@sandia.gov S.Manzato Siemens Industry Software, Leuven, Belgium e-mail: simone.manzato@siemens.com © The Society for Experimental Mechanics, Inc. 2022 G. Kerschen et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-77135-5_1 1

2 E. Robbins et al. Fig. 1 (Left) Demo aluminum aircraft test setup from [3]; (right) pylon subassemblies (marked by red boxes) even provide opportunities for improved performance in design. Adequate modeling relies thoroughly on experimental as well as computational techniques [2]. Cooper et al. studied the nonlinear dynamics of a demonstration aluminum aircraft in [3] with intentionally designed, localized nonlinearities. This investigation utilized finite element modeling and experimental test-based identification to extend linear analysis techniques to develop a nonlinear model of the system, following the approach described in [4]. This approach was applied to the demo aircraft depicted in Fig. 1. On the wings of this structure, the subcomponents representing engine pylon subassemblies were mounted as shown in the red boxes in Fig. 1, and consisted of two “block” components, a “thin beam” and a swinging “tip mass.” Potential sources of nonlinearity marked on the right of Fig. 1 include (1) geometric nonlinearity of the thin beam, (2) contact with the blocks, and (3) friction in the bolted connections. During the experimental procedures presented in [3], the identification of the nonlinear elements of the pylon proved difficult due to the high modal density of the aircraft structure and the influence of the nonlinearity on the global modes. The results of the initial study motivated further investigations to identify the nonlinearity by removing the pylon, and hence the localized nonlinearity, from the aircraft assembly, and attaching it to a more rigid test fixture with less modal density in the frequency range of interest. The study by Ligeikis et al. performed system identification of this isolated pylon by executing stepped-sine and free decay experiments and post-processing the results to identify the frequency and damping backbones of the pylon [5]. The pylon assembly was mounted within a stiff box fixture structure as shown in Fig. 2. Figure 2(a) shows the overall view of the experimental setup, while Fig. 2(b) shows a more detailed photograph of the accelerometer locations during the tests. Data collected from these tests were used to develop and validate a single-degree-of-freedom nonlinear model of the pylon and provided motivation for the current research presented in this paper. This paper describes the identification of the localized nonlinearity of the isolated pylon structure for a multi-degree-offreedom (MDOF) representation of the structure. A detailed, linear finite element model (FEM) was created of the fixturepylon setup in Fig. 2, from which a Hurty/Craig-Bampton (HCB) superelement was then created with physical degreesof-freedom at the location of the nonlinearity [6, 7]. This nonlinear HCB superelement was calibrated using the frequency backbone curves extracted from the experimental results in [5]. The nonlinear normal modes (NNMs) [8] of the HCB model were computed using the multi-harmonic balance (MHB) approach [9], from which the calculated frequency-amplitude curves were correlated with the test data. Different constitutive model forms were explored, and parameters were optimized to determine which model best replicated the test data. The pylon model was further verified by comparing stepped-sine simulations to the experimental stepped-sine response. The calibrated, nonlinear HCB model of the pylon was next coupled to a linear HCB wing model in order to gain new insight into the behavior of the next-level wing-pylon assembly. The rest of the paper is organized as follows. Section 2 provides a brief overview of HCB and NNM theory used throughout the identification and analysis efforts. Section 3 presents the post-processing of the stepped-sine experimental data from Ligeikis et al. and its utilization in the development and validation of the nonlinear MDOF model of the isolated pylon assembly. Section 4 discusses the results of the next-level assembly study when mounting the calibrated pylon to a winglike structure. The influence of the nonlinearity on the global modes of the wing is discussed in the context of the resulting NNMs of the assembly. Finally, Section 5 summarizes the conclusions and future work.

