Nonlinear Structures & Systems, Volume 1: Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 (Conference ... Society for Experimental Mechanics Series) 3030771342, 9783030771348

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Conference Proceedings of the Society for Experimental Mechanics Series

Gaetan Kerschen Matthew R. W. Brake Ludovic Renson   Editors

Nonlinear Structures & Systems, Volume 1 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021

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

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

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

Gaetan Kerschen • Matthew R. W. Brake • Ludovic Renson Editors

Nonlinear Structures & Systems, Volume 1 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021

Editors Gaetan Kerschen Structures & Sys Lab, Bldg B52/3 University of Li`ege, Space Li`ege, Belgium

Matthew R. W. Brake Department of Mechanical Engineering Rice University Houston, TX, USA

Ludovic Renson Department of Mechanical Engineering Imperial College London London, UK

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-77134-8 ISBN 978-3-030-77135-5 (eBook) https://doi.org/10.1007/978-3-030-77135-5 © The Society for Experimental Mechanics, Inc. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

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 Houston, TX, USA London, UK

Gaetan Kerschen Matthew R. W. Brake Ludovic Renson

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Contents

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

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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

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Nonlinear Dynamic Analysis of a Shape Changing Fingerlike Mechanism for Morphing Wings . . . . . . . . . Aabhas Singh, Kayla M. Wielgus, Ignazio Dimino, Robert J. Kuether, and Matthew S. Allen

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Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moheimin Khan, Patrick Hunter, Benjamin R. Pacini, Daniel R. Roettgen, and Tyler F. Schoenherr

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A Non-Masing Microslip Rough Contact Modeling Framework for Spatially and Cyclically Varying Normal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Justin H. Porter, Nidish Narayanaa Balaji, and Matthew R. W. Brake

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Finite Elements and Spectral Graphs: Applications to Modal Analysis and Identification . . . . . . . . . . . . . . . . Nidish Narayanaa Balaji and Matthew R. W. Brake

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Effects of the Geometry of Friction Interfaces on the Nonlinear Dynamics of Jointed Structure . . . . . . . . . Jie Yuan, Loic Salles, and Christoph Schwingshackl

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Bifurcation Analysis of a Piecewise-Smooth Freeplay System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

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Insights on the Bifurcation Behavior of a Freeplay System with Piecewise and Continuous Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

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Model Updating and Uncertainty Quantification of Geometrically Nonlinear Panel Subjected to Non-uniform Temperature Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kyusic Park and Matthew S. Allen

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On Affine Symbolic Regression Trees for the Solution of Functional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. D. Champneys, N. Dervilis, and K. Worden

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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

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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 Karaa˘gaçlı 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

Contents

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: [email protected]; [email protected] T. Schreiber Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] A. Malla Virginia Polytechnic Institute and State University, Blacksburg, VA, USA e-mail: [email protected] B. R. Pacini () · R. J. Kuether · D. R. Roettgen Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected]; [email protected]; [email protected] S. Manzato Siemens Industry Software, Leuven, Belgium e-mail: [email protected] © 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

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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

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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 Mu¨ + 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



This results in the transformation into reduced coordinates:       η¨ fi ηfi 0 MCB = FCB (t) + KCB + u¨ b ub fnl (ub )

(2)

(3)

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where MCB =  CB T M CB KCB =  CB T K CB FCB (t) =  CB T F(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: h  u  cx sk sin (kωt) + cuk cos (kωt) u(t) = √0 + 2 k=1

(5)

Nh  f   c f f sk sin (kωt) + ck cos (kωt) fnl (u) = √0 + 2 k=1

(6)

N

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: ⎤ A z + b (ω) (z)    (k=1) ⎥ ⎢ R (z, ω) = ⎣ T z z ⎦ V − ω ω ⎡

(8)

where V represents the tangent prediction vector. Each value of z and ω that 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-pylon assembly (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

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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 first 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 HCB model.

Mode 1 (fn = 7.3 Hz)

(a)

(b)

Fig. 3 Fixture-pylon CAD assembly; (a) general view; (b) natural frequency and mode shape for mode 1

2

1

Virtual nodes

Fig. 4 Nonlinear element in pylon block

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Virtual nodes

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b)

a)

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

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Fig. 6 Comparison of gap and cubic spring models to experimental backbone (s1 location); parameters: kNL = 4e10 N/m, kpen = 7e4 N/m, and xgap = 0.68 mm

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 f or δij > xgap 0 otherwise

(10)

where kpen was the linear spring constant of the penalty springs, δ ij = 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

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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.

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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

Mode 1 (fn = 7.3 Hz)

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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 the fixturepylon 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.

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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)

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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)

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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. Butlin, T., Woodhouse, J., Champneys, A.R.: The landscape of nonlinear structural dynamics: an introduction. Phil. Trans. R. Soc. A. 373, 20140400 (2015). https://doi.org/10.1098/rsta.2014.0400 2. Noel, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017) 3. Cooper, S.B., et al.: Investigating Nonlinearities in a Demo Aircraft Structure Under Sine Excitation. Springer International Publishing, Cham (2020) 4. Cooper, S.B., DiMaio, D., Ewins, D.J.: Integration of system identification and finite element modelling of nonlinear vibrating structures. Mech. Syst. Signal Process. 102, 401–430 (2018) 5. Ligeikis, C., et al. Modeling and Experimental Validation of a Pylon Subassembly Mockup with Multiple Nonlinearities. In: 38th International Modal Analysis Conference (IMAC XXXVIII), Houston, TX (2020) 6. Hurty, W.C.: Dynamic analysis of structural systems using component modes. 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. Segalman, D.J.: Modelling joint friction in structural dynamics. Struct Contr Health Monitor. 13(1), 430–453 (2006) 15. Peeters, M., Kerschen, G., Golinval, J.C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330, 486–509 (2011). https://doi.org/10.1016/j.jsv.2010.08.028

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: [email protected]; [email protected]; [email protected] R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © 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

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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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microslip joint nonlinearity. However, at higher amplitudes, both modes exhibit a decrease in damping, and Mode 2 shows a clear increase at even higher amplitudes. For Mode 1, though the trends for damping are quite different, the repeatability observations for the 2017 beam still hold. At lower amplitudes before passing the peak damping, the curves are quite repeatable between DP1 and DP5. DP3 was not used in this test because it was shown to give nearly identical results to DP1 from the last set of tests. Mode 2 again shows a larger spread in possible damping values depending on the drive point. As with the 2017 beam, when excited at DP4, the results show a repeatable damping curve, at least if some of the results at low amplitudes can be disregarded. This again reinforces the assertion that when Mode 2 is isolated from Mode 1, the damping will be the most repeatable. DP1, with a significant Mode 1 and Mode 2 excitation, again shows a larger spread in damping over a range of amplitudes. Although the damping curves are quite different for the two beams in general, the results show that the mode coupling relationship between the two beams is similar.

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4 Conclusion The experimental results here present a case where the common assumption of uncoupled modes is not accurate. While the damping of Mode 1 was independent of the amplitude of Mode 2, the damping of Mode 2 changed by an order of magnitude depending on the drive point used, so mode coupling was very important for that mode. The authors are not aware of any models that can capture this behavior, so this presents an interesting challenge for the community. Interestingly, even though the damping curves for the two structures were very different, the mode coupling was qualitatively similar. This suggests that the mode coupling depends more on the structure of the system, i.e., the mode shapes and natural frequencies, which are similar between the beam sets, rather than on the details of stick and slip at the interfaces, which were very different between the beams. Acknowledgments This material is based in part on the work supported by the National Science Foundation under Grant Number CMMI1561810. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The views expressed in the article do not necessarily represent the views of the US Department of Energy or the US Government. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2020-13620 C. The authors wish to thank Iman Zare for the excellent work that he did to explore methods for measuring preload, which were instrumental in developing the experimental setup used here. The authors would also like to thank Adam Brink for providing the finite element models of the 2020 beams used throughout the study.

References 1. Beards, C.: Damping in Structural Joints. Shock Vib. Dig. 11(9), 35–41 (1979) 2. Feldman, M.: Non-linear system vibration analysis using Hilbert transform-II. Forced vibration analysis method ‘FORCEVIB’. Mech. Syst. Signal Process. 8(3), 3 (1994) 3. Deaner, B.J., Allen, M.S., Starr, M.J., Segalman, D.J., Sumali, H.: Application of viscous and Iwan modal damping models to experimental measurements from bolted structures. J. Vib. Acoust. 137(2), 021012 (2015). https://doi.org/10.1115/1.4029074 4. Roettgen, D.R., Allen, M.S.: Nonlinear characterization of a bolted, industrial structure using a modal framework. Mech. Syst. Signal Process. 84, 152–170 (2017). https://doi.org/10.1016/j.ymssp.2015.11.010 5. Eriten, M., Kurt, M., Luo, G., Michael McFarland, D., Bergman, L.A., Vakakis, A.F.: Nonlinear system identification of frictional effects in a beam with a bolted joint connection. Mech. Syst. Signal Process. 39(1–2), 1–2 (2013). https://doi.org/10.1016/j.ymssp.2013.03.003 6. Jewell, E., Allen, M.S., Zare, I., Wall, M.: Application of quasi-static modal analysis to a finite element model and experimental correlation. J. Sound Vib. 479, 115376 (2020) 7. Singh, A., et al.: Experimental characterization of a new benchmark structure for prediction of damping nonlinearity. Presented at the 36th International Modal Analysis Conference (IMAC XXXVI), Orlando, FL. http://sd.engr.wisc.edu/wp-uploads/2018/04/31_sin.pdf (2018) 8. Brink, A.R., Kuether, R.J., Fronk, M.F., Witt, B.L., Nation, B.L.: Contact stress and linearized modal predictions of as-built preloaded assembly. ASME J. Vib. Acoust. 142(5) (2020) 9. Ames, N.M., et al.: Handbook on Dynamics of Jointed Structures. Sandia National Laboratories (2009)

Nonlinear Dynamic Analysis of a Shape Changing Fingerlike Mechanism for Morphing Wings Aabhas Singh, Kayla M. Wielgus, Ignazio Dimino, Robert J. Kuether, and Matthew S. Allen

Abstract Morphing wings have great potential to dramatically improve the efficiency of future generations of aircraft and to reduce noise and emissions. Among many camber morphing wing concepts, shape changing fingerlike mechanisms consist of components, such as torsion bars, bushings, bearings, and joints, all of which exhibit damping and stiffness nonlinearities that are dependent on excitation amplitude. These nonlinearities make the dynamic response difficult to model accurately with traditional simulation approaches. As a result, at high excitation levels, linear finite element models may be inaccurate, and a nonlinear modeling approach is required to capture the necessary physics. This work seeks to better understand the influence of nonlinearity on the effective damping and natural frequency of the morphing wing through the use of quasi-static modal analysis and model reduction techniques that employ multipoint constraints (i.e., spider elements). With over 500,000 elements and 39 frictional contact surfaces, this represents one of the most complicated models to which these methods have been applied to date. The results to date are summarized and lessons learned are highlighted. Keywords Morphing wings · Quasi-static modal analysis · Joints · Modal analysis · Hysteresis

1 Introduction A morphing wing has great potential to improve the design and operation of future generations of aircraft. Due to the adaptive nature of the mechanism, the aircraft wing geometry may transform into optimal shapes given the flight condition. Camber morphing technology, for instance, may enhance aircraft high-lift performance during takeoff and landing by reducing friction drag and aerodynamic noise. During high-speed flight, the last portion of the flap may be also adjusted to reduce the overall aerodynamic drag by reducing fuel consumption and emissions [1–3]. Among the wide range of aerodynamic benefits, the morphing wing may also suppress the need for flap track fairings used to hide the flap deployment mechanism, with additional benefits on aerodynamic efficiency in cruise. NASA and Boeing adapted this concept to Mission Adaptive Wing F111 program [4, 5] in the 1980s. The F111 morphing wing mechanism allowed the outer wing of the aircraft to flex from high to low camber to adjust to flight conditions, resulting in performance benefits in all flight phases, ranging from 7

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. A. Singh () · M. S. Allen University of Wisconsin–Madison, Madison, WI, USA e-mail: [email protected]; [email protected] K. M. Wielgus University of Washington, Seattle, WA, USA e-mail: [email protected] I. Dimino Italian Aerospace Research Center, Capua, Italy e-mail: [email protected] R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © 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_3

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Fig. 1 Example of trailing edge rib mechanism developed in the framework of the SARISTU project [20]

to 20% reduction in drag [6]. More recently, in 2016 a full-size morphing wing was flight-tested by a consortium made of NASA, US Air Force Research Lab, Gulfstream, and FlexSys [7]. With the rising interest of utilizing morphing aero-structures, there come several challenges. For instance, the added degrees-of-freedom (DOFs) generate systems with increased modal density, producing a more complex aeroelastic behavior which may lead to flutter instabilities. Two kinds of architectures are currently studied to implement the morphing wing capability based on either compliant or kinematic layouts. Compliant layouts involve the controlled deformation of subcomponents to smoothly modify the overall shape of the assembly [8–10]. This involves tailoring the structural stiffness to ensure enough compliance to accommodate large deformations and enough robustness to preserve a given shape under external aerodynamic loads. Likewise, kinematic layouts, also referred to as fingerlike mechanism-based morphing structures, target the design of skeleton-like articulations with multi-hinge arrangements to enable shape adaptation of large aircraft lifting surfaces [11–13]. More specifically, the inner structure is articulated with different rigid parts moving according to mechanical law; the shape change is obtained through the activation of a mechanism that consists of a loadbearing actuator and a transmission line able to withstand aerodynamic loads. Finally, a morphing external skin envelopes the skeleton to preserve geometrical smoothness during shape changing. In traditional modeling strategies typically used in the aerospace industry, the morphing wing’s individual subsystems and components such as torsion bars, bushings, bearings, and joints are assumed rigid. In simulations, inexpensive rigid connectors may be used to replace fully discretized hinges and bushing to reduce computational costs while globally capturing the macroscopic response. This approach may be valid for relatively simple layouts (as in the case of traditional wing flap systems) but may not be valid for adaptive systems involving several mechanical parts. Nonlinear joints and frictional interfaces cause more complex structures to have nonlinear damping and nonlinear stiffness. These nonlinearities make their behavior difficult to model accurately with traditional simulation approaches. For instance, structures have shown to exhibit amplitude-dependent damping and natural frequencies that change with excitation amplitude and may even vary depending on the actual shape or configuration. As a result, when nonlinear effects are present at high excitation force levels, operative response predictions using linear finite element models may be inaccurate, and detailed, nonlinear modeling approaches should be developed to capture the necessary physics associated with the joints and interface conditions. In this work, a single adaptive rib of the morphing trailing edge device developed in the framework of the SARISTU project is considered as benchmark [1, 18, 25]. Inspired by a “fingerlike” mechanism, as shown in Fig. 1, this mechanism was already successfully validated on both a full-scale morphing wing trailing edge [14, p. 22], [15, p. 23], [16] and aileron demonstrators [17–19]. However, for the purposes of this research, the morphing trailing edge mechanism was reengineered to allow for easier manufacturing and assembly in the absence of the morphing skins and other parts while remaining fully representative of the actual subassembly design. The morphing mechanism consists of four consecutive hinge-connected blocks, referred to as B0, B1, B2, and B3, whose relative rotations enable the trailing edge camber morphing. Block B0 is rigidly connected to a test fixture, while all other blocks are free to rotate around the hinges on the camber line, thus physically turning the camber line into an articulated chain of consecutive segments. Linking rod elements (L1, L2), hinged on nonadjacent blocks, force the camber line segments to rotate according to a specific gear ratio compliant with the shapes to be achieved. The resulting system is a single-degree-of-freedom (SDOF) architecture; if rotation of any of the blocks is prevented, no change in shape can be obtained. On the contrary, if an actuator moves any of the blocks, all the other blocks follow the movement accordingly.

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Due to the large number of joints and contact interfaces involved in the morphing wing design, nonlinearities are associated with friction, clearance, or bilinear stiffness between the preloaded contacting surfaces. These nonlinearities introduce variations in the damping ratio and natural frequencies, which are dependent on excitation amplitude. Quasi-static modal analysis (QSMA) [21] can be utilized to extract the amplitude-dependent frequency and damping curves from the finite element model using only quasi-static simulations. With this approach, the structure is statically loaded with a force proportional to a vibration mode of interest, and the static response is computed at various load levels. Modal hysteresis loops are then calculated from the load-displacement curves to evaluate the nonlinear behavior of each mode. With over 500,000 elements and 39 frictional contact surfaces in the finite element model, this represents one of the most complicated models to which these methods have been applied to date. Additionally, a reduced order model is developed using the whole joint approach [22] to examine the effect of the rotational spring stiffness on the particular mode of interest. The following sections highlight the theory employed and the modeling efforts of this work. Section 2 provides a theoretical background of the QSMA approach for nonlinear finite element models. Section 3 outlines the finite element modeling process for the morphing wing model, and Sect. 4 utilizes QSMA and model reduction techniques in an effort to characterize the frictional nonlinearity in the structure.

2 Theory: Quasi-Static Modal Analysis To gain insight into the nonlinear behavior of a structure, dynamic transient simulations are able to predict the response to various loading scenarios at different load levels, analogous to simulated experiments. Transient simulations provide data to infer the change of the modal characteristics due to the presence of nonlinearity, but are often too computationally expensive [23] because of the cost associated with time-marching algorithms. Quasi-static modal analysis provides an alternate approach which utilizes quasi-static loading to determine the modal frequency and damping with respect to excitation amplitude. These quantities are obtained at a reduced computational cost relative to transient simulations but are only applicable to models with frictional nonlinearities in microslip. The method used is a variation to the one developed by Festjens et al. [24] which was extended to whole joint models by Lacayo and Allen [21] and later to nonlinear finite element models in [23]. A brief overview is presented here but the reader is referred to [21, 23] for additional details and limitations. Consider the equation of motion for a multi-degree of freedom system as given by Eq. 1 with M and K as the N × N mass and stiffness matrices and u as the N × 1 displacement vector and the dot notation noting the derivatives with respect to time. FJ and Fext are the N × 1 vectors of frictional contact forces and external applied loads, respectively. The joint force is represented by a model of internal sliders where θ is a vector to capture the stuck/slip state and the displacement of each slider. It is assumed that the joint forces depend nonlinearly on the displacements: Mu¨ + Ku + F J (u, θ ) = F ext

(1)

At low amplitudes, the joint force can be linearized by evaluating the derivative of the frictional contact force about the equilibrium position, u0 : KT =

 ∂F J  ∂u u0

(2)

The vibration modes are computed about the linearized state (e.g., after applying a preload) by solving the eigenvalue problem where ωr is the rth natural frequency and φr is the rth mode shape vector:   K + K T − ωr2 M φr = 0

(3)

Following the linearized modal analysis, QSMA applies a load in the shape of a mode of interest for incrementing load levels, α, on the static equation of motion as given by Ku + F J (u) = F pre + Mφr α

(4)

The N × 1 vector Fpre represents the preload force that was used to linearize the system. The static response u(α) can be found by solving Eq. 4 at each load increment, from which the modal amplitude is calculated by using an appropriate

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Fig. 2 A hysteresis curve generated using Masing’s rule [21]

Fig. 3 (left) Finite element model of the morphing wing experimental structure, (right) exploded view of section B1 depicting the internal components

modal filter. Next the initial loading curve is computed in the modal subspace, and Masing’s rule reconstructs the full modal hysteresis loop, assuming that the force-deflection hysteresis cycle is symmetric about the origin [25]. The nonlinear natural frequency and damping ratio can be calculated from this hysteresis curve as a function of α; see Eqs. 12–17 in [21] for complete details. Figure 2 depicts the QSMA process and the utilization of Masing’s rule to generate a full and a quarter cycle from an initial quarter cycle.

3 Finite Element Model of Morphing Wing This work applies the QSMA process on a finite element model of a fingerlike mechanism morphing wing, as shown in Fig. 3. The goal is to evaluate the QSMA algorithm on a complex model with many contact interactions and degrees-of-freedom, to evaluate this approach for a realistic aerospace structure. The model is discretized with 510,819 fully integrated hexahedron elements. The adaptive wing subassembly is composed of many different parts consisting of five different materials. These materials and their corresponding properties are provided in Table 1. The ribs and links are aluminum, fixture and bolts are steel, and various washers are plastic. An exploded view of Block B1 is shown in Fig. 3. Unlike SARISTU [26], this single rib mechanism is coupled with a fixture designed to allow various configurations for experimental testing in place of using electromechanical actuation. For the purposes of this work, the structure is locked such that the chord line is parallel to ground. The full-order finite element model consists of a total of 72 contact interfaces. To reduce the computational cost and complexity of the model, the interfaces were all considered, and 33 of the interfaces were identified that are not expected to contribute significantly to energy dissipation through friction. These 33 interfaces were tied within the model using multipoint constraints. These surfaces included stiff interactions such as tightened bolts and rigid connections between the ribs. The remaining 39 contact interfaces include contacts between rib surfaces and hinge joints that allow the joints to rotate to a

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Table 1 Material properties of morphing wing parts Material

Young’s modulus (psi)

Density (

6061-T6 aluminum Alloy steel PFTE plastic Acetal plastic Stainless steel

1E7 2.9E7 1E5 4E5 2.9E7

0.0975 0.284 0.0723 0.0509 0.284

lb ) in3

Poisson’s ratio

Coefficient of friction against aluminum

0.3 0.29 0.46 0.37 0.29

0.4 0.22 0.04 0.2 0.2

Fig. 4 Section B2 of the structure to show three types of joints

given wing shape during actuation. These nonlinear interfaces were modeled with Coulomb frictional contact elements with the assumed static friction coefficients listed in Table 1. Figure 4 depicts Block B2 with a rib moved to see the internal components, showing three types of joints: (1) bolted joints, (2) shoulder joints, and (3) pinned connections. All quasi-static contact simulations were conducted using implicit integration schemes using the Sierra Solid Mechanics [27] finite element solver. During flight, the joints in this morphing wing structure would be preloaded by the wing skin and by aerodynamic forces. In the laboratory tests that are planned with this prototype hardware, the joints would be preloaded via gravity loads or via a point load at the tip of the wing. This later case was simulated in the results presented in this paper. To eliminate rigid body motion, the base of the fixture in Fig. 4 is constrained to have no displacement. Once the tip preload is applied and the contact solution converges, the solution can be used to determine which portions of each interface are in contact. These regions are then bonded to one another, and linearized modal analysis is performed about the preloaded state for the next step in the QSMA process. The contact areas found in the preload step are also used to generate a Hurty/CraigBampton [28] model for the system as explained later.

4 Simulated Results of Morphing Wing Fem 4.1 Static Preload The tip load used for the results presented in this work was given a magnitude of 3 lbf and distributed across four nodes at the tip of the structure. The peak tip displacement as a function of tip force is shown in Fig. 5. The load was incrementally ramped using a ramp-cosine function and then held steady for 100 timesteps to ensure that the equilibrium is accurately reached. The force-displacement curve reveals the nonlinear response of the wing structure due to the various frictional

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Fig. 5 Tip load versus peak displacement over the 100 preload steps

Fig. 6 Stress in [psi] before the snap-through on the back (row 1) and front (row 2) of the link connecting B2 and B3, (right) stress after snapthrough

joints in the structure. Initially, the curve appears to exhibit softening behavior as the slope decreases. At a force level of around −0.3 lbf, there appears to be a sudden jump in response. Beyond this point, at higher load levels, the slope appears to increase with displacement suggesting a hardening behavior. The jump in displacement at about −0.3 lbf in Fig. 5 was investigated further and determined to be the result of a snapthrough/buckling phenomenon during loading. Figure 6 shows the axial stress in the link pin connecting blocks B2 and B3 before and after the event; these stress changes occurred over one load step increment. Prior to the snap-through, the pin

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Fig. 7 The resulting contact status at the end of the preload simulation

Fig. 8 Contact status of a joint where contact is allowed

was mostly in a compressive stress state. The sudden jump changed the stress state to a combined axial and bending stress, suggesting that the pin had buckled under a combined compressive load and load applied from the link. Upon investigation of the model, this was the only quantity observed to change suddenly at these load levels, suggesting that this link was the source of the jump in displacement observed in the quasi-static loading curves. This buckling phenomenon is not thought to be physical, but an artifact of the model. A cause of this jump could be due to the coarse mesh of the hinge joints, leading to a phenomenon known as “facet locking” when rotating about a hinge joint. This highlights the challenges associated with detailed, nonlinear finite element analysis of complex mechanisms with many frictional contact interfaces throughout. The contact statuses in all of the joints are shown in Fig. 7 for the instant when the tip preload is maximum. Some ribs have been removed to better visualize the internal components. A value of 1 (red) for contact status indicates the portion of the surface is in contact. A value of 0.5 (green) indicates that the surface was defined as a contact surface but is not computed to be in contact. Surfaces in blue are not defined as contact surfaces and thus are not capable of supporting contact loads. Figure 8 provides a close-up view of a shoulder joint, where the preload causes a portion of the hinge (red) to come into contact, while the opposite side (green) does not. The state of contact within the joints influences both the stiffness of the joint during the linearized modal analysis and the energy dissipation of the joint during QSMA.

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4.2 Quasi-Static Modal Analysis Following the preload step, the next step in the QSMA process is to compute the linearized modes of vibration about the preloaded state. The modal analysis step was accomplished using the Sierra Structural Dynamic (Sierra/SD) [29] finite element code, which is able to import the deformation and stress state directly from the Sierra Solid Mechanics preload simulation. Within Sierra/SD, tied multipoint constraints can be defined based on the normal contact traction magnitudes calculated from the preload step. It should be noted that only fully tied interfaces were used within the modal analysis, i.e., the preloaded joints were constrained in both normal and tangential directions. For the linearized modal analysis step, the chosen normal contact pressure cutoff value was 0 psi. Figure 9 depicts the mode of interest for the morphing wing structure, referred to as the first stiff direction bending mode, with a natural frequency of 166 Hz. The final step of the QSMA process is to apply a body force to the structure proportional to the shape of the bending mode in Fig. 9. The modal force was appliedto the structure such that the tip had a positive displacement, opposite of the tip preload step. The solver was allowed to settle for 50 additional iterations prior to applying the modal force to ensure that the model was in the equilibrium state. After the modal filter was applied to the displacement fields, the nonlinear frequency and damping curves were computed and shown in Fig. 10. The modal force amplitude ranges were chosen such that the linearized model had a tip displacement in a prescribed range. In the results shown here, two separate simulations are shown for a tip displacement range of 0.005– 0.05 inch, and 0.01–0.1 inch. The frequency and damping curves are plotted against the tip displacement (peak) to quantify

Fig. 9 Stiff bending mode of the morphing wing structure

Fig. 10 Frequency (left) and damping (right) as a function of displacement amplitude estimated using QSMA

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the relevant deflection in the mechanism when it vibrates in this mode. The two results appear to overlay each other quite well, suggesting that the method is repeatable for different forcing levels. The frequency curves in Fig. 10 show softening behavior as frequency initially decreases rapidly, and then appears to plateau to a lower value in the range of 60–70 Hz. The QSMA results should converge to the linearized frequency of 166 Hz at low displacement amplitudes, but this was not the case for the results shown here. This suggests that the smallest loads applied here were already large enough to cause a very significant frequency shift. The damping is similarly computed from the load displacement data. The damping ratio curves shown in Fig. 10 achieved a peak value of almost 50% damping but decreased at higher displacement levels. Taken together, the QSMA curves appear to have the features of macro-slip behavior, where both the frequency and damping decrease after slip has initiated in the joints. Microslip behavior typically shows increasing damping ratios with displacement level, and slight decreases in the natural frequency as small regions of the joint begin to slip. It was observed that the peak tip displacement resulting from the modal force was much greater than the prescribed tip displacement from the linearized analysis, up to eight times the prescribed level. These results suggest that even the lowest loads applied were causing significant slip in the joints, more than would be expected for the linearized model where the preloaded interfaces were assumed to be fully tied (or stuck). Efforts were made to apply even lower force amplitudes with the QSMA approach; however, the implicit solver was unable to converge and produce reasonable results. When the magnitude of the modal force was small relative to the preload force, the response to the preload seemed to dominate, hence resulting in a noisy response to the modal force and unreasonable results with QSMA. There are several assumptions that were made that could be questioned at this point. Error could arise from the mesh discretization within the frictional interfaces. Considering that the contact happens on cylindrical surfaces, the model likely needs more refinement in the area of the hinge joints in order to properly capture the contact and slip. However, the model already has over 5 M DOF. This highlights a key challenge for the analysis of jointed structures; the physics of interest may require extreme levels of refinement locally, far more than is typically needed for accurate stress analysis or to predict the natural frequencies well. Another assumption of the QSMA approach is that the joints obey Masing’s hypothesis such that the hysteresis curves can be computed from the initial loading curve. If this assumption is not satisfied, one must compute the full hysteresis curve quasi-statically. This hypothesis can be directly evaluated by computing the modal hysteresis curve for the mode of interest and comparing it directly to the hysteresis curve reconstructed using Masing’s rule. To do this, the modal force was applied to the structure in a forward cycle discretized from 0 ➔ α ➔ −α ➔ α, as well as a reverse cycle discretized from 0 ➔ −α ➔ α ➔ −α. The resulting modal hysteresis curves are shown in Fig. 11. The first observation from these data is that the hysteresis curves are different for the forward and reverse loading cycles. Additional loading cycles would be needed to reach steady state and close the hysteresis loops, indicating either that the computational model is not realistic or that the behavior of the structure is far more complicated than expected. To explore this further, in Fig. 12 the reverse loading cycle is compared to hysteresis loop generated from the initial quarter cycle of the reverse load using Masing’s rules.

Fig. 11 Moderate (left) and high (right) modal force vs. peak displacement hysteresis loop

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Fig. 12 QSMA vs. direct static hysteresis loop for the modal force step

The two results are quite different in terms of shape (i.e., stiffness) and the area enclosed (damping). Although the quarter cycle initially follows the direct hysteresis loop, it deviates at higher displacements. At this point, it is unclear whether these results are reliable or not. It has been shown that if the contact pressure varies over a loading cycle, then a model with Coulomb friction can violate Masing’s rules [21], and considering the low preload in these joints, this could be the case here. Conversely, the physics observed could be spurious and due to an inadequate mesh in the contact regions. While this case study cannot resolve these issues, it does raise some important issues that one must be aware of when applying QSMA to a complex structure.

4.3 Reduced Order Model In addition to the full-order model, a reduced order model (ROM) of the morphing wing structure was developed using the whole joint approach [22]. The ROM was derived from the full-order model using the following process. First, contact surfaces at the joints are defined within the full-order model, and a preload analysis was performed to determine the contact status of all contact interfaces. From the preload results, a subset of each contact surface was defined to include only the nodes/faces on the surface in contact. This subset of each interface was then assigned to a multipoint constraint, or “spider,” to tie all these nodes on the surface to a single, virtual node using either averaging or rigid bar elements. Next, two spider joints at a contact interface were connected by a whole joint model in between the virtual nodes, each whole joint having six independent DOFs. The joints can be assigned any constitutive element to represent the physics of interest. In this study, the model contains linear spring elements whose stiffness may be easily altered to approximate the stiffness of the contact interface. This spidering method, as shown schematically in Fig. 13, was performed on all hinged joint and frictional contact interfaces in the morphing wing model. There is no way to predict the stiffness that a whole joint model such as this should have, and so some sort of optimization was used. Toward this end, a rotational stiffness sensitivity study was conducted on the morphing wing subassembly. The rotational stiffness about the x-axis, denoted as KRx, which is the same for all the hinged joints in the current model, was varied from 1e1 to 1e9 in-lbf, while all other stiffnesses (rotational and axial) were held constant at 1 + E08 in-lbf or lbf/in, depending on whether it is a rotational or translational DOF. Modal analysis was performed on the reduced order model

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Fig. 13 Spidering process example for a rib-to-washer contact interface

Fig. 14 Mode 2 frequency with increasing rotational stiffness, KRx

at each KRx stiffness value to determine how the frequency of the mode of interest changes. The second mode natural frequency was found to vary significantly with changing rotational stiffness, as shown in Fig. 14. At low stiffness values, the mode converges to around 50 Hz. As the rotational stiffness increases, the frequency increases and converges to an upper bound of around 110 Hz. The lower bound on the mode frequency is within the range of the lower frequency bound of the QSMA results in Fig. 10. This lower bound would correspond to the response when the joint has slipped, and no rotational stiffness is provided by the joint. The upper bound was smaller than the mode frequency with all joints stuck, and so presumably this spring would need to be set in conjunction with others to cause the model to agree with the linear eigenvalue results.

5 Conclusions and Future Work High-fidelity nonlinear finite element models can provide new insights to guide the design and predict the performance of complex aerospace structures, yet those models can be extremely expensive to simulate and this severely limits our ability to understand their dynamics. This work utilized the QSMA framework to study an industrial scale structure of a morphing wing and provided a preliminary view of how this algorithm may scale to a large-scale model. The amplitude-dependent natural frequency and damping ratio curves from the full-order model were found to show significant loss of stiffness in the joint, as well as significant levels of energy dissipation, even when the structure responds at lower amplitudes for the mode of interest. The QSMA results were investigated by directly calculating the modal hysteresis curves, and the results suggested that the mode under study violated Masing’s hypothesis since the reconstructed hysteresis loop did not overlay with the directly computed curves. This seems to be a valuable check of the validity of QSMA for complex structures. The results to date are promising but show that more effort is needed to scale the methodology to general large-order structural models. One potential area requiring further attention is the mesh density at the localized regions of the joint; for this method

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to be practical on very large models, it seems that one needs a convenient way to increase the mesh resolution in very small regions near interfaces. The present study was only able to initiate a reduced order model for this system, but there wasn’t sufficient time to determine whether the ROM could be tuned to agree with the full-order model for the modes of interest. In future studies, the authors hope to explore this and better quantify the advantages and disadvantages of using spiders in this framework to model a structure. Additionally, experimental data should be acquired to supplement the modeling work to provide validation data as well as provide key insights into the modeling assumptions used throughout this study. Acknowledgments This research was conducted at the 2020 Nonlinear Mechanics and Dynamics (NOMAD) Research Institute supported by Sandia National Laboratories in partnership with the University of New Mexico. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. The authors would also like to thank Jack Heister and Rick Garcia from Sandia National Laboratories for their support in the design and mesh creation of the morphing wing device presented throughout this work. SAND2020-13181C.

References 1. Pecora, R., Concillo, A., Dimino, I., Amoroso, F., Ciminello, M.: Structural design of an adaptive wing trailing edge for enhanced cruise performance. Presented at the 24th AIAA/AHS Adaptive Structures Conference, San Diego, CA, 2016 2. Dimino, I., Concilio, A., Pecora, R.: An adaptive control system for wing TE shape control. Presented at the SPIE International Conference on Smart Structures, San Diego, CA, 2013 3. Pecora, R., Amoroso, F., Noviello, M.C., Dimino, I., Concilio, A.: Aeroelastic stability analysis of a large civil aircraft equipped with morphing winglets and adaptive flap tabs. In: Proc. SPIE 10595, Active and Passive Smart Structures and Integrated Systems XII, 105950L, vol. 10595. https://doi.org/10.1117/12.2300173 (15 March 2018) 4. Bonnema, K.L.L AFTI/F-111 mission adaptive wing briefing to industry. AFWAL Technical Report, Oct 1988 5. Webb, L.D., McCain, W.E., Rose, I.A.: Measured and predicted pressure distributions on the AFTI-F-111 mission adaptive wing. In: NASA Technical Memorandum TM-100443, National Aeronautics and Space Administration, Edwards, Ed., (1988) 6. Calzada, R., N.A.S.A.: https://www.nasa.gov/centers/dryden/multimedia/imagegallery/F-111AFTI/F-111AFTI_proj_desc.html. Accessed 17 Apr 1998 7. N.A.S.A.: Past project: adaptive compliant trailing edge flight experiment (2014) 8. Bilgen, O., Kochersberger, K.B., Inman, D.J., Ohanian, O.J.: Novel, bidirectional, variable-camber airfoil via macro-fiber composite actuators. J. Aircr. 47(1), 303–314 (2010). https://doi.org/10.2514/1.45452 9. Gaspari, A.D., Riccobene, L., Ricci, S.: Design, manufacturing and wind tunnel validation of a morphing compliant wing. J. Aircr. 55(6) (2018) 10. Wildschek, A., et al.: Multi-functional morphing trailing edge for control of all-composite, all-electric flying wing aircraft. In: The 26th Congress of ICAS and 8th AIAA ATIO, American Institute of Aeronautics and Astronautics 11. Vecchia, P.D., Corcione, S., Pecora, R., Nicolosi, F., Dimino, I., Concilio, A.: Design and integration sensitivity of a morphing wing trailing edge on a reference airfoil: the effect on high-altitude long-endurance aircraft performance. J. Intell. Mater. Syst. Struct. 20(28) (2017) 12. Pecora, R., Amoroso, F., Lecce, L.: Effectiveness of wing twist morphing in roll control. J. Aircr. 49(6), 1666–1674 (2012) 13. Pecora, R., et al.: Actuator device based on a shape memory alloy, and a wind flap assembly fitted with such an actuator device. US Patent US 8348201, 2018 14. Dimino, I., Amendola, G., Pecora, R., Concilio, A., Gratias, A., Schueller, M.: Chapter 22—On the experimental characterization of morphing structures. In: Concilio, A., Dimino, I., Lecce, L., Pecora, R. (eds.) Morphing Wing Technologies, pp. 683–712. Butterworth-Heinemann (2018) 15. S. Kuzmina et al., “Chapter 23—Wind tunnel testing of adaptive wing structures,” in Morphing Wing Technologies, A. Concilio, I. Dimino, L. Lecce, and R. Pecora, Eds. Butterworth-Heinemann, 2018, pp. 713–755 16. Schorsch, O., Lühring, A., Nagel, C.: Elastomer-based skin for seamless morphing of adaptive wings. In: Smart Intelligent Aircraft Structures (SARISTU), Cham, pp. 187–197 (2016) 17. Amendola, G., Dimino, I., Pecora, R., Concilio, A., Amoroso, F.: Preliminary design of an adaptive aileron for the next generation regional aircraft. J. Theor. Appl. Mech. 55, 307–316 (2017) 18. Amendola, G., Dimino, I., Concilio, A., Magnifico, M., Pecora, R.: Numerical Design of an Adaptive Aileron, Las Vegas, NV (2016) 19. Amendola, G., Dimino, I., Amoroso, F., Pecora, R.: Experimental Characterization of an Adaptive Aileron: Lab Tests and FE Correlation, Las Vegas, NV (2016) 20. Pecora, R., Amoroso, F., Magnifico, M., Dimino, I., Concilio, A.: KRISTINA: kinematic rib-based structural system for innovative adaptive trailing edge. In: Industrial and Commercial Applications of Smart Structures Technologies 2016, vol. 9801, p. 980107. https://doi.org/10.1117/ 12.2218516 (2016) 21. Lacayo, R.M., Allen, M.S.: Updating structural models containing nonlinear Iwan joints using quasi-static modal analysis. Mech. Syst. Signal Process. 118, 133–157 (2019) 22. Segalman, D.J.: Modelling joint friction in structural dynamics. Struct. Control. Health Monit. 13, 430–453 (2006). https://doi.org/10.1002/ stc.119

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23. Jewell, E., Allen, M.S., Zare, I., Wall, M.: Application of quasi-static modal analysis to a finite element model and experimental correlation. J. Sound Vib. 479, 115376 (2020) 24. Festjens, H., Chevallier, G., Dion, J.-L.: A numerical tool for the design of assembled structures under dynamic loads. Int. J. Mech. Sci. 75, 170–177 (2013). https://doi.org/10.1016/j.ijmecsci.2013.06.013 25. Masing, G.: Eigenspannungen und verfestigung beim messing (self stretching and hardening for brass). pp. 332–335 (1926) 26. C. I. R. A.-CIRA: SARISTU project adaptive wing. http://futuroremoto.cira.it/saristu_eng.html 27. SIERRA Solid Mechanics Team: Sierra/SolidMechanics 4.56.2 User’s Guide. Sandia National Laboratories, Albuquerque, NM, SAND20205362, May 2020 28. Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3, 4 (1965). https://doi.org/10.2514/3.2947 29. Sierra Structural Dynamics Development Team: Sierra/SD – User’s Manual – 4.56. Sandia National Laboratories, Albuquerque, NM, SAND2020-3028, Apr 2020

Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure Moheimin Khan, Patrick Hunter, Benjamin R. Pacini, Daniel R. Roettgen, and Tyler F. Schoenherr

Abstract Calibrating a finite element model to test data is often required to accurately characterize a joint, predict its dynamic behavior, and determine fastener fatigue life. In this work, modal testing, model calibration, and fatigue analysis are performed for a bolted structure, and various joint modeling techniques are compared. The structure is designed to test a single bolt to fatigue failure by utilizing an electrodynamic modal shaker to axially force the bolted joint at resonance. Modal testing is done to obtain the dynamic properties, evaluate finite element joint modeling techniques, and assess the effectiveness of a vibration approach to fatigue testing of bolts. Results show that common joint models can be inaccurate in predicting bolt loads, and even when updated using modal test data, linear structural models alone may be insufficient in evaluating fastener fatigue. Keywords Bolted joint · Joint stiffness · Modal analysis · Model calibration · Fatigue

1 Introduction Bolted joints are commonly used to connect parts and assemblies but are still a source of error in analytical structural models. Various joint modeling techniques can be utilized, depending on the application and quantities of interest. For structures subjected to dynamic loading, it is important to consider the frequency response of the structure. When fastener fatigue is a concern, predicting failure requires a joint model that can accurately represent the loading through the bolt. In order to evaluate and improve current joint modeling techniques, enhance fatigue analysis, and advance testing capabilities, an experimental structure is developed. The structure is designed to fail a bolt in fatigue using a modal shaker forced at the axial resonant mode of the structure. Free-free and fixed base mode shapes of the structure are obtained, and finite element (FE) analyses are performed using the SIERRA finite element code suite developed at Sandia National Laboratories [1]. Several different joint models are explored in this work, and analysis results along with test data are used to compare these models and evaluate the viability of using the structure to perform fatigue testing of bolts.

2 Hardware Description A structure is designed for use in joint model calibration and fatigue testing for a single ¼ inch bolt (¼-20 UNC). It consists of a large mass with a small support structure that is attached to a plate, shown in Fig. 1. The large mass, denoted the Kettlebell (KB), directly attaches to the Adapter Plate (AP), which has eight bolts to attach to a seismic mass. The structure was designed such that its axial mode imparts a large force on the single bolt when the KB is driven with a modal shaker,

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. 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. M. Khan () · P. Hunter · B. R. Pacini · D. R. Roettgen · T. F. Schoenherr Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © 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_4

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KB

AP

a)

b)

Fig. 1 Assembly isometric view (a) and bottom view (b)

KB Strainsert Bolt

AP

Fig. 2 Assembly cross section with Strainsert bolt

ideally resulting in the bolt failing in fatigue. Two sets were fabricated, with one having the KB heat treated to a hardness value of Rockwell C (HRC) 32 for increased fatigue strength. Both plates were also heat treated.