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 3 Fig. 2 Isolated fixture-pylon assembly test setup from [5]. The (a) refers to the left-most picture and the (b) the two right-most pictures 2 Theory A brief overview of the theory is presented in the following subsections. Section 2.1 describes the HCB methodology deployed to generate the nonlinear ROM of the pylon subassembly. Section 2.2 provides a brief overview of MHB and its use for calculating periodic orbits, or NNMs, of conservative systems. 2.1 Nonlinear Hurty/Craig-Bampton Reduction Hurty/Craig-Bampton reduction is a method often used to reduce large-scale finite element models to a lower-order and more manageable scale. It retains the physical coordinates at the interface (boundary) of a structure, which lends itself well to adding nonlinear constitutive elements that can be readily parameterized. The remaining DOFs in the model are reduced with a fixed-interface modal basis. Hurty provided the first development based on fixed-interface and constraint modes [6]. Craig and Bampton [7] simplified Hurty’s method, which has been widely adopted due to its accuracy, ease of implementation, and computational efficiency. The Craig-Bampton method is detailed in [10] and summarized here. The undamped equations of motion for the full physical system with a conservative nonlinear forcing term is written as M¨u+Ku+fnl (u) =F(t) (1) The transformation matrix, CB, transforms the full physical space DOFs, u, to a reduced space containing fixed-interface modal coordinates, ηfi, and retained boundary DOFs, ub: u= CB ηfi ub (2) This results in the transformation into reduced coordinates: MCB ¨ ηfi ¨ub + KCB ηfi ub + 0 fnl (ub) = FCB(t) (3)

4 E. Robbins et al. where MCB = CB TM CB KCB = CB TK CB FCB(t) = CB TF(t) (4) In this study, the reduced mass matrix, MCB, and stiffness matrix, KCB, are obtained for the fixture-pylon and wing-pylon assemblies using Sierra Structural Dynamics [11] finite element codes. With the undamped equations of motion in reduced coordinates (3), a nonlinear restoring force can be added to any boundary DOF, ub, to obtain the nonlinear undamped equations of motion. Two types of elements were used in Sect. 3 to explore the effect of the nonlinearity on the NNM backbone curve predicted using MHB, namely, cubic spring elements and linear penalty springs. 2.2 Multi-harmonic Balance The MHB method is a Fourier-Galerkin mathematical technique to solve for periodic solutions for nonlinear equations of motion [9]. The technique approximates the displacements with periodic solutions represented by a finite number of harmonics in a Fourier series: u(t) = cx 0√ 2 + Nh k=1 su k sin(kωt) +c u k cos(kωt) (5) fnl (u) = cf 0√ 2 + Nh k=1 sf k sin(kωt) +c f k cos(kωt) (6) Note that the displacement field, u(t), can be any set of DOF to describe the dynamics of a system (i.e., physical DOF, modal DOF, etc.). Projecting the Fourier basis onto the nonlinear equations of motion, such as those in Eqns. (1) and (3), and performing a Galerkin projection onto the periodic functions produces the frequency-domain equations of motion: A(ω)z +b(z) =0 (7) where z is the collection of Fourier coefficients, ω is the period of the harmonic frequency, A(ω) is the linear dynamic stiffness matrix, and b(z) is the nonlinear restoring force. The algorithm is coupled with pseudo-arclength continuation to follow a branch of periodic solutions which is initialized by a starting guess based on the low-energy, linearized modes of the system [12]. The pseudo-arclength continuation technique is used with a Newton solver to find periodic solutions by satisfying a residual function: R(z,ω) = ⎡ ⎢⎣ A(ω)z +b(z) VT z ω − z ω (k=1) ⎤ ⎥⎦ (8) where Vrepresents the tangent prediction vector. Each value of z and ωthat solve R(z, ω) =0 represents an NNM solution along the branch. 3 Fixture-Pylon Model Calibration This section describes the calibration efforts of the nonlinear HCB model of the fixture-pylonassembly (Sect. 3.1). This was accomplished by extracting the amplitude-dependent frequency backbone curve from the experimental stepped-sine data of the fixture-pylon assembly from [5] (Sect. 3.2). This data was used with the nonlinear HCB model of the test assembly to evaluate different nonlinear element model forms and select the most appropriate (Sect. 3.3). The calibrated nonlinear pylon model was further validated by comparing the experimental stepped-sine data to the simulated response of the model (Sect. 3.4).

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 5 3.1 Fixture-Pylon Finite Element Model A detailed finite element model (FEM) of the fixture-pylon assembly was used to create an initially linear HCB model. The mesh of the finite element model was generated using CUBIT [13], and the Sierra Structural Dynamics codes [11] were used for the eigenvalue analysis and HCB reduction. An eigenvalue analysis was performed on the fixture-pylon assembly with a fixed base to determine the linear natural frequencies and mode shapes, such as the first mode shown in Fig. 3b. This f rist mode is the “swinging pendulum” mode of the pylon, with a natural frequency of 7.3 Hz. This was the target mode for the experiments conducted in Ligeikis et al. [5] and was used to characterize the nonlinearity between the thin beam and block. The linear ROM was generated from an HCB reduction with 16 fixed-interface modes and retained seven physical DOFs (drive point, accelerometer s1, accelerometer s2, and four virtual nodes). To account for the nonlinearity, a whole joint modeling approach [14] was used to constrain the finite element nodes along the contact edge of the block to a single, virtual node as shown in Fig. 4; an analogous whole joint is created along a node line along the thin beam. The nonlinearity localized within the pylon block was modeled as a 1-D constitutive element between the virtual node pairs, resulting in the nonlinear HCBmodel. (a) Mode 1 (fn = 7.3 Hz) (b) Fig. 3 Fixture-pylon CAD assembly; (a) general view; (b) natural frequency and mode shape for mode 1 2 1 3 4 Virtual nodes Virtual nodes Fig. 4 Nonlinear element in pylon block

6 E. Robbins et al. a) b) Fig. 5 Phase (a) and magnitude (b) response spectra at s1 accelerometer point. Dashed line represents quadrature value of 90◦ in (a) and the backbone curve marked in (b) 3.2 Extracting Backbone Curves from Experimental Stepped-Sine Data Experimental data was used from stepped-sine excitation tests presented in [5]. These tests recorded the system’s steadystate response to sinusoidal forcing over a range of discrete frequencies and forcing amplitudes, testing a single frequency and amplitude at a time. This results in a nonlinear force response (NLFR) curve for each forcing amplitude. Compared to other techniques such as broadband/burst random excitation, this method results in a higher quality NLFR to observe the influence of nonlinearity on the resonant modes of interest. The experimental stepped-sine data was initially recorded by accelerometers s1 and s2 (labeled in Fig. 2) into a set of accelerance NLFR curves including both real and imaginary components. The data was used to calculate the magnitude and phase angle response at the s1 location for each of the 11 forcing amplitudes (0.5–20 N); these plots are shown in Fig. 5. The phase resonance condition for nonlinear systems [15] occurs when there is a 90◦ phase difference between the input force and output response. By tracking where this phase quadrature criterion is satisfied between the forcing phase and s1 response phase for each of the forcing amplitudes, the amplitude-dependent resonant frequency of the first bending mode of the pylon can be extracted. The backbone curve is the interpolated curve connecting the quadrature points for the range of forcing amplitudes. It is important to note that perfect quadrature (90◦ phase angle) was not achieved during the experimental testing since this was not the original objective of the test efforts. Thus, the points with phase angle closest to 90◦ were used when constructing the backbone curve. The exact phase angle of these closest points ranged from 81 to 96◦. The final acquired backbone curve, shown in Fig. 5b, displays an initial weak softening nonlinearity at low forcing amplitudes (0.5–7 N), which transitions to a strong hardening behavior as the force level continues to increase (7–20 N). This transition coincides with the forcing level at which the thin beam element begins to contact the block elements as the mass swings at higher amplitudes [5]. Reflecting this transition was vital when developing the nonlinear model of the isolated pylon system. 3.3 Calibration of Nonlinear Elements The interaction between the pylon “block” and “thin beam” components is a significant source of nonlinearity in the fixturepylon system, as evidenced by the sudden stiffening observed in the experimental backbone curve. A nonlinear constitutive element was added to the HCB model and two constitutive elements were considered for this connection: a cubic spring element and a gap spring element. These elements were chosen as candidates since both are capable of producing a strong hardening behavior for the frequency backbone curve. The initial softening behavior was neglected for this effort, as it occurred over a small frequency range that was negligible compared to that of the hardening behavior. The cubic spring element was connected to the virtual nodes in Fig. 4 with a restoring force defined as fNL(x1,x2) =kNL(x2 −x1) 3 (9)