2.1 Assembly Hardware The following hardware is used in the assembly: • 4340 Steel Kettlebell—approximately 4.5 × 4.5 × 6.5 (inch) size and 26 lbf weight – 0.4375 thickness at joint • 4340 Steel Adapter Plate—approximately 10 × 10 × 1.4 size and 35 lbf weight – 0.4375 thickness at joint • ¼-20 UNC Strainsert Model SXS-FB force-sensing bolt with embedded strain gauge, 1.75 length and 0.875 shank • Standard SAE washers (0.65 diameter and 0.0625 thick) • Nylon-insert locknut Figure 2 shows a cross section of the assembly with the Strainsert bolt visible. The Strainsert bolt was selected to ensure proper preload in the joint and provide joint force time history data during modal testing and eventually the fatigue testing. The Strainsert bolt was preloaded when assembled in the structure. Preload is commonly specified using 75% of proof strength [2], and for ¼-20 UNC SAE grade 5 fasteners with Spr = 85 ksi proof

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strength, and tensile stress area, At , the required preload force, Fi , is   Fi = 0.75Spr At = (0.75) (85 ksi) 0.0318 in2 = 2027.25 lbf

(1)

The nut factor, K, relates applied torque to axial load in the bolt [2], and for a general value of K = 0.2, the required torque to obtain the preload is T = KdFi = (0.2) (0.25 in) (2826 lbf ) = 141.3 in· lbf

(2)

This value of torque was used as a starting point for preload, but the Strainsert force gauge was used to ensure the target value calculated in Eq. (1) was reached during testing.

3 Experimental Data and Analysis Free-free and fixed base modal testing was performed for the structure. The test results were used to characterize the structure and bolted joint in order to calibrate the finite element model and compare the various joint modeling techniques.

3.1 Free-Free Modal Data The structure was instrumented with 11 triaxial accelerometers and free-free modal testing was performed. Figure 3 shows the accelerometer layout and coordinates. The test assembly was suspended using bungee cords to approximate a free-free boundary condition and modal testing was performed using a hammer input. Figure 4 shows the experimental setup and a close-up of the Strainsert bolt. The Strainsert was preloaded to approximately 144 in·lbf , corresponding to the required preload calculated in Eq. (1). Table 1 summarizes the initial free-free test results. A second set of free-free data was obtained with the Strainsert bolt preloaded to approximately 100 in·lbf instead of the 144 in·lbf . This was done because the limit of the Strainsert force sensor was reached with the initial installation torque alone, leaving little to no overhead for dynamic measurements. Table 2 lists the second set of data and includes a comparison to the first set. The second test was performed using input from a modal shaker attached to the top of the KB.

Fig. 3 Accelerometer layout (a) and coordinates (b) in inches

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Fig. 4 Free-free test setup (a) and Strainsert bolt close-up (b) Table 1 Initial free-free test results (144 in·lbf preload) Free-free test 1 mode 1 2 3 4 5 6 7

Description First bending in Z First bending in Y Torsion about X Second bending in Y Axial in X Plate twist Second bending in Z

Frequency (Hz) 194 356 371 1202 1307 1517 1566

Damping (%) 0.52 0.40 0.73 0.06 0.21 0.33 0.16

Table 2 Second set of free-free test results (100 in·lbf preload) and comparison to the first set Free-free test 2 mode 1 2 3 4 5 6 7

Description First bending in Z First bending in Y Torsion about X Second bending in Y Axial in X Plate twist Second bending in Z

Frequency (Hz) 233 307 408 1213 1260 1520 1645

Frequency % difference 20.1 −14.2 8.2 1.2 −4.0 0.2 20.1

Damping (%) 0.38 0.33 0.18 0.05 0.13 0.23 0.09

Damping % difference −19.1 6.5 −35.7 −64.3 −27.8 −30.3 −19.1

3.2 Fixed Base Modal Data The structure was bolted to a seismic mass to approximate a fixed base boundary condition and testing was performed using a modal shaker as shown in Fig. 5. The Strainsert bolt was preloaded to approximately 100 in·lbf as in the second free-free test. Table 3 summarizes the fixed base test results.

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Fig. 5 Fixed base setup Table 3 Fixed base test results Fixed base test mode 1 2 3 4 5 6

Description First bending in Z First bending in Y Torsion about X Second bending in Y Axial in X Second bending in Z

Frequency (Hz) 101 189 339 1137 1254 1512

Damping (%) 8.61 9.38 0.12 0.14 2.42 1.82

3.3 Additional Fixed Base Testing and Characterization of Axial Mode In addition to functioning as a case study to evaluate joint modeling techniques, one of the objectives of the test structure is to fail a bolt in fatigue by dwelling at the axial mode of the structure. Due to nonlinear characteristics of bolted joints, as excitation levels increase, the resonant frequency of the axial mode is expected to shift. A preliminary characterization of this changing resonance was conducted using band-limited random excitation in the fixed base configuration. The linear modal results for the fixed base configuration (see Table 3) showed that the second bending mode in Y was relatively close in frequency to the axial mode. The mode shapes are shown in Fig. 6. Since the desired loading during the fatigue test is purely axial, an additional drive point was added to the top center of the KB to minimize the excitation of the second bending mode. This new drive point was labeled 1001 (see Fig. 5). Excitation was applied from 850 Hz to 1650 Hz at various forcing root-mean-square (RMS) amplitudes with the axial mode and second bending modes in Y and Z being active in the bandwidth. Figure 7 shows the imaginary portion of the drive point frequency response function (FRF) results for force levels ranging from 0.14 to 17.99 lbf RMS. As the force level increased, the drive point FRF showed the axial mode at around 1250 Hz shifting down in frequency to the second bending mode in Y near 1130 Hz. The damping initially increased with higher excitation force as indicated by the decrease in the imaginary FRF peak amplitude, but was then decreased with further increases in force beginning around 5 lbf RMS. This reduction in damping results in a nearly 500% increase in the amplification seen at the peak FRF amplitude for the axial mode between low and high forcing. From Table 3, the second bending mode in Y has significantly smaller

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1137 Hz

a)

1254 Hz

b)

Fig. 6 Fixed base second bending mode in Y (a) and axial mode (b)

Fig. 7 Fixed base drive point FRF results

damping than the axial mode. Since the latter mode softens to the same frequency as the former, there was the potential that at the high force levels the two modes were coupling resulting in the drastic decrease in damping of the axial mode. To investigate this potential coupling, a singular value decomposition (SVD) analysis was performed by decomposing the measured FRFs, H, at each force level: U SV + = svd(H )

(3)

From the linear modal analysis of this system, there are six modes below 1650 Hz. Therefore, the first six columns of U and first six singular values σ of S were retained for each force level. The modal assurance criterion (MAC) value between these U vectors (hereafter called “SVD Shapes”) and the linear mode shapes for the second bending mode in Y and the axial mode were computed. This provides an indication as to whether the deflection shapes of the structure change with forcing amplitude, and Fig. 8 summarizes the results. Figure 8a shows that the MAC between the linear mode shape for the second bending mode in Y and that the SVD Shape associated with the second singular value for all force levels is above 97.6, while the maximum for all other SVD Shapes is 1.8. Similarly, Fig. 8b shows that the minimum MAC value between the SVD Shape for the first singular value and the axial mode is 96.7, while the maximum for all other SVD Shapes is 2.7. Therefore,

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Fig. 8 MAC at different force levels between SVD Shapes and linear shapes, (a) second bending mode in Y and (b) axial mode

Energy Composition for SVD Shapes vs Force SVD Shape1 SVD Shape2 SVD Shape3 SVD Shape4 SVD Shape5 SVD Shape6

80

Energy Ratio (%)

70 60 50 40 30 20 10 0 0

2

4

6

8

10

12

14

16

18

Force (lbfRMS) Fig. 9 Energy ratios for the first six singular values

the SVD Shapes for these two singular values remain distinct and do not significantly change, indicating that there is no substantial change in the deflection shapes of the test structure as the force levels increase. Moreover, the largest singular value is solely associated with the axial mode and that for the second singular value is the second bending mode in Y. At low force, the drive point FRF indicates that the second bending mode in Y is minimally excited. However, as the drive point FRF peak decreases toward the linear natural frequency of the second bending mode, there is a potential that this mode becomes more easily excited. To investigate this, SVD results were used to determine the contribution of each mode to the overall energy of the structure at each force level. This was accomplished through energy ratios, ri , defined for each singular value, σ i , as σi r i = N

i=1 σi

(4)

where N is the number of FRFs at each force level. From Fig. 8, the first and second singular values correspond to the axial and second bending modes, respectively, for all force levels. Therefore, r1 is the ratio of the total energy associated with the axial mode and r2 that for the second bending mode. Figure 9 shows the energy ratios in terms of the percentage of total energy (100 × ri ) for the first six SVD Shapes. At low force levels, the majority of the energy (approximately 75%) is due to the axial mode, while that for the second bending mode in Y is only 15%. This agrees with the visual inspection of the drive point FRF at low level since it shows that the second bending mode should not be well excited. However, as the excitation

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force increases, the second bending mode comprises increasing measures of the total system energy, increasing to 35% of the total system energy, while that for the axial mode decreases to 55%. Therefore, as the axial mode softens, the second bending mode in Y does become more excited and contributes more to the overall response. For a purely linear system, these ratios would not change as the relative excitation between the two modes would remain constant. However, due to the nonlinearity introduced by the bolted joint, there appears to be coupling between these two modes that results in a decrease in apparent damping and increased excitation of the second bending mode in Y. Therefore, at high force levels, the response of the structure is no longer mostly axial, but has significant lateral motion due to the activation of the second bending mode. From a fatigue testing standpoint, this means that a multiaxial load will be applied to the bolt, which limits the viability of the structure to consistently test bolts to failure. There may be experimental methods to reduce lateral motion using another shaker, but further research would be needed. Although traditional joint models would not be able to account for this nonlinear modal coupling, calibration to modal test data can still be performed to develop useful approximations. The following sections detail the linear joint models developed and model calibration performed for the structure.

4 Structural Dynamic Modeling Four different FE models were created, each with a distinct representation of the bolted joint. All meshes were created using the CUBIT meshing software and consisted mainly of ten-noded tetrahedral (TET10) elements. The KB, washers, and AP were modeled using these TET10 elements, with the bolt representation dependent on the modeling technique used. For fixed base analysis, the bottom surface of the AP was fixed. An image of the model is shown in Fig. 10.

4.1 Model 1: Solid Bolt and Tied Joint Interface The first model uses a solid bolt with washers and a nut. The bolt head, washers, and nut are connected to the KB and AP using Tied Data functionality, which constrains nodes from different meshes, effectively “gluing” them together. For the models presented here, preload is not included and instead the load transfer between the bolt and members is represented with spring elements. In Model 1, the interface between the KB and AP is connected using the Tied Joint capability in SIERRA Structural Dynamics (SIERRA/SD) [1]. The Tied Joint model creates a virtual spring element with six user-defined stiffness parameters (three translational and three rotational), which are used to connect two surfaces. The contact radius was adjusted based on later model calibration steps. Figure 11 shows the Model 1 configuration.

Fig. 10 FE model of structure

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Interface Tied Joint

Tied Data --

Merged Nodes

Fig. 11 Model 1 configuration

Bolt Tied Joint Surfaces Interface Tied Joint

Fig. 12 Model 2 configuration

4.2 Model 2: Tied Joint Bolt and Interface In Model 2, the bolt and washers are replaced with a Tied Joint element that is attached to an equivalent area under the washers on each side. Again, preload is excluded but a representative spring model is created with the Tied Joint. The interface stiffness values remain the same as the previous model, so Model 2 has a total of two Tied Joints connected in parallel. By removing the solid mesh of the bolt and washers, the joint model is simplified while still retaining stiffness through the second spring. Figure 12 shows this configuration.

4.3 Model 3: Single Tied Joint In Model 3, the second Tied Joint that represented the bolt in Model 2 is removed, and only a single spring element represents both the member and bolt stiffness, as shown in Fig. 13. This model is a further simplification that only requires definition of one set of stiffness parameters to define the entire joint.

4.4 Model 4: Tied Data In Model 4, there are no spring elements and instead the interface is connected using Tied Data, with the same surface definition as Model 3 in Fig. 13. This is the simplest of the four models, but it contains no individual representation of either the bolt or member stiffness. It has use in analyses where reduced element count is necessary and where the bolted joint is not a critical feature, but does not account for any load transfer to a bolt.

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Single Interface: Tied Joint or Tied Data

Fig. 13 Model 3 and 4 configurations

Overall, the four different models showcased in this section are representative of typical joint models used in linear structural dynamic analysis. There are limitations, such as the lack of frictional contact, exclusion of preload, and the artificial rigidity at the interface due to the Tied Data and Tied Joint models. Although the spring elements generally approximate member and joint stiffness, future work is needed to fully study the effects with preload included.

5 Model Updating Using the test data presented in Sect. 3, model calibration and updating was done with the solid bolt model, Model 1. This reference model was used as a baseline and mode shapes and frequencies were compared to the other three models and test data.

5.1 Initial Study Modal analysis was done for each of the four models using the SIERRA/SD linear structural dynamics code. An initial study was done using the Model 1 configuration discussed in Sect. 4.1 and shown in Fig. 11. As a starting point, the full interface surface between the KB and AP was used for the Tied Joint and the axial member stiffness, kx , was calculated using [3]. Stiffness parameters for this initial Model 1 configuration (Model 1a) were kx = 6.13e6 lbf /in, ky = kz = 1e6 lbf /in, and krx = kry = krz = 1e7 in·lbf /rad. Mode shape and frequency results for the initial comparison to test data are shown in Fig. 14a and Table 4. In general, mode shapes between the model and test were similar with the lowest MAC value being 0.949, but there was swapping of the order of modes 5, 6, and 7. In addition, frequencies differed by as much as 51%, indicating that the preliminary Tied Joint parameters needed adjustment. The same comparison was done with the second set of free-free data, shown in Fig. 14a and Table 4. Results for the second set of data were similar, although modes 1 and 2 had a lower MAC value under 0.9. Since the first set of free-free data had a better mode shape and frequency correlation, it was used for the calibration. Starting from the preliminary Model 1a, the stiffness parameters of the Tied Joint and the interface contact radius were adjusted to better match the shapes and frequencies to the test data. Eventually, another modified model (Model 1b) was developed with updated parameter values, summarized in Table 5. The resulting MAC and frequency comparisons for Model 1b and the free-free data are shown in Fig. 15 and Table 6. With Model 1b, the MAC values improved slightly, and frequencies matched much better with the test, with the largest error of 4.2%. Figures 16 and 17 show an overlay of the test and FE free-free mode shapes. The fixed base mode shapes were similar, except that the plate twisting mode 6 was not present since the AP was bolted to a seismic mass.

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a)

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b)

Fig. 14 Model 1a initial study MAC plot, free-free test 1 (a) and free-free test 2 (b) Table 4 Model 1a free-free comparison (free-free test 1) Test mode 1 2 3 4 5 6 7

Test 1 Freq (Hz) 193.6 356.4 371.2 1201.5 1307.3 1517.2 1565.8

Test 2 Freq (Hz) 233.2 307.3 407.7 1212.7 1259.5 1520.0 1644.5

Model 1a Mode 1 2 3 4 6 7 5

Model 1a Freq (Hz) 291.6 404.0 448.4 991.4 1230.2 1507.6 1130.8

Freq Err 1% 50.6 13.4 20.8 −17.5 −5.9 −0.6 −27.8

Freq Err 2% 25.0 31.4 10.0 −18.2 −2.3 −0.8 −31.2

MAC 1 0.980 0.949 0.969 0.953 0.992 0.999 0.989

MAC 2 0.896 0.857 0.986 0.935 0.962 0.973 0.964

Table 5 Model 1 interface Tied Joint stiffness parameters Parameter kx ky kz krx

Initial Model 1a 6.13e6 1.00e6 1.00e6 1.00e7

Modified Model 1b 5.00e7 9.50e6 9.50e6 5.00e6

Units lbf /in lbf /in lbf /in in·lbf /rad

kry

1.00e7

1.40e6

in·lbf /rad

krz

1.00e7

1.00e8

in·lbf /rad

Interface contact size

Full surface (1.25 × 1.25)

1.125 (diameter)

in

5.2 Free-Free Comparison Taking Model 1b as the baseline calibrated model, mode shapes and frequencies were calculated for each of the other three models outlined in Sect. 4, and results were compared to test data, summarized in Table 7. Comparing the results, MAC values were similar between all cases, though slightly lower for Model 4b. Frequency results were close as well, except that Model 2b and 4b had larger error in modes 1 and 3. Model 3b matched best with the calibrated Model 1b and the test. Results from the calibration study show that a solid mesh representation of the bolt may not be required for structural dynamic models, since the single Tied Joint model (Model 3) was able to provide almost identical results. In addition, the two-spring model (Model 2) does not provide any additional advantages. Furthermore, the Tied Data model (Model 4) is the least accurate in terms of frequency and may not be suitable, especially if a model of the fastener is required. Overall, the study indicates that modeling bolted joints with a single Tied Joint spring element appears to be a low fidelity yet accurate technique that exhibits similar dynamic behavior to the a fully detailed bolt with a Tied Joint interface.

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Fig. 15 Updated Model 1b MAC Table 6 Calibrated Model 1b comparison Test 1 mode 1 2 3 4 5 6 7

Test 1 Freq (Hz) 193.6 356.4 371.2 1201.5 1307.3 1517.2 1565.8

Fig. 16 Free-free modes test-model comparison

Model 1b mode 1 2 3 4 5 6 7

Model 1b Freq (Hz) 201.5 371.0 382.4 1183.3 1346.5 1496.8 1500.4

Freq Err % 4.1 4.1 3.0 −1.5 3.0 −1.3 −4.2

MAC 0.982 0.958 0.972 0.983 0.998 0.999 0.997

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Fig. 17 Free-free mode 5 (axial) test-model comparison Table 7 Free-free model and test comparison Model 1b Freq (Hz) 201.5 371.0 382.4 1183.3 1346.5 1496.8 1500.4

% Diff 4.1 4.1 3.0 −1.5 3.0 −1.3 −4.2

MAC 0.982 0.958 0.972 0.983 0.998 0.999 0.997

Model 2b Freq (Hz) 221.3 373.7 401.9 1189.2 1354.6 1496.9 1529.4

% Diff 14.3 4.9 8.3 −1.0 3.6 −1.3 −2.3

MAC 0.981 0.958 0.971 0.983 0.998 0.999 0.996

Model 3b Freq (Hz) 201.3 370.7 382.2 1183.1 1343.4 1496.8 1500.0

% Diff 4.0 4.1 3.0 −1.5 3.0 −1.3 −4.2

MAC 0.982 0.958 0.972 0.983 0.998 0.999 0.997

Model 4b Freq (Hz) 255.5 362.2 427.8 1212.9 1314.6 1491.8 1675.6

% Diff 32.0 1.6 15.2 1.0 0.6 −1.7 7.0

MAC 0.978 0.957 0.970 0.982 0.997 0.999 0.986

% Diff 14.0 3.5 10.5 −0.9 3.2 −5.0 14.0

MAC 0.987 0.982 0.994 0.986 0.979 0.997 0.987

Model 3b Freq (Hz) 104.1 194.0 356.1 1121.1 1274.3 1407.8 104.1

% Diff 3.1 2.5 5.1 −1.4 1.7 −6.9 3.1

MAC 0.987 0.982 0.994 0.986 0.979 0.998 0.987

Model 4b Freq (Hz) 134.6 190.0 398.7 1149.9 1260.7 1578.8 134.6

% Diff 33.2 0.4 17.6 1.1 0.6 4.4 33.2

MAC 0.986 0.983 0.992 0.985 0.978 0.993 0.986

Table 8 Fixed base model and test comparison Model 1b Freq (Hz) 104.2 194.2 356.3 1121.3 1280.0 1408.0 104.2

% Diff 3.2 2.6 5.1 −1.4 2.1 −6.9 3.2

MAC 0.987 0.982 0.994 0.986 0.979 0.998 0.987

Model 2b Freq (Hz) 115.2 195.9 374.5 1126.8 1293.9 1435.8 115.2

5.3 Fixed Base Comparison Fixed base comparisons were also calculated by fixing the bottom surface of the AP and frequencies and mode shapes were compared as in the previous section. The trends for each model were similar to the free-free comparisons and are summarized in Table 8. Overall, both the free-free and fixed base comparisons between all models show that joint dynamics can be adequately represented with a simplified single Tied Joint model. Since member stiffness dominates in most typical joints, it is reasonable to expect this result. However, for analyses where bolt force is important or where bolt stiffness is large relative to the members, the above joint models may need to be adjusted. For example, in Model 3, the Tied Joint stiffness parameters directly affect the loading behavior of the overall joint and not necessarily the bolt itself. In cases where bolt force is important, such as vibration or fatigue analyses, these joint models may not be accurate.

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6 Initial Fatigue Assessment Once a calibrated joint model is developed and the dynamics of the joint are calculated using FE analysis, the next step is to evaluate the joint under external loading. For dynamic loading scenarios, a fatigue analysis can be performed. For the joint models presented in this work, a brief fatigue assessment was done in order to show the importance of joint modeling.

6.1 Background Fatigue failure is caused by cyclic loading and occurs well below the tensile strength of the material. One metric that is typically used in fatigue analysis is the endurance limit, which is the ideal theoretical stress value below which a material has infinite fatigue life. In general, for steel it can be estimated as half the ultimate tensile strength [3]. Correction factors related to geometry, loading, and surface finish can subsequently be multiplied on top of this value to obtain a more realistic endurance limit. Bolts have lower endurance limits than bulk steel due to factors such as stress concentrations caused by the threads [3]. Literature has shown that the endurance limit for fasteners can be as low as 4–5% of the ultimate tensile strength [4, 5]. In both [6] and [7], the effect of preload on fastener fatigue life was studied. Results from [6] showed endurance limits around 5% of ultimate strength for a bolt with low preload, and 7% for a bolt with high preload. In [7], both coarse and fine threads were considered, and resulting endurance limits were in the 4–5% range. It should be noted that all these results were conducted in the axial-only load condition except [4], which experimentally verified that the fatigue life of bolts is higher for bending loads. In addition, all bolts tested in the literature were M8.8 (SAE grade 5) and above. For the structure discussed in this work to be viable as a fatigue testing apparatus, the endurance strength of the KB and AP components must exceed that of the bolt being tested.

6.2 Evaluating Uniaxial Fatigue To evaluate fatigue for the structure, endurance limits for the KB, AP, and bolt can be compared. For the KB and AP, 4340 steel with a minimum tensile strength of 150 ksi was selected. The material choice was influenced by cost, heat treatment capability, and the availability of fatigue data, which was obtained from [8]. Due to the nature of the axial loading planned for fatigue testing, data for an R-value of −1.0 for fully reversed loading was used. Unnotched data was used for most of the structure and notched data for the stress concentrations at the fillets. An axial loading factor, k1 , of 0.85 and a machined surface factor, k2 , of 0.7 can be used as recommended from [3] to  modify the endurance limit, Se , and obtain the corrected value for the structure: Se = k1 k2 Se 

(5)

SeS,unnotched = (0.85)(0.7)(65) = 38.68 ksi

(6)

SeS,notched = SeS = (0.85)(0.7)(30) = 17.85 ksi

(7)

As mentioned in the background, 7% of the bolt tensile strength can be used as a rough approximation of the endurance limit. For a grade 5 bolt, Seb ≈ 0.07Sut = (0.07) (120 ksi) = 8.4 ksi

(8)

Once the endurance strengths are determined, the fatigue factors of safety for the assembly and bolt can be evaluated assuming constant amplitude cyclic loading. For variable amplitude loading, a cycle counting method such as Miner’s law would be needed [3], but for simplicity constant amplitude loading was used. Factors used in design for theoretical infinite

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life are presented here. Assuming a uniaxial state of stress at a critical point in the structure, the corresponding fatigue safety factor for the structure is given by [3] as  nf,structure =

σaS σmS + S SeS Sut

−1 (9)

For the bolt, the safety factor can be calculated using the following equation: nf,bolt =

 b  − σi Seb Sut   b σ b + Sb σ b − σ Sut i a e m

(10)

In Eqs. (9) and (10), Sut is the ultimate tensile strength, σ i is the preload stress, σ m is the midrange stress, and σ a is the alternating stress. Superscripts designate the structure (KB or AP) values (S ) or the bolt (b ) values. For axial loading, the external force is shared between the bolt and joint members. This is represented using the joint stiffness constant, C, which indicates the portion of the load taken by the bolt. Equations for the alternating and midrange stresses in the bolt use this stiffness constant [3]: C (Pmax − Pmin ) 2At

(11)

C (Pmax + Pmin ) Fi + 2At At

(12)

σa =

σm =

Here Pmax is the maximum value of the applied load, Pmin is the minimum value, At is the tensile stress area of the bolt, and Fi is the preload force. Equations (9) through (12) highlight that even for the simple case of uniaxial fatigue analysis in the time domain, an accurate value for C is needed to determine the bolt factor of safety for cyclic loading. Overall, these equations used in fatigue design and analysis show that this parameter is important, and the next section demonstrates how different FE joint model techniques can affect this stiffness constant.

7 Joint Stiffness Comparison Since the alternating and midrange stresses used in Eqs. (11) and (12) depend on the stiffness constant, C, it is important to ensure that this value can be accurately determined. Analytical calculations from literature are performed for the structure geometry, and both linear and nonlinear FE analyses are done to show how this stiffness factor can vary depending on the model used.

7.1 Analytical Calculations First, linear structural analysis is done to compare the natural frequencies using the calibrated and analytical joint stiffness. These stiffness values are calculated using [3]. The bolt stiffness can be calculated using a series of springs representing the shank and threaded portion: kb =

Ad At E Ad lt + At ld

(13)

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In Eq. (13), Ad is the shank area, E is Young’s modulus, ld is the shank length, and lt is the length of the threaded portion in the grip. From [3] the member stiffness is based on a typical pressure cone angle α = 30◦ and washer or bearing face diameter, dw (1.5 times the nominal bolt diameter, d), and total grip length l: π Edtanα

⎡

km =

2 ln ⎣

⎤

(14)

(ltanα+dw −d)(dw +d) ⎦ (ltanα+dw +d)(dw −d)

Substituting in the bolt, KB, and AP joint thickness dimensions and material properties, the stiffnesses can be obtained and the theoretical joint stiffness constant, C, can be calculated using C=

kb km + kb

(15)

Equation (14) was applied to obtain the calibrated FE Model 1b member stiffness parameters, discussed in Sect. 5 and listed in Table 5. Similarly, Eqs. (13) through (15) were used to compute the stiffness constant and values for the ¼-20 UNC bolt and joined members. Results are summarized in Table 9 for both a solid shank and partially threaded bolt. Using these values as a baseline, different joint models were studied for comparison. The next section presents linear analysis of the bolted joint and comparison of the stiffness constant and axial mode frequencies.

7.2 Linear Structural Dynamic Analysis An additional free-free modal analysis was performed on Model 1b detailed in Sect. 5.1. In this case, instead of using the calibrated stiffness (kx = 5e7), the Tied Joint axial member stiffness value was chosen based on the analytical solution presented in the previous section (kx = 6.13e6), calculated with Eqs. (13) to (15). All other Tied Joint stiffness parameters were kept the same. The resulting axial mode frequency and computed stiffness constant were compared to the original Model 1b, and Table 10 summarizes the frequency and stiffness constant results. The resulting frequency of the axial mode is close to 200 Hz lower than the calibrated model. This is due to the difference in axial stiffness that was chosen. Since kx = 5e7 for the calibrated model, the member stiffness is about an order of magnitude larger than the analytical solution. Using the same approximate bolt stiffness of 1.33e6, there is over a factor of 5 difference in C. Using preliminary hand calculations for the joint may give the theoretical stiffness constant and loading behavior, but it results in a much lower frequency of the axial mode. It should also be noted that all the models used an interface contact of 0.5625 inches, which was chosen based on the calibration to modal test data. This value can be compared to the analytical solution of the pressure cone diameter at the interface [3]: dinterface = dw + l tan α

(16)

Table 9 Calculated stiffness constants for structure joint Configuration Partially threaded bolt Full solid shank

Joint dimensions ld = 0.875, lt = 0.125 ld = 0.875, ld = 0

Bolt stiffness (lbf /in) 1.33e6 1.63e6

Member stiffness (lbf /in) 6.13e6 6.41e6

Calculated stiffness constant, C 0.179 0.202

Table 10 Effect of axial stiffness on mode frequency and stiffness constant Model Calibrated Model 1b Model 1b adjusted with analytical stiffness

Tied Joint axial stiffness (lbf /in) 5.00e7 6.13e6

Approximate stiffness constant, C 0.026 0.179

Free-free axial mode frequency (Hz) 1346 1165

Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure

49

Using α = 30◦ , l = 0.4375 in., and dw = 0.375 in., results in dinterface = 0.628 in., so the interface diameter used in the models is comparable. In [9], the interface contact for a single bolt lap joint was also calculated using Eq. (16) and good frequency match was obtained for various bolt models. Thus, having a calibrated linear structural model does not imply that the fastener or load transfer in the joint is necessarily accurate. However, until the joint is statically tested, or more detailed analysis is done, it is not immediately apparent if either value of the stiffness constant in Table 10 is correct, since the linear models presented here do not accurately account for important factors such as preload or contact. By performing a nonlinear analysis that includes the effects of frictional contact and preload, the joint stiffness can be studied further.

7.3 Nonlinear Solid Mechanics Analysis In order to better understand the joint stiffness and loading behavior, SIERRA Solid Mechanics (SIERRA/SM) was used for the analysis. SIERRA/SM is a nonlinear FE code that is suited to handling bolted joints, contact, and preload. Implicit quasi-static analysis was done to further investigate the joint and examine the accuracy of the linear structural dynamic models. Three separate loading cases were set up to evaluate the contribution of nonlinearity and the effect of loading planes, loading distance from bolt axis, and washer face diameter, since each of these influences the resulting joint behavior [10]. Case 1 was the most representative of the laboratory loading case in fixed base testing. The load was applied over a small annular region on the KB top surface and the AP bottom surface was fixed, as shown in Fig. 18. Case 2, shown in Fig. 19a, moved the loading to the KB surface beneath the bolt head washer to isolate the effects of the KB deformation as the joint is pulled. In addition, the AP surface beneath the nut washer was fixed. For Case 3, the joint was simplified, and the washers were removed to apply boundary conditions as close to the bolt axis as possible, as in Fig. 19b. For each loading case, a coefficient of friction of 0.3 was used and preload was applied to a separate portion of the bolt shank using artificial strain, as shown in Fig. 20. The required SAE grade 5 torque of 2027.25 lbf was applied and the load was linearly ramped up from 0 to 3000 lbf to achieve joint separation. The internal bolt force and the interface contact force throughout the analysis were calculated.

Applied Load

Fixed Fig. 18 Case 1 loading configuration

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Applied Load

Applied Load

Fixed

Fixed

Fig. 19 Cases 2 (a) and 3 (b) loading configurations

Fig. 20 Bolt mesh with separate shank section for preload

The resulting plots in Fig. 21 show the bolt internal force and the interface contact force as a function of the applied load. By studying the load plots in Fig. 21, the separation force, Fs , at the change in slope was used to determine the equivalent stiffness constant using the following eqs. [3]: Fs =

Fi 1−C

(17)

Fi Fs

(18)

C =1−

Results for each case along with analytical solutions are summarized in Table 11. For the FE analysis, the stiffness constant was computed with Eq. (17) based on the separation load in Fig. 21. For the analytical solutions, the separation load was

Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure Bolt Force

2500

2500

2000

2000

1500

1000

500

Case 1 Case 2 Case 3

1500

1000

500

Case 1 Case 2 Case 3

0

0 0

a)

Interface Contact Force

3000

Contac t Forc e (lbf)

Internal Forc e (lbf)

3000

51

500

1000

1500

2000

2500

Applied Load (lbf)

3000

0

500

b)

1000

1500

2000

2500

3000

Applied Load (lbf)

Fig. 21 Quasi-static loading results, bolt force (a) and member interface contact force (b) Table 11 Separation load and joint stiffness comparison Model Case 1 Case 2 Shigley, (α = 30◦ ) Shigley, (α = 33◦ ) Case 3 (no washers) Shigley, no washers (α = 30◦ ) Shigley, no washers (α = 33◦ )

Preload value (lbf ) 2023 2016 2027 2027 2021 2027 2027

Separation load (lbf ) 2103 2135 2470* 2435* 2403 2541* 2503*

Approximate stiffness constant, C 0.038* 0.056* 0.179 0.167 0.159* 0.202 0.190

computed with Eq. (18) based on a joint stiffness constant value from Eq. (15). Table 11 summarizes the results, where (*) are the computed values. These results show that the stiffness constant can be highly variable and dependent on the loading geometry. For Case 1, the nearly constant bolt force until separation is due to the gradual deformation of the KB support structure as it is pulled. Nassar et al. [11] explain the phenomenon, stating that the separating force results in bending of the clamped member, which causes the joint to gap starting at the outer edges as the members are pried apart. This gapping explains the nonlinear variation in the bolt force until separation. Traditional methods and linear models cannot account for the interface compression and separation behavior. Additionally, although Case 1 gives a similar C value to the calibrated structural model shown in Table 10, that model does not consider frictional contact. The nonlinear force relationship for Case 1 shows a slight decrease, and then gradual increase until the separation, and a linear ramp may be inappropriate even if the separation loads are similar. Cases 2 and 3 have linearly increasing bolt force before separation, but Case 2 has a lower slope. This indicates that the washer face diameter has a large effect on the joint load behavior. Case 3 is the closest behavior to an ideal linear joint, although the stiffness constant value of 0.159 is still slightly below the analytical solution of 0.179. Since most joints exhibit some amount of this nonlinearity due to the location of loading, a linear approximation may not be accurate. In the context of fatigue analysis using a linear structural model, these results indicate that the stiffness constant and any stress calculations done using Eqs. (11) and (12) combined with analytical equations would not be accurate for this joint. Even though the model was calibrated to modal test data, these results indicate that the fastener load behavior may not be properly represented. Additionally, using Eq. (17) in designing a joint against separation could result in underconservative calculations since the separation load in the nonlinear analysis was observed to be about 400 lbf lower. Although the significance of these effects could vary depending on the joint geometry, caution must be used in applying these common formulae.

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8 Conclusion In this work, an experimental structure was designed to assess and improve current joint models, enhance fatigue analysis, and advance testing capabilities. Modal testing was conducted, FE analyses were performed on four separate joint models, and model calibration was performed. A brief fatigue study was presented and analytical calculations for joint stiffness were compared to the calibrated model and results from nonlinear FE analysis. Of the four linear joint models presented, the simplified single Tied Joint spring model (Model 3) was found to accurately represent joint dynamics. Good agreement was obtained for mode shapes and frequencies compared to the calibrated solid bolt model (Model 1) and test data. Although the models matched test data well, additional nonlinear analysis showed that the linear models cannot account for the nonlinear effects that occur as the joint is loaded and eventually separates. Joint stiffness calculations showed that traditional analytical equations and linear modeling techniques can be inaccurate at representing fastener stiffness and loading behavior due to this phenomenon, and that resulting fatigue calculations can be affected. Additionally, fatigue testing using the designed structure was considered, but nonlinearity of the joint means that it will not function ideally as a fatigue testing device due to coupling of bending and axial modes along with the nearly constant bolt load until separation. Moreover, the dynamic behavior of jointed structures tends to vary between unique instances of nominally identical assemblies (unit-to-unit variability), so more data is needed to assess the modal behavior relative to fatigue testing. Further joint characterization is needed, and an updated design could address some of these issues, so future work will investigate this. Acknowledgments Notice: This manuscript has been authored by National Technology and Engineering Solutions of Sandia, LLC, under Contract No. DE-NA0003525 with the US Department of Energy/National Nuclear Security Administration. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US Government purposes.

References 1. Crane, N.K., Day, D.M., Hardesty, S., Lindsay, P., Stevens, B. L.: Sierra/SD – User’s Manual - 4.56. United States: n. p., (2020). https://doi.org/ 10.2172/1673458 2. Brown, K.H., Morrow, C., Durbin S., Baca, A: Guideline for Bolted Joint Design and Analysis: Version 1.0. United States: N. p., (2008). https:/ /doi.org/10.2172/929124 3. Budynas, G.R., Nisbett, K.J.: Shigley’s Mechanical Engineering Design: Ninth Edition. McGraw Hill, New York (2011) 4. Wentzel, H., Huang, X.: Experimental characterization of the bending fatigue strength of threaded fasteners. Int. J. Fatigue. 72, 102–108 (2015) 5. Bickford, J.H.: Introduction to the Design and Behavior of Bolted Joints: Non-Gasketed Joints: Fourth Edition. CRC Press (2007) 6. Shahani, A.R., Shakeri, I.: Experimental evaluation of the effect of preload on the fatigue life of bolts. Int J Steel Struct. 15(3), 693–701 (2015) 7. Stephens, R.I., et al.: Fatigue of high strength bolts rolled before or after heat treatment with five different preload levels, SAE Technical Papers (2005) 8. MMPDS-11: Chapter 2 Steel Alloys. Metallic Materials Properties Development and Standardization (MMPDS). Battelle Memorial Institute (2016) 9. Kim, J., Yoon, J., Kang, B.: Finite element analysis and modeling of structure with bolted joints. Appl. Math. Model. 31, 895–911 (2007) 10. He, P., Liu, J., Zhang, C., Liu, Z.: Analytical modeling of axial stiffness of tensile bolted joints under concentric external load. J. Mech. Sci. Technol. 33(11), 5285–5295 (2019). https://doi.org/10.1007/s12206-019-1020-8 11. Nassar, S.A., Yang, X., Gandham, S.V.T., Wu, Z.: Nonlinear deformation behavior of clamped bolted joints under a separating service load. J. Press. Vessel. Technol. 133 (2011). https://doi.org/10.1115/1.4002674

A Non-Masing Microslip Rough Contact Modeling Framework for Spatially and Cyclically Varying Normal Pressure Justin H. Porter, Nidish Narayanaa Balaji, and Matthew R. W. Brake

Abstract The development of predictive computational models of joints is an ongoing challenge within the community. Unlike monolithic structures, the addition of friction in joints introduces nonlinearities in the vibration response of the structure. Frictional contact models can be applied to reproduce the nonlinear behavior, but the best predictive modeling framework is not clear. Elastic dry friction is a popular choice for predictive modeling, but recent work has highlighted its inability to recreate experimental behavior. As an alternative, several microslip rough contact models have been derived from distributions of asperity heights. Unlike elastic dry friction, these models have a smooth transition from sticking to slipping allowing them to capture smoother experimental trends. However, these models have often used the Masing assumptions and constant (over the interface and a cycle) normal pressures. The assumption of constant normal pressures neglects the kinematics of jointed interfaces, while the Masing assumptions do not generally hold for normal pressures that vary throughout a cycle. The present work seeks to further develop a microslip rough contact modeling framework without the simplifying assumptions to realize more physical simulations. Experiments on a benchmark structure, along with interfacial scans, are used to assess the validity of the proposed modeling framework. Keywords Frictional systems · Rough contact · Hysteretic modeling · Nonlinear modal analysis · Jointed interface

1 Introduction Bolted joints are used in many engineering applications motivating the development of accurate computational models. However, practical applications of bolted joints require experimental tests to obtain parameters for computational models. Predictive models based on surface features present an opportunity to reduce expensive testing but require further development to achieve satisfactory predictions. One of the most commonly used predictive models is elastic dry friction [1], which prescribes a tangential stiffness followed by a Coulomb friction law. These parameters can easily be obtained from fretting tests [2] or estimated from surface scans [3]. However, a recent model fitting effort has shown that even with sufficient degrees of freedom representing the interface and optimized model parameters, elastic dry friction is unable to satisfactorily reproduce the dynamic behavior of a jointed structure [4]. An alternative approach is to use surface scan data to model asperity interactions statistically resulting in a microslip model of the joint [5–9]. However, these studies applied simplifying assumptions including the Masing assumptions or considering only monotonic loading [6–9], spatially constant normal tractions [5–9], and predicted relationship between bolt torque and normal prestress [5–7, 9]. In practical applications, the Masing assumptions are not guaranteed to be valid since normal pressure can vary throughout a cycle [4]. Furthermore, the normal pressure distribution throughout the interface due to a tensioned bolt is not, in general, constant [3]. Finally, there is high uncertainty in the torque–tension relationships of bolts [10]. This work addresses these assumptions through the development of a new friction model without using the Masing assumptions, a refined interface representation that can capture the full kinematics, and measured bolt strains to accurately capture the normal prestress.

J. H. Porter () · N. N. Balaji · M. R. W. Brake Department of Mechanical Engineering, Rice University, Houston, TX, USA e-mail: [email protected]; [email protected]; [email protected] © 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_5

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2 System Modeling The present study seeks to predict experimental nonlinear frequency and damping trends of the Brake-Reuß Beam (BRB) [3]. Nonlinear frequency and damping trends are extracted with a quasi-static modal analysis technique termed Rayleigh Quotient-based Nonlinear Modal Analysis (RQNMA) [11]. In addition to the hysteretic damping from the friction interactions at the interface, a viscous damping factor of 1e−3 is added to all of the computational results. To keep computation times reasonable, a reduced order model with 49 fixed interface modes and 152 zero-thickness elements (ZTEs), each with three translational degrees of freedom (DOFs) is used to represent the BRB [12]. The remeshed interface for the reduced order model is shown in Fig. 1a. The focus of this work is the friction model that is applied to the ZTEs representing the BRB interface. The friction model calculates forces from an asperity contact model and integrates the forces over a probability distribution for the nominal gap between the contacting asperities to predict the frictional traction within a ZTE (see Fig. 1b). Parameters for the asperity contact model are obtained from processing surface scan data and are presented in Table 1.

(a)

(b)

Fig. 1 Modeling approach: (a) remeshed interface from model reduction [12] and (b) procedure for calculating friction traction for a single element

A Non-Masing Microslip Rough Contact Modeling Framework for Spatially and Cyclically Varying Normal Pressure

55

Table 1 System parameters used friction model Parameter E ν R R  Re η α β zmax F2 (e) x y

Description Elastic modulus Poisson’s ratio Major principal relative radius of curvature [13] Minor principal relative radius of curvature [13] Equivalent radius [13] Asperity density First parameter of beta distribution for normalized gap Second parameter of beta distribution for normalized gap Maximum value of gap Function of eccentricity of contact [13] Ellipsoid tangential adjustment for x direction [14] Ellipsoid tangential adjustment for y direction [14]

Value 192.85 GPa [3] 0.29 [3] 9.393 mm 0.0925 mm 0.932 mm 2.902e6 Asperities/m2 6.091 7.230 0.0459 mm 1 (spheres), 2.039 (ellipsoids) 1 (spheres), 2.701 (ellipsoids) 1 (spheres), 3.287 (ellipsoids)

3 Asperity Contact Modeling Three asperity contact models are considered for both sphere on sphere and ellipsoid on ellipsoid contacts. All of the asperity contact models use Hertzian normal contact [15] and assume that the material properties of both sides of the interface are identical. For a normal displacement un and gap between surfaces z, the asperity interference w is w = un − z.