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 7 Fig. 6 Comparison of gap and cubic spring models to experimental backbone (s1 location); parameters: kNL =4e10N/m, kpen =7e4N/m, and xgap =0.68mm Here kNL is the nonlinear spring constant and x1 and x2 are the displacements of the virtual nodes located on the left side of the thin beam. Since there was no directional dependence on this element, only a single cubic spring was modeled to capture the stiffening effect. Gap elements were modeled by two linear penalty springs with one each attached to the left and right virtual node connections in Fig. 4. These springs only applied a restoring force when the relative displacement between the virtual node pairs was sufficient to close the gap. The restoring force for the penalty springs can thus be expressed as fgap xi,xj = kpen δij −xgap for δij >xgap 0 otherwise (10) where kpen was the linear spring constant of the penalty springs, δij =xi −xj, and xgap is the gap distance of the contact element. The elements were placed between nodes with displacements x1 and x2, as well as between x3 and x4 to represent the restoring force on the beam/block on each side. By adding the described nonlinear constitutive elements to the HCB model, a nonlinear reduced order model of the fixture-pylon subassembly was developed using each type of element. Frequency backbone curves were computed for both models using MHB to calculate the NNMs. Figure 6 shows the comparison between the experimental backbone curve and the nonlinear HCB ROMs corresponding to the best fit cubic spring and penalty spring elements. Note that these curves are plotted versus the displacement amplitudes at the s1 location and the frequencies are normalized to their respective linear natural frequencies. A parametric study was conducted to determine the set of nonlinear parameters for each model that minimized the error to the experimental backbone. The penalty spring was able to better match the experimental backbone curve and was thus selected as the constitutive element to represent the nonlinearity of the pylon. The penalty spring element was calibrated to the s1 location, and the plot in Fig. 7 shows the correlation of the backbone at the s2 location, again showing good agreement with both sets of experimental data. 3.4 Stepped-Sine Validation Using the calibrated penalty spring elements, a stepped-sine simulation was performed to validate the nonlinear HCB model’s ability to reproduce the stepped-sine experimental data. Rayleigh mass and stiffness proportional damping was used to calculate the damping matrix for the model based on linear damping ratios for modes 1 and 3 [5]. The steppedsine simulation was conducted by inducing a constant amplitude harmonic force on the fixture drive point node at various oscillating frequencies. The model was integrated using MATLAB’s ode15s solver to steady state and the response amplitude was recorded as a single point in the NLFR curve. The frequency was incremented with positive frequency steps until the

8 E. Robbins et al. Fig. 7 Comparison between experimental and NNM backbones for calibrated penalty spring element Fig. 8 Stepped-sine test results of fixture-pylon (- - -) experimental (—) simulation final frequency was reached. This was repeated for several force amplitudes corresponding to the experimental results. The drive point DOF was on the stiff fixture and the output DOF was at the s1 location. The results are shown in Fig. 8 for the experiment (- - -) and simulation (—) for 7 of the 11 forcing amplitudes. The comparison plots reveal that the amplitude of the simulated data matches well with the experimental data, but the jump-down frequency seems to be in slight disagreement for most amplitudes. The 17 N forcing in the nonlinear response nearly produced an identical response between the simulation and experiment. It can be seen in Fig. 8 that the linear resonances in the test data were consistently occurring around 1.03, with a slight softening behavior, whereas the linear resonances in the simulation were occurring around 1.045. Some of the most significant differences between the simulation and experimental results may be attributed to the difference in the damping formulation in the model. Here a constant damping ratio is assumed for each mode; however, the experiments in [5] reveal that the damping backbone curves are amplitude dependent. The damping is known to influence the resonance condition for NLFR curves, so it is likely that the model is missing the physics to capture this dependence.

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 9 4 Wing-Pylon NNM Computation This section describes the simulations performed on the nonlinear HCB model of the wing-pylon-fixture assembly (Sect. 4.1). The calibrated nonlinear pylon model from Sect. 3 was attached to a linear HCB ROM of the wing structure and the NNMs of the next-level assembly are calculated. The first NNM under investigation (Sect. 4.2) corresponds to the localized mode of the pylon when connected to the wing. The second NNM (Sect. 4.3) corresponds to the first bending mode of the wing. 4.1 Wing-Pylon Finite Element Model The mesh of the wing-pylon-fixture assembly was generated using CUBIT [13], and the Sierra Structural Dynamics codes [11] were used for the eigenvalue analysis and HCB reduction. This assembly has free-free boundary conditions. The linear ROM of the wing-pylon-fixture assembly was generated with an HCB reduction that used 30 fixed-interface modes and retained physical DOF for various drive points along the wing and fixture in addition to the same accelerometer and virtual nodes in Sect. 3.1. The calibrated penalty spring elements between the virtual nodes in the pylon were added to the linear HCB ROM to generate the nonlinear HCB model. A linear eigenvalue analysis was performed on the wing-pylon-fixture assembly to obtain the linear natural frequencies and mode shapes without the inclusion of the penalty springs. Figure 9 shows the elastic modes of interest for the model, in which modes 1 and 2 are the starting points for the NNM computations in Sects. 4.2 and 4.3, respectively. The first mode of the wing-pylon-fixture assembly (7.3 Hz) is a localized first bending mode of the pylon and is the same as the first mode of the fixture-pylon assembly that was used to calibrate the nonlinear element. The second mode is a combination of bending in the wing and swinging of the pylon mass at a resonant frequency of 22.2 Hz. The seventh mode imparts torsional motion (a) Mode 1 (fn = 7.3 Hz) (b) Mode 2 (fn = 22.2 Hz) (c) Mode 7 (fn = 102.1 Hz) (d) Fig. 9 Wing-pylon CAD assembly (a) and mode shapes/frequencies for mode 1 (b),mode 2 (c), and mode 7 (d)

10 E. Robbins et al. in the wing and higher order bending of the pylon at 102.1 Hz. This mode is included to help explain the modal interactions that occur within the next-level assembly. 4.2 NNM 1 Computation Figure 10 shows the corresponding simulated data for NNM 1 which continues from the linearized mode at 7.3 Hz at a low energy level. The NNMs were calculated using MHB with up to the seventh harmonic in the Fourier approximation, and the frequency-energy plot (FEP) is shown in Fig. 10(a, b). The FEP for NNM 1 reveals that the penalty spring does not introduce nonlinearity into the dynamic response until about 1E-02 J, at which point the frequency begins to stiffen. As the energy in the NNM increases, the backbone frequency increases until an internal resonance occurs around 7.4 Hz. The tongue occurs just where the NNM 2 FEP (the frequency divided by an integer of three) crosses the NNM 1 FEP, indicating that this is a 3:1 modal interaction between NNM 1 and 2. The displacement time histories shown in Fig. 10(c, d) show the higher harmonic content of the response on the tongue as the wing tip completes three oscillations during the fundamental period of motion. In this case, the higher frequency content produced by the nonlinearity at the pylon block strongly excites mode 2. Figure 10(e, f) shows the frequency content of the displacement time histories for the wing tip and the pylon block, thus confirming the spectral content of the particular periodic orbit. It is worth noting here the difference between this NNM and the NNM of the first bending mode computed from thefixturepylon model in Sect. 3. This localized mode of the pylon produces nearly equivalent solutions along the main backbone curve; however, the dynamics of the next-level assembly clearly influence whether or not the mode can interact with other modes of the system. The introduction of the 3:1 modal interaction is strictly due to the dynamics of the wing structure, thus highlighting the importance of the fixturing when predicting NNMs of a subcomponent. A stiff or rigid frame may simplify the nonlinear dynamics of the structure by avoiding any modal interactions (as motivated by the system in Sect. 3). This approach may not necessarily reveal the potentially damaging exchanges of energy that could occur within the system where the nonlinearity introduced into the next-level assembly can introduce global nonlinear effects. 