(1)

If w ≤ 0, then the asperity is not in contact and all forces associated with the asperity are zero. For a positive interference, the normal force N is √ 2E Re N= w 3/2 , (2) 3(1 − ν 2 ) [F2 (e)]3/2 where Re is the equivalent radius as defined by Johnson [13] and F2 (e) (defined by Johnson [13] and included in Table 1) is a function of the eccentricity of the contact ellipse. The eccentricity of the contact ellipse is defined as e = (1 − b2 /a 2 )1/2 ,

(3)

which is independent of normal load. In addition, Hertzian contact gives expressions, dependent on normal load, for the major and minor semi-axes of contact a and b, respectively (for the case of sphere on sphere contact, a = b). Based on surface scans, the major and minor semi-axes are aligned with the y and x axes, respectively, so the following equations only consider this case. Hertzian normal contact has been extended to tangential displacements for spheres by Cattaneo [16] and Mindlin [17] and for ellipsoids by Mindlin [17] and Deresiewicz [14]. These analytical solutions are simplified to create the three types of tangential asperity contact models shown in Fig. 2. All three friction models apply independent forces in the x and y directions. The first two asperity models use the elastic dry friction model to represent the full asperity with a tangential q stiffness kt dependent on the normal load. For direction q (chosen as x or y), the tangential force ft given a tangential q displacement ut is

q

ft =

⎧ q q q ⎪ f + kt (N )(ut − ut,0 ) ⎪ ⎨ t,0 

q

|ft,stuck | < μN (4)

q

ft,stuck ⎪ ⎪ ⎩ q μNsgn[ft,stuck ] q

Otherwise q

for friction coefficient μ, force at previous instant ft,0 , and displacement at previous instant ut,0 . For the first friction model, referred to as “tangent,” the tangential stiffness is taken to be the slope of the analytical partial slip solution at

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1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.5

-1

-0.5

0

0.5

1

1.5

Fig. 2 Example hysteresis loops for different asperity models compared to analytical partial slip solution at constant normal load. The axes are nondimensional so the hysteresis loops are applicable to contact of spheres or ellipsoids with displacements aligned with either principal axis

zero displacement or kt,T angent (N ) =

4Ga . (2 − ν)q

(5)

Here, q is defined for the x and y directions as x =

! " ν νE(e) 4 1 − 2 K(e) + π(2 − ν) e e2

(6)

and " ! ν νE(e) 4 y = , 1 − ν + 2 K(e) − π(2 − ν) e e2

(7)

where K(e) and E(e) are the elliptic integrals of the first and second kind, respectively. For spherical contact, x and y are defined to be 1. The values of x and y for the present problem are included in Table 1. For the second asperity contact model, the secant stiffness of the analytical solution at the inception of complete slip is used as the tangential stiffness in the elastic dry friction model. This model will be referred to as the “secant” model. This stiffness is q

kt,Secant (N ) =

8Ga . 3(2 − ν)q

(8)

The third asperity contact model is an Iwan model with individual sliders representing concentric elliptical rings. Expressions over the contact area are converted to a reference domain of a unit circle parameterized with radius ρ as ρ2 =

x2 y2 + . b2 a2

(9)

A Non-Masing Microslip Rough Contact Modeling Framework for Spatially and Cyclically Varying Normal Pressure

57

The traction at a given radius is calculated with an elastic dry friction model of the form ⎧q q q ⎪ t (ρ) + kt,I wan (N )(ut − ut,0 ) ⎪ ⎨t,0  q q tt (ρ) = tt,stuck (ρ) ⎪ ⎪ ⎩ q ts (ρ)sgn[tt,stuck (ρ)]

q

|tt,stuck (ρ)| < ts (ρ) (10) Otherwise.

q

Here, tt,0 (ρ) is the traction at the given radius at the previous instant, noting that the value is taken based on the traction at the same radius in the physical domain at the previous instant and not necessarily at the same value of ρ. In (10), the slip traction is # 3N ts (ρ) = μ 1 − ρ2 (11) 2π ab based on the analytical pressure distribution [15], and the tangential stiffness is !

q kt,I wan

4G = q (2 − ν)π b

" (12)

corresponding to the analytical tangential stiffness at zero displacement being evenly distributed spatially over the contact area. Finally, the tractions at each radius are integrated over the reference domain as q ft

$

1

= 2π ab 0

q

tt (ρ)ρdρ,

(13)

where the integral is evaluated numerically. From Fig. 2, it is clear that the tangent and secant models bound the analytical solution, while the Iwan model most closely approximates the analytical behavior for the case of a constant normal load. In addition, the use of these three models derived from elastic dry friction allows for variations in the normal load throughout a cycle, which the analytical solution does not handle for the most general cases.

4 Results Figure 3 shows several example backbones using different asperity contact models and different friction coefficients. Though all of the models show notable errors compared with the experimental data, comparisons of the models provide key insights into different aspects of the modeling procedure. Figure 3a shows a comparison of the different asperity contact models for treating each asperity as either contacting spheres or ellipsoids. From the comparison, using ellipsoids instead of spheres for the asperities increases the low amplitude frequency by 3.2 Hz and thus reduces the error compared to the experimental behavior. Note that all of the adjustment factors for the ellipsoid case reduce the forces at a given displacement from the spherical case. However, since the system is prestressed to a constant normal bolt tension rather than a constant normal displacement, the net effect of changing from spheres to ellipsoids is to increase the frequency. For both ellipsoids and spheres, the tangent and Iwan asperity models achieve nearly identical frequency results. The secant asperity contact model produces lower frequencies than the other models for the same type of contact. This suggests that if only the frequency behavior is of interest, then using an elastic dry friction model for the asperity contact is sufficient and may be preferable due to the reduced computational complexity. While the tangent and secant models bound the frequency behavior of the Iwan models, the Iwan models for both the spheres and the ellipsoids produce more damping than either the tangent model of the secant model for the respective asperity types. This is expected since the Iwan models dissipate energy at all amplitude levels unlike the linear stiffness regimes of the other models. Overall, this increase in damping is relatively minor and only visible for a few of the intermediate amplitude levels. Figure 3b shows the behavior of the system for different values of the friction coefficient μ. While the present work seeks to produce a predictive model, the friction coefficient cannot yet be predicted. In Fig. 3b, increasing the friction coefficient generally decreases the damping and reduces the drop-off in frequency at higher amplitudes. Here, the cases of μ = 0.01 and

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180

180

170

170

160

160

150

150

10

-2

10

-3

10 -7

10 -6

10 -5

(a)

10 -4

10

-2

10

-3

10 -7

10 -6

10 -5

10 -4

(b)

Fig. 3 Modal backbones for (a) different asperity representations for μ = 0.05 and (b) different values of the friction coefficient μ with the ellipsoid Iwan asperity model

0.05 bound the experimental damping factor for most amplitude levels. In addition, the frequency drops by 21.7 and 7.82 Hz over the amplitude range for the cases of μ = 0.01 and 0.05, respectively. Thus, these two cases bound the experimental frequency drop of 16.4 Hz. Further development of the contact model is required to address the error in low amplitude natural frequency and dissipation. However, the presented results illustrate the importance of considering the eccentricity of contact asperities and provide promising behavior by bounding the frequency shift and the damping factor with friction coefficients of μ = 0.01 and 0.05. Since the time of publication, an error was discovered with the calculation of the elliptical contact parameters, so the effect of using ellipsoids on the frequency is likely less than shown here but in the same direction. Acknowledgments This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number(s) DE-SC0021110. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

References 1. Yang, B.D., Menq, C.H.: Modeling of friction contact and its application to the design of shroud contact. J. Eng. Gas Turbines Power 119(4), 958–963 (1997). https://doi.org/10.1115/1.2817082 2. Schwingshackl, C.W., Petrov, E.P., Ewins, D.J.: Measured and estimated friction interface parameters in a nonlinear dynamic analysis. Mech. Syst. Signal Process. 28, 574–584 (2012) 3. Balaji, N.N., Chen, W., Brake, M.R.W.: Traction-based multi-scale nonlinear dynamic modeling of bolted joints: formulation, application, and trends in micro-scale interface evolution. Mech. Syst. Signal Process. 139, 106615 (2020). https://doi.org/10.1016/j.ymssp.2020.106615 4. Porter, J.H., Balaji, N.N., Brake, M.R.W.: A quantitative assessment of the model form error of friction models across different interface representations for jointed structures. Mech. Syst. Signal Process. Under review 5. Eriten, M., Polycarpou, A.A., Bergman, L.A.: Physics-based modeling for fretting behavior of nominally flat rough surfaces. Int. J. Solids Struct. 48(10), 1436–1450 (2011). ISSN 0020-7683. https://doi.org/10.1016/j.ijsolstr.2011.01.028 6. Li, W., Zhan, W., Huang, P.: A physics-based model of a dynamic tangential contact system of lap joints with non-Gaussian rough surfaces based on a new solution. AIP Adv. 10(3), 035207 (2020). ISSN 2158-3226. https://doi.org/10.1063/1.5143927 7. Zhan, W., Huang, P.: Physics-based modeling for lap-type joints based on the Iwan model. J. Tribol. 140(5) (2018). ISSN 0742-4787. https:// doi.org/10.1115/1.4039530. Publisher: American Society of Mechanical Engineers Digital Collection 8. Zhan,W., Huang, P.: Modeling tangential contact based on non-Gaussian rough surfaces. In: Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology (2018). https://doi.org/10.1177/1350650118758742. SAGE Publications, London, England

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9. Chen, J., Zhang, J., Hong, J., Zhu, L.: Modeling tangential contact of lap joints considering surface topography based on Iwan model. Tribol. Int. 137, 66–75 (2019). ISSN 0301-679X. https://doi.org/10.1016/j.triboint.2019.04.031 10. Ruan, M.: The variability of strains in bolts and the effect on preload in jointed structure. Masters thesis, Rice University, Houston, Texas (2019) 11. Balaji, N.N., Brake, M.R.W.: A quasi-static non-linear modal analysis procedure extending Rayleigh quotient stationarity for non-conservative dynamical systems. Comput. Struct. 230, 106184 (2020). ISSN 0045-7949. https://doi.org/10.1016/j.compstruc.2019.106184 12. Balaji, N.N., Dreher, T., Krack, M., Brake, M.R.W.: Reduced order modeling for the dynamics of jointed structures through hyper-reduced interface representation. Mech. Syst. Signal Process. 149, 107249 (2021). ISSN 0888-3270. https://doi.org/10.1016/j.ymssp.2020.107249 13. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985) 14. Deresiewicz, H.: Oblique contact of nonspherical elastic bodies. J. Appl. Mech. 24, 623–624 (1957) 15. Hertz, H.: Über die berührung fester elastischer körper (On the contact of elastic solids). J. Reine Angew. Math. 92, 156–171 (1882) 16. Cattaneo, C.: Sul contatto di due corpi elastici: Distribuzione locale degli sforzi. Rend. Accad. Naz. Lincei 27, 342–348, 434–436, 474–478 (1938) 17. Mindlin, R.D.: Compliance of elastic bodies in contact. ASME J. Appl. Mech. 16, 259–268 (1949)

Finite Elements and Spectral Graphs: Applications to Modal Analysis and Identification Nidish Narayanaa Balaji and Matthew R. W. Brake

Abstract There have been several studies developing calculus on graph domains, defining and generalizing the concepts of differential and integral operators on discrete domains. This chapter considers potential applications for such ideas in the field of modal analysis and identification. The thesis of the chapter lies in generalizing the “weak form” integral equations of the wave equation on a weighted graph domain using said developments (graph operations, Lebesgue integrals, etc.), leading to the definition of a parametric finite element model with sufficient flexibility to allow for model identification. There exist several results in the mathematical discipline of graph theory with regard to the physical interpretations of graphs and its subsets based on the relative weight distributions of the graph members (nodes and edges). The chapter will consider if there is merit for considering such ideas in the context of structural dynamics, specifically modal testing. Keywords Modal testing · System identification · Spectral Graphs

1 Introduction Classical frequency-domain techniques of parametric linear system identification [6] focus on identifying a parametric model of an input–output transfer function. Mathematically, the identification problem with a parameterized form such as U (ω) = G(ω; θ ), F (ω)

(1)

where ω denotes frequency, and U (ω) and F (ω) denote the “response(s)” (say displacement(s)) and “input(s)” (say force(s)) spectra, respectively,1 and θ (real or complex) denote the parameters, is to identify the “best” parameters θ to describe given data Z(ωk ) = [U (ωk ) F (ωk )] at a discrete sequence of frequencies {ωk }k=1,...,F . Table 1 provides a simplified overview of these methods. It goes without saying that all of these methods can be adapted (with varying amounts of complications), for multi-input (MI) and multi-output (MO) applications. One main issue with the above methods is that the physical meaning of the identified parameters (and by extension, the model) is not always readily apparent. This is partly addressed through finite element model updating, wherein similar identification techniques are used to refine parameters used for constructing the matrices of a finite element model [4]. Finite element techniques require the specification of (a) the governing equations and (b) the model domain/structural geometry. Taking inspiration from the fact that a finite element model is just the corresponding wave equation discretized over the continuum geometry, this chapter constructs a discretization of the wave equation over a graph network and demonstrates its application for linear identification of the wave equation. Unlike a discretized continuum domain, a graph domain is a discrete domain by definition and is characterized by the measures along the vertices and edges therein. The graph measures as well as terms in the assumed partial differential equation are estimated using an output-error optimization framework. Numerical

1 Classical

notation employs Y (ω) & U (ω) as outputs and inputs instead of U (ω) and F (ω) here. The disparity in notation is regretted.

N. N. Balaji () · M. R. W. Brake Department of Mechanical Engineering, Rice University, Houston, TX, USA e-mail: [email protected]; [email protected] © 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_6

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Table 1 Overview of classical parameterizations Name Rational form [6, 9]

Partial fraction expansion [2]

Parameterization (G(ω; θ)) nb r r=0 br ω  na 1 + r=1 ar ωr p q   Lr Sr + ω − σr r=−p ω − λr r=1

Remarks

C(iωIna − A)−1 B + D

Based on an underlying state-space representation: X˙ = AX + BF

r=0

State-space representation [5, 8, 9]

Most generic SISO transfer function model. Several variations exist. Assumes simple poles; appropriate for Experimental Modal Analysis (EMA)

U = CX + DF

results are presented for the identification of noisy frequency response data from a 1D bar discretized using regular finite elements.

2 Formulation and Approach: Discretization of the Wave Equation Mathematically, a wave equation is written on a domain  as ¯ = f (x) ¯ u(t, ¨ x) ¯ − c2 ∇ 2 u(t, x)

x¯ ∈ 

(+B.C.s)

u, f ∈ H(R+ × ).

(2)

(H is some function space.) The current section deals with the discretization of the above. Discussions will be restricted to scalar-valued functions for simplicity. First, classical finite elements are briefly reviewed, in order to develop motivation for graph discretization.

2.1 Quick Recap of Linear Finite Elements Assuming that the domain  ⊂ R3 allows for a finite element discretization as follows. The domain is discretized into nodal E locations {x¯n }n=1,...,N and finite elements over regions {k }k=1,...,NE (s.t. ∪N k=1 k = ). The solution approximation is written in terms of nodal values {u¯n }n=1,...,N using appropriate shape functions N(x) ¯ as uh (x) ¯ = N(x) ¯ u¯ with N :  → RN . h h 3×N ¯ is expressed as ∇u (x) ¯ = B(x) ¯ u, ¯ where B :  → R , denoting ∇N. Employing a Galerkin The gradient of u (x) projection in the function space yields the finite element discretized system, !$

" " $  !$ NT Nd u¯¨ + BT c2 Bd u¯ = NT f d + b.c.s .          M

(3)

F¯

K

The N × N matrices M and K are the mass and stiffness matrices of the system, and F¯ is the forcing vector. For a 1D wave equation (domain is (0, L)) with finite elements of length Le , represented using piecewise linear shape functions, the element-level matrices are ! " Le 2 1 M = 6 12 e

! " c2 1 −1 , K = Le −1 1 e

(4)

which can be “stitched” to represent their globally integrated counterparts. In summary, at the level of Eq. (3), if one may construct N and B for a given domain, operators M and K may be written down. If the domain is discretized, we should expect discrete operators, i.e., matrices.

Finite Elements and Spectral Graphs: Applications to Modal Analysis and Identification

63

2.2 Discretization on the Graph Domain The theoretical descriptions in this section are mostly based on [3,7] (with key differences that will be pointed out), to which the interested reader is referred for more details. A graph domain is denoted as G = (V , E), with V denoting the set of vertices it contains and E denoting undirected edges specified by pairs of vertices. The length of each edge in the mapping (interpreted as a “local” geometric realization) is taken to be the edge measure Ee . Figure 1 depicts a schematic element from a graph consisting of two vertices {1, 2} with measures V1,2 and the edge e with measure Ee . The figure also depicts the reference element that the edge interior is mapped to. Lebesgue integrals will be used in the following definitions for the development of the variational form. The following two measures are defined on the geometric realization of G: • A discrete vertex measure V s.t. V(v) = 0 iff v ∈ V . • An edge measure E coinciding with the Lebesgue measure (length) on the reference mapping of the edge interior (see Fig. 1). % % Using these measures, integrals over the vertices . . . dV and over the edges . . . dE may be realized as Lebesgue integrals. Integral over the whole graph may be defined by introducing integrating factors d = αdV + βdE such that the resulting  is still a measure. Ref. [7] specifies α : V → R and β : L2 (G \ V ). They will simply be chosen as indicator functions α(g) = 1(g ∈ V ) and β = 1(g ∈ G \ V ) for present purposes. Therefore, integral of a function u over the graph domain can be written as $ $ $ ud = uα dV + uβ dE G

$ =

 uV dV +

$

 uG\V dE.

(5)

  Here, uV and uG\V are restrictions of the function over the set of vertices and the set of edge interiors, respectively.

In order to approximate a function, say u, denote its values at vertices using a vector u¯ ∈ R|V | (with |V | denoting the cardinality of V ) and define its approximation on the reference element for the graph element in Fig. 1 as uhe (x)

 ! " ! " x x 1(x ∈ (0, Ee )) + 1(x ∈ {0}) {u¯1 } + = 1− 1(x ∈ (0, Ee )) + 1(x ∈ {Ee }) {u¯2 } Ee Ee   "  ! x x  u¯1  u¯1 1(x ∈ (0, Ee )) + 1(x ∈ {0}) 1(x ∈ {Ee }) . = 1− Ee Ee u¯2 u¯2

(6)

The restrictions of the approximation may be written as  

"  ! x x u¯1 (x) = 1 − Ee Ee u¯2 G\V      u¯1  uhe  (x) = 1(x ∈ {0}) 1(x ∈ {Ee }) . u¯2 V

uhe 

Fig. 1 Schematic graph element (illustrating edge-interior mapping)

(7) (8)

V1

Ee

V2

1

e

2

x 0

Ee

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Mathematically, the approximation uh may be termed to be edgewise linear, i.e., its restriction to each edge interior is a linear function. The gradient of the approximation may easily be written as ∂uhe (x)

   u¯1 1  . = −1 1 u¯2 Ee

(9)

Note that this is identical to the definition in [3] (only in slightly different notation). In this form, the gradient at the vertices, if estimated as continuous extensions of the gradients in the edge interiors, will assume multiple values at the vertices. This is analogous to the gradient discontinuity encountered in linear finite elements. Using the same ideas as before, the Galerkin projection of the wave equation on the graph domain now yields, over a single graph element, the following mass and stiffness matrices: " ! "" (1 − r)2 (1 − r)r V1 0 dr + (1 − r)r r2 0 V2 0 " ! " ! Ee 2 1 V 0 = + 1 0 V2 6 12  $ 1! ! ! " " " V1 1 −1 V2 1 −1 1 −1 2 Ee Ke = c dr + 2 + 2 Ee2 0 −1 1 Ee −1 1 Ee −1 1  ! " V1 + V2 c2 1 −1 1+ . = −1 1 Ee Ee

! $ Me = Ee

1!

(10)

(11)

The forcing vector, restricting the discussion to vertex-concentrated forces alone (with no inhomogeneous boundary conditions), becomes $ ¯ Fv = f (x)1(x ∈ {v})dV = f (v)Vv ⎡

V1

⎢ ⎢

⇒ {F¯ } = ⎢ ⎣ 

⎫ ⎤⎧ f (1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎥⎨ V2 ⎥ f (2) ⎥ .. ⎪ . .. ⎦⎪ . ⎪ . ⎪ ⎪ ⎪ ⎩ ⎭ VN f (N ) 

(12)

B

Here, the function f : V → R is the forcing function analogous to the continuous function f (x) in Eq. (3), and the vector {F¯ } ∈ RN is the discretized forcing vector. Upon assembly of matrices for a given graph, the wave equation (Eq. (2)) is written as Mu¨¯ + (αM + βK) u˙¯ + Ku¯ = BF¯v .

(13)

In the above, Rayleigh proportional damping terms have been inserted in addition to the inertial and stiffness terms from the undamped wave equation. More detailed linear damping methods may be considered (see [1] for an exhaustive treatment), but only the above two terms are selected for simplicity here. The graph model identification problem now becomes a parametric optimization problem with respect to any objective of choice. Several approaches exist, but the errors-in-output formulation is employed here. A trust region implementation of Newton–Conjugate Gradient implemented in scipy optimize was found to perform satisfactorily for the current investigation. The unknowns for the problem are the vertex and element weights (Vi , Ei ), the wave-speed on the graph c, and the Rayleigh damping coefficient (α, β).

Finite Elements and Spectral Graphs: Applications to Modal Analysis and Identification

(a)

(c)

65

(b)

(d)

(e)

Fig. 2 (a) The frequency response of the true model with the fitted response from the graph models of different numbers of vertices; the spectral embedding of the identified graphs in Cartesian space using the Fiedler vector for the graph models with (b) 2, (c) 3, (d) 4, and (e) 5 vertices

3 Results and Conclusions Identification of an SDOF spring-mass-damper is attempted with increasing numbers of vertices in the graph. Figure 2 presents the frequency response of the fitted models along with the original data for number of vertices N set to two through five. It can be seen that for N = 2, 3, the models result in very good fits, but for higher number of vertices, the identification starts showing over-fitting tendencies. This is due to the fact that initially all the vertices are assumed to be connected to all other vertices in the graph and the search space is rather broad for the optimization solver. This issue could be mitigated through the use of more modern techniques such as LASSO, L1 regularization, etc., wherein the number of unknowns will be progressively restricted based on some measure of their magnitudes. The figure also plots the identified graphs through their Fiedler vector embedding in Cartesian space. The edges and nodes are sized based on their weights from the algorithm. Although no direct observations can be made at this point, it can clearly be seen that at least one edge can be eliminated in the cases corresponding to N = 3 and upward, showing that there is room for improvement in the optimization approach taken.

4 Future Work Exploration of such graph-based methods for SISO (single-input single-output) as well as MIMO (multi-input multi-output) multi-DOF systems is presently ongoing work. Furthermore, the use of this approach for the isolation of nonlinearities in nonlinear systems with localized nonlinearities is a promising area that warrants exploration.

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References 1. Adhikari, S.: Structural Dynamic Analysis with Generalized Damping Models: Analysis. Wiley, Hoboken (2013). ISBN: 978-1-118-57205-4 2. Ewins, D.J.: Modal Testing: Theory and Practice, vol. 15. Research Studies Press, Letchworth (1984) 3. Friedman, J., Tillich, J.-P.: Calculus on Graphs (2004). Preprint arXiv cs/0408028 4. Friswell, M., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics, vol. 38. Springer Science & Business Media, Dordrecht (2013) 5. Mayo, A., Antoulas, A.: A Framework for the solution of the generalized realization problem. In: Linear Algebra and Its Applications, vol. 425, pp. 634–662, 2–3 Sept 2007. ISSN: 00243795. https://doi.org/10.1016/j.laa.2007.03.008 6. Pintelon, R., Schoukens, J.: System Identification: A Frequency Domain Approach. Wiley, Hoboken (2012) 7. Solomon, J.: PDE Approaches to Graph Analysis (2015). Preprint, arXiv:1505.00185 8. Van Overschee, P., De Moor, B.: Subspace identification for linear systems. Theory, implementation, applications. Incl. 1 disk. In: Springer Science & Business Media, vol. xiv, pp. xiv + 254 (1996). ISBN: 978-0-7923-9717-5. https://doi.org/10.1007/978-1-4613-0465-4 9. Verhaegen, M., Verdult, V.: Filtering and System Identification: A Least Squares Approach. Cambridge University Press, Cambridge (2007)

Effects of the Geometry of Friction Interfaces on the Nonlinear Dynamics of Jointed Structure Jie Yuan, Loic Salles, and Christoph Schwingshackl

Abstract Friction interfaces are commonly used in large-scale engineering systems for mechanical joints. They are known to significantly shift the resonance frequencies of the assembled structures due to softening effects and to reduce the vibration amplitude due to frictional energy dissipation between substructural components. It is also widely recognized that the geometrical characteristics of interface geometry have a significant impact on the nonlinear dynamical response of assembled systems. However, the full FE modeling approaches including these geometrical characteristics are extremely expensive. In this work, the influence of geometry of friction interfaces is investigated by using a multi-scale approach. It consists in integrating a semi-analytical contact solver into a high-fidelity nonlinear vibration solver. A highly efficient semi-analytical solver based on the boundary element method is used to obtain the pressure and gap distribution from the contact interface with different geometrical characteristics. The static pressure and gap distribution are then used as input for a nonlinear vibration solver to evaluate nonlinear vibrations of the whole assembled structure. The effectiveness of the methodology is shown on a realistic “Dogbone” test rig, which was designed to assess the effects of blade root geometries in a fan blade disk system. The friction joints with different interface profiles are then investigated. The obtained results show that the effects of the surface geometrical characteristics can have a significant impact on the damping and resonant frequency behavior of the whole assembly. Keywords Friction interface · Nonlinear vibration · Multi-scale analysis · Contact mechanics · Nonlinear modal analysis

1 Introduction Friction interfaces are widely used in large-scale engineering system that need to connect different sub-components. It is well-known that these interfaces have a significant impact on the dynamics of the assembled structure by shifting the resonance frequencies and effectively decreasing the vibration amplitude via strong energy dissipation. Because of the large damping effects, friction interfaces have been also widely exploited for the design of mechanical dampers, particularly in turbomachinery, where other types of dampers are generally infeasible due to the high temperature. Underplatform dampers (UPDs) [1] are one of these typical examples which are usually placed in the groove under the platforms of two adjacent blades. The energy dissipation from friction interfaces can effectively reduce the vibration level and mitigate the risk of high-cycle fatigue failures. Similar examples can also be found in the design of ring dampers for blisks [2]. However, an accurate prediction of the dynamics of the structure with friction interfaces remains a challenge due to complex physics at the friction interface [3]. Significant research efforts have been made in the last decade to better understand the role of friction interfaces in the nonlinear dynamical response, particularly for UPDs. Petrov [4] and Krack [5] carried out a sensitivity analysis of UPD designs to contact parameters at the blade-damper interface including the normal loads. Tang and Epureanu [6] investigated the effects of geometric parameters of a V-shaped friction ring damper on the turbine blade. Panning et al. studied the influence of the contact geometry on the damping effectiveness by parametrically varying both the geometry of the blade platform and the damper [7]. Hüls investigated the effects of geometric parameters of turbine friction dampers on the nonlinear vibration response [8]. To consider the fretting wear effects, Gallego carried out multi-scale computation of fretting

J. Yuan () · L. Salles · C. Schwingshackl Vibration University Technology Centre, Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected]; [email protected]; [email protected] © 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_7

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wear at the interfaces in a bladed disk system [9]. Armand implemented a multi-scale approach to quantify the effects of the surface roughness and fretting wear at the friction interface on the nonlinear dynamic response [3]. Delaune et al. also investigated the impact of fretting wear and showed that a large variety of vibration behaviors can be observed when wear effects are considered [10]. Recently, Gastaldi et al. experimentally investigated the effect of surface finish on the functioning of UPD showing a large impact of the contact interface conditions on the dynamic response [11]. All these studies have indicated that the topology of friction interfaces has a big influence on the dynamics of structures with friction interfaces. However, most of the previous studies have focused on the effects of the contact parameters or macro-scale geometries. The effects of surface geometry at the contact interface have not been thoroughly investigated due to high computational expense, because an extremely dense mesh would be required to model these interface profiles making the nonlinear dynamic analysis infeasible. To address this problem, this paper employs a multi-scale modeling approach to study the effects of the interface geometry and improve the dynamic design of jointed structures. The main idea of this approach is that, instead of directly using a dense mesh in FEA, a semi-analytical solver is used for the highly refined nonlinear contact analysis of the interface with different profiles. In this way, one can significantly reduce the high computational cost by avoiding using dense meshes for the FE analysis. The paper will firstly present the methodology of this multi-scale approach; it is followed by the presentation of a realistic test case, where four different interface profiles will be evaluated and compared.

2 Methodoloy The general structure of the multi-scale modeling approach is described in Fig. 1. This approach includes nonlinear dynamic/static analysis based on a macro-scale model and nonlinear contact analysis based on a “micro-scale” interface. The “micro-scale” refers to the interface with extremely fine mesh that is sufficient to represent the interface profiles. The nonlinear static analysis (with a flat-on-flat contact interface) is firstly performed to evaluate overall contact loads on the friction interfaces. With these contact loads, refined contact analysis is then carried out based on a semi-analytical solver, which can effectively consider the variation of interface profiles. The normal and gap distribution on the friction interfaces is then obtained at a very low computational cost. The distributions of these contact loads are then interpreted and then upscaled as the input for further nonlinear dynamic analysis of the macro-scale FE model. In this way, the effects of different interface profiles on the nonlinear dynamics can be evaluated efficiently without the need to re-mesh the contact surface for each profile. The nonlinear dynamic analysis can be still performed on the same macro-scale model to save the computational cost. This approach is mainly based on an assumption that the variation of micro-scale surface geometries does not change the overall contact loads at the contact interfaces.

Nonlinear static analysis

Macro-scale model

Micro-scale interface

Overall Static contact loads

Fig. 1 A general structure of the multi-scale approach

Nonlinear dynamic analysis

Normal load and gap distribution Refined Nonlinear contact analysis with micro-scale profiles

Effects of the Geometry of Friction Interfaces on the Nonlinear Dynamics of Jointed Structure

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2.1 Step 1: Nonlinear Static Analysis A nonlinear static analysis is initially performed on a macro-scale FE model to identify the nonlinear static equilibrium condition. The purpose of this analysis is to obtain the overall contact loads at the contact interfaces. As shown in Eq. (1), K represents the linear stiffness of the joint stiffness; fnl is the nonlinear contact force from the mechanical joints, which is dependent on the displacement and velocity of the interface DOFs; Fs (t) is the static loading applied to the system. This nonlinear contact analysis can be carried out using any commercial software, e.g., Abaqus, where the surface-on-surface hard contact is fined on the interface. As a result, the distribution of normal contact loads on the contact load can be extracted:   ˙ KX(t) + fnl X(t), X(t) = Fs (t)

(1)

The identified loads from the interface will then be used for a refined contact analysis, including different micro-scale interface profiles.

2.2 Step 2: Refined Contact Analysis A refined contact analysis with a dense mesh is then performed by the means of an already available semi-analytical boundary element solver [3]. The contact solver is essentially based on the projected conjugate gradient method and a discrete-convolution fast Fourier transform which speeds up the computation. The half-space assumption allows the use of the Boussinesq and Cerruti potentials to compute the surface elastic deflections in the normal and tangential direction. Thanks to the computational speed of this contact solver, a very refined contact mesh can be used, which allows to include roughness and surface profiles into the analysis. The normal displacement uz caused by a pressure distribution p is described in Eq. (2), whereas Eq. (3) is its discretized form on a regular grid of Nx × Ny points:

uz (x, y) =

1 − v2 πE

uz (i, j ) = Kzz

$

)

+∞ $ +∞

−∞

p=

−∞

#

Ny Nx  

p (ξ, η)

dξ dη

(2)

(ξ − x)2 + (η − y)2

p (k, l) Kzz (i − k, j − l)

(3)

k=1 l=1

* where E and v are Young’s modulus and Poisson ratio of the material, respectively, is the discrete convolution product, and Kzz (i, j) are the discrete influence coefficients which are used to obtain the normal displacement resulting from unit pressure on the element (i, j). Initially, the normal contact problem is solved using the conjugate gradient method, after which the tangential problem is solved with the Coulomb friction law as a bound to the shear distribution in the slipping region. As a result, the distribution of normal contact loads and gap between contact surfaces can be effectively obtained for the contact surface including the effects of micro-scale interface profiles. These distributions are then interpolated for the macro-scale FE model for nonlinear dynamic analysis with respect to Newton’s third law making sure the force and moment are equivalent on the whole interface.

2.3 Step 3: Nonlinear Dynamic Analysis Nonlinear dynamic analysis is then carried out for the macro-scale FE model with the input contact loads from interpolated micro-scale profiles. The equation of motion for the dynamic analysis is shown in Eq. (4):   ¨ ˙ M X(t) + KX(t) + fnl X(t), X(t) = Fd (t)

(4)

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where Fd (t) is the dynamical loading in the system and M is the linear mass matrix of the jointed structure. 3D node-to-node contact element is used to model the contact friction between the surfaces, which combines two Jenkins elements (in contact plane), and a unilateral spring is used to describe the normal contact forces between the contacting surfaces [12]. Instead of using forced frequency analysis, damped nonlinear modal analysis (dNNM) is employed. The advantage of using dNNM analysis is that it can directly and efficiently calculate resonance frequency and damping with a wide range of energy levels. In this study, the extended periodic motion concept, proposed by Krack [13], is used for dNNM analysis. In contrast to complex nonlinear modal analysis [14], the used approach is also suitable when the system has a large damping value. As shown in Eq. (5),     ¨ ˙ ˙ a M X(t) − 2ζ ω0 M X(t) a =0 + KX(t) + fnl X(t), X(t),

(5)

An artificial damping ζ is introduced to make the nonconservative system periodic by balancing the energy dissipated from the nonlinear contact force; a is the modal amplitude indicating the level of input energy in the system. A wellestablished HBM framework [15] is then used to compute the steady-state dynamics for different modal amplitudes. The modal amplitude can then be interpolated as an equivalent force level Fd by using the extended-energy balance method [17].

3 Case Study The presented FE joint model for the case study is based on an available blade root test rig setup [16]. In this assembly, a particular blade root design is machined onto both ends of a beam (“Dogbone”), which is then fitted between two clamps, and put into tension to simulate the in-rotation centrifugal loading that occurs due to rotation. The Dogbone test rig shown in Fig. 2(a) consists of two main components: a set of identical solid root-block disks and a set of “bones” for different root designs. The disks shown in Fig. 2(b) are 16-sided with currently five adjacent sides used to house different blade root designs. A rounded and hardened central bore in the disk provides a near-point contact with a hardened U-shaped hook which holds the clamping disk in place. This arrangement was chosen in an attempt to minimize the damping introduced by the supporting structure. In this case study, a root design similar to a dovetail joint in a fan blade disk is considered, which is shown in Fig. 2(c). In terms of FE modeling, the disks on both sides are simplified as one cyclic sector to reduce the computational expense. There are four contact interfaces where two are on the top disk and the other two on the bottom disk. The size of friction interface is 18 mm × 1.6 mm. For each friction interface, there are 28 nonlinear elements involving 40 contact nodes. Figure 2(d) shows the first bending mode of the test rig from a linear dynamic analysis, which will be the focus of this study. Figure 3(a) shows the root geometry of the bone in Fig. 2, where the contact surface is in blue. Figure 3(b–d) shows three different “micro-scale” interface geometries which will be investigated in this study. They will be referred to as Y-wise bump,

(a)

(b)

(c)

Fig. 2 (a) Experiment setup, (b) 3D full-scale FE model, (c) zoomed dovetail joint, (d) first bending mode

(d)

Effects of the Geometry of Friction Interfaces on the Nonlinear Dynamics of Jointed Structure

x

71

z y

(a)

(b)

(c)

(d)

Y(mm)

Fig. 3 (a) Root geometry; (b) Y-wise bump; (c) Center bump; (d) Y-wise concave

X (mm) (a)

(b)

Fig. 4 Nonlinear static analysis: (a) pressure distribution in joints, (b) pressure distribution on the contact surface (Pa)

center bump, and Y-wise concave respectively. The maximum height of these interface geometries is around 1 μm. These profiles are introduced to the blade (bone) interface, while the friction interfaces on the disks remain flat. The flat-to-flat contact interface is also investigated for reference. All four friction interfaces on the bone have the same profile.

4 Results Figure 4(a) shows the results from a nonlinear static analysis of the Dogbone test rig with a flat-to-flat interface. The simulation was performed in Abaqus using surface-to-surface hard contact. In terms of pre-loading, the end of one disk is fully fixed, and a pulling loading of 1000 N is applied at the end of the other disk, which is equivalent to the loading applied during the experiment. Figure 4b) shows the distribution of the extracted normal pressure loads at two friction interfaces from the disk on the bottom. It is clear that the normal contact pressure is localized on the downward sides of the interfaces, which is because the blade root is opening up a bit when the Dogbone is pulled up. It is also found that there is no gap appearing on these two friction interfaces. The pressure distribution is the same for all four friction interfaces (top and bottom) due to the symmetry of the test rig and loading conditions. All the pressure distributions are summed up to form an equivalent force vector applied at the center of the interface for the analysis in SAM, including forces in three directions and bending moment in three directions. Figure 5 shows the pressure distribution from the refined contact analysis with the semi-analytic solver for different interface profiles described before. Herein, it is assumed that the overall contact load would remain unchanged for different micro-scale contact interfaces. They are all computed at a refined grid of 250 × 125 points for each friction interface instead of 8 × 5 points used in FE model. The computation for each profile only takes around 20s. As expected, for the Y-wise bump profile, the pressure is localized in the middle of interface; the pressure is centralized for the central bump profile, while the Y-wise concave profile leads to edgewise localized pressures. The pressure distributions are then interpolated to the much rougher nonlinear dynamic mesh to form the inputs for nonlinear dynamic analysis.

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Fig. 5 Pressure distribution at the friction interface

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Figure 6 shows the results from the dNNM analysis using harmonic balance method. Instead of showing the modal amplitudes, the equivalent force level is calculated by the extended-energy balance method assuming the forcing position is applied in the middle of the bone and at the out-of-plane direction. As expected, the global trend is similar for these four interface profiles. The resonance frequencies decrease in all four cases with the forcing level due to increasing softening effects at the friction interface, and the damping ratio increases with the excitation levels for all four profiles. However, the gradient of these two quantities to force levels during the stick-slip transition is very different for these four interface profiles. For the flat and Y concave profile, both the resonance frequency and damping ratio change rapidly once the slipping

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Fig. 7 Energy dissipation on the friction interface at the equivalent force level of 2 N

condition is trigged. It is mainly because the normal pressure of these two geometries is both localized on the edges of the interface. In comparison, they change much gradually for the Y and C bump as the contact pressure spreads over a wider contact zone leading to a lower pressure peak. One can also see that the starting resonance frequency for Y and C bump is much lower than Y concave and flat interface. This is because the bump profiles would leave a lot of gaps or much lower pressure on the edges, which will reduce the joint stiffness. One can also observe some difference in damping in Fig. 6b. The bump interfaces have a steady damping increase with excitation levels, while the other two have a steep and unsteady increase at low excitation levels. Figure 7 shows energy dissipation on the contact interface for these four interface profiles at an excitation level of 2 N. As expected, the distribution of dissipated energy varies quite a lot for different surface geometries and is consistent with pressure distribution shown in Fig. 5. Overall, the frequency drop due to the interface geometry can be up to 5.7% and the damping increase can be up to 0.02 with different interface profiles. It means that the variation of these nonlinear dynamic parameters is highly sensitive to the micro-scale interface topology through its significant impact on the distribution of normal pressure and gap at the contact interfaces.

5 Conclusion The objective of the work was to present a multi-scale-based methodology to efficiently evaluate the effects of micro-scale interface geometries on the nonlinear dynamical response of a structure with frictional interfaces. This approach is based on combining a macro-scale FE model for nonlinear dynamic analysis with a micro-scale friction interface model for the quasi-static nonlinear contact analysis. The main advantage of the approach is that, instead of directly using FE models to include often minute interface profiles, a highly efficient semi-analytical contact solver is used. It can efficiently evaluate the influence of interface profiles on the contact pressures at a much lower cost. The resulting contact pressure and gap distribution are then fed as inputs into the macro-scale FE model to evaluate the nonlinear dynamical response with an HBM solver. The proposed approach was applied to the design of a new blade root Dogbone test rig, to evaluate the effects of blade root geometries on the overall dynamic response of the system. Four different interface profiles were investigated. The

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obtained results have shown that the effects of micro-scale interface profiles can have a significant impact on the damping and resonant frequency behavior, which should not be ignored in the dynamic prediction. The surface geometry may also be used as the design parameters to achieve robust and better dynamic performance of complex and nonlinear systems. The multi-scale approach has proven to be a very efficient method to simulate the effects of interface profiles on the nonlinear dynamical response. Acknowledgments The authors would like to acknowledge the financial support from the EPSRC under SYSDYMATS project, Grand Ref: EP/R032793/1.

References 1. Yuan, Y., Jones, A., Setchfield, R., Schwingshackl, C.W.: Robust design optimisation of underplatform dampers for turbine applications using a surrogate model. J. Sound Vib. 494, 115528 (2020) 2. Sun, Y., Yuan, J., Denimal, E. and Salles, L.: Nonlinear modal analysis of frictional ring damper for compressor Blisk. In: Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers (2020) 3. Armand, J., Salles, L., Schwingshackl, C.W., Süß, D., Willner, K.: On the effects of roughness on the nonlinear dynamics of a bolted joint: a multiscale analysis. Eur J Mech A Solid. 70, 44–57 (2018) 4. Petrov, E.P.: Direct parametric analysis of resonance regimes for nonlinear vibrations of bladed disks. J. Turbomach. 129(3), 495–502 (2006) 5. Krack, M., Tatzko, S., Panning-von Scheidt, L., Wallaschek, J.: Reliability optimization of friction-damped systems using nonlinear modes. J. Sound Vib. 333(13), 2699–2712 (2014) 6. Tang, W., Epureanu, B.I.: Geometric optimization of dry friction ring dampers. Int J Nonlinear Mech. 109, 40–49 (2019) 7. Panning, L., Sextro, W., Popp, K.: Optimization of the contact geometry between turbine blades and underplatform dampers with respect to friction damping. In: Proceedings of the ASME Turbo Expo 2002: Power for Land, Sea, and Air. Volume 4: Turbo Expo 2002, Parts A and B. Amsterdam, The Netherlands, pp. 991–1002 8. Gallego, L., Fulleringer, B., Deyber, S., Nelias, D.: Multiscale computation of fretting wear at the blade/disk interface. Tribol. Int. 43(4), 708–718 (2010) 9. Hüls, M., Panning-von Scheidt, L., Wallaschek, J.: Influence of geometric design parameters onto vibratory response and high-cycle fatigue safety for turbine blades with friction damper. J. Eng. Gas Turbines Power. 141(4) (2019) 10. Delaune, X., de Langre, E., Phalippou, C.: A probabilistic approach to the dynamics of wear tests. J. Trib. 122(4), 815–821 (2000) 11. Gastaldi, C., Berruti, T.M., Gola, M.M.: The effect of surface finish on the proper functioning of underplatform dampers. J. Vib. Acoust. 142(5) (2020) 12. Salles, L.C., Blanc, L., Thouverez, F., Gouskov, A.M., Jean, P.: Dynamic analysis of a bladed disk with friction and fretting-wear in blade attachments. ASME Turbo Expo 2009: Power for Land, Sea, and Air, Jun 2009, Orlando, vol. 48876, pp. 465–476 (2009) 13. Krack, M.: Nonlinear modal analysis of nonconservative systems: extension of the periodic motion concept. Comput. Struct. 154, 59–71 (2015) 14. Laxalde, D., Thouverez, F.: Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. J. Sound Vib. 322(4–5), 1009–1025 (2009) 15. Krack, M., Salles, L., Thouverez, F.: Vibration prediction of bladed disks coupled by friction joints. Arch Comput Methods Eng. 24(3), 589– 636 (2017) 16. Schwingshackl, C.W., Zolfi, F., Ewins, D.J., Coro, A., Alonso, R.: Nonlinear friction damping measurements over a wide range of amplitudes. In: Proceedings of the international modal analysis conference XXVII, Orlando (2009) 17. Sun, Y., Vizzaccaro, A., Yuan, J., Salles, L.: An extended energy balance method for resonance prediction in forced response of systems with non-conservative nonlinearities using damped nonlinear normal mode. Nonlinear Dynamics. 103, 3315–3333 (2020)

Bifurcation Analysis of a Piecewise-Smooth Freeplay System Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

Abstract Physical systems that are subject to intermittent contact/impact are often studied using piecewise-smooth models. Freeplay is a common type of piecewise-smooth system and has been studied extensively for gear systems (backlash) and aeroelastic systems (control surfaces like ailerons and rudders). These systems can experience complex nonlinear behavior including isolated resonance, chaos, and discontinuity-induced bifurcations. This behavior can lead to undesired damaging responses in the system. In this work, bifurcation analysis is performed for a forced Duffing oscillator with freeplay. The freeplay nonlinearity in this system is dependent on the contact stiffness, the size of the freeplay region, and the symmetry/asymmetry of the freeplay region with respect to the system’s equilibrium. Past work on this system has shown that a rich variety of nonlinear behaviors is present. Modern methods of nonlinear dynamics are used to characterize the transitions in system response including phase portraits, frequency spectra, and Poincaré maps. Different freeplay contact stiffnesses are studied including soft, medium, and hard in order to determine how the system response changes as the freeplay transitions from soft contact to near-impact. Particular focus is given to the effects of different initial conditions on the activation of secondary- and isolated-resonance responses. Preliminary results show isolated resonances to occur only for softer-contact cases, regions of superharmonic resonances are more prevalent for harder-contact cases, and more nonlinear behavior occurs for higher initial conditions. Keywords Bifurcation analysis · Nonlinear dynamics · Contact · Freeplay · Piecewise-smooth

1 Introduction Dynamical systems subject to some form of intermittent contact or impact are very common across engineering fields. In many vibro-contact systems, the contact is predictable and can easily be expressed mathematically based on the critical displacement for contact to occur. A common type of easily expressible contact is freeplay, which occurs when there is a clearance between parts in a system. When parts sufficiently displace enough to cause contact, the resulting contact force can be modeled as a piecewise-smooth force-displacement curve. Examples of freeplay include backlash between gears and control surfaces in aeroelastic systems [1, 2]. Freeplay and other contact behaviors cause a dynamical system to behave nonlinearly; some notable behaviors that can occur includes isolated or subharmonic resonances, chaos, and discontinuityinduced bifurcations. When undesired, these can lead to unexpected aircraft flutter instability or premature wear and failure of parts.

B. E. Saunders () · A. Abdelkefi Mechanical and Aerospace Engineering Department, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected] R. M. G. Vasconcellos Campus of São João da Boa Vista, São Paulo State University, São João da Boa Vista, Brazil e-mail: [email protected] R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 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_8

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Past researchers have studied relatively simple vibro-contact systems to gain a fundamental understanding of how contact can affect the nonlinear response of a system. Shaw [3] studied a spring-mass oscillator with freeplay in which the contact was rigid-body hard impact with variable coefficient of restitution. deLangre et al. [4] developed an experimental setup of the nonlinear Duffing oscillator but used a rigid stopper and contact springs of relatively soft stiffness. Researchers have also extensively studied the contact problem for aeroelasticity applications [5, 6], often including bifurcation analyses to determine how the contact can cause transitions in the systems’ responses. Literature review on more general, variablestiffness contact systems has turned up little focus on bifurcation analysis. This would be useful to fundamentally determine how a variable contact nonlinearity can affect a vibrating system and even how it can interact and couple with other present nonlinearities. Thus, in this work, bifurcation analysis is carried out on a forced Duffing oscillator with freeplay. Modern methods of nonlinear dynamics including phase portraits, frequency spectra, and Poincaré maps are used to characterize the transitions in the system response. Particular attention is paid to how the freeplay transitions from soft contact to hard impact and to the effects of different initial conditions on the activation of secondary resonances.

2 System’s Modeling The equations of motion for the Duffing-freeplay system as used by deLangre et al. [4] are given by: ⎧ x < −j1 ⎨ Kc (x + j1 ) , α 3 Fc p 2 x¨ + 2ωn ζ x˙ + ωn x + x + = cos (ωt) , Fc = − j1 < x < j2 0, ⎩ m m m x < j2 Kc (x − j2 ) ,

(1)

®

where α is the cubic stiffness nonlinearity, Kc is the contact stiffness, and j1 , j2 are the freeplay gap boundaries. MATLAB ode45 with Event Location is used to perform simulations to accurately capture the switching points between freeplay regions. Figure 1(a) shows a schematic of the system. The freeplay gap is kept symmetric about the origin, so j1 = j2 .

3 Nonlinear Characterization Figure 1(b) presents 3D bifurcation diagrams showing all peaks in the steady-state time history at a given forcing frequency, for different values of contact stiffness. Results show that frequency bands of superharmonic resonance and chaos widen as contact stiffness hardens, but regions of subharmonic resonance will first appear and then disappear as contact stiffness hardens. More chaos occurs for hard contact, and intuitively hard contact strongly limits the maximum amplitude of the system to the span of the freeplay gap. This also implies the cubic stiffness is less influential for hard contact than for soft contact. Further, initial displacements x0 near the origin do not always impart enough energy for the system to activate superor subharmonic resonances compared to initial displacements closer to the freeplay boundaries.

4 Conclusions In this work, bifurcation analysis was performed on a forced Duffing oscillator with freeplay nonlinearity. The effects of contact stiffness on the transitions in the system were studied. Results indicated that the system’s secondary resonances are highly affected by harder contact stiffnesses and that the contact nonlinearity is generally more dominant over the system response than the cubic nonlinearity. The contact also causes a strong dependence on initial conditions.

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(a)

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Fig. 1 (a) Schematic of the Duffing-freeplay system and (b) 3D bifurcation diagrams showing the effects of contact stiffness on the Duffingfreeplay system for m = 5 kg, ωn = 5 Hz, ζ = 0.03, α = 7 ∗ 108 N/m3 , p = 4 N, j1 = j2 = 0.4 mm, x0 = 0.32 mm Acknowledgments The authors B. Saunders and A. Abdelkefi gratefully acknowledge the support from Sandia National Laboratories. This study describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the US Department of Energy or the US Government. This study is 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. SAND2020-13605 C. R. Vasconcellos acknowledges the financial support of the Brazilian agency CAPES (grant 88881.302889/2018-01).

References 1. Yang, Y., Xia, W., Han, J., Song, Y., Wang, J., Dai, Y.: Vibration analysis for tooth crack detection in a spur gear system with clearance nonlinearity. Int. J. Mech. Sci. 157–158, 648–661 (2019). https://doi.org/10.1016/j.ijmecsci.2019.05.012 2. Dai, H., Yue, X., Yuan, J., Xie, D., Atluri, S.N.: A comparison of classical Runge-Kutta And Henon’s methods for capturing chaos and chaotic transients in an aeroelastic system with freeplay nonlinearity. Nonlinear Dyn. 81, 169–188 (2015). https://doi.org/10.1007/s11071-015-1980-x 3. Shaw, S.W.: The dynamics of a harmonically excited system having rigid amplitude constraints, Part 1: Subharmonic motions and local bifurcations. ASME J Appl Mech. 52(2), 453–458 (1985). https://doi.org/10.1115/1.3169068 4. De Langre, E., Lebreton, G.: An experimental and numerical analysis of chaotic motion in vibration with impact. In: ASME 8th International Conference on Pressure Vessel Technology, Montreal, Quebec, Canada (1996) 5. Vasconcellos, R., Abdelkefi, A., Hajj, M.R., Marques, F.D.: Grazing bifurcation in aeroelastic systems with freeplay nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1611–1625. https://doi.org/10.1016/j.cnsns.2013.09.022 6. Vasconcellos, R.M.G., Abdelkefi, A., Marques, F.D., Hajj, M.R.: Characterization of Grazing bifurcation in airfoils with control surface freeplay nonlinearity. In: Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal (2014)

Insights on the Bifurcation Behavior of a Freeplay System with Piecewise and Continuous Representations Brian Evan Saunders, Rui M. G. Vasconcellos, Robert J. Kuether, and Abdessattar Abdelkefi

Abstract Dynamical systems containing contact/impact between parts can be modeled as piecewise-smooth reduced-order models. The most common example is freeplay, which can manifest as a loose support, worn hinges, or backlash. Freeplay causes very complex, nonlinear responses in a system that range from isolated resonances to grazing bifurcations to chaos. This can be an issue because classical solution methods, such as direct time integration (e.g., Runge-Kutta) or harmonic balance methods, can fail to accurately detect some of the nonlinear behavior or fail to run altogether. To deal with this limitation, researchers often approximate piecewise freeplay terms in the equations of motion using continuous, fully smooth functions. While this strategy can be convenient, it may not always be appropriate for use. For example, past investigation on freeplay in an aeroelastic control surface showed that, compared to the exact piecewise representation, some approximations are not as effective at capturing freeplay behavior as other ones. Another potential issue is the effectiveness of continuous representations at capturing grazing contacts and grazing-type bifurcations. These can cause the system to transition to high-amplitude responses with frequent contact/impact and be particularly damaging. In this work, a bifurcation study is performed on a model of a forced Duffing oscillator with freeplay nonlinearity. Various representations are used to approximate the freeplay including polynomial, absolute value, and hyperbolic tangent representations. Bifurcation analysis ® results for each type are compared to results using the exact piecewise-smooth representation computed using MATLAB Event Location. The effectiveness of each representation is compared and ranked in terms of numerical accuracy, ability to capture multiple response types, ability to predict chaos, and computation time. Keywords Bifurcation analysis · Nonlinear dynamics · Freeplay · Piecewise-smooth · Continuous representation

1 Introduction Vibro-contact and vibro-impact dynamical systems are very common across engineering fields, with examples ranging from large aeroelastic structures [1] to small energy harvesters [2]. Various numerical methods and models have been used and developed to represent contact/impact behavior. In reduced-order models, an important consideration is how to adequately represent the contact force(s), particularly because contact can induce very strong nonlinearities into a dynamical system. A realistic contact representation is a piecewise-smooth force curve; e.g. in a freeplay system, there is a gap between parts and there is no contact until the parts’ displacement is larger than the gap size. The non-smooth behavior, however, can lead to numerical problems and roundoff error if the switching points from no-contact to contact are not captured. Contact points ® can be accurately captured if Henon’s method [3] or another numerical scheme (e.g., MATLAB Event Location) is used, but this often increases computation time.

B. E. Saunders () · A. Abdelkefi Mechanical and Aerospace Engineering Department, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected] R. M. G. Vasconcellos Campus of São João da Boa Vista, São Paulo State University, São João da Boa Vista, Brazil e-mail: [email protected] R. J. Kuether Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 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_9

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Other representations that are continuous and fully smooth have been developed, which may be based on combinations of absolute value and polynomial functions [2, 4], the hyperbolic tangent function [1, 5], or similar functions/combinations. These continuous contact-force representations allow for low computational costs and remove the accumulating roundoff error, but there are trade-offs with other sources of error. For example, the contact force within the freeplay gap may be nonzero, or the contact stiffness beyond the gap may inappropriately harden or soften with increasing displacement. Vasconcellos et al. [1] found that, when used in the model of an aeroelastic system with control surface freeplay, some continuous representations are unable to capture all of the nonlinear behavior that may be present. This includes potentially dangerous responses, such as grazing contact and grazing bifurcations. The goal of this work is to study how different contact-force representations can affect a more general nonlinear system, namely, a forced Duffing oscillator with freeplay [6]. Bifurcation analysis results for each representation are compared ® to results using the exact piecewise-smooth representation, computed using MATLAB ode45 with Event Location. The effectiveness of each representation is compared and ranked in terms of numerical accuracy, ability to capture multiple response types, ability to predict chaos, and computation time. This work intends to explore under what conditions a smooth contact-force representation may be used instead of the exact piecewise representation. Results for one representation (hyperbolic tangent) are presented in this extended abstract for brevity; results for other representations (absolute value, polynomial, etc.) are reserved for the final conference presentation.

2 System Modeling The equations of motion for the Duffing-freeplay system [6] are given by:\vspace*{-3pt} ⎧ x < −j1 ⎨ Kc (x + j1 ) , p α F c = cos (ωt) , Fc = x¨ + 2ωn ζ x˙ + ωn2 x + x 3 + − j1 < x < j2 0, ⎩ m m m x < j2 Kc (x − j2 ) ,

(1)

where α is the cubic stiffness, Kc is the contact stiffness, and j1 , j2 are the freeplay gap boundaries. To solve the system with ® piecewise-smooth representation, MATLAB ode45 with Event Location is used to accurately capture the switching points between freeplay regions. Figure 1(a) shows a schematic of the system. Figure 1(b) presents frequency response curves for the system using both the piecewise representation and the hyperbolic tangent representation studied in [1], given by: 

Peak Amplitude (m)

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1 1 [1 − tan h (e (x + j1 ))] (x + j1 ) + [1 + tan h (e (x − j2 ))] (x − j2 ) + P 2 2

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where e is a “tolerance” parameter; in the limit e → ∞, the representation approaches the piecewise representation. System parameters are m = 5 kg, ωn = 5 Hz, ζ = 0.03, α = 7 ∗ 108 N/m3 , p = 4 N, j1 = 0 mm, j2 = 0.8 mm, Kc = 1.4 ∗ 104 N/m.

3 Effectiveness of Continuous Representations for System with Freeplay Nonlinearity Figure 1(b) indicates that results can significantly diverge as forcing frequency increases past the primary resonance peak if a large enough value of e is not used, meaning regions of subharmonic resonance may be inaccurately predicted. A graph of the contact force versus displacement for even the coarsest value of e = 104 used in Fig. 1(b) appears acceptable (omitted from this extended abstract for brevity), though, and does not indicate that frequency-response results will diverge. Thus, a convergence analysis is necessary. The low-frequency superharmonic resonances and chaotic behavior seem to be relatively unaffected, in addition to the primary resonance peak, for all values of e. However, a good agreement in frequencyresponse results is not always a good indicator that results agree globally and that system physics are not lost [7]. Nonlinear characterization (omitted for brevity) is also performed to determine how well the continuous representations can capture the overall physics of the system response.

4 Conclusions In this work, bifurcation analysis was carried out on a forced Duffing oscillator system with freeplay nonlinearity for different mathematical representations of the freeplay contact force. Results using a hyperbolic tangent representation indicated good frequency-response agreement after a parameter convergence analysis was performed. This convergence was required because the contact-force displacement may look acceptable for an unconverged model, but the frequency response significantly diverges as forcing frequency increases past the primary resonance peak. This is particularly dangerous because subharmonic resonances often lead to high-amplitude responses which can be damaging. Results for other representations (absolute value, polynomial, etc.) are reserved for the final conference presentation. Acknowledgments The authors B. Saunders and A. Abdelkefi gratefully acknowledge the support from Sandia National Laboratories. 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 paper is 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. SAND2020-13766 C. R. Vasconcellos acknowledges the financial support of the Brazilian agency CAPES (grant 88881.302889/2018-01).

References 1. Vasconcellos, R., Abdelkefi, A., Marques, F.D., Hajj, M.R.: Representation and analysis of control surface freeplay nonlinearity. J Fluids Struct. 31, 79–91 (2012). https://doi.org/10.1016/j.jfluidstructs.2012.02.003 2. Zhou, K., Dai, L., Abdelkefi, A., Zhou, H.Y., Ni, Q.: Impacts of stopper type and material on the broadband characteristics and performance of energy harvesters. AIP Adv. 9, 035228 (2019). https://doi.org/10.1063/1.5086785 3. Henon, M.: On the numerical computation of Poincaré maps. Physica D. 5(2–3), 412–414 (1982) 4. Paidoussis, M.P., Li, G.X., Rand, R.H.: Chaotic motions of a constrained pipe conveying fluid: comparison between simulation, analysis, and experiment. ASME. J. Appl. Mech. 58(2), 559–565 (1991). https://doi.org/10.1115/1.2897220 5. Alcorta, R., Baguet, S., Prabel, B., Piteau, P., Jacquet-Richardet, G.: Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances. Nonlinear Dynamics. 98, 2939–2960 (2019). https://doi.org/10.1007/s11071-019-05245-6 6. De Langre, E., Lebreton, G.: An experimental and numerical analysis of chaotic motion in vibration with impact. In; ASME 8th International Conference on Pressure Vessel Technology, Montreal, Quebec, Canada (1996) 7. Saunders, B.E., Vasconcellos, R., Kuether, R.J., Abdelkefi, A.: Importance of event detection and nonlinear characterization of dynamical systems with discontinuity boundary. In: AIAA Sci Tech 2021, virtual forum (2021)

Model Updating and Uncertainty Quantification of Geometrically Nonlinear Panel Subjected to Non-uniform Temperature Fields Kyusic Park and Matthew S. Allen

Abstract Thin structures comprising the skin panels of advanced aircraft will experience extreme thermal stresses as well as dynamic loads at hypersonic speeds, leading to highly nonlinear behaviors such as buckling. In order to determine whether a model correctly captures changes in the dynamics due to heating, the linearized natural frequencies can be compared between test and the FE model at a certain thermal state. This is considerably more difficult if the panel is subjected to localized heating. This work presents a case study in model updating for a non-uniformly heated, geometrically nonlinear panel and evaluates the effect of uncertainty. A curved panel was subjected to localized heating, and measurements of the temperature distributions and of the initial shape were mapped to the FE model and parameterized to use in model updating. The model was then updated for the baseline thermal state, after which the updated model was used to compute the linear natural frequencies and mode shapes with respect to varying temperature fields and those were compared with the experimental data, revealing that the modal properties are highly sensitive to the model’s design parameters. It proved difficult to find an exact correlation by deterministic model updating. The uncertainties in some of the design parameters were then evaluated using a Monte Carlo simulation. The results suggest that even modest uncertainties in the model parameters cause large changes in the natural frequencies, so that the uncertain model bounds the range of the measured natural frequencies. Keywords Nonlinear dynamics · Geometric nonlinearity · Model updating · Thermal loading · Localized heating · Uncertainty quantification

1 Introduction An advanced hypersonic aircraft will be exposed to not only severe acoustic pressure but also extreme thermal stress during its operation [1]. Thin panels of the high-speed vehicle experience large compressive in-plane loads due to thermal stresses, and if those are large enough the panels can exhibit highly nonlinear behaviors such as buckling [2–4]. Numerous experiments and simulations have been conducted to explore geometrical nonlinearities and consider them in the actual design of the thermally load structures. K.D. Murphy et al. [5] presented an experimental study that characterized the snap-through and buckling behavior of a thermally excited plate. The finite element (FE) nonlinear response of a heated clamped plate was obtained and the corresponding multi-axial fatigue life was estimated in [6]. Gordon et al. [7] and Radu et al. [8] applied reduced order modeling (ROM) to accurately predict the nonlinear response of the thin skin panels subjected to combined acoustic thermal pressure. A recent work of VanDamme and Allen [9] investigated the geometrically nonlinear behavior of a clamped beam by computing the nonlinear normal modes (NNMs) about thermal equilibrium states. While all the studies mentioned above analyzed the nonlinear response of the thin structures subjected to uniformly distributed temperature fields, Ehrhart et al. [10] experimentally examined the effect of localized heating on the nonlinear response of a curved panel. The modal properties were shown to vary in a complicated way, and interactions between the linear modes were observed under some heating conditions.

K. Park () Department of Mechanical Engineering, University of Wisconsin–Madison, Madison, WI, USA e-mail: [email protected] M. S. Allen Department of Engineering Physics, University of Wisconsin–Madison, Madison, WI, USA e-mail: [email protected] © 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_10

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Given that thin structures exhibit highly complex dynamics when thermally stressed, model updating of geometrically nonlinear systems may also need to account for thermal effects to achieve accurate response prediction and life estimation. Model updating techniques, which aim to correlate between the simulated predictions and experimental measurements from an actual structure, have been investigated with a significant amount of attention over the past few decades [11, 12]. A majority of existing methods use linear modal properties for the model correlation, such as natural frequencies and mode shapes, which are load invariant and easily computed from both FE model and experiments [13]. Recently, a few methods of model updating based on the nonlinear normal modes (NNMs) have been introduced [14–16], which enable to correlate the nonlinear system using both linear and nonlinear range of deformations. However, considerably less attention has been paid to a model correlation subject to thermal loading. Cheng et al. [17] proposed a hierarchical approach to update the dynamic system in a high-temperature environment, which consists of a two-stage procedure: a temperature field updating and a dynamic structural model updating. While they applied a non-uniform temperature distribution for the temperature field updating, its resulting thermal effects were not accounted for in the structural parameter correlation. Sun et al. [18] identified the temperature-dependent properties of a thermo-elastic structure, such as Young’s modulus and spring stiffness, in a time-varying temperature environment. They demonstrated a numerical example with a simply supported beam subjected to a time-varying uniformly distributed temperature field to reveal that the temperature-dependent properties were identified with high accuracy. A recent study of Yuan et al. [19] also explored temperature-dependent model correlation but differed in their strategy that the FE parameters were updated simultaneously with respect to multiple uniform temperature fields. This work demonstrates model updating of a curved panel subjected to non-uniform thermal loading. To the best of the authors’ knowledge, this work provides a first trial of model correlation between the actual test data and the simulated prediction of a geometrically nonlinear structure with multiple localized temperature fields. In this work, the finite element model of a curved panel is correlated to the experimental measurements obtained in [10]. Ehrhardt et al. measured the temperature distribution on the panel, its deformed shape, and the natural frequencies and mode shapes at several levels of heating. In this work, the measured temperature distributions are aligned to the FE model and the nonlinear static response is found at the thermal equilibrium states to compute the corresponding modal properties of the curved panel. In the first step, deterministic model updating is applied wherein one minimizes the error between a single FE model and the test data of a single physical structure [11]. In the case study considered here, the deterministic updating procedure produces general agreement between the modal properties of the updated panel model and the measurement data with a few noticeable mismatches. It is also shown that the dynamic behavior of the locally heated structure is highly sensitive to the thermal effect so that it may be impractical to find the exact match to the experimental results deterministically. For this reason, this work also explores the effect of uncertainties in the structural parameters of the panel by utilizing a Monte Carlo simulation. This reveals that small model uncertainty considerably expands the range of the predicted natural frequencies at each temperature, so the uncertain model bounds the range of the measured frequencies versus temperature. This chapter is organized as follows. Section 2 briefly explains the modal analysis of FE model about a thermal equilibrium state. In Sect. 3, a case study of model correlation and uncertainty quantification for the curved panel subjected to the localized heating is described in detail. Section 4 summarizes the conclusion and some aspects of future work.

2 Theory 2.1 Geometrically Nonlinear Finite Element Model A geometrically nonlinear FE model for an n-DOF system subjected to thermal effects can be expressed as M¨x + C˙x + (K + Kσ (x, T ))x + fnl (x) = fext (t) + fT (x, T )

(1)

where M, C and K are the n x n mass, damping and stiffness matrices and fext (t) is the n x 1 external force. The n x 1 nonlinear restoring force fnl captures the geometric nonlinearity of the model, which is a function of the displacement vector x. There are two terms that capture the thermal effect: Kσ is the thermal stiffness matrix induced by internal thermal stress and fT is the thermal load vector from external thermal sources. They are thermally coupled terms, which depend on both displacement and temperature change.

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2.2 Modal Analysis About Thermal Equilibrium The linear modes of the thermally coupled geometrically nonlinear system can be evaluated about the static thermal equilibrium state in the following equation: (K + Knl (x) + Kσ (x, T ))x = fext (t) + fT (x, T )

(2)

where Knl (x) is the tangent stiffness matrix that approximates the nonlinear restoring force fnl . At given temperature T , the static responses x can be updated to find the thermal equilibrium state of the system. Once the equilibrium is achieved, the r-th natural frequency ωr and its mode shape r can be identified by solving the following eigenvalue problem: (K + Knl (x) + Kσ (x, T ) − ωr2 M)r = 0

(3)

All the numerical procedures, including modal analysis about thermal equilibrium, were implemented by a MATLABbased nonlinear FEA code called OSFern (https://bitbucket.org/cvandamme/osfern, [9]). It has many benefits over commercial software (e.g. Abaqus) in that the user can directly access and compute nonlinear terms such as tangent stiffness matrix, thermal stress matrix, and thermal load vector to evaluate the thermal equilibrium.

3 Model Correlation of Curved Panel 3.1 Target Experimental Data Ehrhardt et al. recently investigated the linear and nonlinear dynamic behavior of a curved panel subjected to combined mechanical–thermal loading [10]. They measured the temperature distribution over the panel and captured at about one million data points using a forward looking infrared (FLIR) camera. They also obtained the dynamic response in three dimensions at 167 points using 3D digital image correlation (DIC). The linear natural frequencies and mode shapes of the panel were identified from the measurements to demonstrate a complex variation of modal properties of the curved panel with respect to the localized temperature fields. In this study, a finite element model of the curved panel is updated based upon the modal information measured when the structure was locally heated near its center up to a peak temperature of 500 ◦ F.

3.2 Temperature Field Mapping to FE Model The initial geometry and material properties of the FE model followed the nominal data of the curved panel used in [7, 10]. The dimensions of the panel were 9.75 × 15.75 in (projected length in the curved direction) with a radius of curvature of 100 in. The Young’s modulus was 28.5 Mpsi, the Poisson’s ratio was 0.3, the density was 0.289 lb-in−3 , and the thermal expansion coefficient was 10.5 × 10−6 ◦ F−1 to approximate stainless steel. The FE model of the panel was composed of a 64 × 40 grid of 4-node shell elements of 0.048 in thickness. The nodes at the boundaries were fixed to approximate the actual experimental setup, of which boundary edges were tightly bolted to the mounting frame. In order to apply the temperature fields to the model, they were first aligned with the FE model geometry. The DIC data and FLIR temperature data captured in the same coordinate system were both used for the alignment. Since the DIC data did not provide the boundary coordinates of the panel, the 2D FLIR data were used to initially translate the coordinates of the FLIR/DIC to match with the FE model. An image was created from the FLIR data and the locations of the boundary edges of the panel were extracted, as can be seen in Fig. 1. Then, both the FLIR and DIC data were translated to overlap with the FE model based on the location of the boundary edges. An additional fine alignment using DIC data was applied using the iterative closest point (ICP) algorithm [20]. The algorithm computed the optimal rigid 3D transformation matrix that minimizes the distance between the DIC data and the FE coordinates. The final alignment of the DIC data to the FE model is shown in Fig. 2. Note that although the two seem to match well, the aligned DIC data are not symmetric about the center position of the panel. Instead, it is slightly rotated counterclockwise about the y-axis. Also, there are deviations as large

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Fig. 1 FLIR measurement at room temperature, used to identify the location of the boundary edges

Fig. 2 The DIC data points and the nominal FE model geometry before the alignment (a) in 3D space and (b) when projected to XZ plane, and after the alignment (c) in 3D space, and (d) when projected to XZ plane

as 0.2 in toward the center of the FE model. These discrepancies after the alignment indicate that there may be significant differences between the nominal geometry and the actual hardware. After the alignment, the non-uniform temperature fields were projected onto the FE geometry. Since a much greater number of points of FLIR data were provided than the FE nodes, a simple “nearest point” algorithm was applied for the

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Fig. 3 Temperature field at each level from measurement in [10], and after mapping using the nearest point method Table 1 Maximum and mean temperature of each temperature field from measurement in [10] (left), and after mapping using the nearest point method (right) Measured Max (◦ F) Mean (◦ F)

T0 75.160 73.357

T1 111.103 75.905

T2 316.367 93.947

T3 514.743 109.949

Mapped Max (◦ F) Mean (◦ F)

T0 75.150 73.301

T1 110.394 75.752

T2 313.231 92.826

T3 509.784 107.992

mapping. The algorithm finds the point among the FLIR data that is closest to each FE node, and the temperature of the FLIR point is directly mapped to the corresponding FE node. Another method was also used, which fit the temperature data to a piecewise linear surface using shape functions that are identical to those used to model the temperature between the FE nodes. This approach was computationally expensive, as it requires a least squares problem with thousands of nodes, but produced very similar results to those obtained using the “nearest point” method for our case study. Therefore, all the numerical results demonstrated in this chapter are based on using the “nearest point” method. Figure 3 illustrates the temperature field mapped onto the FE model at four different levels of thermal loading from T0 to T3. The mean and maximum temperatures at each level are presented in Table 1. The mean percentage errors of the maximum and mean values of the mapped temperature fields were both less than 1% with respect to those of measurement.

3.3 Model Updating of Curved Panel at Room Temperature The FE model of the curved panel was initially correlated to the measurement data at room temperature (T0). The first four natural frequencies and mode shapes were measured from the experiment in [10]. Their natural frequencies served as the target data and were compared with the frequencies of the FE panel model in the model updating procedure. The updating design variables were chosen among the structural parameters that have considerable uncertainty and a significant effect on the dynamic behavior of the panel model, such as Young’s modulus and radius of curvature. Also, asymmetry of the curved panel was applied and updated in the form of non-zero skewness constant cs in Eq. (4), which adds a half cycle of a sinusoid to the half of the curved panel length (lx ). 

2π x z = z + cs × sin lx

 , iff x >

lx 2

(4)

Lastly, the pre-stress of the clamped panel, which could not be identified in the experiment, was accounted for by adding a uniform temperature field to the unheated model. Some other parameters, such as boundary stiffness, could be further considered as the design variables, but they are not covered in the chapter due to their negligible impact on the result. The model updating procedure applied the interior-point method as an optimization algorithm [21], implemented in the R MATLAB function, fmincon. The optimization routine is used to minimize the error between the modeled frequencies and the measured frequencies of the first four modes until it reaches a local minimum. After running 50 iterations of the model updating, the mean frequency error of the FE panel model significantly reduced, as presented in Table 2. The mode shapes after the model updating in Fig. 4 also show considerably improved agreement compared to those of nominal FE model.

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Table 2 Natural frequencies of the curved panel at room temperature (T0) from the measurement, and the natural frequencies computed by the nominal and updated FE model Mode 1 2 3 4 Average

Measured (Hz) 211.200 257.000 297.900 385.100 −

Nominal FEM Freq (Hz) 255.558 272.690 347.800 425.996 −

% Error 21.003 6.105 16.751 10.620 13.620

Updated FEM Freq. (Hz) 211.203 257.060 297.904 385.074 −

% Error 0.002 0.023 0.001 0.007 0.008

Fig. 4 The first four mode shapes of the curved panel at room temperature (T0) from the measurement, and the mode shapes computed by the nominal and updated FE model Table 3 The initial (nominal) and updated values of design variables of the curved panel model at room temperature (T0) Design variable Initial Updated

Young’s modulus (psi) 2.850 × 107 2.561 × 107 (−10%)

Radius of curvature (in) 100 122.529 (+23%)

Skewness 0 0.0223

Pre-stress field (◦ F) 0 −3.224

These results indicate that the updated model can predict the modal properties at room temperature much more accurately compared to the nominal model. However, note that the accurate match of linear natural frequencies did not necessarily result in the same level of agreement in terms of mode shapes. For example, the second mode shape of the updated model exhibited a slight shift for its peak from the center, and the fourth mode also showed a small difference in the localized shapes from those of measurement. There could be errors in the actual measurements of either the natural frequencies or the mode shapes that cause the mismatch, or the model may still be missing some features. The updated design variables are compared with their nominal values in Table 3. The Young’s modulus reduced by 10% and the radius curvature increased by 23% from the nominal values. Asymmetry was also added to the structure by the skewness term, and pre-stress was found to be equivalent to applying negative uniform temperature by −3 ◦ F to the panel. The significant changes of the design variables suggest that the actual curved panel test article varies from the nominal structure in terms of geometry and material properties, presumably due to inaccuracies in the manufacturing process. Also, there could be error in DIC measurements or calibration that causes some of the discrepancies. The uncertainties arising from both the experimental data and the FE model become more noticeable when the localized heating is considered in the model correlation, as will be further discussed in the following section.

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3.4 Modal Analysis of Curved Panel Subjected to Localized Central Heating Nonlinear static analysis was performed on the updated FE model subjected to the localized temperature fields in order to identify the corresponding complex modal behavior of the curved panel. (The temperature fields from T0 to T3 were mapped to the updated FE model using the nearest point method, as described in Sect. 3.2.) The natural frequencies of the updated panel model at each of the temperature fields are compared with the measured frequencies from [10] in Fig. 5. Compared to the frequency curves computed with the nominal FE model, the frequency curves of the updated model provide considerably improved agreement to the measured curves, also following the general trend of increasing frequencies with respect to the temperature fields. However, although the updated model matched the linear natural frequencies of the first four modes very well, that did not cause an equivalent match when the localized temperature fields were applied. As the localized fields were applied, the frequencies were not highly accurate and deviated from the measured frequencies, having a mean frequency error of 3.487 Hz. The Mode 3 curve showed a substantial difference from the measured curve and failed to capture the modal interaction with Mode 2. The mode shapes depicted in Fig. 6 also support these discrepancies. The localized shapes of Mode 2 at T2 and T3 significantly differ from those of measurement, and the modal interaction, which was observed in the measurement, was not reproduced by the updated model. In summary, the updated model catches the general trends observed in the experiments but was not highly accurate in terms of the standards typically applied in linear model updating.

Fig. 5 Natural frequencies of the curved panel model with respect to each of the temperature fields compared with the measured frequencies. (a) Frequencies computed with the nominal FE model and (b) frequencies computed with the updated FE model

Fig. 6 The first three mode shapes of the curved panel with respect to each of the temperature fields (a) from the measurement and (b) computed with the updated FE model

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3.5 Model Updating of Curved Panel Subjected to Localized Central Heating To further investigate this, the model updating procedure was repeated, but this time using more modal information simultaneously. To be specific, the first four natural frequencies measured at four temperature fields from T0 to T3 were used as the target data to match in the optimization routine. The four parameters updated at room temperature in the previous section were used as the design variables. The initial and final parameter values after 30 iterations are presented in Table 4, which show a relatively small change compared to the previous model updating result at room temperature. There was a noticeable difference in the agreement between the measured and FE model natural frequencies at the various temperatures, as shown in Fig. 7, but the agreement is not dramatically better. This shows that the modal properties at the various temperatures are quite sensitive to the design parameters of the curved panel. Note that additional design variables could be further considered to enlarge the design space of the panel model. For example, the thermal expansion coefficient could be applied as one of the design variables that directly determines the impact of thermal stress. The boundary condition of the panel model might also be better modeled by adding axial springs at the boundaries rather than approximating them as fixed. The transformation coefficients (rotation and scaling coefficients) used for the temperature field alignment could also be considered as design variables to find the optimal mapping. However, various trials were attempted with these additional design parameters and none of them resulted in an improved correlation, and so the results are not included in this chapter.

Table 4 The design variables of the curved panel model after the model updating using multiple temperature fields from T0 to T3 Design variable Nominal Initial Updated

Young’s modulus (psi) 2.850 × 107 2.561 × 107 2.700 × 107

Radius of curvature (in) 100 122.529 124.035

Pre-stress field (◦ F) 0 −3.224 −2.272

Skewness 0 0.0223 0.0255

550 500 450

o

Temperature ( F)

400 350 Mode 1 (Exp.) Mode 2 (Exp.) Mode 3 (Exp.) Mode 4 (Exp.) Mode 5 (Exp.) Mode 6 (Exp.) Mode 1 (FEM) Mode 2 (FEM) Mode 3 (FEM) Mode 4 (FEM) Mode 5 (FEM) Mode 6 (FEM)

300 250 200 150 100 50 200

250

300

350

400

450

500

550

600

Frequency (Hz) Fig. 7 Natural frequencies of the curved panel model with respect to each of the temperature fields after the model updating using multiple temperature fields from T0 to T3

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3.6 Effect of Uncertainty in the Panel Model Subjected to Localized Heating The results of the model updating procedure applied in the previous section suggest that it can be challenging to find one set of parameter values that brings the model for this complicated structure into exact agreement with measurements. Furthermore, even if such a set of parameters was identified, the actual behavior of the panel might be very sensitive to the test conditions or to manufacturing variations. As a result, it seems wise to approach the model correlation probabilistically to evaluate the sensitivity of the model to uncertainties. In this section, the Monte Carlo method was applied to the design parameters of the curved panel to quantify the uncertainties in the resulting natural frequency. The four design parameters were assumed to be random variables with Gaussian distributions. The mean values were the updated parameter values at room temperatures in Sect. 3.3, and the standard deviation of each parameter was 10% of its mean value for the Young’s modulus, radius of curvature, and skewness, while the standard deviation for the pre-stress was 5% of its mean value. The Monte Carlo simulation was performed using 1000 sampled sets (n = 1000), and the corresponding distributions of the random design parameters are shown in Fig. 8 compared to the nominal distribution. A thousand randomly sampled parameter sets were simulated, and in each case, modal analysis was performed about the static equilibrium state of the non-uniform temperature fields from T0 to T3. The average of each frequency for the first six modes is shown in Fig. 9a, along with their ±1 sigma bands. The small uncertainty in the design parameters translates into commensurate spread in the natural frequencies; for example, the first mode has a standard deviation of about 25 Hz, or 13% of the mean value, which is a little larger than the average uncertainty in the individual model parameters. The uncertainties in the natural frequencies are of similar order as the shifts in the frequencies with increasing temperature. It is also interesting to note that the measured frequencies were mostly bounded by the ±1-standard deviation bands around the mean frequencies of the model. Considering that the model spans the measurements even with such small uncertainties, one could argue that the mean model used in the simulation is as good as could be expected and no further model updating is needed. To evaluate this further, the 1000 models were evaluated to find the sample that produced the frequencies that best matched the measurements, and its resulting frequencies versus temperature are plotted in Fig. 9b. They show a similar level of agreement compared to that of the mean frequency curves in Fig. 9a, indicating that the mean model produced by model updating is as optimal model as can be obtained with the given design variables. It is important to mention that the target curve of Mode 3 still could not be bounded by the populated samples of the four design parameters. This implies that these parameters are not sufficient to fully

Fig. 8 Distribution of each of random design parameters when having 1000 samples (n = 1000)

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Fig. 9 Natural frequencies of the curved panel model with respect to each of the temperature fields compared with the measured frequencies. (a) Frequency distributions after the Monte Carlo simulation. (b) Frequencies of the 239th sample that have the best match to the measured frequencies

explain the modal properties revealed from the experiment, and some additional parameters would need to be considered and added in order to capture this behavior.

4 Conclusion This study investigated model updating of a curved panel subjected to a set of localized temperature fields that were measured experimentally. The model updating was initially conducted at room temperature and the result showed that the natural frequencies of the panel model could accurately match the measured natural frequencies by updating a few effective design parameters. The errors in the natural frequencies at room temperature were all less than 0.03%. In contrast, when the localized temperature fields were taken into account, the level of agreement decreased dramatically with errors greater than 10% for some modes/temperatures. The modal properties were highly sensitive to the design parameters of the curved panel as the non-uniform temperature field evolved, and it became a very difficult problem to achieve an exact correlation by the deterministic approach. On the other hand, the updated model did at least produce similar trends to those that were measured, and perhaps such a model would still predict the limit stresses adequately to be useful; this should be explored in future works. The probabilistic approach was then applied to the FE model to evaluate the effect of model uncertainty on the dynamic response when subjected to non-uniform thermal loading. The results from the Monte Carlo simulation revealed that when small uncertainties were applied to the design parameters, the model could bound the resultant natural frequencies at most temperatures. This would suggest that the updated model is a good representation of the mean panel, but that significant deviations can be expected in practice due to typical manufacturing tolerances. In an ideal case, one would test multiple nominally identical panels to estimate the distribution of the measurements. If this could be done, then model correlation could be used to identify the optimal distribution (mean and standard deviation) of the FE model parameters using a stochastic model updating procedure, as suggested in [22, 23]. The computational cost of the model updating procedure was significant as a nonlinear static solution was needed for each thermal field and after each change in the design parameters. The Monte Carlo simulation with 1000 samples took about 120 h to complete on a desktop computer with an Intel Core i7-7700K 4.2 GHz quad-core computer with 64 GB of RAM. To alleviate the computational expense, a reduced order model (ROM) could possibly be used. Gordon and Hollkamp [7] introduced two methods of generating ROMs for a heated structure: a cold mode method that uses the linear modes at room temperature and a hot mode method that directly uses the linear modes at the temperature of interest. While the hot mode method may not reduce the computational expense much, the cold mode method could dramatically reduce the cost of model updating and should be considered in a future work.

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Acknowledgments This work was supported by the Air Force Office of Scientific Research, Award Number FA9550-17-1-0009, under the Multiscale Structural Mechanics and Prognosis program managed by Dr. Jaimie Tiley.

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On Affine Symbolic Regression Trees for the Solution of Functional Problems M. D. Champneys, N. Dervilis, and K. Worden

Abstract Symbolic regression has emerged from the more general method of Genetic Programming (GP) as a means of solving functional problems in physics and engineering, where a functional problem is interpreted here as a search problem in a function space. A good example of a functional problem in structural dynamics would be to find an exact solution of a nonlinear equation of motion. Symbolic regression is usually implemented in terms of a tree representation of the functions of interest; however, this is known to produce search spaces of high dimension and complexity. The aim of this chapter is to introduce a new representation—the affine symbolic regression tree. The search space size for the new representation is derived, and the results are compared to those for a standard regression tree. The results are illustrated by the search for an exact solution to several benchmark problems. Keywords Symbolic regression · Genetic programming · Solutions of differential equations · Search space analysis

1 Introduction Differential equations are among the most fundamental tools available for scientific analysis. No other mathematical object has the same ability to describe change in the physical systems. It is therefore no wonder that differential equations have been so extensively studied. Tireless investigation into the specification, existence and uniqueness of solutions to differential equations has borne some of the most eminently useful techniques in engineering. Exact solutions to linear differential equations have enabled modal analysis, an invaluable analysis tool for linear dynamic systems. However, as the materials, geometries, demands and capabilities of engineering structures become ever more complex, a linear description of the physics becomes less and less applicable. By far, the majority of nonlinear differential equations are without exact solution. This reality hamstrings the development of powerful analysis tools that might be viewed as nonlinear alternatives to modal analysis. In the place of pure mathematical solutions, powerful heuristic methods have been developed. Heuristic methods fall into a number of distinct categories based on the form of the yielded solution. Numerical methods provide approximations to the true solution at a finite number of points in the domain. Analytical approximate methods such as the shooting method [1], or harmonic balance [2], provide analytic approximations, often by the consideration of series expansions. A third class of approaches, referred to hereafter as pure heuristic methods, offer the possibility of both an approximate and an exact solution. Such approaches are often (but not always [3, 4]) based on a standard symbolic regression (SR), a specific application of the more general method of Genetic Programming (GP) [5]. A large number of authors have proposed SR for solving differential equations in the last three decades. Highlights include the earliest suggestions of the approach in [5], a reverse-Polish notation-based tree structure in [6], a hybrid analytical approach in [7], a grammar-based approach with a benchmarking set in [8], and a Cartesian Genetic Programming (CGP) approach in [9].