4.3 NNM 2 Computation The simulated data for NNM 2 is shown in Fig. 11. The NNMs were again calculated using MHB with up to the seventh harmonic in the Fourier approximation, and the frequency-energy plot (FEP) is shown in Fig. 11(a, b). The backbone for NNM 2 does not begin to stiffen until approximately 1 J. Over the entire energy range of the computed mode, the penalty spring produces a minimal frequency shift over the operating range, going from 22.2 Hz to about 22.25 Hz, or about 0.2% increase. This suggests that the localized nonlinearity in the pylon does not significantly shift the frequency of the wing bending mode. A more interesting observation comes from the tongue that emanates along the backbone. The backbone crossings of NNM 2 with NNM 7 (divided by a frequency integer of five) are shown in Fig. 11(a, b), indicating a 5:1 modal interaction. The displacement time histories shown in Fig. 11(c, d) correspond to the tip of the internal resonance where the pylon completes five oscillations in one oscillation on the wing tip. Figure 11(e, f) shows the frequency content of the displacement time histories for the wing tip and the pylon block, thus highlighting the dominant frequency content of the motion. It is interesting to observe the relative effect (or lack thereof) of the penalty spring on the backbone frequency of the wing bending mode, i.e., NNM 2. This mode could be effectively assumed to be linear in practice if not for the presence of the modal interaction with NNM 7. NNM theory provides the theoretical foundation to understand the conditions required for modes to interact, and thus reinforces that it does not necessarily require the mode to have a significant shift in frequency. A modal interaction may occur when a higher frequency NNM of the system has a significant shift, thus satisfying the condition that resonant frequencies are commensurate at a given energy level. This highlights the importance of investigating the NNMs of the next-level assembly and how fixturing decisions can introduce (or eliminate) complex behavior in the system. The wing bending mode was demonstrated here to exchange energy with the higher order mode of the pylon (i.e., NNM 7). Additionally, this mode was also able to receive energy exchange from NNM 1 due to the presence of the 3:1 modal interaction in Sect. 4.2.

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 11 (a) (b) (c) (d) (e) (f) Fig. 10 FEP plots for NNM 1 (a, b) and the corresponding (red point) displacement time histories of the s1 node (c) and the wing tip (d). The frequency content of the displacement time histories shown for s1 node (e) and wing tip (f)

12 E. Robbins et al. (a) (b) (c) (d) (e) (f) Fig. 11 FEP plots for NNM 2 (a, b) and the corresponding (red point) displacement time histories of the s1 node (c) and the wing tip (d). The frequency content of the displacement time histories shown for s1 node (e) and wing tip (f)

Pre-test Predictions of Next-Level Assembly Using Calibrated Nonlinear Subcomponent Model 13 5 Conclusions This research is built upon the experimental study of the isolated fixture-pylon assembly from previous research. A nonlinear reduced order model of their test assembly was used to identify the nonlinearity localized in the pylon when the thin beam contacts the surrounding support blocks. A penalty spring element was used to describe this nonlinear contact behavior and produced frequency backbone curves that agreed well with measured results. This model was validated through comparison of displacement response from stepped-sine simulations to those measured during the previous tests. A nonlinear reduced order model of the next-level assembly comprising the pylon, wing, and fixture block was created using this calibrated nonlinear pylon model to generate pre-test predictions in the form of frequency-energy curves for the first two nonlinear normal modes. These were both shown to have internal resonances due to the dynamics of the next-level assembly, providing valuable insight into the design of future experiments and potential nonlinear phenomena to be observed in the data. The results presented on the wing-pylon-fixture reveal the complex physics associated with the dynamics of the next-level assembly and fixturing. The NNM framework combined with nonlinear system identification can serve as a useful design tool to understand potential regimes in response when modes can interact. Depending on the objective of the structure, the tools utilized throughout this study can be used to tailor the dynamics of the system for the intended needs, i.e., either exploit or eliminate the modal interactions. Future work will seek to validate these findings on test hardware with a variable length wing. Acknowledgments This research was conducted at the 2020 Nonlinear Mechanics and Dynamics (NOMAD) Research Institute supported by Sandia National Laboratories and hosted by the University of New Mexico. This paper 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 research was also 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. The authors would like to thank Amy Chen of Sandia National Laboratories for her efforts in creating the finite element meshes used throughout this study. References 1. 