M. D. Champneys () Industrial Doctorate Centre in Machining Science, Advanced Manufacturing Research Centre with Boeing, University of Sheffield, Sheffield, UK Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected] N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK © 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_11

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Despite some progress, no single approach has yet broken through as a significant step forward. A potential reason is that these methods suffer from inherent difficulties arising from the complexity of the underlying search problems. A recent body of work has focussed on ways that the performance of SR algorithms applied to the solution of differential equations might be improved. This chapter opens with a thorough mathematical description (based in part on the work of Seaton et al. in [9]) of the search problem underpinning SR-based solution of differential equations. Of the elements defined in this description, the current study is focussed on the encoding space. The main contribution of this chapter is a novel scheme for representing expressions in SR—the affine regression tree. The sizes of the search spaces for this new representation, as well as two existing schemes, are enumerated. The potential of the new representation is demonstrated with a benchmark case study.

2 Symbolic Regression Is a Search Problem Symbolic regression casts the specification of an analytic relationship of interest as an optimisation problem. This approach has been undertaken by many authors attempting to develop heuristic solution methods for differential equations. In this section, notation is presented that describes the underlying search problem formally. The search problem of locating a solution f ∗ (x) to a general differential equation (f, x) is defined as the set of the following elements (the definitions in parentheses are the equivalent terms used in the GP literature): • • • • •

an encoding space (Genotype): E; a decoding function (Phenotyping function): P : E → ; a functional interpretation function: S :  → Cr ; an objective function (Fitness function): J : Cr → R; one or several search operations (Genetic operations): Mi : E → E.

A brief definition of each of the above is offered here. The encoding space E represents the set of trial expressions f . This space can be thought of as the overall search space. Each point in the space E is a structurally (but not necessarily mathematically) unique representation of a mathematical expression. Points in the encoding space are mapped via a decoding function P into the space of representable expressions (denoted as  following [9]). Each point in  is a structurally unique mathematical expression. However, this is still not enough to guarantee mathematical uniqueness. Consider the expressions (10x + 2) 2

(1)

5x + cos2 (x) + sin2 (x)

(2)

and

The above expressions can be trivially manipulated to demonstrate equality, but they have different expression structures and rely on different bases of operations in their representation. In order to test for mathematical equality, a further projection S is required. Intuitively, this mapping is the semantic realisation of an expression in the function space. Since solutions to differential equations must necessarily be differentiable up to the order of the differential equation r, the relevant function space is denoted by Cr , as is conventional. The subspace ω ⊆  is additionally defined by u ∈ ,

S(u ∈ ω) = f ∗

(3)

as the subset of expressions in  that are mathematically equivalent to the solution to the differential equation. An objective function J maps trial functions to values on the real line. In the case of a minimisation problem, J is constructed such that J (f ∗ ) = 0,

J (f = f ∗ ) > 0

(4)

The final ingredients in the problem specification are the search operations Mi , and these are perturbing functions that move points around in the expression space. The structure of the spaces is summarised in Fig. 1. It is clear that the appropriate selection of these elements is instrumental in controlling the overall difficulty of the search problem. An ongoing body

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Encoding space

ej

E

Mi : E → E

P :E→Ω

Mi (ej )

Representable expressions 2x + 1 Ω

ω √

sin t

Function space S : Ω → Cr f∗ Cr

J : Cr →

0 Fig. 1 Visualising the underlying search problem

of work investigates heuristics for optimal selection of these components. This chapter will focus on the selection of the encoding space E. Seaton et al. define a heuristic metric for comparison of encoding spaces termed unguided complexity in [9]. The method relies on estimating the ratio k=

μ() μ(ω)

(5)

by enumerating the search space and sampling from the space of representable expressions by some search method. Here, μ is the counting measure from measure theory used to evaluate the size of the spaces. The quantity k −1 can be interpreted as the probability of sampling the solution at random from all representable expressions. However, in the absence of methods for direct enumeration of μ(ω) or an efficient method for uniformly sampling from ω, this metric is limited to compact expressions and small encoding spaces. Because of this issue, the analysis conducted in this chapter will focus on a direct comparison of μ() for different representation schemes.

3 Enumerating Search Spaces In order to evaluate the unguided complexity, one must first be able to evaluate μ(). In the current work, this is accomplished by considering the encoding space E. A common choice [5, 10, 11] for E is to use a tree structure. This approach has the nice

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property that all mathematical expressions can be mapped in an isomorphic way to a tree object, Etree  tree

(6)

μ(Etree ) = μ(tree )

(7)

and therefore the spaces have the same measure,

The size of the search space of trees is discrete but naturally not finite. Practically, one places a restriction on the size of tree structures that are permitted. Several approaches are possible including limits on tree depth or internal nodes [3]. However, the approach here will be to restrict the total number of nodes in the expression tree. This has the advantage of not placing any bias upon either very deep or very wide trees. Another advantage of using the total number of nodes is that it is equivalent to maximum tree depth by the relation n = mh , where n is the number of nodes, m is the maximum number of child nodes connected to any given node (hereafter referred to as the arity of that node), and h is the tree depth. The maximum arity of any node defines the arity of the tree structure. The number of unlabelled m-ary trees with exactly n nodes is given by the Fuss–Catalan numbers [12], Cn =

  1 mn (m − 1)n + 1 n

(8)

However, the nodes in an expression tree representation are not unlabelled. Instead, each node of arity i ∈ [0, . . . , m] derives a label from set fi , and the edges are unlabelled. Thus, the overall label set F = {f0 , f1 , . . . , fi , . . . , fm }

(9)

is defined as the search basis. In the literature, the case i = 0 is sometimes considered separately with 0-ary nodes referred to as terminals or leaves and f0 = T referred to as the terminal set. For compactness, this notation will not be used here. Implementations of the search problem must necessarily select a basis that contains all the mathematical objects needed to describe the solution, or else only approximate solutions will be possible. For example, when considering linear ODE problems, it is essential to include sinusoids and exponential terms. The size of the tree space clearly depends on the maximum arity m, the maximum number of nodes n and the basis set F . Let Tj = μ (T(m, j, F ))

(10)

be the number of possible node-labelled m-ary trees with basis F comprising exactly j nodes. By considering trees of all sizes up to n, one has μ(Etree (m, n, F )) =

n 

Tj

(11)

j =0

In order to derive this quantity, consider the case n = 1. Since this can only be a single node with no children, the number of possible labels (and therefore trees) is T1 = f0

(12)

Next, consider the case n = 2; there is still only a single possible tree structure (one with one root and one child). The number of such trees is equal to the number of label combinations, T2 = f1 f0

(13)

This process is displayed graphically up to n = 4 in Fig. 2. A recurrence relation can now be derived for the case n = k. In order to simplify the notation, the following derivation will continue in the case m = 2. For basis functions of analytical expressions, this is a realistic restriction as there are few common analytical expression operations with an arity greater than

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n

m

Possible trees

1

0

f0

2

1

f1 f0

3

2

f2 f02 + f12 f0

4

3

f3 f03 + f2 f1 f02 + f2 f1 f02 + f13 f0

Fig. 2 Possible labelled trees with up to n = 4 nodes

k−1

k−p

p−1

Fig. 3 Possible trees with m = 2, n = k, p ∈ [2, k − 1]

two (an example might be a summation with limits on indices—technically a trinary operation, but these are not often seen in exact solutions to differential equations). In the case m = 2, n = k, there are two important cases to consider as shown in Fig. 3. In the first, the root node is unary. Its single argument is a subtree with k − 1 nodes. In this case, there are Tk,unary = f1 Tk−1

(14)

possible trees. In the other case, the root node is binary and the number of nodes in its arguments sums to n−1. In performing this sum, one has Tk,binary = f2 {Tk−2 T1 , Tk−3 T2 , . . . , T1 Tk−2 } = f2

k−1 

Tk−p Tp−1

(15)

p=2

The required relation is now the sum of these two possibilities, Tk = Tk,unary + Tk,binary = f1 Tk−1 + f2

k−1 

(16)

Tk−p Tp−1

p=2

Given T1 = f0 and T2 = f1 f0 from above, one can construct a recursive scheme to calculate any μ(Etree (2, n, F )) and thus the number of structurally unique expressions μ(tree ), μ(Etree ) = μ(tree ) =

n  k=1

⎡ ⎣f1 Tk−1 + f2

k−1 

⎤ Tk−p Tp−1 ⎦

(17)

p=2

Another common representation scheme is a grammar-based structure [8, 13]; these are employed by several authors alongside techniques such as modular arithmetic to encode expressions as vectors of integers. The search space of grammarbased representations can conveniently be enumerated by considering the number of unique tree structures. This process gives a better estimate of μ(grammar ) than simply raising the maximum integer value to the power of the vector length (i.e.

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μ(Egrammar )). This advantage arises because the modular arithmetic removes the need for a maximum integer value resulting in a many-to-one mapping via the decoding function. Practically, this means that unlike the expression tree representation, μ(Egrammar )  μ(grammar )

(18)

Another advantage is that an equivalent tree enumeration allows better comparison with other representation schemes. Since grammar-based structures sometimes include trinary operations, an additional term in Eq. (16) is required. The number of distinct trees with k nodes is now ⎡ ⎤ k−1 k−2 k−1    ⎣Tq−1 Tk−p Tp−1 + f3 Tk−p Tp−q ⎦ (19) μ(Tgrammar (3, k, F )) = f1 Tk−1 + f2 p=2

q=2

p=q+1

However, there is some subtlety here; several of the nodes included in the grammar do not alter the underlying function mathematically and only act as placeholders for the decoding function P . Such nodes include the ‘expression’ and ‘operation’ nodes. In considering μ(grammar ), these meta-nodes can be safely collapsed (i.e. ignored) in the computation. Figure 4 is a plot of μ(tree ), μ(Tgrammar ) and μ(grammar ) with increasing n. The tree space data are generated with a basis set of operations given by Ftree = {{x, [0, 9]}, {sin, cos, log, exp},

(20)

{+, −, ×, ÷}} for the expression trees (indicative of the approach suggested by [10]). The grammar space data are generated with m = 3 and basis set,

Fig. 4 Size of tree space with increasing n

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Fgrammar = {{x, [0, 9]}, {sin, cos, log, exp, < expr >, < digit >}, {+, −, ×, ÷, < func >},

(21)

{< op >}} for the grammar representation (indicative of the study by Tsoulos et al. in [8]). The meta-operations (in angle brackets) are ignored in the computation of μ(grammar ) resulting in an identical basis. As can be seen in Fig. 4, the size of these spaces grows extremely quickly with the number of nodes in the tree. The number of trees in grammar-based approaches appears to grow more quickly due to the trinary operation. However, it can be seen that the number of expressions representable by these encodings is equivalent.

4 Affine Regression Trees A novel encoding representation is proposed here—the affine symbolic regression tree. This encoding is essentially an extension of the expression tree with additional structure at each node. The new approach is inspired by two observations. First, the exact discovery of constants in symbolic regression schemes is an open problem with few approaches (there are several notable techniques that specify constants approximately—the reader is directed to [14–16] for examples). The second observation is that adjacent points in E are unlikely to be adjacent in R once projected there via the objective function. Put plainly, it is often not trivial for the SR algorithm to increment towards the true solution despite a high degree of semantic similarity (for example, a constant that is wrong by a small integer/rational value). To this end, the affine symbolic regression tree is defined in the same manner as the expression tree. However, each node η is now a 3-tuple, η = {a, f, b}

(22)

where a and b are termed constants. During execution, nodes take the affine form, η = af + b

(23)

In this regard, the representation bears some similarity to the multiple regression approach in [14], whereby linear combinations of all tree subexpressions are used during a meta-optimisation step. However, the current approach differs both in that an affine combination is used and in that the constants are included as a part of the tree structure itself. Constants have their own structure and are represented by a 4-tuple, θ = {s, p, q, r}

(24)

where p and q are integers in the range of [0, z] and [1, z], respectively. r is a member of the set of permitted exponents R with a default value of 1. s is an element from the permitted transcendental constants S, also with a default value of 1. Upon execution, constants are evaluated as  r p θ =s q

(25)

Upon initialisation, ‘a’ constants are given a value of 1 ({1, 1, 1, 1}) and ‘b’ constants a value of 0 ({0, 1, 1, 1}). Initialising the constants in this way provides a bias towards sparse solutions. Care is taken to ensure that q = 0 so that illegal divisions are avoided by design. A more granular approach is also possible for the specification of the exponents. In this case, the constants are represented by the 5-tuple θ = {s, p, q, r1 , r2 } and are evaluated as

(26)

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Table 1 Constant mutation operations proposed by the current study Operation m0 (θ) m1 (θ) mi (θ)

Action θ →0 θ →1 θ →θ ±1

Description Zero constant Unit constant Increment/decrement

mr1 (θ)

θ→

p±1 q

Increment/decrement numerator

mr2 (θ)

θ→

p q±1

Increment/decrement denominator

ms (θ) mr (θ) mpm (θ)

θ → sθ, s ∈ S θ → θr, r ∈ R θ → −θ

Multiply by transcendental constant Raise to exponent Flip sign of constant

{a, ×, 0}

× ×

{1, exp, 0} {1, sin, 0} {−b, t, 0}

{c, t, −d}

sin

a

−

exp ×

× −b

t

−b

d t

Fig. 5 Comparison between compact affine and expression trees for the expression f = ae−bt sin(ct − d)

θ =s

  r1 p r2 q

(27)

However, this increase in granularity comes at the cost of a larger search space and is not the approach considered in this chapter. Constants are mutated during the run of the search by a number of constant mutation operations that are defined in addition to more orthodox tree-based search operations (the reader is directed to [11] for a reference). Several of these constant mutation operations are described in Table 1. The advantages of this representation are several; consider the expression tree in Fig. 5. The affine tree is far more compact requiring only a fraction of the number of nodes to represent the same expression. In fact, with affine trees, there is no need to include integers or other constants in the basis set F at all. Constant discovery is handled entirely by the affine constant objects. The size of the search space can readily be estimated by extending the analysis of expression trees above. The first step is to enumerate the number of possible values an affine constant θ ∈ A can take. This is achieved by considering the elements in the tuple, μ(A) = μ(P )μ(Q)μ(R)μ(S) = z(z + 1)μ(R)μ(S)

(28)

Since there are two constants per node, the expression for μ(affine ) can be written as μ(affine ) =

n    (z(z + 1)μ(R)μ(S))2k Tj

(29)

k=0

or explicitly, μ(affine ) =

n  k=0

⎡

⎡

⎣(z(z + 1)μ(R)μ(S))2k ⎣f1 Tk−1 + f2

k−1  p=2

⎤⎤ Tk−p Tp+1 ⎦⎦

(30)

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Adding this result to the previous data in Fig. 4 seems to suggest that this approach is strictly worse (in terms of search space size) than expression trees. However, this is a false equivalence. While the affine tree representation is at least as compact as the expression tree representation, in many cases it will be significantly more so. In practice, one is able to select a lower value of n when working with the affine representation and still maintain the same coverage of Cr . Figure 7 compares several scenarios in which the affine representation is more compact and plots them against the number of nodes required for an affine representation. The scenarios shown are illustrated in Fig. 6. The results of Fig. 7 indicate that the affine representation has a smaller search space for expressions that require four times as many nodes to represent as an expression tree. For trivial expressions with only a few elementary integer constants, such a ratio is perhaps unrealistic. However, the authors would argue that for more realistic engineering expressions such as the one depicted in Fig. 5, a 1:4 ratio is more likely. The authors would note that the tree representations in Fig. 5 assume that the constants a, −b, c, −d are specified in the basis. In reality, an expression tree would require further subtrees to represent these values adding yet more complexity. The advantages of the affine tree representation extend beyond the compactness of the search space. The constant mutations described in Table 1 permit a pseudo-gradient between expressions of the correct form (only errors in constants) and the true solution. Work is ongoing in developing a formal mathematical argument, but a rudimentary explanation is thus. Consider any two affine trees of the same structure α1 and α2 (but with different constant values) with objective function values j1 and j2 , respectively, such that j2 < j1 . There must then be a chain of constant mutation operations [M1 , . . . , Mv ], and the successive application of which will map α1 onto α2 . It is furthermore argued that each application of the mutation operations in the chain is more likely to result in a monotonically decreasing objective score than an equivalent path based on orthodox search operations. The affine representation has additional advantages in terms of extendibility. As noted in [9], the application of the rules for differentiation tends to result in constants appearing in multiple places in a single expression. A future study examines an extension of the affine tree method such that constants are sampled from a finite pool. The successful outcome of this work would dramatically reduce the size of the required search space.

Best case 1:1

{1, x, 0}

Affine case 1:4

{a, x, b}

x

+ × a

Worst case 1:17

b x

{s( pq )r , x, s( pq )r }

+

×

× × s

x

s ÷

ˆ ÷

p

p q

ˆ

q

r

r

Fig. 6 The potential savings offered by an affine structure over a standard expression tree in terms of nodes in the tree structure

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Fig. 7 Comparison of affine and expression trees in terms of search space size versus minimum number of nodes required for affine representation Table 2 ODE problems (from [8]) studied in the current investigation Problem

ODE

1

f −

2t−f t

2

f −

1−f cos(t) sin(t)

3

f  + 15 f − e

4

f  + 100f = 0

5

f 

6

f  + 15 f  + f + 15 e

7

f 

8

tf  + (t − 1)f  + f = 0

9

− 6f 

Domain

Subject to

Exact solution

t ∈ [0.1, 1]

f (0.1) = 20.1

f =t+

=0

t ∈ [0.1, 1]

f (0.1) =

f =

cos(t) = 0

t ∈ [0, 1]

f (0) = 0

f =e

t ∈ [0, 1]

f (0) = 0, f  (0) = 10

f = sin(10t)

=0

−t 5

+ 9f = 0 −t 5

cos(t) = 0

+ 100f = 0

f  + 15 f  + f + 15 e

−t 5

cos(t) = 0

2.1 sin(0.1)

0, f  (0)

t ∈ [0, 1]

f (0) =

=2

t ∈ [0, 2]

f (0) = 0, f  (0) = 1 sin(10), f  (0)

t ∈ [0, 1]

f (0) =

t ∈ [0, 1]

f (0) = 1, f (1) = 0

t ∈ [0, 1]

f (1) =

sin(0.1) 1 e5

2 t

t+2 sin(t) −t 5

sin(t)

f = 2te3t f =e =0

, f  (0) = 1

−t 5

sin(t)

f = sin(10t) f =1−t f =e

−t 5

sin(t)

5 Illustration with a Benchmark Set This section illustrates the potential efficacy of the affine tree representation with a simple case study. In order to make comparisons, the benchmark set of ODE problems described in [8] is used. The problem set is outlined in Table 2. An affine tree representation with a maximum n value of 10 is compared to a standard regression tree representation with maximum n values of 10 and 40. For all representations, the symbolic regression is undertaken by a genetic algorithm (GA) with properties described in Table 4. An objective function is selected with four terms. The function is defined as the weighted sum of the mean squared error (MSE) over the ODE, the trial solution f , its first derivative f  and initial or boundary conditions. The target quantities fˆ and fˆ  are computed in advance of the run by a fixed-step fourth-order Runge– Kutta scheme with a step size determined by the domain of the target problem. For first-order problems, the MSE over f  is not used. The overall objective function is therefore

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Table 3 Remaining properties of the search problem in the case study Description Decoding function Functional interpretation Objective function Search operations

Notation P S J M

Value ‘Sympy’ computer algebra system d = 100 point even sampling over the domain See (31) Reproduction, tree crossover, random tree, subtree mutation

Table 4 Parameters of the genetic algorithm used in the case study

J (f ) =

Parameter Population size Maximum generation Initialisation procedure Selection procedure Initial maximum tree depth Learning period

Value 400 500 ‘Grow’ [5] Tournament (k = 2) 3 5

d   1  λ1 ((fˆ) − (f ))2 + λ2 (fˆi − fi )2 + λ3 (fˆi − fi )2 + + λ4 (Ij (fˆ) − Ij (f ))2 d i

(31)

j

where (f ) is the ODE of interest and the Ij are the initial or boundary conditions applicable to that problem. The λi are the weights and are all set to unity with the exception of λ4 which is set to 100. A computer algebra engine (in this case, the symbolic library available in the python language sympy) is used for the evaluation of derivatives and assessment of exact solutions. The remaining elements of the search problem are described in Table 3. For both representations, a basis set given by F = {{t, }, {sin, cos, log, exp},

(32)

{+, −, ×, ÷}} is used. For the expression tree representations, the digits [0, 9] are added to f1 . Initialisation, mutation and application of affine constants are as described in the previous section. In order to simplify the search, the sets of exponents R and transcendental constants S are set to {1} and the corresponding affine mutations are excluded. The application of affine mutations and search operations is handled by an adaptive process inspired by [17]. The procedure samples search operations randomly according to method probabilities pi that are initialised as equal but updated every L generations. This interval is referred to as the learning period. The probabilities are updated by the rule, pi =

oi2 si

(33)

where si is the number of times the i th search operation is selected and oi is the number of successful entrants to the population produced by that method during the last learning period. The raw probabilities are then scaled such that they sum to unity. Adaption of affine mutations is handled separately to search operations. For the affine representation, up to 5 affine mutations are applied during the production of a new individual. The procedure is the following. First, mutations are applied with a probability of 0.9. If successful, then a number in the range [1, 5] is sampled at random and that many mutation operations (Table 4) are sampled and applied sequentially. In order to speed up the search, some additional techniques are employed; checks for illegal operations such as infinite terms or division by zero are employed before the evaluation of the objective function. In addition, to restrict the size of the search space, nested transcendental functions (sines, cosines and exponentials) are disallowed. In the case that such a nesting is produced by the action of the search operation, the individual is given a null fitness score and loses tournament selections automatically. The GA was run 10 times for each problem and representation. In order to compare results, the expected number of  function calls required for the solution to be found E ∗ | is estimated. Because the GA does not find the exact solution

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  in every run, a right-censored estimator of E ∗ | is employed. Assuming an exponential distribution for the underlying probability of finding the solution after any given assessment of the objective function, one retrieves the maximum likelihood estimate of the right-censured mean as   E[ ] =  ∗ f ∗

(34)

which is the sum of all function calls divided by the number of times an exact solution is found. A precision of 400 function evaluations applies to the results, as the exact number of function calls within the final generation is not stored by the current implementation. A 90% confidence interval for the estimates of E[∗ |] is also estimated using a bootstrapping approach with 1000 samples. Although the number of samples is perhaps low for rigorous estimation of confidence intervals, the same approach is used for each case to aid in the visualisation of uncertainty. These data are plotted alongside the results of the individual runs in Fig. 8. Uncertainty in the grammar encoding is due to uncertainty regarding the population sizes used in the original study. To ensure the most competitive comparison, the lower bound of these estimates is used for comparison with the proposed approach. The results show varied performance, and the affine method performs better on several problems including ODE1 and ODE8. However, there are also several problems for which the affine method was not able to locate the exact solution in any of the 10 runs. Overall, both the affine and expression tree representations perform better than the grammarbased encoding of [8], often dramatically so. However, in problems 3, 6 and 9, the tree-based approaches were unable to find the solution with any regularity. It is unlikely to be a coincidence that all three of these problems share the same functional solution and that this solution has the least compact tree representation of any in the problem set. Investigation of the suitability of the encoding schemes for problems of differing compactness is to be the subject of the future work. The benchmark problems are not without their own limitations. One such limitation is the apparent variation in difficulty of the ODE problems within it. Another is the limited domains over which most of the solutions are monotonic. An interesting avenue for further investigation is the construction of a new benchmarking suite of problems for which the difficulty can be controlled by selecting problems with increasing unguided complexities. Such a benchmark would more readily enable useful comparisons between candidate approaches.

6 Conclusions In this chapter, a formal description of the search problem underlying an SR-based approach to the solution of differential equations is presented. This description is used to compare several existing representation schemes for which the sizes of the search spaces are enumerated and compared. A novel expression encoding—the affine regression tree—is also presented. The search space of this representation is enumerated and found to be at least as small as that of expression trees for problems with an affine structure and considerably more compact than the worst case. The new encoding is demonstrated in a case study where the optima of interest are the solutions to a benchmark suite of ODE problems. Despite mixed results, the proposed approach shows similar or better performance in a number of problems. However, in problems with less compact solutions (such as ODE3, ODE6 and ODE9), both affine and tree-based representations perform poorly. An ongoing body of work seeks to address this and the outstanding issue of exact constant discovery in symbolic regression.

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Fig. 8 Results of the benchmarking study ODE problem plotted against number of function evaluations before the exact solution is found. Estimates of E ∗ | are displayed as diamonds with estimated 90% confidence intervals. Individual samples of successful runs are plotted as points, and unsuccessful runs are plotted as crosses. Results for individual runs [8] are not reported and so these are omitted from the figure

Acknowledgments MDC would like to acknowledge the support of the EPSRC grant EP/L016257/1.

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3. Guillaume, L., Charton, F.: Deep learning for symbolic mathematics. In: Proceedings of the 2019 International Conference on Learning Representations, 2019 4. Jin, Y., Fu, W., Kang, J., Guo, J., Guo, J.: Bayesian symbolic regression (2019). Preprint. arXiv:1910.08892 5. Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992) 6. Burgess, G.: Finding approximate analytic solutions to differential equations using genetic programming. Technical report, Defense science and technology organisation, Canbera (1999) 7. Kirstukas, S.J., Bryden, K.M., Ashlock, D.A.: A hybrid genetic programming approach for the analytical solution of differential equations. Int. J. General Syst. 34, 279–299 (2005) 8. Tsoulos, I.G., Lagaris, I.E.: Solving differential equations with genetic programming. Genet. Programm. Evol. Mach. 7, 33–54 (2006) 9. Seaton, T., Brown, G., Miller, J.F.: Analytic solutions to differential equations under graph-based genetic programming. In: European Conference on Genetic Programming, pp. 232–243. Springer, New York (2010) 10. Lobão, W.J.A., Dias, D.M., Pacheco, M.A.C.: Genetic programming and automatic differentiation algorithms applied to the solution of ordinary and partial differential equations. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 5286–5292. IEEE, New York (2016) 11. Poli, R., Langdon, W.B., McPhee, N.F., Koza, J.R.: A Field Guide to Genetic Programming. Lulu Enterprises, Essex (2008) 12. Graham, R.L., Knuth, D.E., Patashnik, O., Liu, S.: Concrete mathematics: a foundation for computer science. Comput. Phys. 3, 106–107 (1989) 13. Mex, L., Cruz-Villar, C.A., Peñuñuri, F.: Closed-form solutions to differential equations via differential evolution. Discr. Dynam. Nat. Soc. 2015(6), 1–11 (2015) 14. Arnaldo, I., Krawiec, K., O’Reilly, U.: Multiple regression genetic programming. In: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, pp. 879–886 (2014) 15. Diver, D.A.: Applications of genetic algorithms to the solution of ordinary differential equations. J. Phys. A: Math. General 26, 3503–3513 (1993) 16. Iba, H., deGaris, H., Sato, T.: A numerical approach to genetic programming for system identification. Evol. Comput. 3, 417–452 (1995) 17. Qin, A.K., Suganthan, P.N.: Self-adaptive differential evolution algorithm for numerical optimization. In: 2005 IEEE Congress on Evolutionary Computation, vol. 2, pp. 1785–1791. IEEE, New York (2005)

Comparative Analysis of Mechanical and Magnetic Amplitude Stoppers in an Energy Harvesting Absorber Tyler Alvis, Mikhail Mesh, and Abdessattar Abdelkefi

Abstract A popular technique to control dynamical systems is the implementation of tuned-mass dampers. Most tunedmass dampers only transfer the mechanical energy of the primary system to a secondary system, but it is desirable to convert the primary systems’ mechanical energy into usable electric energy. A piezoelectric energy harvester is used in this study. Furthermore, amplitude stoppers are included to possibly generate a broadband region by causing a nonlinear interaction. Mechanical stoppers have been investigated to sufficiently widen the response of piezoelectric energy harvesters. The effectiveness of the stoppers type is also investigated by comparing magnetic stoppers to mechanical stoppers. A nonlinear reduced-order model using Galerkin discretization and Euler-Lagrange equations is developed. The goal of this study is to maximize the energy harvested from the absorber without negatively affecting the control of the primary structure. Keywords Nonlinear dynamics · Energy harvesting absorber · Mechanical stoppers · Magnetic forces

1 Introduction The control of unwanted vibrations is one of the most important aspects when designing a system. These vibrations are the most harmful when the excitation frequency matches the natural frequency of the structure, causing large oscillations that could damage the structure. A popular way to combat harmonic oscillations is a tuned-mass damper. Most tuned-mass dampers dissipate the energy as mechanical energy, but in some cases like dangerous to reach locations, it is desirable to convert the energy into usable electrical energy to power sensors or other devices, thus avoiding potentially dangerous battery replacements. An efficient way to accomplish this is by using a piezoelectric energy harvester as the tuned-mass damper [1]. Due to the harmonic response of the system, the operable range of the harvester to generate power is quite small. To further increase the width of the operable range of the energy harvesting absorber, amplitude stoppers are included to the energy harvesting absorber. Contact with the amplitude stoppers may generate a nonlinear response that is evaluated to improve the energy harvesting absorber’s ability to generate power as well as maintain control of the primary structure’s oscillations. In addition to mechanical stoppers, magnetic stoppers are also investigated to obtain the optimal design.

2 Nonlinear Reduced-Order Modeling A nonlinear reduced-order model is developed. The primary system is modeled as a spring-mass-damper system with stiffness k and damping coefficient c. The piezoelectric energy harvesting absorber consists of a steel substrate partially covered with bimorph piezoelectric layers with a tunable tip mass, as shown in Fig. 1. For the mechanical stoppers, a smoothened trilinear model is considered [2]. For the magnetic stoppers, the dipole-dipole representation by Tang and Yang [3] is used. It should be noted that this representation is only accurate for gaps larger than 7 mms. The expressions of these

T. Alvis () · A. Abdelkefi Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected] M. Mesh Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © 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_12

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Fig. 1 Schematic of the dynamical system under base excitations

w2(x,t)

w1(t)

M1

k

c

forces are given by: Fmechanical = k

+∞  i=1



ϕi Lf



1 ri (t) − 2

Fmagnetic = −

 ∞  ,3  ∞             ϕi Lf ri (t) + d  −  ϕi Lf ri (t) − d       i=1

(1)

i=1

3τ a1 a2   4  2π d − w2s − ∞ i=1 ϕi Lf ri (t) 

(2)

where Lf denotes the distance along the beam where the stoppers are placed, d is the initial distance between the beam and the stoppers, τ is the vacuum permeability, a1 and a2 are the moment of the magnetic dipoles, and w2s represents the static position of the absorber. More details about the magnetic force, mode shapes, and static position can be found in [4]. In this study, permanent magnets are investigated, but in future work electromagnets will be investigated to tune their effects on the system. Following the approach used in [5] to develop a reduced-order model for the primary structure and energy harvesting absorber and considering the external forces from the stopper forces, the final equations of motion can be written as: 

   %L      M + Mt + Mb1 L1 + Mb2 L − L1 w¨1 + cw˙1 + kw1 + ∞ i=1 Mt ϕi (L) + Mt Lc ϕi (L) + Mb 0 ϕi (x)dx r¨i = F cos ωf t + φ

(3)  $ r¨i + 2ξi ωi r˙i + ωi ri + Mt ϕi (L) + Mt Lc ϕi (L) + Mb

L

   ϕi (x)dx w¨1 − θi V = ϕi Lf Fstopper

(4)

0

∞

1  Cp V˙ + + θi r˙i = 0 R

(5)

i=1

where the mass of the beam Mb1 = bρ s hs + 2bρ p hp and Mb2 = bρ s hs , the ith mode of the energy harvester is ϕi (x), the   1 capacitance of the harvester is Cp = 2 ε33hbL , and the piezoelectric coupling term is θ = Ep d31 b hp + hs ϕi (L1 ). p

3 Analysis Three different cases are investigated: mechanical stoppers with a soft stiffness of 5 ∗ 103 N/m, mechanical stoppers with a hard stiffness of 5 ∗ 107 N/m, and magnetic stoppers that are both repulsive to the magnets in the tip mass. Figure 2 shows these three cases effects on the primary structure’s amplitude. The most important aspect while analyzing the primary structure’s amplitude is the stoppers’ effect on the control of the system, or the mitigation of its large oscillations. Mechanical stoppers with soft stiffness show great control of the primary structure, with a reduction of 62% or greater. Mechanical stoppers with a hard stiffness show a total loss of control, with a best-case scenario of a 23% reduction. Also, apparent is the onset of aperiodic oscillations where the second peak deteriorates due to the root mean square average. When studying the time histories of this region, the aperiodic region is apparent, and it occurs due to when the

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Fig. 2 Primary structure amplitude: (a) soft mechanical stoppers, (b) hard mechanical stoppers, and (c) magnetic stoppers

Fig. 3 Average harvested power: (a) soft mechanical stoppers, (b) hard mechanical stoppers, and (c) magnetic stoppers

absorber contacts the stoppers with zero velocity. Magnetic stoppers’ effects seem to be between the soft and hard mechanical stiffnesses, where medium-size gaps show an adequate control of the structure, but small gaps show a loss of control and aperiodic regions. Figure 3 shows the three cases effects on the average power generated. While investigating the energy harvested, peak power and the bandwidth are considered. Mechanical stoppers with a soft stiffness show great results. As the gap gets smaller, the peak power increases. The first peak shifts to the right increasing the power over the operable range of harvesting energy. Mechanical stoppers with hard stiffness show a decrease in power when there is any contact. Small gaps show the generation of a broadband region. This is desirable to generate more energy, but the extreme decrease in amplitude nullifies any benefit the broadband region would provide. A gap of 50 millimeters for magnetic stoppers shows a slight increase in power, but smaller gaps show a decrease in power. Again, a broadband region is present but is outweighed by the extremely small amplitudes.

4 Conclusions The results showed that mechanical stoppers with a soft stiffness are the most promising. The primary structure’s amplitude remains controlled while increasing the amount of energy harvested. Mechanical stoppers with a hard stiffness are not desirable. This case loses all control of the primary structure regardless of gap size if there is contact and has a great decrease of energy harvested. Magnetic stoppers with a medium gap are beneficial with regard to control of the primary structure and energy harvested, but smaller gaps show similar results to the hard stiffness case. Acknowledgments T. Alvis and A. Abdelkefi acknowledge the funding support from Sandia National Laboratories which is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND202013824 A.

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References 1. Abdelmoula, H., Dai, H.L., Abdelkefi, A., Wang, L.: Control of base-excited dynamical systems through piezoelectric energy harvesting absorber. Smart Mater. Struct. 26(9), 095013 (2017) 2. Paidoussis, M.P., Li, G.X., Rand, R.H.: Chaotic motions of a constrained pipe conveying fluid: comparison between simulation, analysis, and experiment. J. Appl. Mech. 58(2), 559–565 (1991) 3. Tang, L., Yang, Y.: A nonlinear piezoelectric energy harvester with magnetic oscillator. Appl. Phys. Lett. 101, 094102 (2012) 4. Abdelkefi, A., Barsallo, N.: Nonlinear analysis and power improvement of broadband low-frequency piezomagnetoelastic energy harvesters. Nonlinear Dyn. 83, 41–56 (2016) 5. McNeil, I.: Nonlinear Reduced-Order Modeling and Vibration Mitigation of Dynamical Systems Through Energy Harvesting Absorbers. New Mexico State University (2019)

NIXO-Based Identification of the Dominant Terms in a Nonlinear Equation of Motion Michael Kwarta and Matthew S. Allen

Abstract While many algorithms have been proposed to identify nonlinear dynamic systems, nearly all methods require that the form of equation of motion is known a priori. Examples of very effective methods of this kind are NIFO, CRP, and NARMAX. Several works have sought to extend NARX or NARMAX to a black-box modeling technique. They have proven to be successful in finding accurate mathematical models for certain types of nonlinear systems, yet no method has proved universally successful. This work presents and evaluates a new black-box identification approach based on a new NIFO/CRP type algorithm called Nonlinear Identification through eXtended Outputs (NIXO). The proposed algorithm expresses the nonlinear part of equation of motion as a polynomial of high order and then removes the terms that are classified (with high probability) as irrelevant in the mechanical system’s response. This division into dominant and irrelevant nonlinear terms relies on the values of two novel indicators that are particular to NIXO. This technique is demonstrated on a numerical case study employing a curved beam. Then, the method will be used to estimate the NLEOM of flat and curved beams that were manufactured using a 3D printer. The experimental results will be validated against those obtained using phase resonance testing, which identifies a nonlinear normal mode (NNM) of the system using a vastly different approach. Keywords Nonlinear system identification · Nonlinear parameter estimation · Black-box methods · Nonlinear Normal Modes · NIXO methods

1 Definition of the -Indicators The objective of this publication is to propose a new technique for black-box nonlinear system identification. This work builds on that presented in [1] where the authors introduced a new frequency-domain system ID algorithms called D1 - and D2 -NIXO, for Nonlinear Identification through eXtended Outputs. The NIXO base formulas are proposed in two versions. The first one estimates the nonlinear coefficients as complex numbers, while the second enforces them to be found as real. The reader can refer to [1] for a detailed derivation of these expressions. While testing NIXO in several case studies, the authors noticed an interesting feature of the algorithm; namely, it was observed that if the tested structure (i) is excited with a swept (co)sine forcing signal and (ii) oscillates at low (but sufficiently high) amplitudes during the experiment, then the nonlinear terms that are not dominant in the system’s response tend to be complex-valued. Hence, if we analyze the structure assuming the most general form of the nonlinear EOM—see, e.g., Eq. (2)—then the NIXO algorithms will point out which nonlinear terms should be kept and which ones could be removed from the equation of motion. The complexity of the result can be quantified by two system identification metrics called ∗ and ∗∗ , which are defined in Eq. (1).   .   Re β cmplx − βreal   ∗ =     βreal

∗∗

 .   .   Re βcmplx  − Im βcmplx   = .   Re βcmplx 

/

∗ <  ∗∗ > 1 − 

(1)

where βreal and βcmplx are the parameters identified by the complex and real versions of the NIXO algorithms, respectively.

M. Kwarta () · M. S. Allen Engineering Physics Dept., University of Wisconsin–Madison, Madison, WI, USA e-mail: [email protected]; [email protected] © 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_13

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As shown in Eq. (1), the ∗ -indicator expresses the relative difference between the coefficient that was enforced to be found as real and the real part of the complex one. Hence, ∗ will be small when the two algorithms produce consistent results. In contrast, ∗∗ is defined as the relative difference between the real and imaginary parts of the complex solution and gives a measure of how large the imaginary part of the solution is. Hence, the nonlinear term is considered to be dominant when its ∗ value is low enough and ∗∗ is close to 1. Note that these two requirements must be satisfied simultaneously. The accuracy thresholds could be specified with a parameter , as presented in Eq. (1c). The value of  should be a small number, say  = 0.05. With the two -indicators defined, we are ready to illustrate the black-box capabilities of the NIXO algorithms. The next sections present a successful black-box identification performed on a numerical model of a curved beam.

2 Simulated Black-Box Identification of a Curved Beam The numerical test is performed on an ICE-ROM of a clamped–clamped curved beam subjected to a uniformly distributed swept cosine forcing signal. The beam has a length of 304.8 mm, a width of 12.7 mm, a thickness of 0.508 mm and a radius of curvature of 11.43 m. It is made of steel with a Young’s modulus of 207.4334 × 1011 GPa, a density of 7850 kg/m3 and a Poisson’s ratio of 0.29. The ICE-ROM consists of the first three symmetric modes, i.e., modes 1, 3, and 5. Their linear natural frequencies and damping ratios are {65.181, 158.636, 385.882} Hz and {0.035, 0.0262, 0.0174}, respectively. The nonlinear equation of motion of the system, including every possible nonlinear term, is presented in (2). Since (1) the nonlinear part consists of the quadratic and cubic parts, and (2) there are three modes present in the ROM, the number of nonlinear terms that can occur in the EOM is at most 16. In each case study run, we assume the most general form of the NLEOM, see Eq. (2); hence, NIXO can point out the terms dominant in the system’s response out of the most general set of 16 terms. This brief publication focuses on identification of the nonlinear mode 1, and thus we presented the nonlinear equation of motion of this mode only; the equations for the two remaining modes are analogous. Note that the subscripts of the nonlinear coefficients correspond to the product of polynomial terms they multiply; e.g., β111 multiplies term q13 , while β123 multiplies term q1 q2 q3 . 1 2 1 1 1 q¨1 + 2ζ1 ω1 q˙1 + ω12 q1 + α11 q + α12 q1 q2 + · · · + β111 q13 + β112 q 2 q2 + · · · = T1 f(t).     1  1  linear part quadratic stiffness part cubic stiffness part

(2)

3 Identification of Mode 1 The beam is excited with swept cosine signals of various magnitudes, such that it oscillates at different response levels in every test. These input/output signals are later provided to the NIXO algorithms, which use them to estimate the underlying linear and the nonlinear parts of the system. Since the first mode occurs at approximately 65 Hz, the authors decided to excite the system with 300-second-long (up and/or down) sweeps, with frequencies ranging from 1 to 115 Hz. The output signals obtained in these numerical tests are illustrated in Fig. 1 and can be grouped into two sets: down- and up-sweeps, which correspond to the force amplitudes of . F0 ∈ 1 × 10−7 , . . . , 8 × 10−4 and F0 ∈ {2.4, . . . , 4.75} × 10−3 newtons, respectively. Note that some of the responses shown are not symmetrical with respect to the equilibrium position. This result was expected, since the beam is curved and the snap-through effect causes this asymmetry. Moreover, this observation explains the importance of including the quadratic stiffness terms in the nonlinear equation of motion.

4 Sample Black-Box ID Outcomes from High-Amplitude Vibration Tests The results from a black-box system identification attempt are presented in this section. The input signals provided to the NIXO algorithms are the sweep cosines with magnitudes of 2.4 × 10−3 (up-sweep) and 5.0 × 10−4 newtons (down-sweep). The corresponding output signals are already shown in Fig. 1. Note that the signals used have very different amplitudes; the proposed algorithm seemed to work best when this is the case.

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10-3

2

4.75e-03 N 4.50e-03 N 4.25e-03 N 4.00e-03 N 3.50e-03 N 3.00e-03 N 2.80e-03 N 2.60e-03 N 2.40e-03 N 8.00e-04 N 7.50e-04 N 5.00e-04 N 2.50e-04 N 1.00e-04 N 1.00e-05 N 1.00e-06 N 1.00e-07 N NNM 5 Harm.

1.5

Disp. of Beam Center [m]

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0

20

40

60

80

100

120

140

Forcing Frequency [Hz] Fig. 1 The NNM curve is overlayed on the response at the beam’s center to swept cosine input forces of varying amplitudes

Linear Frequency Response Function

10-4 True FRF NIXO-D1 NIXO-D2

10-5

0

20

40

60

80

100

120

Frequency [Hz] Fig. 2 Underlying linear system estimated successfully by the NIXO algorithms

The outcomes from the case study are presented in Fig. 2 and Table 1. Figure 2 compares the true linear Frequency Response Function to those returned by NIXOs. The FRFs match perfectly, showing that the underlying linear system was successfully identified. Table 1 presents the four nonlinear terms (out of sixteen assumed beforehand) that were pointed out by NIXO algorithms as dominant in the system response (the coefficients that does not meet the accuracy criteria (1c) are not presented in the table). These terms correspond to the following parameters: α11 , α12 , β111 and β112 . The nonlinear equation of motion including these four terms only does not lose much accuracy when compared to the true NLEOM containing the full set of 16 terms (for more details, see the section below, where the results are validated). Since this is the smallest model that captures most of the structure’s dynamics, these four nonlinear terms must be dominant in the mechanical system. Naturally, if more nonlinear coefficients have to be identified, then the signals from tests where the structure oscillates at higher amplitudes should be provided to NIXO. Then, besides estimating accurately the four parameters itemized above, the NIXO algorithms will additionally point out the nonlinear terms that are next on the importance list.

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Table 1 Estimated values of the nonlinear coefficients obtained using NIXO methods. Parameters marked with green satisfy the accuracy criteria. Parameters marked with blue are close to satisfying these requirements. The parameters that do not meet the accuracy criteria specified in Eq. (1c) are not shown in this table D1 –NIXO

D2 –NIXO

Re{·cmplx }

Im{·cmplx }

·real

Δ∗ [%]

Δ∗∗ [%]

Re{·cmplx }

Im{·cmplx }

·real

Δ∗ [%]

Δ∗∗ [%]

α11

3.42E+09

-4.33E+06

3.42E+09

0.16

99.87

3.34E+09

-4.62E+07

3.44E+09

2.99

98.62

α12

3.67E+09

2.38E+08

3.94E+09

6.89

93.51

4.29E+09

2.36E+08

3.73E+09

14.96

94.50

β111

2.04E+13

1.68E+11

2.01E+13

1.51

99.18

1.94E+13

-4.71E+11

2.03E+13

4.71

97.57

β112

6.88E+13

5.48E+12

6.58E+13

4.65

92.04

6.47E+13

-2.59E+12

6.28E+13

3.00

95.99

2

10-3

Amplitude of Beam's Center [m]

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 True NNM EOM with 4 nonlin. terms EOM without 4 nonlin. terms

0.2 0 40

60

80

100

120

140

Frequency [Hz]

Fig. 3 A comparison of the true NNM curve to the one computed using four nonlinear terms pointed out by the NIXO algorithms

5 Validation of Results To prove that the polynomial terms pointed out by NIXO are dominant in the system, we compare the true NNM curve with the one computed for the EOM with the reduced set of the nonlinear terms pointed out by NIXO, see Eq. (3). The values of coefficients α11 , α12 , β111 , and β112 are presented in Table 1. 1 2 1 q1 + α12 q1 q2 + q¨1 + 2ζ1 ω1 q˙1 + ω12 q1 + α11 1 1 β111 q13 + β112 q12 q2 = T1 f(t).

(3)

The NNMs were computed using the Multi-Harmonic Balance algorithm using 5 harmonics and their comparison is shown in Fig. 3. The curve obtained using only the dominant terms matches well with the true NNM over a large range of the motion amplitudes (up to two times the beam thickness). In contrast, if the dominant terms are excluded, then the normal mode is a straight vertical line. This indicates that the structure behaves linearly when these dominant terms are removed from the EOM.

6 Conclusion and Future Work This work briefly discussed a capability of the NIXO algorithms, which could allow them to become an effective black-box nonlinear identification tool. The results presented here prove that the methods can successfully determine the smallest set of

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terms that should be kept in the nonlinear equation of motion. Since the dynamics of the structure described with the reduces and full sets of terms are comparable, the terms pointed out by NIXO can be considered as dominant in the mechanical system. In the future work, the method will be employed experimentally to identify the dominant nonlinear terms in an equation of motion of flat and curved beams. Then, the NNMs of the identified models will be computed and validated against those collected using well-established phase resonance tests.

Reference 1. Kwarta, M., Allen, M.S.: Extensions to NIFO and CRP to estimate frequency-independent nonlinear parameters. In: Proceedings of the 38th International Modal Analysis Conference (IMAC), Houston, TX (2020)

Nonlinear Dynamics and Characterization of Beam-Based Systems with Contact/Impact Boundaries M. Trujillo, M. Curtin, M. Ley, B. E. Saunders, G. Throneberry, and A. Abdelkefi

Abstract In this work, the contact/impact problem in a mechanical system with various locations of stoppers is investigated. Finite element simulations and analytical verification in the form of Euler-Bernoulli beam theory are utilized to initially understand the ideal linear system and identify an expected frequency range of interest. The complex nonlinear behavior of varying stopper’s location on the characteristics of a cantilever beam system with a tip mass is experimentally highlighted through the analysis of time histories and frequency response functions. Experimental analyses using free and forced vibration methods, such as impulse, harmonic, and random vibration tests, are performed to extract linear and nonlinear dynamic characteristics of the system. This work seeks to increase awareness of the effects of nonlinearities in the design of dynamical systems and expand the understanding of how these effects can be positively exploited. Results can aid in prevention of premature wear and extend the overall lifetime of systems by avoiding ranges of frequencies that exhibit chaotic responses. Keywords Contact/impact · Stopper stiffness · Finite element modeling · Freeplay nonlinearity · Experiments

1 Introduction Understanding the intricacy of contact/impact nonlinearities and the effects projected onto a dynamical system is a highly motivated subject and is widely researched as a result. Previous explorations of contact and impact on a cantilever beam include investigation of the effects of stopper placement on a beam, in particular, softening/hardening nonlinear characteristics. Chen et al. [1] observed a cantilever beam with two rigid stops set at different asymmetric gaps to create various impact scenarios experimentally. These experimental results were used to verify novel numerical solutions obtained through nonlinear system identification. To gain insight on the nonlinear behavior of a pylon apparatus, Ligeikis et al. [2] studied freeplay nonlinearities using multiple forms of environmental testing and noted that sensor cables introduce nonlinear damping. Zhou et al. [3] created a theoretical mathematical model to predict the nonlinearities of several stopper’s cases, modeling stoppers as cubic nonlinear spring terms which impact the cantilever beam with tip mass system. Their novel mathematical model was verified through experimental testing. Adding stoppers to a system results in hardening effects that increase the bandwidth of the resonant frequency range, which is ideal for energy harvesting purposes. In another paper, Zhou et al. [4] studied a similar energy harvesting system considering the effects of rigid and soft stoppers; soft stoppers were found to have an increased output voltage and larger frequency bandwidth making them most ideal for energy harvesting purposes. To address a wide range of contact/impact problems, the experimental system described in this work is designed with the ability to be reconstructed into infinite configurations. Initially, the contact/impact system is configured using a cantilever beam with an added tip mass. This configuration is investigated to first understand the basic system response before analyzing more complex configurations.

M. Trujillo, M. Curtin, and M. Ley contributed equally to this work. M. Trujillo () · M. Curtin · M. Ley · B. E. Saunders · G. Throneberry · A. Abdelkefi Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © 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_14

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(a)

(b)

Fig. 1 (a) Experimental configuration consisting of a cantilever beam with tip mass and stopper on the shaker and (b) equipment used during experiments

2 Experimental Setup and Testing This experiment investigates the effects of contact/impact through addition of two symmetric following stoppers with durometer hardness of 75A, located 4/5 of the beam’s length away from the fixture. The linear system in this study consists of tip mass weighing 6.3 g added to the top of a corner of a 6061-T6 Aluminum cantilever beam. The system is observed at resonance initially to determine the maximum displacement of the beam; this distance is denoted as xmax , and is measured to be 1 cm. Experimental tests are then conducted using symmetric stopper gaps at distances xmax , 0.5*xmax , and 0.125*xmax . The beam has measurements of 20.32 × 5.08 × 0.127 cm and the tip mass has measurements of 1.016 × 1.016 × 1.016 cm. Finite element and theoretical modeling are compared to the experimental results of the linear system to validate the models and to ensure the experimental setup did not significantly alter the expected results. Modal testing is performed in order to extract the system’s damping ratio and natural frequencies. Displayed in Fig. 1 is the experimental system along with laboratory equipment which imitates realistic environmental conditions and provides controlled, consistent modal testing.

3 Nonlinear Analysis of Beam with Hard Stoppers Modal analysis of a continuous system involves exciting a system, obtaining time history data, and manipulating the data to ® obtain desired results. By the use of Siemens Testlab FRF post-processing, a stepped sine experiment is performed and the time acceleration data is transformed from the time domain into the frequency domain, normalizing the output response by the input excitation. These results are shown in Figs. 2 and 3. The stepped sine excitation excited the system (1) beginning at 5 Hz and increasing to 60 Hz and (2) beginning at 60 Hz and decreasing to 5 Hz, both with a frequency step of 0.1 Hz. The frequency range of interest is identified through analytical and numerical methods and is found to be between 17 Hz and 25 Hz. As seen in Fig. 2, the linear fundamental frequency FRF amplitude decreases and shifts to a higher broadband range of frequencies when stoppers are introduced. When stoppers are placed at xmax , the system behaves similar to the linear case, with a slight decrease of the fundamental frequency range. As the symmetrical following stopper gaps decreases (in the case of 0.5*xmax , and 0.125*xmax ), for experiment (1), the fundamental frequency broadband range increases and the average FRF amplitude decreases; for experiment (2), as the symmetrical following stopper gaps decreases, the frequency broadband

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Fig. 2 Increasing/decreasing FRF response for varying contact/impact boundaries

Fig. 3 Time histories obtained using stepped sine excitation using 0.125 cm symmetrical following stopper case

range slightly increases and the FRF amplitudes decrease similar to the amplitudes for experiment (1). In Fig. 3, acceleration time history data of the system response for experiment (1) with 0.125 cm stopper gaps is found to display aperiodic behavior within the broadband resonant frequency range, which is the range during which the system experiences maximum impact. Deeper investigations are needed to determine the reason of the transition from periodic to aperiodic responses and vice versa.

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4 Conclusions In this work, effects of contact/impact on a cantilever beam system with tip mass were investigated through analysis of time histories and FRF. The dynamic response of the beam under stepped sine excitation was studied. Results indicated that the system’s fundamental resonant frequency is increasingly affected as stopper gaps increase. The contact nonlinearity dominantly affects the system response, resulting in a broadband hardening nonlinear response. Acceleration time history data of the system response with added stoppers was found to display aperiodic behaviors within the range of maximum contact. It is also found that the response of the system with added stoppers depends heavily upon the initial excitation conditions; responses varied depending on whether the exciting frequency was increasing or decreasing. A deeper investigation consisting of bifurcation analysis will be conducted to understand the source of nonlinearities within systems containing contact/impact. Acknowledgments The authors would like to acknowledge the financial support from Los Alamos National Laboratory. A special thanks to Dr. Thomas Burton for his fruitful discussions.

References 1. Chen, H., et al.: Nonlinear system identification of the dynamics of a vibro-impact beam. In: Topics in Nonlinear Dynamics - Proceedings of the 30th IMAC, A Conference on Structural Dynamics, pp. 287–299 (2012) 2. Ligeikis, C., et al.: Modeling and experimental validation of a pylon subassembly mockup with multiple nonlinearities. In: Nonlinear Structures & Systems, vol. 1, pp. 59–74. Springer, Cham (2019) 3. Zhou, K., et al.: Theoretical modeling and nonlinear analysis of piezoelectric energy harvesting with different stoppers. Int. J. Mech. Sci. 166, 105233 (2020) 4. Zhou, K., et al.: Impacts of stopper type and material on the broadband characteristics and performance of energy harvesters. AIP Adv. 9, 035228 (2019)

Experimental Modal Analysis of Geometrically Nonlinear Structures by Using Response-Controlled Stepped-Sine Testing Taylan Karaa˘gaçlı and H. Nevzat Özgüven

Abstract The everlasting competition in the industry to achieve higher performance in aircraft, satellites, and wind turbines encourages lightweight design more than ever, which eventually gives birth to more flexible engineering structures exhibiting large deformations in operational conditions. Accordingly, continuously distributed geometrical nonlinearity resulting from large deformations is currently an important design consideration. Being guided with this motivation, this paper investigates the performance of a recently developed promising nonlinear experimental modal analysis method on a clamped-clamped beam structure which exhibits geometrical nonlinearity continuously distributed throughout the entire structure. The method is based on response-controlled stepped-sine testing (RCT) where the displacement amplitude of the excitation point is kept constant during the frequency sweep. In this study, the nonlinear beam structure is instrumented with multiple accelerometers at several different locations along its length and is excited at a single point. Tests are conducted at energy levels where no internal resonance occurs, yet the beam structure exhibits strong stiffening nonlinearity which results in jump phenomenon in the case of classical constant-force sine testing. Nonlinear modal parameters are experimentally identified as functions of modal amplitude by applying standard linear modal identification methods to quasi-linear frequency response functions (FRFs) measured with RCT. Validation of the identified modal parameters is accomplished by comparing the constant-force FRFs synthesized using the identified modal parameters with the ones obtained from constant force testing and also with the ones extracted from the harmonic force surface (HFS). Keywords Distributed geometrical nonlinearity · Nonlinear experimental modal analysis · Response-controlled stepped-sine testing · Harmonic force surface · Unstable branch

1 Introduction Although the vibration theory of geometrically nonlinear beams, plates, and shells is well established in the existing literature [1], the experimental identification of geometrical nonlinearities is still in its infancy even for these simple structures. Stateof-the-art nonlinear system identification techniques [2, 3] which have been developed over the last two decades are in practice applicable to structures with localized nonlinearities and almost none of them are suitable for the identification of distributed nonlinearities occurring due to large amplitude oscillations. However, the increasing competition in the industry to achieve lightweight and eventually more flexible engineering structures seems to give some thrust to the experimental investigation of geometrical nonlinearities. For example, in a recent study [4], the method of multiple scales [5] is used to parametrically identify non-ideal boundary conditions of a geometrically nonlinear clamped-clamped beam structure. In another recent work [6], the identification of distributed nonlinearities is based on a reduced order model constructed by using the normal form theory [7] and the concept of nonlinear modes. In this method, the free parameters of the reduced order model corresponding to a particular nonlinear mode are determined by using the backbone curve of the same mode

T. Karaa˘gaçlı () Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey ˙ The Scientific and Technological Research Council of Turkey, Defense Industries Research and Development Institute, TÜBITAK-SAGE, Ankara, Turkey e-mail: [email protected] H. N. Özgüven Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey e-mail: [email protected] © 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_15

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measured with phase-locked-loop (PLL) control strategy [8]. In [6], the proposed method is successfully applied to a circular plate and a Chinese gong but fails in the case of a piezoelectric cantilever beam where the backbone curves are almost straight lines which cannot be represented with quadratic and cubic coefficients. Another interesting technique is proposed in [9], where a clamped-clamped beam is identified by applying an ad hoc version of the nonlinear subspace identification (NSI) algorithm [10, 11] to the experimental data measured under broadband Gaussian excitation. This technique takes into account the modal interactions; however, it requires the experimental extraction of linear normal modes which may not always be determined accurately by using low-excitation level tests especially in the case of friction type of nonlinearity which is widely encountered in many engineering systems comprising mechanical joints (e.g., bolted connections). Another method worth mentioning is proposed in [12], where a diesis-like structure with geometrical nonlinearities is identified by correlating the nonlinear responses measured during high-amplitude testing with the dynamics of the underlying linear system obtained by low-amplitude testing to extract the residuals which are eventually used to estimate the nonlinear stiffness and damping coefficients. This approach also relies on linear normal modes measured at low vibration levels. The review given above indicates that state-of-the-art techniques developed for the identification of geometrical nonlinearities are quite a few in number and they are generally restricted to simple structures, i.e., beam, plates, and shells, which exhibit a conservative type of nonlinearity. As an alternative to these techniques, the nonlinear experimental modal analysis method recently proposed by the authors of this paper [13] has the potential to be applied to a wider range of engineering structures. In this method, nonlinear modal parameters are determined as functions of modal amplitude by applying standard linear modal identification methods to quasi-linear FRFs measured with RCT. These identified modal parameters can then be used to synthesize near-resonant frequency responses of nonlinear systems corresponding to untested constant-amplitude harmonic forcing scenarios. In [13], the proposed method was successfully applied to a beam structure with a localized strong stiffening nonlinearity and on a real missile structure which exhibits considerable damping nonlinearity mostly due to several bolted joints. In this current paper, the applicability of the method to identify geometrical nonlinearities is demonstrated on a clamped-clamped metal strip which exhibits strong distributed nonlinearity due to large amplitude oscillations. The advantage of the method proposed in [13] lies in its simplicity and generality. The main restriction of the method is the assumption of no-internal resonance, which is also a common issue for almost all of the previously mentioned state-of-the-art techniques.

2 Nonlinear Experimental Modal Analysis with RCT and HFS Experimental modal analysis of nonlinear structures by using response-controlled stepped-sine testing (RCT) and harmonic force surface (HFS) concept is a novel technique recently proposed by the authors in [13], where detailed theoretical derivations are already given. However, for the sake of completeness, a summary of the important features of the proposed method is also given here. The main steps of the proposed experimental methodology are summarized in Fig. 1. The first step consists of measuring quasi-linear FRFs and harmonic excitation force spectra at several different constant displacement amplitude levels. Quasilinearization of constant-response FRFs stems from the following receptance expression derived by using the Nonlinearity Matrix concept [14] and the single nonlinear mode theory [15] as explained in [13]:

αj k (ω, qr ) =

Aj kr (qr ) ω2r (qr ) − ω2

+ i2ξ r (qr ) ωωr (qr )

(1)

where Aj kr (qr ), ωr (qr ), and ξ r (qr ) are the modal constant, natural frequency, and modal damping ratio corresponding to the rth nonlinear mode, respectively. All these modal parameters are functions of a single parameter; the modal amplitude qr . αj k indicates the near-resonant receptance corresponding to the displacement at point j for a given excitation at point k, and ω denotes the excitation frequency. Equation (1) clearly indicates that if the modal amplitude is kept constant throughout the stepped-sine testing, the measured FRFs turn out to be quasi-linear. In the case of a single point excitation, the constant modal amplitude can be achieved by keeping the displacement amplitude of the excitation point constant [13]. The displacement amplitude can be kept constant indirectly by using an accelerometer as the control sensor and by feeding the closed-loop controller with an appropriate acceleration profile. The good thing is that in the absence of internal resonance, this control strategy can easily be achieved via standard equipment (e.g., LMS Test Lab).

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Fig. 1 Flowchart of the experimental modal analysis with RCT and HFS

The second step of the proposed methodology consists of processing the measured experimental data. Quasi-linear FRFs corresponding to each constant displacement amplitude level can easily be processed by using standard linear modal analysis techniques to extract modal parameters. Repeating the same procedure at different amplitude levels ultimately gives the modal parameters as functions of the displacement (or equivalently modal) amplitude. In parallel, harmonic excitation spectra measured at different displacement amplitude levels are collected together to build up the so-called HFS. In the third step, the constant-force FRF corresponding to any force level, including untested force levels, can be independently obtained by two different approaches. It can either be synthesized by using the identified nonlinear modal parameters in the Newton-Raphson scheme with arc-length continuation algorithm or directly be extracted by cutting the HFS, which is purely experimental data, with the corresponding constant force plane. Ideally, synthesized and extracted constant-force FRFs must match perfectly, which constitutes the self-validation measure of the proposed experimental methodology. More importantly, this approach provides a modal model of a nonlinear structure, which can be used in further analysis of the structure, including structural coupling type of studies.

3 Experiment The nonlinear experimental modal analysis technique proposed in [13] is validated in this paper on a geometrically nonlinear structure shown in Fig. 2. The test rig consists of a very thin clamped-clamped beam structure which exhibits distributed nonlinearity due to large amplitude oscillations. The material of the beam is stainless steel and its dimensions are 502 × 19 × 0.8 mm3 . The beam is instrumented with seven miniature accelerometers (Dytran 3225 M23) along its length, numbered from left to right as shown in Fig. 2. Positions of the accelerometers are given in Table 1. The structure is excited at the position where the first accelerometer is located by using a push-rod attached to a B&K shaker as shown in Fig. 2. The harmonic excitation spectrum is measured via a Dytran 1022 V force transducer attached to the push-rod. In this experiment, all the data acquisition and closed-loop control tasks are accomplished by using the LMS SCADAS Mobile data acquisition system and the LMS Test Lab. software package. The focus of this study is the first elastic mode of the clamped-clamped beam which exhibits a hardening type of nonlinearity. The upper and lower frequency limits of the stepped-sine tests covering the first mode of the structure are determined based on FRF data measured with preliminary broadband random excitation. During the response-controlled and force-controlled stepped-sine tests, the frequency step is taken to be 0.125 Hz. Measurement results obtained from the test campaign are shown in Fig. 3. Constant-force FRFs measured by the classical force-control strategy in the sweep-up direction at three different excitation levels (0.125 N, 0.150 N, and 0.175 N) are

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Fig. 2 The geometrically nonlinear beam experimental setup Table 1 Positions of the accelerometers from the left end of the beam Accelerometer # Position (mm)

1 50

2 110

3 180

4 250

5 320

6 390

7 460

shown in Fig. 3(a). Since the experimental data of all seven accelerometers are very similar, only the data corresponding to accelerometers 1, 4, and 6 are illustrated in the figure. The shift of FRF peaks towards higher frequencies with increasing excitation level and the jump phenomenon observed at all excitation levels clearly indicate the strong hardening nonlinear behavior. On the other hand, constant-response FRFs measured at nine constant displacement amplitude levels of the driving point (i.e., accelerometer 1 in Fig. 2) by using the RCT strategy are given in Fig. 3(b). The first important observation that can be made from Fig. 3(b) is that FRFs turn out to be quasi-linear, which makes it possible to apply classical linear modal analysis techniques at each displacement amplitude level to determine the modal parameters as functions of displacement (or equivalently modal) amplitude. It is interesting to note that although the constant-response FRFs shown in Fig. 3(b) cover the excitation levels shown in Fig. 3(a), they do not exhibit unstable branches. This is related to the fact that points which would appear on the unstable branches of constant-force FRFs are visited at different times during response-controlled stepped-sine tests conducted at different displacement amplitude levels. Two final observations worth mentioning about Fig. 3(b) are the shift of resonance peaks towards the higher frequencies with increasing displacement level, which confirms hardening nonlinearity determined from constant-force FRFs, and the separation of half-power points (increasing bandwidth) with increasing displacement level that indicates increasing damping nonlinearity. Before getting into further analysis, it is necessary to check whether the effects of higher harmonics are negligible compared to the fundamental harmonic, which is the main assumption of the nonlinear experimental modal analysis with RCT and HFS. Accordingly, examples of fast-Fourier transform (FFT) of the time data recorded at the driving point during force-controlled and response-controlled stepped-sine tests of the geometrically nonlinear beam around the resonance region are illustrated in Fig. 4. It is clearly observed that, in both cases, there is more than an order of magnitude difference between the fundamental harmonic and higher harmonics, which validates the single harmonic assumption for this application. After being sure that no internal resonance occurs in the energy range of interest, constant-response FRFs measured by using the RCT control strategy are subjected to linear modal analysis with the PolyMAX module of the LMS Test Lab. As an example, Fig. 5 shows the stabilization diagram of the constant-response FRFs measured at 0.275 mm displacement amplitude level. As can be seen from the figure, the PolyMAX finds one and only one stable root within the frequency range of interest, which clearly indicates the quasi-linearization of the measured constant-response FRFs. Analytical FRFs synthesized from the modal parameters identified at 0.275 mm amplitude level are compared with the original experimental data obtained from the first 6 accelerometers in Fig. 6. It is clearly observed that the match between the synthesized and measured FRFs is perfect, which demonstrates the power of the RCT strategy.

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Fig. 3 Frequency response functions of the geometrically nonlinear clamped-clamped beam: (a) Constant-force FRFs measured by classical constant-force testing, (b) quasi-linear constant-response FRFs measured with RCT

Repeating the above procedure for all nine displacement amplitude levels of the driving point, ranging from 0.05 to 0.275 mm, gives the modal parameters of the geometrically nonlinear beam structure as functions of the modal amplitude as shown in Figs. 7 and 8. The parametric models shown in these figures are obtained by fitting third order polynomials to the experimentally extracted nonlinear modal parameters. The first important observation made from Fig. 7 is the considerable increase of the natural frequency with increasing modal amplitude, which points out the hardening stiffness nonlinearity of the structure. The second important observation is the order of magnitude increase of the modal damping ratio from about 0.5% up to 4% with increasing modal amplitude. So, it is very interesting to note that the geometrically nonlinear clampedclamped beam does not exhibit only conservative type nonlinearity but also considerable damping nonlinearity mostly due to jointed interfaces at the boundaries. In order to test the performance of the nonlinear modal model of the geometrically nonlinear beam structure obtained from RCT, constant-force frequency response curves synthesized from the parametric models of the identified nonlinear modal parameters by using Newton’s method and arc-length continuation algorithm are compared with the corresponding experimental data obtained from classical constant-force stepped-sine testing in Figs. 9, 10, and 11. The agreement between

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Fig. 4 FFTs of time data recorded at the driving point of the geometrically nonlinear beam: (a) Response-controlled test (0.150 mm amplitude level), (b) force-controlled test (0.175 N amplitude level)

Fig. 5 Stabilization diagram obtained by the PolyMAX to extract modal parameters based on the measured constant-response FRFs at 0.275 mm displacement amplitude level

the computational and experimental results is found to be quite satisfactory. The match between the synthesized and experimental frequency response curves is almost perfect at moderate and high force levels (0.150 N and 0.175 N). The discrepancy between some of the FRF pairs (mostly at the lowest force level 0.125 N) is mainly related to the variability of the nonlinear dynamics (which might be due to the variation in support conditions in time) between the RCT and constantforce test campaigns. This point is further addressed at the end of this section by illustrating the perfect match between the synthesized constant-force frequency response curves and the ones extracted from the HFS, which is a purely experimental surface obtained from RCT. In Figs. 9, 10, and 11, it is interesting to note that although the frequency response curves measured using the constantforce testing exhibit jump, as expected, the unstable branches of the frequency response curves synthesized from the identified

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Fig. 6 Comparison of the constant-response FRFs synthesized by using the modal parameters identified from the PolyMAX (green) at 0.275 mm displacement amplitude level with the ones (red) measured during RCT

Fig. 7 Variation of the modal parameters corresponding to the first mode of the geometrically nonlinear beam with modal response level: (a) natural frequency, (b) viscous modal damping ratio

nonlinear modal parameters are predicted by virtue of the arc-length continuation algorithm. This is a very typical drawback of the classical force control approach encountered in the case of strongly nonlinear systems. In the context of this paper, the strong nonlinearity concept is used to refer to nonlinear systems with overhanging unstable branches in the frequency response curves. This does not necessarily imply internal resonance as illustrated in Fig. 4 where the higher harmonics are negligible compared to the fundamental harmonic. The harmonic force surface (HFS) concept proposed in [13] provides a very convenient way of validating the unstable branches computed by the nonlinear modal model. HFS is basically constructed by combining harmonic excitation force spectra measured during RCT with linear interpolation as shown in Fig. 12. Cutting the HFS with a constant-force plane

Fig. 8 Variation of the modal constants corresponding to the first mode of the geometrically nonlinear beam with modal response level

Fig. 9 Comparison of the constant-force frequency response curves synthesized by using the identified nonlinear modal parameters of the geometrically nonlinear beam with the ones obtained from classical force-control testing at F = 0.125 N force level

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Fig. 10 Comparison of the constant-force frequency response curves synthesized by using the identified nonlinear modal parameters of the geometrically nonlinear beam with the ones obtained from classical force-control testing at F = 0.150 N force level

gives the frequency response curve corresponding to that force level with accurately identified turning points and unstable branches. Figure 13 shows the comparison of the driving-point frequency response curves extracted from the HFS with the ones synthesized from the identified nonlinear modal parameters and also with the ones measured by classical constant-force stepped-sine testing. It is important to emphasize that HFS is purely experimental data. Figure 13 indicates that the constantforce frequency response curves synthesized from the identified nonlinear modal parameters, including the turning points and unstable branches, perfectly agree with the ones extracted from the HFS. From Fig. 13 it can also be observed that there is a very good match between the constant-force frequency response curves directly measured from force-controlled testing and the corresponding curves extracted from the HFS at two force levels (0.150 N and 0.175 N). However, at the lowest force level (0.125 N) the match is not so perfect, even though both data are purely experimental. The effect of higher harmonics is not significant as shown in Fig. 4, and consequently it cannot be the main reason for the discrepancy between the two types of test results. The main reason for the discrepancy is seemingly poor repeatability, i.e., the variability of the nonlinear dynamics, most probably due to changes in support conditions, between RCT and constant-force test campaigns. During the tests, it was observed that repeating even the same (response-controlled or force-controlled) test does not give exactly the same frequency response curve. The accurate identification of turning points and unstable branches of frequency response curves, which is a considerable issue even for the new generation experimental continuation techniques, is the prominent feature of the HFS technique. This feature was successfully used in [16] to experimentally determine the backbone curves of strongly nonlinear structures.

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Fig. 11 Comparison of the constant-force frequency response curves synthesized by using the identified nonlinear modal parameters of the geometrically nonlinear beam with the ones obtained from classical force-control testing at F = 0.175 N force level

Fig. 12 (a) Harmonic excitation force spectra of the geometrically nonlinear beam measured during RCT, (b) HFS of the driving point constructed by combining harmonic excitation force spectra with linear interpolation

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Fig. 13 Comparison of the driving-point constant-force frequency response curves of the geometrically nonlinear beam extracted from HFS with the ones synthesized by using nonlinear modal parameters and also with the ones measured by classical constant force stepped-sine testing

4 Discussions and Conclusions Over the last two decades, although extensive literature has accumulated, the nonlinear system identification in structural dynamics still retains its toolbox philosophy. In other words, no general method applicable to a wide range of nonlinear structures is available yet. Most of the state-of-the-art techniques are in practice applicable to structures with localized nonlinearities and the identification of distributed nonlinearities due to large amplitude oscillations is still a challenging problem. The increasing interest of the industry towards the design of lightweight flexible engineering structures gave some thrust to the experimental investigation of geometrical nonlinearities in recent years. However, state-of-the-art techniques developed for the identification of geometrical nonlinearities are still quite a few in number and, more importantly, they are restricted to simple structures, i.e., beam, plates, and shells, which exhibit a conservative type of nonlinearity. This paper demonstrates the successful application of a recently proposed nonlinear experimental modal analysis method developed by the authors of this paper to a geometrically nonlinear clamped-clamped beam structure. The method is simple and general in the sense that it does not put any theoretical restriction on the type of nonlinearity or the engineering complexity of the test subject. The main limitation is the absence of internal resonance, which is a common restriction for most of the state-of-the-art techniques. The method is used, firstly, to identify the nonlinear modal parameters of the structure as functions of the modal amplitude by simply applying standard linear modal analysis techniques to the quasi-linear FRFs measured by response-controlled stepped-sine testing (RCT). Identification results indicate that the structure exhibits both considerable hardening stiffness nonlinearity and considerable damping nonlinearity. Identified modal parameters are then used to synthesize near-resonant constant-force frequency response curves which are compared with the ones measured by classical constant-force stepped-sine testing. Quite satisfactory results are obtained. Unstable branches of the frequency response curves are also computed by applying Newton’s method with arc-length continuation algorithm. The accuracy of the unstable branches and turning points is validated by comparing computational results with the ones experimentally extracted from the harmonic force surface (HFS), which is a novel concept and a part of the proposed experimental methodology RCT.

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˙ Acknowledgments The provision of TÜBITAK-SAGE for modal testing and analysis capabilities is gratefully acknowledged.

References 1. Nayfeh, A.H., Pai, P.F., Linear and Nonlinear Structural Mechanics, 2004 2. Kerschen, G., Worden, K., Vakakis, A.F., Golinval, J.C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20, 505–592 (2006) 3. Noël, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017) 4. Claeys, M., Sinou, J.J., Lambelin, J.P., Alcoverro, B.: Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 19, 4196–4212 (2014) 5. Malatkar, P., Nayfeh, A.H.: A parametric identification technique for single-degree-of-freedom weakly nonlinear systems with cubic nonlinearities. J. Vib. Control. 9, 317–336 (2003) 6. Denis, V., Jossic, M., Giraud-Audine, C., Chomette, B., Renault, A., Thomas, O.: Identification of nonlinear modes using phase-locked-loop experimental continuation and normal form. Mech. Syst. Signal Process. 106, 430–452 (2018) 7. Jezequel, L., Lamarque, C.: Analysis of non-linear dynamical systems by the normal form theory. J. Sound Vib. 149(3), 429–459 (1991) 8. Peter, S., Leine, R.I.: Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation. Mech. Syst. Signal Process. 96, 139–158 (2017) 9. Anastasio, D., Marchesiello, S., Kerschen, G., Noël, J.P.: Experimental identification of distributed nonlinearities in the modal domain. Mech. Syst. Signal Process. 458, 426–444 (2019) 10. Marchesiello, S., Garibaldi, L.: A time domain approach for identifying nonlinear vibrating structures by subspace methods. Mech. Syst. Signal Process. 22, 81–101 (2008) 11. Noël, J.P., Kerschen, G.: Frequency-domain subspace identification for nonlinear mechanical systems. Mech. Syst. Signal Process. 40, 701–717 (2013) 12. Wang, X., Hill, T.L., Neild, S.A.: Frequency response expansion strategy for nonlinear structures. Mech. Syst. Signal Process. 116, 505–529 (2019) 13. Karaa˘gaçlı, T., Özgüven, H.N.: Experimental modal analysis of nonlinear systems by using response-controlled stepped-sine testing. Mech. Syst. Signal Process. 146 (2021) 14. Tanrıkulu, Ö., Kuran, B., Özgüven, H.N., Imregün, M.: Forced harmonic response analysis of nonlinear structures using describing functions. AIAA J. 31(7), 1313–1320 (1993) 15. Szemplinska-Stupnicka, W.: The modified single mode method in the investigations of the resonant vibrations of nonlinear systems. J. Sound Vib. 63(4), 475–489 (1979) 16. Karaa˘gaçlı, T., Özgüven, H.N.: Experimental identification of backbone curves of strongly nonlinear systems by using response-controlled stepped-sine testing (RCT). Vibration. 3(3), 266–280 (2020)

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness T. Gowdridge, N. Dervilis, and K. Worden

Abstract Analytical solutions to nonlinear differential equations—where they exist at all—can often be very difficult to find. For example, Duffing’s equation for a system with cubic stiffness requires the use of elliptic functions in the exact solution. A system with general polynomial stiffness would be even more difficult to solve analytically, if such a solution was even to exist. Perturbation and series solutions are possible but become increasingly demanding as the order of solution increases. This chapter aims to revisit, present and discuss a geometric/algebraic method of determining system response which lends itself to automation. The method, originally due to Fliess and co-workers, makes use of the generating series and shuffle product, mathematical ideas founded in differential geometry and abstract algebra. A family of nonlinear differential equations with polynomial stiffness is considered; the process of manipulating a series expansion into the generating series follows and is shown to provide a recursive schematic, which is amenable to computer algebra. The inverse Laplace–Borel transform is then applied to derive a time-domain response. New solutions are presented for systems with general polynomial stiffness, for both deterministic and Gaussian white noise excitations. Keywords Generating series · Nonlinear system modelling · Polynomial stiffness · Gaussian white noise

1 Introduction One of the main problems in the discipline of nonlinear structural dynamics is that the equations of motion concerned very very rarely admit closed-form exact solutions, and this usually forces a dependence on approximate solutions [1]. The most common approximation methods are series solutions, often perturbation expansions in the coefficients of the nonlinear terms, which must be ‘small’ in order that the series converge and that low-order truncations are useful. An alternative series formulation is provided by the Volterra series [2, 3], which is not formally defined in perturbative terms, but as a functional series. One of the attractive features of the Volterra series is that the generalised coefficients of the series have physical interpretations; the coefficients are actually linear and nonlinear impulse responses, and their Fourier transforms can be regarded as Higher-dimensional Frequency Response Functions (HFRFs) [1]. Unfortunately, calculations with the Volterra series are very demanding in algebraic terms and rapidly become intractable (at least by hand), as the order of the expansion increases. One can bring computer algebra to bear on the problem, in order to automate calculations, but this does require a careful reformulation of the problem. Fortunately, such a reformulation exists in the form of the generating series; this is a geometric–algebraic nonlinear system analysis developed in the 1980s, by Michel Fliess and co-workers [4–9]. The groundbreaking idea involved the representation of the Volterra series—a sequence of high-dimensional integrals—as a purely algebraic expansion. The cost of the approach was that the expansion variables were non-commutative. The overriding benefit of the approach was that the operations in the series expansion could be implemented in computer algebra. Like the Laplace and Fourier transforms, the generating series offered a duality between time responses mediated by differential equations and a purely algebraic approach based on polynomials in the transform variables. Once solutions were established in the transform domain, they could be taken back into the time domain using the Laplace–Borel transform [10, 11]. Although the theory of the generating series is extremely elegant, it sadly did not gain a great deal of traction in the structural dynamics community. However, with interest turning back towards HFRFs and concrete calculations using the Volterra series [1], it is arguably time to revisit the approach.

T. Gowdridge () · N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected] © 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_16

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The modest objective of this chapter is to give an accelerated tutorial on the generating series and illustrate its use. A novel result presented here will be from the analysis of a family of nonlinear systems with general polynomial terms; in the notation of [4], the equation of motion is n  i=0

 di y(t) + εi y i (t) = u(t) i dt m

li

(1)

i=2

where the first term is a general linear differential operator. For a single-degree-of-freedom (SDOF) oscillator, n = 2, and the first and second derivatives represent damping and inertia terms, respectively. (The equation can always be scaled so that ln = 1.) The second term on the LHS is the polynomial stiffness term. With n = 2 and the order of nonlinearity m = 3, the equation becomes that of the asymmetric Duffing oscillator [1, 12], my¨ + cy˙ + ky + k2 y 2 + k3 y 3 = x(t)

(2)

in the standard notation where the overall (mass) scale has been restored; y is the displacement (response) and x is the force (excitation). Setting k2 = 0 gives the symmetric Duffing oscillator, my¨ + cy˙ + ky + k3 y 3 = x(t)

(3)

Previous papers using the generating series showed results for systems with a single nonlinear term; in this chapter, results will be given for the asymmetric quadratic–cubic equation (2). Responses for the system under both deterministic and random excitations will be considered. The layout of the chapter is as follows: Sect. 2 will provide the basic terminology and definitions in order that one can motivate the generating series. Section 3 sketches the basic of related perturbation theory and shows how to construct a diagrammatic representation of functional expansions. Section 4 outlines the main ideas of the generating series in the context of the asymmetric Duffing oscillator system, and Sect. 5 outlines how the series can be used to compute system responses. Section 6 discusses how one calculates responses to white noise excitations, and the chapter then concludes.

2 Background Theory and Definitions 2.1 Algebraic Structure As mentioned above, the generating series is an expansion in non-commuting variables; this clearly means that the variables themselves are not standard real or complex numbers and/or the product of the variables is not as standard. In fact, both of these complexities are present in the algebra of interest. The ‘basis’ of the algebra is provided by a set X = {x0 , x1 , . . . , xn } of symbols called the alphabet; the elements in X are called letters. The alphabet X generates a set X∗ , which is called the free monoid over X, whose elements are sequences of the form xjv . . . xj0 and are called words [7, 13]. X∗ is thus the set of all words formed from the letters of the alphabet X. The ‘polynomial’ terms of the generating series will be sets of words in some X∗ , to be elaborated later. Having defined the variables in the algebra of interest, it remains to specify how they are multiplied together; this is by using the shuffle product. The shuffle product is a binary operator that represents the sum of all the interleaved products formed from a riffle shuffle over the two operands. The product is best explained in terms of a number of basic identities, which are essential in working with the generating series: ∃

∃ ∃

1. 1 1 = 1 2. 1 w = w 1 = w 3. xj w xj w  = xj (w xj w  ) + xj (xj w   n n x . 4. x k x n−k = k

∃

∃

∃

w )

∃

Of these identities, the most useful is arguably the third that provides a recursive means of evaluating the product when the words concerned are specified in terms of their constituent letters. An example of the product showing the basic riffle nature is

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness

d) = abcd + acbd + acdb + cabd + cadb + cdab

∃

cd) + c(ab

∃

cd = a(b

∃

ab

137

In the shuffle product, the terms inside each argument do not change their order. For example, in the third case listed above, xj always appears before w and xj always before w  , where {xj , xj , . . .} ∈ X and {w, w  , . . .} ∈ X∗ [13]. This recursive definition for the shuffle product of two generating series is discussed and explored in greater detail in Sect. 4.1. The recursion for the shuffle product is completed when at least one argument is reduced to the identity element (above in cases 1 and 2). While this algebra may seem rather strange and counter-intuitive, it will be shown that it arises naturally in automating the transition between the nonlinear differential equation of interest and the corresponding Volterra series and thence to the generating series. The algebra arises because of the presence of iterated integrals.

2.2 Iterated Integrals There is nothing mysterious about iterated integrals; they are simply multiple integrals with the individual integrations carried out in a prescribed order. It is well known that changing the order of integration in a multiple integral will change the integrand, and this is what will cause non-commutativity here. As a matter of notation, iterated integrals will be denoted here like $ t dxn . . . dx0 (4) 0

and the convention will be that individual integrations will work inwards from the right in terms of variables; here, the first integral will be over x0 and the last will be over xn . In this example, all the limits on the integrals are the same and so the integrals are represented by a single symbol; in the general case, each integral would have its own symbol and limits and these would also be traversed working inwards, this time from the left. In the generating series algebra, single integrals will correspond to the letters and multiple integrals to the words of the corresponding free monoid. It is possible to see how the shuffle product might arise on such an algebra; consider a product of two iterated integrals, $ 0

 $

t

dξjv . . . dξj0



t

(5)

dξkμ . . . dξk0

0

After a certain amount of standard calculus, one finds that integration by parts results in $ 0

 $

t

dξjv (τ )

 $

τ 0

dξjv−1 . . . dξj0

τ 0

 $ t  $ dξkμ . . . dξk0 + dξkμ (τ ) 0

0

 $

τ

dξjv . . . dξj0

0



τ

dξkμ−1 . . . dξk0

(6)

and this is very suggestive of the relation for the shuffle product, w ]

∃

(x  w  )] + x  [(xw)

∃

∃

(xw) (x  w  ) = x[w

(7)

In fact, this is evidence of the correspondence with nonlinear differential equations; the nonlinear terms in such equations engender products of iterated integrals which map to shuffle products in the algebra of the generating series. In the more robust language of Fliess and Ree [5, 14]: Theorem 2.1 The product of two analytic causal functionals is again an analytic causal functional of the same kind, and the generating power series of which is the shuffle product of the two generating power series. Formally, this represents g2

∃

y1 × y2 ⇔ g1

(8)

This theorem can be extended to higher-order products of terms. In the differential equation, terms of the form y n , interpreted as y × · · · × y, n times, map directly to ‘powers’ in the generating series where the product is the shuffle. The simplest way to demonstrate this will be via the concrete examples to be pursued shortly.