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AIAA J. 3(4), 678–685 (1965) 7. Craig Jr., R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analysis. AIAA J. 6(7), 1313–1319 (1968) 8. 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 9. Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems. Springer (2009). https://doi.org/10.1007/978-3-030-14023-6 10. Craig Jr., R.R., Kurdila, A.J.: Fundamentals of Structural Dynamics, 2nd edn. Wiley (2006) 11. Sierra Structural Dynamics Development Team Sierra/SD – User’s Manual – 4.56. SAND2020–3028 (2020) 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. CUBIT Development Team. CUBIT 15.6 User Documentation. SAND2020–4156 W 14. 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Nonlinear Variability due to Mode Coupling in a Bolted Benchmark Structure Mitchell P. J. Wall, Matthew S. Allen, and Robert J. Kuether Abstract This paper presents a set of tests on a bolted benchmark structure called the S4 beam with a focus on evaluating coupling between the first two modes due to nonlinearity. Bolted joints are of interest in dynamically loaded structures because frictional slipping at the contact interface can introduce amplitude-dependent nonlinearities into the system, where the frequency of the structure decreases, and the damping increases. The challenge to model this phenomenon is even more difficult if the modes of the structure become coupled, violating a common assumption of mode orthogonality. This work presents a detailed set of measurements in which the nonlinearities of a bolted structure are highly coupled for the first two modes. Two nominally identical bolted structures are excited using an impact hammer test. The nonlinear damping curves for each beam are calculated using the Hilbert transform. Although the two structures have different frequency and damping characteristics, the mode coupling relationship between the first two modes of the structures is shown to be consistent and significant. The data is intended as a challenge problem for interested researchers; all data from these tests are available upon request. Keywords Mode coupling · Bolted joint · Backbone curve · Hilbert transform · Damping 1 Introduction Bolted joints present a challenge for characterizing and modeling the dynamics of a structure. As a structure is loaded, frictional slip at the joint contact interface results in a decrease in stiffness and an increase in damping. Although this is typically considered a weak nonlinearity, bolted joints can account for up to 90% of the damping in some structures [1]. This type of nonlinearity is referred to as a microslip nonlinearity because only part of the contact region in the joint begins to slip, while some part of it remains fully stuck. This work is concerned with experimentally quantifying the nonlinearity in a bolted structure. Only the damping nonlinearity is discussed here, but similar results are seen for the frequency nonlinearity as well, although the frequency shifts are smaller than the changes in damping. The Hilbert transform is used to calculate the damping of the mode as a function of amplitude, which was originally proposed by [2] and has been developed further by [3]. When possible, it is very convenient to treat the modes of a jointed structure as uncoupled so that the linearized eigenvectors can be used to decompose the model into a set of uncoupled, nonlinear equations of motion. Previous studies have demonstrated that this assumption holds for some systems [4, 5]. This work presents a set of test results where mode coupling has a significant effect on the damping of the structure. Multiple drive points are used on two sets of nominally identical beams to excite different combinations of modes to different amplitudes, In each case, the modal damping is found, and mode coupling is observed when the damping of a mode changes due to the amplitude of another mode. The two structures used in this work will be referred to as the 2017 beam, which was previously used in [6, 7], the 2020 beam which was previously used in [8]. M. P. J. Wall ( ) · M. S. Allen Department of Engineering Physics, University of Wisconsin–Madison, Madison, WI, USA e-mail: mwall4@wisc.edu; msallen@engr.wisc.edu; matt.allen@byu.edu R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: rjkueth@sandia.gov © The Society for Experimental Mechanics, Inc. 2022 G. Kerschen et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-77135-5_2 15