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2.3 Volterra Series As mentioned in the introduction, another key ingredient in methodology here is the Volterra series; this is essentially a functional Taylor series for the response of a nonlinear input–output system [2, 3]. The series generalises the Duhamel integral for a linear system x(t) −→ y(t), given by Worden and Tomlinson [1], $ y(t) =

+∞ −∞

h(τ )x(t − τ )dτ

(9)

where h(τ ) represents the impulse response of the system. For a nonlinear system, one obtains instead an infinite series y(t) = y0 (t) + y1 (t) + y2 (t) + · · · + yi (t) + · · ·

(10)

where the general term is $ yi (t) =

$

+∞

−∞

...

+∞

−∞

hi (τ1 , . . . , τi )x(t − τ1 ) . . . x(t − τi )dτ1 . . . dτi

(11)

which is, of course, an iterated integral. The functions hi (τ1 , . . . , τi ) are the ‘coefficients’ in the functional expansion and have a direct physical interpretation as higher-dimensional impulse response functions [1]; they are termed Volterra kernels. Clearly, the problem of establishing a Volterra series is that of determining the kernels for a given nonlinear system. The multi-dimensional Fourier transforms also have a physical interpretation as higher-dimensional frequency response functions (HFRFs). One way to find the Volterra kernels is by determining the HFRFs directly and then using an inverse Fourier transform; although this sounds rather indirect, it is possible because the HFRFs can be found from the nonlinear equations of motion by harmonic probing as introduced in [15]. In fact, a variant of the strategy just described will be used in this chapter to find the Volterra kernels and system responses. Rather than computing objects in the Fourier domain and using an (inverse) Fourier transform to find time-domain objects, the idea will be to compute objects in the algebra of the generating series and transform back; the relevant transform is called the inverse Laplace–Borel transform. Like the Laplace transform, the forward map is easier to find than the inverse, so the usual means of inversion is to use a table of inverse Laplace–Borel transforms [6, 8, 16].

3 The Consolidated Expansion and a Diagrammatic Representation As mentioned earlier, the best way of illustrating the difficult concepts here is via concrete examples. With this in mind, this section will single out the asymmetric Duffing oscillator (Eq. (2)) as the system of interest. The system is actually of considerable practical interest as it represents the lowest-order approximation to a general SDOF system with both odd and even nonlinearities. Before proceeding, it is important to note that the form of equation can be simplified without losing generality. By scaling the independent (t) and dependent variables (x, y), the number of parameters can be reduced, so that the equation becomes y¨ + a y˙ + y + ε1 y 2 + ε2 y 3 = x(t)

(12)

and it is this form that is considered from now on, in order to simplify the notation and algebra. Note that there are parameters associated with each of the nonlinear terms, ε1 (quadratic) and ε2 (cubic); in a standard perturbation approach, these would be the expansion parameters, and this will also be the case here. It is useful at this point to look at the perturbation approach, even if it will not be pursued directly here; the formulation will show the complexity of the problem and also allow the construction of a useful and intuitive diagrammatic representation.

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139

3.1 Developing the Diagrammatic Representation The first stage in the analysis here is to pass to the frequency domain via the Fourier transform. The standard operations on Eq. (12) yield, via the convolution theorem, (1 + iαω − ω2 )Y (ω) + ε1 (Y ∗ Y )(ω) + ε2 (Y ∗ Y ∗ Y )(ω)

(13)

where $ (Y ∗ Y )(ω) =

∞ −∞

Y (ω − )Y ()d

(14)

and $ (Y ∗ Y ∗ Y )(ω) =

∞$ ∞

−∞ −∞

Y (ω − )Y ( −  )Y ( )dd

(15)

These latter expressions are cumbersome and unsymmetrical and can be rewritten as symmetrical integrals, $ (Y ∗ Y )(ω) =

∞$ ∞

−∞ −∞

Y (ω1 )Y (ω2 )δ(ω − ω1 − ω2 )dω1 dω2

(16)

and $ (Y ∗ Y ∗ Y )(ω) =

∞$ ∞$ ∞ −∞ −∞ −∞

Y (ω1 )Y (ω2 )Y (ω3 )δ(ω − ω1 − ω2 − ω3 )dω1 dω2 dω3

(17)

With these modifications and a little more rearrangement, Eq. (13) becomes $ Y (ω) = H (ω)X(ω) − ε1 H (ω)

∞ −∞

$ Y (ω1 )Y (ω2 )dμ2 − ε2 H (ω)

∞

−∞

Y (ω1 )Y (ω2 )Y (ω3 )dμ3

(18)

where the integral signs have been coalesced and the measures are dμ2 = δ(ω − ω1 − ω2 )dω1 dω2 and dμ3 = δ(ω − ω1 − ω2 − ω3 )dω1 dω2 dω3 . Furthermore, H (ω) = 1/(1 + iαω − ω2 ), which is the standard FRF of the underlying linear system (ε1 = ε2 = 0). Note that the equation is recursive: i.e., Y (ω) is expressed as a nonlinear function of itself. In the case that ε1 and ε2 were small perturbation parameters, the equation could be used to compute Y (ω) iteratively, starting from the response of the linear system. With this observation in mind, it makes sense to compare the result with the actual two-parameter perturbation expansion. In the time domain, one has y(t) = y00 (t) + ε1 y10 (t) + ε2 y01 (t) + ε12 y20 (t) + ε11 ε21 y11 (t) + · · · =

j ∞  

j −i

ε1i ε2 yi,j −i (t)

(19)

j =0 i=0

and the Fourier transform is Y (ω) = Y00 (ω) + ε1 Y10 (ω) + ε2 Y01 (ω) + ε12 Y20 (ω) + ε11 ε21 Y11 (ω) + · · · =

j ∞   j =0 i=0

with the obvious notation.

j −i

ε1i ε2 Yi,j −i (ω)

(20)

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Now equating Eqs. (20) and (18) at each level of perturbation, one obtains an infinite sequence of equations; the first nine, corresponding to a truncation at third nonlinear order, are O(ε10 ε20 ) : O(ε11 ε20 ) : O(ε10 ε21 ) : O(ε12 ε20 ) : O(ε11 ε21 ) : O(ε10 ε22 ) : O(ε13 ε20 ) : O(ε12 ε21 ) : O(ε11 ε22 ) : O(ε10 ε23 ) :

= H (ω)X(ω)% = −ε1 H (ω) % Y00 (ω1 )Y00 (ω2 )dμ2 = −ε2 H (ω) % Y00 (ω1 )Y00 (ω2 )Y00 (ω3 )dμ3 = −ε1 H (ω) % 2Y00 (ω1 )Y10 (ω2 )dμ2 % = −ε1 H (ω) % 2Y00 (ω1 )Y01 (ω2 )dμ2 − ε2 H (ω) 3Y00 (ω1 )Y00 (ω2 )Y10 (ω3 )dμ3 = −ε2 H (ω) % 3Y00 (ω1 )Y00 (ω2 )Y01 (ω3 )dμ3 = −ε1 H (ω) % [2Y00 (ω1 )Y20 (ω2 ) + Y10 (ω1 )Y10 (ω2 )]dμ2 = −ε1 H (ω) % [2Y00 (ω1 )Y11 (ω2 ) + 2Y10 (ω1 )Y01 (ω2 )]dμ2 −ε2 H (ω) % [3Y00 (ω1 )Y00 (ω2 )Y20 (ω3 ) + 3Y00 (ω1 )Y10 (ω2 )Y10 (ω3 )]dμ3 Y12 (ω) = −ε1 H (ω) % [2Y00 (ω1 )Y02 (ω2 ) + Y01 (ω1 )Y01 (ω2 )]dμ2 −ε2 H (ω) % [6Y00 (ω1 )Y10 (ω2 )Y01 (ω3 ) + 3Y00 (ω1 )Y00 (ω2 )Y11 (ω3 )]dμ3 Y03 (ω) = −ε2 H (ω) [3Y00 (ω1 )Y00 (ω2 )Y02 (ω3 ) + 3Y00 (ω1 )Y01 (ω2 )Y01 (ω3 )]dμ3 Y00 (ω) Y10 (ω) Y01 (ω) Y20 (ω) Y11 (ω) Y02 (ω) Y30 (ω) Y21 (ω)

where the range of each integral is (−∞, ∞). j The expansion has been carried so far in order to show the contribution of multiple cross terms ε1i ε2 . Setting ε1 = 0 j (respectively, ε2 = 0) or collecting only the terms corresponding to ε1i ε20 (respectively, ε10 ε2 ) generates the expansion for a lone quadratic (respectively, cubic) nonlinearity. As in all perturbation expansions, each term is computable from previously evaluated terms, although the effort quickly becomes large. In fact, one can see that Yi,j = ε1 Yi−1,j + ε2 Yi,j −1 . Although the algebraic representation provided here—referred to as the consolidated expansion in [17]—is cumbersome, the authors of that reference proposed a diagrammatic representation analogous to the Feynman representation of perturbation expansions in quantum field theory [18]. The representation was also adopted in [8]; however, it appears to have only been applied in the case of a single nonlinearity in previous work. The correspondence between the algebraic expansion and the diagrammatic form is encoded in a set of rules, which allow each term to be represented by a tree diagram. In the case of a quadratic–cubic system, the expansion is depicted in Fig. 1; the conventions are corresponds to multiplication by H (ω). corresponds to multiplication by Y0 (ωi ), where the subscript i can equal any positive integer. corresponds to multiplication by ε1 . corresponds to multiplication by ε2 The rules associated with individual terms Yij are Rule 1: Rule 2: Rule 3: Rule 4: Rule 5:

The tree(s) will have i vertices with 3 incident branches and j vertices with 4 incident branches. There will be i + j nodes and i + j solid lines in the tree. A tree will have i + 2j + 1 dashed branches. Any two distinct vertices are connected by a single path. Frequency is conserved at a vertex. The sums of the frequencies either side of a vertex are equal.

The diagrammatic representation (Fig. 1) does not add anything to the algebraic expansion; however, exactly as it does in quantum field theory, it helps considerably in doing calculations by hand. Up to this point, the analysis has not really strayed beyond classical theory—although Fliess did draw upon the diagrammatic representation in [8]. The generating series will be properly introduced in the next section via its calculation for the asymmetric Duffing oscillator.

4 Generating Series for the Asymmetric Duffing Oscillator As discussed above, the analysis will concentrate on the system specified in Eq. (1) at first but then specialise to the asymmetric (quadratic–cubic) Duffing oscillator. The first stage in the analysis is to manipulate the equation into integral

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness

141

+

Y11 = 2

Y10 = Y01 =

3

Y02 = 3

Y20 = 2

+

Y30 = 4

Y21 = 4

+

6

+

+6

+

3

+

Y12 = 6

+6

+

+

2

6

9

Y03 = 9

+

3

Fig. 1 Diagrammatic representation of first nine terms of the consolidated expansion for the quadratic–cubic oscillator

form; one integrates n times in order to remove all the differential operators, with result $

t

y(t) + ln−1 +

m 

$

0

τ2

dτn . . . 0

$

t

$

$

τn

dτn 0

$

t

εi

i=2

$ y(τ1 ) + . . . + l0

0

y(τ1 )dτ1 0

$

t

y(τ1 )dτ1 =

τ2

dτn . . .

u(τ1 )dτ1

0

0

τ2

dτn−1 . . .

(21)

0

if one assumes that all relevant initial conditions are zero. The equation of motion now consists of iterated integrals and can be converted into the generating series domain. The symbol g will denote the with y(t) here; % t generating series associated %t the two basis letters associated with the free monoid will be denoted: x0 = 0 y(τ )dτ and x1 = 0 x(τ ). The rules for the transformation follow from Fliess’ fundamental formula and the Peano–Baker formula [19], as detailed in [4]. The main formal rules for the transformation are as follows: 1. The transform acts on linear combinations linearly. 2. Just as differentiation in the Fourier transform is represented by pre-multiplication by iω, integration in the g-domain is represented by pre-multiplication by the letter x0 . 3. nth powers of y will transform to n-fold shuffle produces of g. Applying these rules in Eq. (21) yields ... 

g ) = x0n−1 x1

∃

∃

g + · · · + εm g 

∃

g + ln−1 x0 g + · · · + l1 x0n−1 g + x0n (ε2 g

mtimes

(22)

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or the more compact form, ⎛

⎞ n−j lj x0 ⎠ g

+ x0n

j =0

m 

i

∃

⎝1 +

n−1 

εi g

= x0n−1 x1

(23)

i=2

∃

∃

∃

where g i = g . . . g, i times. Equation (23) can be further simplified by factorising the polynomial in x0 that multiplies g as follows: 1+

n−1 

n−j

lj x0

j =0

=

p 4

(1 − ai x0 )αi , α1 + α2 + · · · + αp = n

(24)

i=0

One can now formally write Eq. (23) as m

i

∃

xn g = g0 + 5p0

i=2 εi g

i=0 (1 − ai x0 )

(25)

αi

where g0 = 5p

x0n−1 x1

i=0 (1 − ai x0 )

(26)

αi

is seen to be the generating series representation of the underlying linear system. The solution to Eq. (22) can now be constructed recursively; starting with g0 , one computes j =2 εi



5p

ν1 +···+νj =i

i=0 (1 − ai x0

gν1

...

gνj

∃

m

∃

gi+1 = −

x0n ×

)αi

(27)

and the representation of the full nonlinear system response is then g = g0 + g 1 + · · · + g i

(28)

At this point, it is important to recall that the algebra of the generating series is not commutative, so objects like Eq. (26) are actually ambiguous. A careful analysis reveals that g0 actually takes the form R1 (x0 )xi1 R2 (x0 )xi2 . . . xip Rp (x0 ), where Rj (x0 ) represents a rational fraction and {i1 , i2 , . . . , ip } ∈ {0, 1} [7]. The quotient in the recursive scheme of Eq. (27) is of a similar form; meaning that all the successive gi terms will also be of this form. In fact, the general form of the terms of interest can be written as 1 1 1 x1 x1 . . . x1 (1 − a0 x0 )α0 (1 − a1 x0 )α1 (1 − ap x0 )αp

(29)

Expressions of this type can be simplified by using the identity 1 1 ax0 = + α α−1 (1 − ax0 ) (1 − ax0 ) (1 − ax0 )(1 − ax0 )α−1

(30)

and decomposed as partial fractions. In this way, by repeated application of the identity, all the exponents in the denominators can ultimately be reduced to unity, and the general object of interest becomes 1 1 1 xi xi . . . xip (1 − a0 x0 ) 1 (1 − a1 x0 ) 2 (1 − ap x0 )

(31)

The general problem of computing shuffle products is thus reduced to that of computing shuffle produces of terms like that above. Summation of such terms is not an issue as the shuffle product is distributive over addition [13].

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness

143

4.1 Shuffle Product of Series Terms Shuffle products of terms of the specific form in (31) can now be considered in more detail. Suppose the two terms of interest are p

g1 =

xip 1 1 1 p−1 xi1 xi2 . . . xip = g1 1 − b0 x0 1 − b1 x0 1 − bp x0 1 − bp x0

(32)

xjq 1 1 1 q−1 xj1 xj2 . . . xjq = g2 1 − d0 x0 1 − d1 x0 1 − dq x0 1 − dq x0

(33)

and q

g2 =

∃

∃

∃

where {p, q} ∈ N and {i1 , . . . , ip , j1 , . . . , jq } ∈ {0, 1}. By assuming that the generating series is represented as a series of products, one can readily compute their shuffle products; results can be defined recursively as shuffle products of lower-order terms. This recursion ends when any of the following terms occur: x 1, 1 x, or 1 1. The process halts as the shuffle product of a term with the identity element simply returns the term itself. bx0 1 can be rearranged into 1 + 1−bx ; this operation can be applied to the highest-order Looking in more detail, the term 1−bx 0 0 fraction to give  " !  " ! bp x0 dq x0 p q p−1 q−1 g1 g2 = g1 xip 1 + g2 xjq 1 + (34) 1 − bp x0 1 − dq x0 ∃

∃

∃

∃

∃

Recalling that the shuffle product is distributive over addition [13], the above product can be expanded to give [7] " " ! ! 1 1 p q p−1 q−1 p−1 q−1 g1 g2 = g1 xip xjq + g1 xi g2 g2 xjq 1 − bp x0 1 − dq x0 p " " ! ! 1 1 1 1 p−1 q−1 p−1 q−1 bp x0 + g1 xiP dq x0 + g1 xip g2 xjq g2 xjq 1 − bp x0 1 − dq x0 1 − bp x0 1 − dq x0 ∃

∃

(35)

(remembering that the order of the terms is important). Careful regrouping of terms gives a compact recursion for the shuffle product of two generating series of the form [7]

=

" q−1

g2

p−1

)xjq + (g1

q

g2 )xip

∃

1 1−dx0

! 1 p (g1 1 − (bp + dq )x0

∃

1 1−bx0

∃

Noting that

q

g2 =

∃

p

g1

(36)

1 1−(b+d)x0 .

∃

∃

∃

∃

Fortunately, the shuffle product is associative [13], i.e., (gj1 gj2 ) gj3 = gj1 (gj2 gj3 ), and this means that Eq. (36) can be extended straightforwardly (if tediously) to higher-order products, because the order in which the pairwise products are taken does not matter. One of the strengths of the generating series approach is that the formalism above is amenable to computer implementation; the calculations presented in this chapter are the result of a Python implementation of the necessary algebra.

4.2 The Asymmetric Duffing Oscillator When Eq. (22) is applied to the asymmetric Duffing equation in the canonical form in (12), the result is g] = x0 x1

∃

g

∃

g] + ε2 x02 [g

∃

g + x0 g + x02 g + ε1 x02 [g

By collecting like terms and factorising the quadratic expression in x0 , this equation can be rearranged into the form

(37)

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T. Gowdridge et al.

x02 g (1 − a1 x0 )(1 − a2 x0 )

g

g

∃

g − ε2

∃

x02 x0 x1 − ε1 g (1 − a1 x0 )(1 − a2 x0 ) (1 − a1 x0 )(1 − a2 x0 )

∃

g=

(38)

where (1 − a1 x0 )(1 − a2 x0 ) = 1 + ax0 + x02 . The generating series solution can now be obtained for the oscillator, and an iterative procedure can be followed where 



[gj1

i1 +i2 =i

g j2

∃

gi2 ] + ε2

g j3 ]

∃

  x0 x0 ε1 [gi1 1 − a1 x0 1 − a2 x0

∃

gi+1 = −

(39)

j1 +j2 +j3 =i

The iteration begins with g1 from the underlying linear system with ε1 = ε2 = 0, g1 =

1 1 x0 x1 1 − a1 x0 1 − a2 x0

(40)

The generating series g is then the sum, g = g0 + g 1 + g 2 + · · · + g n + · · ·

(41)

allowing for the possibility for a constant offset g0 in the response. For the calculation here, only the first two iterations will be displayed. The additional nonlinear term here causes the iterations to have exponentially more terms compared to the single nonlinearities considered in other works [5, 7]. For a more compact notation, the terms in the generating series, as shown in Eq. (32), for example, can be expressed in the form of a (2 × p) array; in the calculation here, for example, one has !

x0 x1 g1 = 1 −a1 −a2

"

where the notation shows a word in the numerator in the first row and the corresponding coefficient in the denominator in the second row. Each column represents a term in the rational fraction, and the overall coefficient/multiplier is found outside of the array. The first iteration of the algorithm gives g0

g0 ]

∃

g0 ] + ε2 [g0

∃

x0 x0 ε1 [g0 1 − a1 x0 1 − a2 x0

∃

g1 = −

.

(noting that the order is unimportant in expressions considering a single letter). Now, expanding the shuffle products using Eq. (36) yields !

" ! " x1 x0 x1 x0 x1 x1 x0 x0 x0 x0 x0 x0 − 4ε1 g1 = −2ε1 −a1 −a2 −2a1 −a1 − a2 −a1 −a2 −a1 −a2 −2a1 −a1 − a2 −2a2 −a2 ! " x1 x0 x1 x0 x1 x0 x0 x0 −6ε2 −a1 −a2 −3a1 −2a1 − a2 −2a1 −a1 − a2 −a1 −a2 ! " x1 x0 x0 x1 x1 x0 x0 x0 −12ε2 −a1 −a2 −3a1 −2a1 − a2 −2a1 −a1 − a2 −2a2 −a2 ! " x0 x1 x0 x1 x1 x0 x0 x0 −24ε2 −a1 −a2 −3a1 −2a1 − a2 −a1 − 2a2 −a1 − a2 −2a2 −a2 ! " x0 x1 x1 x0 x1 x0 x0 x0 −12ε2 −a1 −a2 −3a1 −2a1 − a2 −a1 − 2a2 −a1 − a2 −a1 −a2 ! " x0 x0 x1 x1 x1 x0 x0 x0 −36ε2 −a1 −a2 −3a1 −2a1 − a2 −a1 − 2a2 −3a2 −2a2 −a2

(42)

On the Application of the Generating Series for Nonlinear Systems with Polynomial Stiffness

145

and for g2 , g1 ]

∃

g0

∃

g0 + g0

∃

g1

∃

g0 + g0

g0

∃

g1 ] + ε2 [g1

∃

g0 + g0

∃

x0 x0 ε1 [g1 1 − a1 x0 1 − a2 x0

∃

g2 = −

.

(43)

!

" x0 x0 x1 x0 x1 x0 x1 x0 x0 x0 g2 = −a1 −a2 −2a1 −a1 − a2 −3a1 −2a1 − a2 −2a1 −a1 − a2 −a1 −a2 ! " x0 x0 x1 x0 x0 x1 x1 x0 x0 x0 2 +8ε1 −a1 −a2 −2a1 −a1 − a2 −3a1 −2a1 − a2 −2a1 −a1 − a2 −2a2 −a2 ! " x0 x0 x0 x1 x0 x1 x1 x0 x0 x0 +8ε12 −a1 −a2 −2a1 −a1 − a2 −3a1 −2a1 − a2 −a1 − 2a2 −a1 − a2 −2a2 −a2 ! " x0 x0 x0 x1 x0 x1 x1 x0 x0 x0 +8ε12 −a1 −a2 −2a1 −a1 − a2 −2a2 −2a1 − a2 −a1 − 2a2 −a1 − a2 −2a2 −a2 ! " x0 x0 x0 x1 x1 x0 x1 x0 x0 x0 2 +8ε1 −a1 −a2 −2a1 −a1 − a2 −3a1 −2a1 − a2 −a1 − 2a2 −a1 − a2 −a1 −a2 ! " x0 x0 x0 x1 x1 x0 x1 x0 x0 x0 2 +8ε1 −a1 −a2 −2a1 −a1 − a2 −2a2 −2a1 − a2 −a1 − 2a2 −a1 − a2 −a1 −a2 ! " x0 x0 x1 x0 x1 x0 x1 x0 x0 x0 +4ε12 −a1 −a2 −2a1 −a1 − a2 −2a2 −2a1 − a2 −2a1 −a1 − a2 −a1 −a2 ! " x0 x1 x0 x0 x1 x0 x1 x0 x0 x0 +4ε12 −a1 −a2 −2a1 −a1 − a2 −2a2 −a2 −2a1 −a1 − a2 −a1 −a2 ! " x1 x0 x0 x0 x1 x0 x1 x0 x0 x0 2 +2ε1 −a1 −a2 −2a1 −a1 − a2 −a1 −a2 −2a1 −a1 − a2 −a1 −a2 ! " x0 x0 x0 x1 x1 x0 x1 x0 x0 x0 2 +4ε1 −a1 −a2 −2a1 −a1 − a2 −3a1 −2a1 − a2 −a1 − 2a2 −a1 − a2 −a1 −a2 6ε12

Only the first 10 terms have been shown for g2 , as there are 360 terms in the full expansion.

5 Determining System Response The analysis up to now has allowed the input–output relationship for the system to be expressed in terms of the generating series algebra. In order to compute an actual response, one needs to substitute for the relevant excitation x(t), as encoded in the letter x1 in the free monoid, and then transform back to the time domain. The analysis has provided terms of a specific form, words in the letters x0 and x1 and rational fractions of them. If the transformation back is made with general x1 , the result will be a Volterra expansion, and one will be able to read off the Volterra kernels. Each term in the series—of the form given by Eq. (31)—corresponds to a specific iterated integral. Each appearance of x0 represents an integration, so a term with g occurrences of x0 represents a q-fold iterated integral. Fliess and co-workers computed the general inverse transform of a q-fold product in the generating series, and it takes the form [7] $ t$

$

τq

... 0

0

0

τ2

α

α

q−1 faα11 (t − τq ) . . . faq−1 (τ2 − τ1 )faqq (τ1 )x(τq ) . . . x(τ1 )dτq . . . τ1

(44)

where faα =

! α−1  α − 1 a j t j " eat j j! j =0

(45)

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This relationship shows how the generating series and Volterra series are so strongly linked. The correspondence between the terms in Eq. (41) and the Volterra terms can be shown to be g0 ⇔ y0 = h0 $ +∞ g1 ⇔ y1 = h1 (τ1 )x(τ1 )dτ1 $ g2 ⇔ y2 =

(46) (47)

−∞

+∞ $ +∞

−∞

−∞

(48)

h2 (τ2 , τ1 )x(τ2 )x(τ1 )dτ2 dτ1

By computing the inverse Laplace–Borel transform of the generating series derived for the oscillators in Sect. 4.2, the Volterra kernels are determined. For specific excitations x(t), the system response can be computed. The relevant inverse transforms are tabulated below. Rather than giving the asymmetric Duffing system response for a deterministic excitation, a little more work will allow characterisation of the response to a random excitation.

6 Response to Gaussian White Noise Clearly, the machinery provided up to now cannot provide a time-series response to a truly random excitation; however, it can be adapted to give output statistics, of the response and this is simplest when the excitation is a Gaussian white noise process. Such a process is specified by a demand that it has zero mean and an auto-correlation function of the form [20] E[x(t)x(τ )] = x(t)x(τ ) = σ 2 δ(t − τ )

(49)

∃

where σ 2 denotes the ‘noise-power’ (one must take care in interpreting this as a variance) and angle brackets denote expectations. The most basic statistic one can estimate is the mean of the response or its expectation y(t). In [15], the authors developed an appropriate form of the Volterra series for random excitation, based on stochastic calculus [21]. This formulation was adapted by Fliess [5], in order to compute statistics from the generating series. The expectations are interpreted as ensemble averages so that one can take the expectations in the transform domain and then map back. In this way, g corresponds to y(t), and the higher-order statistics y(t)n  are obtained by mapping back the shuffle products g n . As usual now, it is sufficient to consider only the calculation for terms of the form shown in Eq. (31); the basic rules are [22] 6 7 ⎧ x0 1 1 ⎪ ⎪ , x . . . xin ⎪ ⎪ 1 − b0 x0 1 − b1 x0 i2 1 − bn x0 ⎪ 6 7 ⎪ ⎨ 6 7 1 1 1 1 x0 1 = σ2 xi1 . . . xin ⎪ , xi3 . . . xin 1 − b0 x0 1 − b1 x0 1 − bn x0 ⎪ ⎪ 1 − bn x0 ⎪ ⎪ 2 1 − b0 x0 1 − b2 x0 ⎪ ⎩ 0,

if i1 = 0 if i1 = i2 = 1

(50)

Otherwise

Note how restrictive this recursion is, many terms will automatically be zero; this is related to the fact that expectations of products of an odd number of Gaussian random variables will average to zero. Once the generating series has been decomposed into the standard terms (and in this case will only contain the letter x0 ), the usual rules allow inversion using a table of Laplace–Borel transforms. The auto-correlation of the response is a little more complicated, which has the form Syy = y(t1 )y(t2 )] and the product of ys will produce a shuffle product in the domain of the generating series.

(51)

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147

For the asymmetric Duffing oscillator under investigation here, performing the ensemble average for all the terms in the generating series, the following result is obtained: ! " σ2 x0 x0 x0 x0 x0 g = −4ε1 −a1 −a2 −2a1 −a1 − a2 −2a2 2  2 2 ! σ x0 x0 x0 x0 x0 x0 x0 +48ε1 ε2 −a1 −a2 −2a1 −a1 − a2 −4a1 −3a1 − a2 −2a1 − 2a2 2  2 2 ! σ x0 x0 x0 x0 x0 x0 x0 +48ε1 ε2 −a1 −a2 −2a1 −a1 − a2 −2a1 −3a1 − a2 −2a1 − 2a2 2 

x0 x0 x0 2a1 −a1 − a2 −2a2 x0 x0 x0 2a1 −a1 − a2 −2a2

" "

+···

(52)

As discussed above, because of the third condition in the recursion (50), many terms are zero and do not contribute. For example, the second significant term in the expansion above is actually the 39th term from the total of 360 in g2 . To move the computation forward, it is necessary to carry out the partial fraction calculations implicit in the terms in Eq. (52). At this point, it is useful to specify numerical values for a1 and a2 , as the partial fraction calculations can be cumbersome when carried out algebraically. In the case of the asymmetric Duffing equation, the system parameters are m, c, k1 , k2 and k3 . The relevant parameters in the generating series can then be calculated via the scaled version of the −1 −1 −1 Duffing equation in Eq. (12); starting with values √ here of 1m = 1√kg, c = 15 Nsm , k1 = 25 Nm , k2 = 625 Nm and 1 −1 k3 = 7500 Nm , one obtains a1 = − 2 (3 + 5), a2 = − 2 (3 + 5), ε1 = 1 and ε2 = 0.5. By decomposing the terms in Eq. (52) into partial fractions and applying the inverse Laplace–Borel transforms as given in Table 1, the mean response of the system E[y(t)] can be computed; the result is 

! 0.02288 − t 0.6315 − t 0.0005835 − t y(t) = −4ε1 0.08332 − e 0.1910 + e 0.3333 − e 2.618 0.1910 0.3333 2.618 "  2! σ 0.03514 − t 0.2409 − t 0.0005778 − t − e 0.3820 + e 1.309 + 48ε1 ε2 0.0001508 + e 0.01667 0.3820 1.309 2 0.01667 σ2 2

t 0.001053 t 0.0002746 − t 0.000006524 − t 0.0005542 − t )e− 1.309 ) − e 0.1910 + e 0.09551 + e 0.3333 (1 − 1.309 0.1910 0.09551 0.3333 1.3092 t 0.003057 − t 0.00002157 t 0.00005400 − t e 2.618 − )e− 0.1910 − e 0.1214 − (1 − 2.618 0.1910 0.1214 0.19102 " t 0.0007346 − t 0.0007298 t 0.001947 − t − 0.3333 0.3820 1.309 + ... e )e e − − (1 − + 0.3820 0.3333 1.309 0.33332

−

(53)

Choosing the somewhat arbitrary value σ = 1, the result in Fig. 2 is obtained. One observes a transient that occurs from ‘switching on’ the excitation at t = 0. In fact, because the asymmetric Duffing oscillator has stationary response if the input is stationary, the expectation will tend to a constant value as t −→ ∞; because the restoring force is asymmetric, that constant value will be non-zero. Table 1 Laplace–Borel transforms of common functions [7, 11]

x(t) Unit step tn n!



n−1  i=0

cos(ωt)

i  ai t i n−1 i!

 eat

g[x(t)] 1 x0n (1 − ax0 )−n (1 + ω2 x02 )

−1

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Fig. 2 Expectation of the response of an asymmetric Duffing oscillator to a Gaussian white noise excitation

7 Conclusions Long conclusions are not warranted here as the aim of this chapter was simply to revisit the generating series of Fliess and co-workers, as an elegant means of nonlinear system analysis. In order to introduce a novel element, the analysis has been extended beyond previous work in order to deal with the case of two nonlinear terms in the equation of motion. This new analysis is also extended to the diagrammatic representation, where the presence of two nonlinearities produces two types of vertices in the ‘Feynman’ rules for the diagrams. Further work on the series is considering how it can be used in an automated manner in order to determine Higher-order Frequency Response Functions for nonlinear structural dynamic systems. Acknowledgments The authors would like to thank the UK EPSRC for funding through the Established Career Fellowship EP/R003645/1 and the Programme Grant EP/R006768/1.

References 1. Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. Institute of Physics Publishing, Bristol (2001) 2. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Blackie & Son, London (1930) 3. Barrett, J.F.: The use of functionals in the analysis of non-linear physical systems. Int. J. Electron. 15, 567–615 (1963) 4. Fliess, M.: Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. Fr. 109, 3–40 (1981) 5. Fliess, M., Lamnabhi-Lagarrigue, F.: Application of a new functional expansion to the cubic an harmonic oscillator. J. Math. Phys. 23, 495–502 (1982) 6. Lamnabhi, M.: A new symbolic calculus for the response of nonlinear systems. Syst. Control Lett. 2, 154–162 (1982) 7. Fliess, M., Lamnabhi, M., Lamnabhi-Lagarrigue, F.: An algebraic approach to nonlinear functional expansions. IEEE Trans. Circuits Syst. 30, 554–570 (1983) 8. Lamnabhi, M.: Functional analysis of nonlinear circuits: a generating power series approach. IEE Proc. H (Microwaves Antennas Propag.) 133, 375–384 (1986) 9. Lamnabhi-Lagarrigue, F.: Application des variables non commutatives à des calculs formels en statistique non linéaire. Ph.D. Thesis, Université Paris-Sud (1980) 10. Lamnabhi-Lagarrigue, F., Lamnabhi, M.: Algebraic computation of the solution of some nonlinear differential equations. In: Proceedings of the European Computer Algebra Conference, pp. 204–211. Springer, Berlin (1982) 11. Li, Y., Gray, W.S.: The formal Laplace-Borel transform, Fliess operators and the composition product. In: Proceedings of the 36th Southeastern Symposium on System Theory, pp. 333–337. IEEE, Piscataway (2004)

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12. Duffing, G.: Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz und ihre Technische Bedeutung. F. Vieweg and Sohn, Braunschweig (1918) 13. Reutenauer, C.: Free Lie Algebras. Elsevier, North-Holland (1993) 14. Ree, R.: Lie elements and an algebra associated with shuffles. Ann. Math. 68, 210–220 (1958) 15. Bedrosian, E., Rice, S.O.: The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proc. IEEE 59, 1688–1707 (1971) 16. Lubbock, J.K., Bansal, V.S.: Multidimensional Laplace transforms for solution of nonlinear equations. Proc. Inst. Electr. Eng. 116, 2075–2082 (1969) 17. Morton, J.B., Corrsin, S.: Consolidated expansions for estimating the response of a randomly driven nonlinear oscillator. J. Stat. Phys. 2, 153–194 (1970) 18. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948) 19. Gantmakher, F.R.: The Theory of Matrices. American Mathematical Society, Providence (2000) 20. Arnold, L.: Stochastic Differential Equations. Wiley-Blackwell, New York (1974) 21. Ito, K.: Stochastic integral. Proc. Imp. Acad. 20, 519–524 (1944) 22. Lamnabhi-Lagarrigue, F., Lamnabhi, M.: Algebraic computation of the statistics of the solution of some nonlinear stochastic differential equations. In: Proceedings of the European Conference on Computer Algebra, pp. 55–67. Springer, Berlin (1983)

A Hybrid Static and Dynamic Model Updating Technique for Structures Exhibiting Geometric Nonlinearity Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

Abstract Finite element methods (FEM) are commonly used to analyze the behavior of various dynamic systems. Several model updating techniques are available that can be used for linear or nonlinear dynamic systems and help reduce discrepancies between characteristics obtained from the FE models and their experimental counterparts. Linear model updating techniques employ eigen solutions or the frequency response functions (FRF) to reduce errors between the discretized mass [M], damping [C], and stiffness [K] terms or FRF terms directly. However, model updating of structures exhibiting nonlinear behavior is seldom straightforward and requires special consideration of various terms contributing to the nonlinear responses when modeling the system. This paper investigates a hybrid static and dynamic model updating technique for structures exhibiting geometric nonlinearity. The static model updating minimizes errors in stiffness modeling and the dynamic model updating minimizes errors in modeling non-stiffness terms of the FE model. Keywords Model updating · Geometric nonlinearity · Hybrid techniques

1 Introduction Finite element model updating (FEMU) in structural dynamics uses one or more techniques to reduce the discrepancy between modal parameters and response characteristics obtained from a finite element model (FEM) and a corresponding experimental data obtained for the system. This topic has gained more prominence in recent years with the introduction of extremely light structural components with complex and varying geometries, material characteristics, etc. Well-established FEMU techniques are popularly used for structures behaving under the linear assumption. Friswell and Mottershead [1, 2] provide the earliest detailed description of both direct and iterative methods using modal data and frequency response function (FRF) data for FEMU. One such popular iterative modal method is the inverse eigen sensitivity method (IESM) formulated by Lin et al. [3, 4]. Methods that formulate the model updating using FRF data include the response function method formulated by Grafe, Lin, and Ewins [5–7]. A comprehensive listing of the further improvements and advancements since is found in [8] along with a comparative study of both methods available in [9]. More recent works on the sensitivity method include [10] where the importance of equation conditioning and quality of model updating is elaborated. Other techniques utilize some form of computational intelligence for FEMU [11] and a few other have utilized popular linear FEMU and then used various other techniques to update nonlinear models as well [12–15]. Most FEMU techniques have their own merits and demerits. These techniques generally formulate a sensitivity analysis problem that is solved using a least-squares approach. A common problem encountered is the balance between total number of variables that require updating and the total available data since most techniques rely only on FRFs obtained during experimental modal analysis and/or modal parameters estimated from these FRFs [16–19]. In order to overcome this issue, an earlier work by the authors [20] discusses a hybrid static and dynamic model updating technique for linear dynamic systems, which updates the linear stiffness matrix of the FE model based on static tests, i.e., update at zero frequency followed by a mass updating performed on this static updated model to obtain a complete model that works for both static and dynamic type tests. Static tests are generally employed prior to a dynamic test to determine linearity of a system, verify global parameters, detect type of nonlinearity in a system, etc. [21–23]. Some FRF-based methods are also used for static

M. Nagesh () · R. J. Allemang · A. W. Phillips Structural Dynamics Research Laboratory (SDRL), Department of Mechanical and Materials Engineering, College of Engineering and Applied Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected]; [email protected] © 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_17

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stiffness and compliance estimation [24, 25]. However, while these techniques discussed in [20] are limited to linear systems alone, a novel way of tackling nonlinear static response is required particularly when this hybrid model updating technique is used for thin and light structures that readily exhibit nonlinear behavior. The use of FEM to determine nonlinear static and dynamic responses are popular. However, when employed in a model updating perspective, a detailed analysis of the entire process is required to perform any updating of parameters. This paper discusses the hybrid static and dynamic model updating technique applied to a geometrically nonlinear structure, where a nonlinear displacement-force relationship obtained from a static test is used along with a traditional linear experimental modal analysis to obtain an updated FE model.

2 Hybrid Testing: Theory A linear dynamic system is represented by a second-order differential equation as given by Eq. (1), where M, C, and K represent the mass, damping (assumed viscous), and stiffness of the system. M x¨ + C x˙ + Kx = F

(1)

Generally, for the system under consideration, a few parameters are known, such as geometry and material specifications (Young’s modulus (E), Poisson’s ratio (ν), density (ρ)), that enable a discretized analytical modeling of the system with Ma , Ca , and Ka representing the analytical mass, damping, and stiffness. This discretization is only an approximation of the physical system based on several assumptions and does not represent the exact parameters of the physical test system. Model updating (FEMU) is employed to reduce errors between the actual mass, damping, and stiffness parameters Mx , Cx , and Kx and their analytical counterparts, the exact characteristics of the actual system being unknown. This updating can be performed directly using the FRFs obtained from the physical system and the corresponding analytical FRFs using methods described in [6, 7] or using the experimental and analytical modal parameters {{ψ x }, λx } and {{ψ a }, λa }, respectively, such as those discussed in [3, 4]. The technique discussed in this paper are based on the latter method of using experimental and analytical modal parameters, but a similar adaptation can be used for model updating using FRFs.

3 Static Test and Stiffness Updating In the absence of inertial forces, Eq. (1) reduces to a static problem represented by Eq. (2). Therefore, if a system is analyzed in the absence of inertial forces, i.e., static configuration, only the stiffness matrix requires updating. This step may be alternately termed as zero-frequency update step. Kx = F

(2)

If the system configuration allows a simple static test to be performed with known external forces Fx applied to the system and its displacement xstatic experimentally recorded, the analytical stiffness [Ka ] can hence be updated by approximating any differences in stiffness values using a correction/update coefficient (β) such that β (Ka ) xstatic = (Kx ) xstatic = Fx

(3)

β [Ka ] {xxs } = [Kx ] {xxs } = {Fx }

(4)

This coefficient β is a real positive dimensionless value. Eq. (3) can be adopted to a finite element (FE) model as shown in Eq. (4), where [Ka ] represents the global stiffness matrix of the analytical system and {xxs } and {Fx } represent the displacement and external force applied to the physical system. The correction coefficient (β) can either be used for correction of global properties such as Young’s modulus (E) and moment of inertia (I) of the entire system or be used to update parameters of element stiffness as shown in Eq. (6) Kelem = βe [Ke ] ; [Ka ] =

n  r=1

f (βr [Kr ]) , where n : Total elements

(5)

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In this paper, element stiffness correction is solved for using the recorded static displacements. If static displacements {xxs } are recorded for more than one set of external forces (total M sets), the correction coefficient for the n elements can be solved in a least-squares approach as shown in Eq. (6) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ ⎤ k1 xxs1 − − − ⎥ ⎢ − k2 xxs2 − − ⎥ ⎥ ⎢ ⎥ From displacements for Force 1 ⎥ ⎣ − − ... − ⎦ ⎥ ⎥ − − − kn xxsn ⎥ ⎥ ...........................................................................................⎥ ⎥ ⎡ ⎤ ⎥ k1 xxs1 − − − ⎥ ⎢ − k2 xxs2 − − ⎥ ⎥ ⎢ ⎥ From displacements for Force m ⎥ ⎣ − ⎦ ⎦ − ... − ⎡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎫ ⎧ β1 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ β2 × = ⎪ ⎪ ⎪ ⎪ ⎪ ...⎪ ⎪ ⎭ ⎩ ⎪ βn n X 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

− − − kn xxsn ⎫ ⎧ ⎫ F ⎪ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ Fx2 ⎪ Force 1 ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ Fxn ⎬ ..................... ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Fx1 ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ Fx2 ⎪ Force m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . . . ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ Fxn Mn X 1

Mn X n

(6)

This method of updating β for each element rather than global parameter updating provides more physical meaning since properties across any physical system are seldom uniform. The estimation of β through this approach is a single-step process. Fewer displacements are generally obtained during a physical test in comparison with the number of FE nodes and associated degrees of freedom. A static reduction-expansion technique such as Guyan reduction [26] of the analytical model is therefore used to expand experimental displacements over the entire model and then FE model is updated. This step using Guyan reduction and expansion is generally performed only for those degrees of freedom that can be measured physically. Other degrees of freedom may be assumed known using the solution obtained from the initial FE model.