16 M. P. J. Wall et al. Fig. 1 Left: Mode 1 and Mode 2 of the S4 beam from a FEM. Experimental setup for the S4 beam with indexes shown for each drive point and the reference coordinate system 2 Experimental Setup and Procedure The experimental setup for the S4 beam is shown in Fig. 1. The 2017 beam is shown but the same setup was used for the 2020 beam. The beams were suspended by bungee cords at two points. The bolt preload on the left bolt was measured by a load washer which uses a strain gauge to measure the tension in the bolt. The torque to reach the desired preload was recorded and then the same torque was used to tighten the other bolt. Five uniaxial accelerometers were used to instrument the beam: one at each end in the Z-direction near the bolts, one at the center in the Z-direction between the two beams, and one on each beam in the center on the bottom facing side. Five drive points were used in the tests, each selected to excite the modes of the structure to different relative amplitudes. The two modes of interest for the S4 beam are Mode 1 and Mode 2, shown from a finite element model (FEM) of the S4 beam from [6] in Fig. 1. The first mode is the first out-of-phase bending mode and the second mode is the first in-phase bending mode. The drive points are named DP1, DP3, DP4, and DP5. DP1 is in the Z-direction in the middle of the beam and will excite Mode 1 and 2. DP3 is 25.4 mm (1 in.) right of DP1, which will excite Mode 1 more than Mode 2. DP4 is in the Z-direction near the bolt, which will not excite Mode 1 but will excite Mode 2. Finally, DP5 is in the Z-direction about 5 inches left from DP1, which is close to a node for Mode 2 based on the FEM, so the Mode 1 amplitude should be greater than Mode 2. The first step of the test procedure was to measure the linear mode shapes of the system. This was done by using small impacts with an amplitude of less than 1 N to excite the structure. The second step of the testing used a large impact hammer with a soft rubber tip, which ranged in impact amplitudes from roughly 100 N to 600 N. The ringdown of the nonlinear impact was recorded, the time data was modally filtered with the linear mode shapes, bandpass was filtered, and the Hilbert transform was used to calculate the damping ratio of the mode of interest versus the peak modal velocity of the mode of interest. 3 Results Results for Mode 1 and Mode 2 of the 2017 beam are shown in Fig. 2. For Mode 1, across all drive points, the damping is very consistent. DP4 is omitted since it did not produce a high enough Mode 1 amplitude. The damping for Mode 1 decreases with increasing amplitude, which is the opposite of what one would expect for a microslip joint nonlinearity [9], suggesting the nonlinearity in this mode is dominated by something other than microslip. Mode 2 shows quite different characteristics. The damping for a given impact can vary by almost an order of magnitude depending on what drive point is used. There is also some correlation between higher damping tests and force amplitude although those details are omitted for brevity. The lowest damping for Mode 2 is achieved at DP4, where there is almost no Mode 1 excitation. This may be considered the baseline damping for Mode 2 when it is isolated from Mode 1. In DP5 Mode 1 is excited more than Mode 2, and the results for this drive point show a relatively repeatable but higher amount of damping. The large variability in damping only occurs at DP1 and DP3, where both Mode 1 and Mode 2 would be excited to relatively high amplitudes. This drive point dependence is the basis for the assertion that mode coupling is a strong contributor to the damping in some modes. Interestingly, this appears to be a one-way coupling relationship, the damping in Mode 1 is independent of the drive point (and hence the amplitude of Mode 2), but Mode 1 greatly affects the damping in Mode 2. Clearly, mode coupling must be accounted for to model or predict the nonlinear damping of Mode 2. The results for Mode 1 and 2 of the 2020 beam are shown in Fig. 3. The damping curves for each mode are distinct from the result for the 2017 beam. The damping increases at lower amplitudes for both modes, in a manner consistent with a

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