4 Nonlinear Static Test and Model Updating Unlike linear FE models where the geometry of deformation is considered not to affect the modeling of the system, a nonlinear FE modeling of a structure to an external load requires careful interpretation of the kinematics and displacements of the deformed geometry which may in turn cause other phenomenon such as stretching and axial loading which is crucial to determine an accurate solution to the nonlinear static problem. An excellent description of the various considerations required and use of various techniques is provided in [27]. To account for the various geometric configurations and associated forces, the nonlinear problem is solved in a multistep manner where the entire external load is applied in different load ratios. A small increment of forces and displacements is predicted, and a Newton-Raphson technique is then employed to obtain the converged solution that accounts for all the internal and external forces before proceeding to the next load level. While using static testing, the initial state, i.e., undeformed state, and final state after applying the static load are known. This does not pose a problem for a linear static updating technique since only the initial and final state is required and hence a model updating can be performed in a single step. This limited information however poses a significant challenge when dealing with nonlinear problems, and a workaround as suggested in Fig. 1 is required to obtain an updated model. This increment of internal forces and its manifestation in the total stiffness of the system is represented by a global tangent stiffness matrix in nonlinear problems. For nonlinear updating of the nondimensional term β using Eq. (6), the global tangent stiffness of the final load step of each of the external forces is utilized to calculate the internal forces and associated terms elementwise according to Eq. (6). In the multistep approach, while the initial and final configuration is known, the intermediate steps are all assumed to be in equilibrium. Hence, the update is performed between the final load step and the penultimate load step alone to determine the nondimensional factor β mentioned in the earlier section. Since displacements are known only for select degrees of freedom, the tangent stiffness matrix from the final load step is utilized for the Guyan reduction/expansion process to obtain

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Fig. 1 Nonlinear static model updating process to estimate β

displacements at other relevant degrees of freedom for the experimental data. This expanded displacement is then utilized to obtain a configuration corresponding to the experimental system in order to estimate β using a least-squares solution.

5 Dynamic Test and Mass Updating Once the corrected stiffness is obtained from the static test, the second-order system described in Eq. (1) is now updated as shown in Eq. (7). An established model updating technique such as the inverse eigen sensitivity method (IESM) and several other popular techniques can be used to update the mass and damping. The IESM technique used in FEMU for updating L number of design variables (p) across m modes of the system is formulated as shown in Eqs. (8)–(10). Ma x¨ + Ca x˙ + Ka = F ; Ka = βK

(7)

  S {Δp} = {Δζ}

(8)

where ⎡

- . - . ∂ ψ 1 ∂ ψ 1 ∂p1 ∂p2 ∂λ1 ∂λ1 ∂p1 ∂p2

⎢ ⎢   ⎢ ⎢ S = ⎢ -. ... ⎢∂ ψ ⎢ m ⎣ ∂p1 ∂λm ∂p1

The eigenvector sensitivity

- . ∂ ψ r = ∂pr

- . ∂ ψ 1 ∂pL ∂λ1 ∂pL

⎤

⎧ ⎫ ⎧ ⎪ ⎫ ⎪ {ψx }1 − {ψa }1 ⎪ ⎪ ⎥ ⎪ ⎪ Δp 1 ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ... ⎨ ⎨ (λx )1 − (λa )1 ⎪ ⎥ ⎬ ⎬ Δp2 ⎥ ; {Δζ } = .- . .. . . . -. . .. ⎥ ; {Δp} = ... ⎪ ⎥ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ∂ ψ m ∂ ψ m ⎥ ⎩ ⎪ ⎪ ⎭ {ψx }m − {ψa }m ⎪ ⎪ ⎪ ⎪ ⎪ ΔpL ∂p2 . . . ∂pL ⎦ ⎩ ⎭ (λx )m − (λa )m ∂λm ∂λm . . . ∂p2 ∂pL ...

m  i=1;i=r

{ψa }i {ψa }Ti (λx )r − (λa )r

!

" ∂ [K] ∂ [M] ∂ [M] ∂ [C] 1 {ψx }r − {ψa }r {ψa }Tr {ψa }r +j − (λx )r ∂pk ∂pk ∂pk 2 ∂pk

(9)

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The eigenvalue sensitivity

∂ [K] ∂ [M] ∂ [C] ∂λr {ψx }r − (λx )r {ψa }Tr {ψx }r + j {ψa }Tr {ψx }r = {ψa }Tr ∂p ∂pk ∂pk ∂pk

155

(10)

For mass updating, an approach similar to stiffness updating using static test is used where the analytical mass has a correction coefficient α as described in Eq. (11) for each element that describes all variations of physical properties such as density and other variations. This coefficient α is a real positive dimensionless value. Mass updating using IESM generally has well-conditioned equations that can be easily solved. Typically normalized normal modes of the undamped harmonic system described in Eq. (7) is used for IESM if the model updating has only mass terms [3, 4]. Melem = αe [Me ] ; [Ma ] =

n 

f (αr [Mr ]) where n : Total elements

(11)

r=1

[C]a = γ [K]a

(12)

A proportional damping model is assumed for this paper. The variations in damping can be solved similar to mass and stiffness by introducing a correction factor elementwise and solving a complex eigen sensitivity problem as described in [3, 4]. This method still has numerical issues if the stiffness proportional damping model is used. An easier alternate used in this paper utilizes the mass only updating using the IESM method to obtain corrected mass coefficients and then use a stiffness proportionality constant γ to describe the damping as shown in Eq. (12). In this approach, the normal modes from the mass and stiffness corrected system are used to obtain system parameters in the modal domain such as modal stiffness. A ratio of modal stiffness and damping estimated from experimental system provides the proportionality constant γ . This method is applied mode-wise since stiffness proportional damping models tend to overdamp higher frequencies. Updating mass correction coefficient α involves an iterative process, and a suitable threshold criterion may be set to check for convergence as used in [3, 4]. However, the proportionality constant γ is only a number obtained for a given mode and is a single-step estimation based on the modal stiffness used during the model updating and modal parameters determined experimentally.

6 Test Setup and Finite Element Modeling A double clamped beam as shown in Fig. 2 is considered for experimentation. A detailed description of the beam is provided in Table 1. For the experimental estimate of parameters such as Young’s modulus, an earlier work by the authors is suggested [28]. For the static test, a physical load is applied at through hole 2 (also S3) in several increments and the displacement data is recorded for Location S1 through S5 using dial gages. Displacements S4 and S5 are considered symmetrical with two other points S2 and S1, respectively, and hence measured for only one location. The static test data is also supplemented with values from a commercial FE software in order to validate the applicability of the hybrid model updating technique. This is particularly done to supplement data at low force levels where the dial gage measurements are found unreliable due to some

Fig. 2 Double clamped beam geometry and FE model description

156 Table 1 Double clamped beam specifications

M. Nagesh et al. Total length (Ltotal ) Clamped beam length (L) Clamping (Lclamp ) Width (b) Thickness (h) Through hole diameter (φ) Material Young’s modulus (estimated)

0.5588 m 0.4572 m 0.0508 m 12.7 mm 1.5875 mm 6.35 mm A36 Low-Carbon Steel 201.11 GPa

Fig. 3 Static displacement-force profile

stiffness associated with these gages that potentially lead to erroneous data collection. The static tests are performed in one direction only, but the characteristics are assumed to be the same in either direction for the double clamped beam. A multi-reference impact testing (MRIT) is performed on the double clamped beam using PCB Piezotronics Model 352C23 miniature accelerometers attached to the beam and using a Model 086E80 miniature instrumented impulse hammer. The response locations are provided as R1 through R7 as shown in Fig. 2. All data acquisition is performed using UC SDRL’s X-Modal software and the modal parameter estimation from the MRIT is performed using the RFP-Z algorithm available in the same software. The double clamped beam is modeled as a 3D Euler-Bernoulli beam undergoing large strains and rotations with 19 nodes and 6 degrees of freedom at each node [29]. Node 1 and 19 are fully fixed in all degrees of freedom. The elemental mass and stiffness for the beam element is provided in the Appendix. The basic nonlinear solver is developed based on the framework provided in [27, 29] with supplementary programs to implement the model updating performed on MATLAB [30]. The first three lateral bending modes of the double clamped beam are chosen for model updating in the dynamic section.

7 Results The nonlinear static displacement-force profile determined experimentally and used for the static model updating is shown in Fig. 3. Three cases of initial model assumptions are considered for updating as shown in Table 2. The element-wise update for the nondimensional parameter β is provided in Fig. 4. The highest correction is applied to elements 3 and 4, and 15 and

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Table 2 Case listing and initial assumed values Case number Case 1 Case 2 Case 3

Initial E (GPa) 200 207 195

Initial ρ (kg/m3 ) 7850 8000 7950

Initial β 1 1 1

Initial α 1 1 1

Fig. 4 Stiffness correction β case-wise

16. The case with assumed Young’s modulus of 195GPa requires the largest updating as seen from Fig. 4. The correction to the dimensionless parameter α is shown in Figs. 5, 6, and 7 mode-wise. From Table 3, it is observed that the stiffness only updated model changes the natural frequency for all three bending modes of the double clamped beam considered. The mass correction α applied to Case 1 and Case 3 for Mode 1 shows some abrupt changes in values that requires further investigation. The model updating performed here is done mode-wise and not considering all modes considered together. This may eliminate such misrepresented values but may not provide a satisfactorily updated model.

8 Conclusion and Future Scope A hybrid static and dynamic model updating technique for structures exhibiting geometric nonlinearity is demonstrated in this paper. An overall framework required to perform such an updating process is elaborated along with some explanations on existing model updating techniques. A double clamped beam is considered, and the static updating is performed using the nonlinear force-displacement profile. A well-established iterative modal technique for FEMU, the inverse eigen sensitivity method (IESM) is employed for mass updating. Overall, satisfactory results were observed with some discrepancies noted for the first bending mode of the double clamped beam. Hence, a FE model with sufficient discretization is henceforth recommended for employing this technique. While this paper focuses on the static updating using a nonlinear force-displacement profile, no satisfactory framework exists to verify a nonlinear dynamic updating of the system. In addition, nonlinear experimentation of such thin and light structures is difficult to obtain satisfactory results that may be utilized for such dynamic nonlinear updating purposes. Some

158

Fig. 5 Mode 1 mass correction α case-wise

Fig. 6 Mode 2 mass correction α case-wise

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Fig. 7 Mode 3 mass correction α case-wise Table 3 Model updating comprehensive results

Case 1

Case 2

Case 3

Mode 1 (Hz) 37.7945 40.8144 41.3062 37.477 39.8823 41.3062 37.8886 40.9013 41.3062

Initial (β update only) Final (β and α update) Experimental (MPE) Initial (β update only) Final (β and α update) Experimental (MPE) Initial (β update only) Final (β and α update) Experimental (MPE)

Mode 2 (Hz) 105.4537 107.0684 107.086 103.9286 107.0709 107.086 105.9028 107.0943 107.086

Mode 3 (Hz) 208.9003 208.1422 207.9784 206.5147 207.8192 207.9784 208.1761 209.8817 207.9784

correlation based on modeling a Duffing-like equivalent system is underway. In addition, damping and damping modeling is another area of concern for such thin and light structures and adequate techniques and methods are required to address these concerns. A compound T-beam comprising multiple such beams exhibiting geometric nonlinearity is also currently under investigation.

Appendix Elemental stiffness [Ke ] and mass matrix [Me ] for 3D Euler-Bernoulli beam element is provided as follows. !

K11 K12 [Ke ] = K21 K22

"

!

M11 M12 | [Me ] = M21 M22

"

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where the individual terms are given below. The terms used are enlisted as follows: ρ: Density E: Young’s modulus L: Element length

⎡

K11 = K22

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

EA L

0 0 0 0 0

0

0 0

12EIz L3

0 0 0 6EIz L2

12EIy L3

0 6EI − L2 y

0 ⎡

M11 = M22

⎢ ⎢ ⎢ ⎢ = ρAL ⎢ ⎢ ⎢ ⎢ ⎣

1 3

0

0 13 35 0 0

0 0

0 0

0

6EI − L2 y

GIx L

0 0

0

13 35

0 0 0 11L 0 0 − 210 0 11L 0 210

⎥ ⎥ ⎥ 0 ⎥ ⎥ | K12 0 ⎥ ⎥ ⎥ 0 ⎦

6EIz L2

4EIz L

0

0 0

⎡

⎤

0

4EIy L

A: Cross-sectional area of element G: Transverse elasticity modulus I: Second moment of area

0 0 0 (Iy +Iz ) 3A

0 0

0 0 − 11L 210 0 L2 105

0

0

− EA 0 0 0 L ⎢ 0 − 12EIz 0 0 ⎢ L3 ⎢ 12EIy ⎢ 0 0 − L3 0 = K  _(21) = ⎢ GIx ⎢ 0 0 0 − ⎢ L ⎢ 6EI 0 − L2 y 0 ⎣ 0 6EIz 0 0 0 L2 ⎤

⎥ ⎥ ⎥ 0 ⎥ ⎥ | M12 = ρAL 0 ⎥ ⎥ ⎥ 0 ⎦

11L 210

L2 105

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 6

0

0 0

9 0 70 9 0 0 70 0 0 0 13L 0 0 − 420 0 13L 0 420

0 0 0

Iy +Iz 6A

0 0

0 0 − 13L 420 0 L2 − 140 0

0 0 6EIy L2

0 2EIy L

0

0

0

⎤

z ⎥ − 6EI L2 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦

2EIz L

⎤

⎥ − 13L 420 ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ L2 − 140

References 1. Friswell, M.I., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics, vol. 38. Kluwer Academic Publishers, Dordrecht (1995) 2. Mottershead, J.E., Friswell, M.I.: Model updating in structural dynamics: a survey. J. Sound Vib. 167(2), 347–375 (1993) 3. Lin, R.M., Lim, M.K., Du, H.: Improved inverse eigensensitivity method for structural analytical model updating. J. Vib. Acoust. Trans. ASME. 117(2), 192–198 (1995) 4. Lin, R.M., Lim, M.K., Du, H.: A new complex inverse eigensensitivity method for structural damping model identification. Comput. Struct. 52(5), 905–915 (1994) 5. Grafe, H.: Model updating of large structural dynamics models using measured response functions. Vib. Univ. Technol. Cent., 184 (1998) 6. Lin, R.M., Ewins, D.J.: Analytical model improvement using frequency response functions. Mech. Syst. Signal Process. 8(4), 437–458 (1994) 7. Lin, R.M., Ewins, D.J.: Model updating using FRF data. In: ISMA, pp. 141–162 (1990) 8. Sehgal, S., Kumar, H.: Structural dynamic model updating techniques: a state of the art review. Arch. Comput. Methods Eng. 23(3), 515–533 (2016) 9. Modak, S.V., Kundra, T.K., Nakra, B.C.: Comparative study of model updating methods using simulated experimental data. Comput. Struct. 80(5), 437–447 (2002) 10. Mottershead, J.E., Link, M., Friswell, M.I.: The sensitivity method in finite element model updating: a tutorial. Mech. Syst. Signal Process. 25(7), 2275–2296 (2011) 11. Marwala, T.: Introduction to finite-element-model updating. In: Marwala, T. (ed.) Finite-Element-Model Updating Using Computional Intelligence Techniques, pp. 1–24. Springer London, London (2010) 12. Canbalo˘glu, G., Özgüven, H.N.: Model updating of nonlinear structures from measured FRFs. Mech. Syst. Signal Process. 80, 282–301 (2016) 13. Canbalo˘glu, G.: Development of a Model Updating Technique for Nonlinear Systems (2015) 14. Kerschen, G.: On the model validation in non-linear structural dynamics. Ph.D. Thesis (2002) 15. Li, Y., Astroza, R., Conte, J.P., Soto, P.: Nonlinear FE model updating and reconstruction of the response of an instrumented seismic isolated bridge to the 2010 Maule Chile earthquake. Earthq. Eng. Struct. Dyn. 46(15), 2699–2716 (2017) 16. Allemang, R.J.: Investigation of some multiple input/output frequency response function experimental modal analysis techniques. PhD Dissertation, University of Cincinnati, p. 358 (1980) 17. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. Wiley (2006) 18. Allemang, R., Brown, D.L.: Experimental Modal Analysis And Dynamic Component Sythesis - Volume 3 - Modal Parameter Estimation, p. 131 (1987)

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19. Allemang, R.J., Brown, D.L.: Correlation coefficient for modal vector analysis. In: Proc. Int. Modal Anal. Conf. Exhib., pp. 110–116 (1982) 20. Nagesh, M., Allemang, R.J., Phillips, A.W.: Finite element model updating of linear dynamic systems using a hybrid static and dynamic testing technique. In: Proc. ISMA 2020 - Int. Conf. Noise Vib. Eng. USD2020 - Int. Conf. Uncertain. Struct. Dyn., pp. 1973–1986 (2020) 21. Pandiya, N.: Design and validation of a MIMO nonlinear vibration test rig with hardening stiffness characteristics in multiple degrees of freedom. MS Thesis, University of Cincinnati, pp. 133 (2017) 22. Chawla, R.D.: Investigation of asymmetric cubic nonlinearity using broadband excitation. MS Thesis, University of Cincinnati, p. 76 (2019) 23. Kolluri, M.M.: Non-parametric nonlinearity detection under broadband excitation. PhD Dissertation, University of Cincinnati (2019) 24. Pasha, H.G.: Estimation of Static Stiffnesses from Free Boundary Dynamic (FRF) Measurements. University of Cincinnati (2014) 25. Young, A.: Validating Automotive Frame Torsion Stiffness Measurement Techniques. University of Cincinnati (2016) 26. Guyan, R.: Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965) 27. Rangel, R.L.: Educational Tool for Structural Analysis of Plane Frame Models with Geometric Nonlinearity. Pontifical Catholic University of Rio de Janeiro, Brazil (2019) 28. Nagesh, M., Allemang, R.J., Phillips, A.W.: Finite element (FE) model updating techniques for structural dynamics problems involving nonideal boundary conditions. In: Proc. ISMA 2020 - Int. Conf. Noise Vib. Eng. USD2020 - Int. Conf. Uncertain. Struct. Dyn., pp. 1937–1949 (2020) 29. Rangel, R.L.: NUMA-TF. MATLAB Central (2020) 30. MATLAB, 9.7.0.1190202 (R2019b). Natick, Massachusetts: The MathWorks Inc. (2018)

Insights on the Dynamical Responses of Additively Manufactured Systems M. Curtin, M. Ley, M. Trujillo, B. E. Saunders, G. Throneberry, and A. Abdelkefi

Abstract Structures and parts are increasingly being fabricated using Additive Manufacturing (AM) due to low material waste, ability to quickly generate prototypes, and potential to create complex geometries that would otherwise be difficult or expensive to manufacture. While there are many benefits to using AM processes, the dynamic behavior of the structure is dependent upon the orientation of the construction. This requires investigation of orthotropic materials which have varying material properties in each of the primary Cartesian axes. Some materials, such as the common 3D printing filament Polylactic Acid (PLA), behave orthotropically. Orthotropic materials have often been represented as isotropic materials to simplify numerical models. While such an assumption may be valid in some cases, this study seeks to characterize PLA beams with a tip mass. Experimental PLA beams are printed in the same orientation and have the same external geometry, but the angle of the filament measured from the major axis of the beam varies. The dynamical characteristics are acquired using free and random vibration experiments. The results show that the angle of filament has a significant effect on the damping ratio and natural frequencies of the system. Keywords Additive manufacturing · Printing orientation · Orthotropic material behaviors · Experimental testing · Modal analysis

1 Introduction There are many benefits to use AM processes, but the material properties and dynamic behaviors of the structure are dependent upon the construction parameters [1]. 3D printing is perhaps the most accessible form of AM which utilizes a technology called Fused Deposition Modeling (FDM), in which material is deposited layer by layer through some sort of heated nozzle. When FDM is used, one must consider printing parameters, such as the part’s geometry, orientation of construction, and extrusion temperature. Many FDM materials behave orthotropically, but this material characteristic is often overlooked during the design process. This dynamic uncertainty is stalling the application of AM and FDM for crucial parts within structures [2]. This requires investigation of anisotropic materials whose stiffnesses vary in the x, y, and z directions. Adkins et al. [2] utilized Digital Image Correlation (DIC) to test the effect of varying the internal lattice structure and print orientation of several cantilever beams printed with Acrylonitrile Butadiene Styrene (ABS) and found that the anisotropic behavior of FDM material is significantly affected by the lattice structure orientation of non-solid parts. Printing parameters such as print orientation, nozzle temperature, and print speed for solid benchmark specimens were tested by Attoye et al. [1] to determine the Young’s modulus through the testing of tensile loading until failure. This research proved that the print orientation of a solid part gives the Young’s modulus an uncertainty range of up to 30 MPa for test specimens oriented along the axes of a Cartesian coordinate system at a print temperature of 200 ◦ C and a print speed of 60 mm/s. This contribution seeks to fill the gap within the literature by dynamically testing a set of solid Polylactic Acid (PLA) beams using constant printing parameters, while varying the printed filament angle, measured from the major axis of the beam.

M. Curtin, M. Ley, and M. Trujillo contributed equally to this work. M. Curtin () · M. Ley · M. Trujillo · B. E. Saunders · G. Throneberry · A. Abdelkefi Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © 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_18

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(a)

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(b)

Fig. 1 (a) Varying filament angles used when printing the experimental beams. (a) 0◦ , (b) 45◦ , (c) 90◦ , (d) alternating layers of 45◦ and 135◦ . (b) Experimental setup

2 Additively Manufactured Beams and Experimental Setup The goal of this study is to fundamentally understand the physics of beam-based systems constructed using orthotropic materials. Although solutions have been proposed to analytically calculate the torsional modes of an orthotropic beam [3], the literature is missing information concerning the modal characteristics of orthotropic systems. The system under investigation consists of a PLA cantilever beam with dimensions 20.32 cm × 5.08 cm × 0.127 cm and a tip mass. This tip mass is 1.016 cm × 1.016 cm × 1.016 cm, weighs 6.3 g, and is located along the central axis of the beam. The beams are printed using a Creality CR-10S Pro V2 3D printer using a 0.4 mm standard brass nozzle. The filament layers are printed with a thickness of 0.2 mm, using a nozzle temperature of 200 ◦ C, a heated bed temperature of 55 ◦ C, and a print speed of 50 mm/s. The part is defined as a solid by defining 3 top and 3 bottom layers to ensure that there is no infill within the part; then the print filament angle (measured from the major axis of the beam) is defined. The printing angles of the beams are varied and the effects on the resonant frequencies, density, and damping factor are monitored. Four unique beams are printed using filament angles of 0◦ , 45◦ , 90◦ , and alternating layers of 45◦ and 135◦ , as depicted in Fig. 1(a). A shaker, combined with use of Siemens® Testlab software shown in Fig. 1(b), provides controlled, consistent modal testing, and imitates realistic environmental conditions. Two types of experiments are performed on each beam. The first is a free vibration experiment which involved creating a small initial displacement at the center of the beam to activate only the first mode of vibration. Using damped time histories to perform logarithmic decrement is a critical step to calculate the damping ratio. Next, the system is excited with random excitations to activate multiple modes in the system. The PLA beams are excited via random excitation; frequencies ranging from 0 Hz to 200 Hz are observed and analyzed.

3 Filament Orientation Effects on the Linear Characteristics of PLA-Beam Systems Free vibrations are induced by initiating a small downward displacement at the geometric center of the top surface. The logarithmic decrement analysis is performed to extract the damping factor. In addition, the beams are weighed to determine the masses and hence their densities. As shown in Table 1, the densities of the beams are varied which is unexpected because the beams all consist of the same external geometry, material, volume, and are printed as solid parts. The 0◦ has the highest density, followed by 45◦ and 45◦ /135◦ , and lastly the 90◦ . Further, the damping factor is also dependent on the filament angle with lower damping factor for the 45◦ . The system is then excited with random excitations to activate all modes of vibrations. The resonant frequencies are recorded for the first five modes and are listed in Table 2. Power spectral density plots for the different filament angles are also presented in Fig. 2. Clearly, it is found that each material orientation provides its own unique natural frequencies.

Insights on the Dynamical Responses of Additively Manufactured Systems Table 1 Variations of density and damping factor with respect to filament orientation

165 Filament angle (◦ ) 0◦ 45◦ 90◦ 45◦ /135◦

Density (Kg/m3 ) 1162.18 1157.77 1148.97 1157.77

Damping factor 0.03308 0.02274 0.04102 0.04772

Table 2 Resonant frequencies identified from random vibration testing Filament angle (◦ ) 0◦ 45◦ 90◦ 45◦ /135◦

Mode 1 (Hz) 7.37 5.91 5.77 7.45

Mode 2 (Hz) 23.48 22.96 22.93 –

Mode 3 (Hz) 47.3 43.67 42.68 45.59

Mode 4 (Hz) 106.89 95.76 94.6 100.38

Mode 5 (Hz) 149.2 148.39 146.6 153.25

Fig. 2 PSD results for different beams with printing filament angles of 0◦ , 45◦ , 90◦ , and 45◦ /135◦

4 Conclusions The effects of angle filament for cantilevered beams with tip mass were highlighted through free and forced vibration analyses. Results indicated that the dynamic responses are highly affected by the angle of 3D printing filament. When analyzing the fundamental frequency of the beams, it was found that the 45◦ /135◦ beam had the highest damping factor, followed by the 90◦ beam, 0◦ beam, and finally the 45◦ beam. There is a complex trend observed for the fundamental natural frequencies, which are dependent upon filament orientation angle and its corresponding density. A new trend was identified through the investigation of higher modes; the beams’ resonant frequencies decrease as the filament angle varies from 0◦ to 90◦ . The 45◦ /135◦ beam has a seemingly random pattern, likely due to the printing complexity introduced by alternating the filament angle. Deeper investigations will be performed analytically and computationally and will be verified experimentally to study the dynamical responses of AM beam systems. Acknowledgments The authors would like to acknowledge the financial support from Los Alamos National Laboratory. A special thanks to Dr. Thomas Burton for his fruitful discussions.

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References 1. Attoye, S., Malekipour, E., El-Mounayri, H.: Correlation between process parameters and mechanical properties in parts printed by the fused deposition modeling process. In: Mechanics of Additive and Advanced Manufacturing, Conf. Proc. SEM Series, vol. 8, pp. 35–41 (2018) 2. Adkins, G., Little, C., Meyerhofer, P., Flynn, G., Hammond, K.: Characterizing Dynamics of Additively Manufactured Parts. https:// link.springer.com/chapter/10.1007/978-3-030-12684-1_17 (2019, May 15). Accessed 25 Nov 2020 3. Miller, A., Adams, D.: An analytic means of determining the flexural and torsional resonant frequencies of generally orthotropic beams. J. Sound Vib. 41(4), 433–449 (1975). https://doi.org/10.1016/s0022-460x(75)80107-6

Characterization of Nonlinearities in a Structure Using Nonlinear Modal Testing Methods Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

Abstract Using phase-locked loop (PLL) controllers for obtaining frequency response curves (FRC) and backbone curves for nonlinear systems is gaining prominence. Such controllers deliberate a phase lag between the response obtained and the excitation provided. The use of such feedback controllers provides many advantages against traditional sine-sweep methods and helps better characterize nonlinear behavior of the test structure. This paper focuses on obtaining the nonlinear frequency response curves (FRC) and backbone curves of an isolated mode of a structural system exhibiting geometric nonlinearity. The capabilities of testing with such PLL controllers are highlighted and other important characteristics such as stability of the system are discussed. A qualitative comparison is provided between nonlinear modal testing using such feedback controllers as against other traditional methods such as sine-sweep methods. Keywords Nonlinear modal testing · Phase-locked loop · Nonlinear frequency response · Geometric nonlinearity

1 Introduction Study of dynamic mechanical systems with nonlinear force-response relationships is often very challenging. Well-established analytical and experimental techniques exist for linear systems [1–5], but such methods are seldom easily adaptable to nonlinear systems. The study of such nonlinear systems and associated phenomenon is vital in current engineering practices due to large-scale application of lightweight and complex materials that intrinsically exhibit nonlinear behavior [6, 7]. Such nonlinear systems typically exhibit one or more types of nonlinearities; geometric nonlinearity is widely observed and requires adequate understanding when characterizing such nonlinear systems. Distinct features of these systems include the variations of its natural frequencies, deflection characteristics, damping, and other important characteristics with variation in forcing levels. Detailed theoretical analyses and explanations for nonlinear behavior of structures are widely available [8–10]. Experimental techniques that can be employed for detecting, identifying, and characterizing such nonlinearities are seldom straightforward. A comprehensive list of experimental techniques accompanied by a vast theoretical background and their historical evolution is available in [6, 7, 11]. Traditional linear modal analysis techniques leads to several discrepancies in identifying and characterizing nonlinear behavior, and traditional sine-sweep techniques for nonlinear systems are challenging in their own ways, [6, 12] describe some novel excitation techniques that utilize a phase resonance approach to nonlinear testing. Other variations of such techniques using control techniques to ensure stability at or near resonances have since been further developed and have gained wide popularity. Two such techniques are in popular use: phase-locked loop (PLL) and control-based continuation (CBC) [13–17]. Phase-locked loop (PLL), widely used in communication and control applications [18, 19] was successfully adapted for nonlinear modal testing as described in [13–17]. The nonlinear normal mode, a nonlinear counterpart to a linear normal mode, is defined as the condition or frequency when external forcing and response are in phase-lag quadrature, i.e., have a 90◦ phase difference [20, 21]. The behavior of the system is purely conservative in such cases and the measurement of this quadrature condition at several excitation levels constitute a backbone curve (BBC) for the nonlinear system depicting the variation of the underlying conservative system with varying excitation levels. In this paper, a software implementation

M. Nagesh () · R. J. Allemang · A. W. Phillips Structural Dynamics Research Laboratory (SDRL), Department of Mechanical and Materials Engineering, College of Engineering and Applied Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected]; [email protected] © 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_19

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[19] of the PLL is elaborated and applied to a structure exhibiting geometric nonlinearity. The procedure to deliberate a phase quadrature between the excitation and response is discussed in detail, and experimentally obtained backbone curves are presented for the structure showing geometric nonlinearity. These backbone curves are accompanied by results obtained from traditional sine-sweep excitations in both sweep-up and sweep-down conditions, where results for nonlinear responses are generally represented as frequency response curves (FRC) that provide responses values (typically RMS) at a given force level, instead of force normalized responses popularly termed as frequency response functions (FRF) used in linear modal analysis.

2 Theoretical Background Dynamic systems can be represented by a second-order differential equation as given in Eq. (1), where M denotes the mass matrix, C denotes damping matrix (assumed viscous), K denotes the linear stiffness matrix, FNL denotes the nonlinear restoring forces (conservative), and Fext denotes external forces acting on the system. M x(t) ¨ + C x(t) ˙ + Kx(t) + F NL (x(t)) = F ext (t)

(1)

[16] provides an excellent explanation of phase resonance testing and derivation of the perfect excitation vector for a m degree-of-freedom dynamic system. The system considered in this paper behaves likes a single degree-of-freedom system that can be approximated to a Duffing-like oscillator with cubic hardening type nonlinearity. Hence Eq. (1) for a Duffing-like oscillator takes the form given in Eq. (2), where M, C, and K terms are the representational mass, viscous damping, and linear stiffness of the single degree of freedom Duffing-like oscillator and β is cubic stiffness coefficient. To extract a response of purely the underlying conservative system as given in Eq. (3), the condition represented by Eq. (4) is compulsory. M x(t) ¨ + C x(t) ˙ + Kx(t) + βx(t)3 = Fext (t)

(2)

M x(t) ¨ + Kx(t) + βx(t)3 = 0

(3)

C x(t) ˙ = Fext (t)

(4)

∞ 

x(t) =

Xn einωt

(5)

n=−∞

Consistent with the approach followed in [16] and since the motion of the underlying conservative system at phase resonance is by definition a nonlinear normal mode motion, this motion of the system can be represented as a periodic motion represented by Eq. (5), where n is the harmonic considered at ω rad/s and the complex Fourier coefficient for the n th harmonic is Xn . From Eqs. (4) and (5), the damping forces are derived as shown in Eq. (6). From Eq. (6) it is clear that at phase quadrature, all harmonics of the damping force are purely imaginary, i.e., the damping force and the displacement have a phase difference of π /2. If the external forces are also assumed as periodic as shown in Eq. (7), then the phase quadrature between damping and displacement can be defined as given in Eq. (8). This signifies that isolating the nonlinear normal mode implies a phase quadrature independently with every harmonic between the damping and displacement. Fext (t) = C

∞ 

∞ 

inωXn einωt = C

n=−∞

Fext (t) =

Vn einωt

(6)

n=−∞ ∞ 

Fn einωt

(7)

Fn = C inωXn = C Vn

(8)

n=−∞

Characterization of Nonlinearities in a Structure Using Nonlinear Modal Testing Methods

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Fig. 1 Phase-locked loop (PLL) schematic

3 Phase-Locked Loop A phase-locked loop (PLL) is a control system where a specific phase relationship exists between its output and input signal. It follows from the corollary that two signals starting from the same reference will have same frequency only when the phase between them is zero. Conversely for dynamic systems, the same principle is used to obtain a phase lag quadrature relationship between the excitation and displacement; the frequency where this quadrature is observed can be assumed to satisfy one or more conditions for a normal mode (linear or nonlinear) as described in the previous section. Traditional sine-sweep type approaches to nonlinear testing lack any such control loops that provide monitoring and control of input and output relationships. In its simplest form a PLL consists of the components as shown in Fig. 1. The phase detector is used to separate the phase information between the reference p(t) and output s(t) signals. In theory, this is achieved by multiplying the input and output signals for single frequency sinusoidal type signals, i.e., q(t) = p(t) * s(t). The resulting DC component of signal can be used to extract the phase information between the two signals. A loop filter, comprising of low pass filter and variations of a PID controller for stability controls, is generally used to extract the phase information. A voltage-controlled-oscillator (VCO) is used to generate a signal that has the required characteristics based on the output of the loop filter. The output of the VCO may be amplified as required to obtain the desired voltage level (excitation level). Detailed descriptions of these components are available in [16]. In nonlinear modal testing, it is generally assumed that the influence of higher harmonics is negligible and only the fundamental frequency is considered for tuning the PLL. The PLL can hence be operated to obtain a phase quadrature for the fundamental harmonic of the excitation frequency. For systems whose fundamental frequency varies with excitation amplitude, the loop circuit will ensure a signal is generated such that a phase quadrature is obtained between excitation and displacement as soon as the excitation levels are altered. The PLL operation can alternately be executed using any software platform [19] that is typically used to operate and acquire data in experimental vibration analysis. Some advantages of using a software PLL (sPLL) are highlighted in the following sections.

4 Software Phase-Locked Loop PLL circuits described above typically operate on a per-wave basis. These variations over longer time durations may be used to determine performance and operation of the PLL circuit. In any experimental analysis of dynamic systems and vibrations, averaging is most crucial in ensuring confidence of data acquired. Moreover, the above-described traditional PLL requires physical components or its equivalent to obtain a nonlinear mode for a given system. The process also involves careful and experienced tuning of the loop filter using additional PID controllers. Alternately, software phase-locked loop (sPLL) provides many advantages and offers more fidelity from a vibration measurement perspective against employing a traditional PLL. The sPLL method used in this paper uses MATLAB [22] and its available features as described in the further sections. A VXI mainframe is used for data acquisition and control of the modal shaker through a power amplifier. The complete schematic for operation of the sPLL is given in Fig. 2. Consistent with [16], two assumptions are made for nonlinear modal testing using sPLL, viz. (1) The fundamental harmonic is used for phase computations and effect of higher harmonics is negligible and (2) The RMS of input and output quantities is mostly dominated by the signal at the fundamental frequency and hence is used for force and response computations. A linear experimental modal analysis is performed on the test system to determine the mode that is considered for investigation. Once the linear modal frequency is determined, a steady sine wave is generated at a frequency very close to this linear modal frequency. At extremely low amplitudes of vibration, the steady sine wave excitation will ideally be in quadrature with the response at the same frequency as the experimentally determined linear modal frequency. To obtain a backbone curve, the response amplitude and frequency of quadrature are essential. The excitation levels can be stepped up either using voltage control of the mainframe or using the modal shaker power amplifier. For nonlinear systems of approximately cubic hardening type nonlinearity, the nonlinear modal frequency increases with increase in vibration levels.

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Fig. 2 Software phase-locked loop schematic

Hence, the phase quadrature condition previously set for low vibration levels is no longer valid. The phase condition of this new excitation is further used to determine the new frequency that must be required to drive the dynamic system towards phase quadrature. In this paper, a steady frequency stepping of 0.5 Hz is used for coarse frequency adjustments (0 < φ < 80 or 100 < φ < 180) and 0.01 Hz is used for fine adjustments (80 < φ < 100). The stepping conditions are selected in this manner keeping in mind the accuracy limitations of excitation provided by the modal shakers for such nonlinear systems and data acquisition capabilities of the VXI mainframe. Moreover, large frequency shifts cause jumps in the modal shaker excitation and are undesirable for measurements. Once the quadrature condition has been reached, the circuit is locked, and no further frequency changes are made for the same excitation levels. In order to ensure the assumption that only the fundamental harmonic is considered for phase calculations, a third-order Butterworth filter with cutoff frequency 1.5 times the fundamental frequency is applied to both the excitation and response, and the phase is calculated based on peak time difference of this filtered excitation and response. These filters attenuate the higher harmonics, particularly the second and third harmonics by 15 dB or more. Applying the filter to both input and output is necessary to ensure characteristics are consistent across the various signals used for the phase calculation. Hence, the assumption of using only the fundamental harmonic for phase calculations is satisfied. The ready-made functions “butter” and “findpeaks” available in MATLAB [22] are used for the filter and phase calculations, respectively. The time window for sPLL can be controlled as required. The window for control can be as close to a single time period of sinusoidal excitation or any larger window. Since the phase computations are based on the timing of the peaks of the sinusoidal waveform, for a larger time window multiple such peaks are available. Therefore, any phase computation provides an averaged value of a large number of cycles of a sinusoidal waveform. This indicates stability of the sPLL and repeated over many ensembles is ideally indicative of a very stable loop. For the results presented in this paper, the time window is over 2.5 s and over 40 peaks in the excitation and response are considered for phase computation over several cycles of a stable loop. Caution must be taken to ensure filter characteristics do not interfere with the phase computation.

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5 Test Rig Details The sPLL described in the previous sections is tested on two configurations of a T-structure. Both configurations comprise a thick cantilevered beam attached at the midpoint of thin double-clamped beam(s); the “ 1X” rig configuration comprises the cantilever beam with one thin double-clamped beam as shown in Fig. 3 and the “ 2X” rig configuration comprises the cantilever beam attached to two thin double-clamped beams as shown in Fig. 4. The details for the individual beams are provided in Table 1. The test structure is excited close to the junction of the cantilevered beam and the double-clamped beams and the output data is measured at the same location. This configuration of a single input single output (SISO) configuration of measurements is approximated as a single DOF Duffing-like oscillator with hardening type cubic nonlinearity [12, 23]. The results of only the first bending mode of the rig structure which is a compound mode of the first bending modes of the individual beams are discussed here in this paper. The combined system of cantilevered beam and double-clamped beam(s) can be approximated to a Duffing-like system using the approach provided by [11]. Under this approximation, for cantilevered and double-clamped beams, the Duffinglike approximation individually for a beam of length L, Young’s modulus E, mass per unit length m, cross-sectional area A, second moment of area I, and under external excitation Fext is given in Eq. (9) and Eq. (10), respectively. For the Duffinglike approximate equations of the compound structure, the individual beam equations are arithmetically added for each of the beams contributing to the full structure. For best comparison of results, the Young’s modulus E, density ρ, and beam geometry are updated using the approach discussed in [24] and the updated parameters are provided in Table 1.

Fig. 3

“ 1X”

rig configuration

3 2EI π 4 EAπ 4 3 mLx¨ + x + x = Fext (t) 8 L3 8L3

(9)

1 EI π 4 EAπ 4 3 mLx¨ + x+ x = Fext (t) 3 4 32L 128L3

(10)

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“ 2X”

rig configuration Table 1 Individual beam specifications

Beam Type

Dimensions = = =

Cantilevered Beam and ) (

= . = . = .

Double Clamped Beam and ) ( Double Clamped Beam only) (

Young’s Modulus ( )

Beam Mass ( )

Other Added Masses ( )

Total Mass Approximation ( )

7813

204.36

300.49

Force Trans = 26 Accel = 0.6

327

= . = . = .

7830

201.11

71.2

Accel = 2 Screw Nut = 6.8

80

= . = . = .

7911

205.65

71.9

Accel = 1

73

Density (

)

6 Results The experimental results obtained from the two configurations of the T-structure are shown in Figs. 5, 6, 7, 8, 9, and 10. The backbone curves obtained experimentally is accompanied by frequency response curves (FRC) for various forcing levels ranging 0.1 N through 1.0 N. The linear mode for each of the configuration obtained from a MPE is shown for reference as well. The linear mode for the rig in 1X configuration is estimated as 41.17 Hz and for the 2X configuration is 42.8 Hz. The FRC in both sweep-up and sweep-down excitation configurations clearly show jump phenomenon at higher forcing levels. At low excitation levels of 0.1 N, the FRC peaks frequency almost coincides with the linear mode.

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Fig. 5 0.1 N FRC and BBC: “ 1X” rig configuration

Fig. 6 0.1 N FRC and BBC: “ 2X” rig configuration

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Fig. 7 0.5 N FRC and BBC: “ 1X” rig configuration

Fig. 8 0.5 N FRC and BBC: “ 2X” rig configuration

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Fig. 9 1.0 N FRC and BBC: “ 1X” rig configuration

Fig. 10 1.0 N FRC and BBC: “ 2X” rig configuration

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Fig. 11 BBC comparison: “ 1X” rig configuration

Fig. 12 BBC comparison: “ 2X” rig configuration

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The advantages of using a PLL for obtaining backbone curves are evident from the results presented. In traditional sinesweep type excitations, FRCs at many forcing levels are required to obtain a locus of points where there is a jump indicating the change in phase of response with respect to the excitation and hence determine the backbone curves approximately. With PLL controllers, a time-efficient method of obtaining the same backbone curve is demonstrated. From Figs. 11 and 12, the Duffing-like equivalent approximation of the T-structure in 1X and 2X configuration indicates good correlation with the experimentally obtained backbone curves. It must be noted that the 1X configuration required a small mass correction for this correlation, but the trajectory of the backbone curve was well captured. A slight initial softening is observed in the backbone curve for the 1X test structure experimentally. The Duffing-like equivalent modeling is incapable of capturing this softening effect. This can be attributed to variations in joint geometry on the side where the double-clamped beam was attached against the empty side for the 1X configuration, and any buckling of the double-clamped beam that cannot be fully accounted for. This effect is not visible in the 2X configuration where both sides have similar joint characteristics.

7 Conclusion and Future Work Implementation of a software-Phase-Locked loop (sPLL) for obtaining backbone curves (BBC) is elaborated in this paper and demonstrated using MATLAB as the software to obtain the loop control. Two configurations of a T-structure are used to check the implementation of this sPLL. The experimentally determined backbone curve is verified with traditional sinesweep frequency response curves (FRC) and with a Duffing-like equivalent system modeling. The efficiency of the controller for various observation windows and a longer time stream of data can provide a better insight into the working of such sPLL(s). Other metrics for measuring efficiencies of such PLL controllers are required to quantify merits and demerits of using sPLL controllers over traditional controllers. The T-structure is approximated as a single DOF system only here, but other dynamics of the system, particularly damping characteristics and modeling, require further investigation.

References 1. Allemang, R.J.: Investigation of some multiple input/output frequency response function experimental modal analysis techniques. PhD Dissertation, University of Cincinnati, p. 358 (1980) 2. Allemang, R.J., Phillips, A.W.: The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proc. 2004 Int. Conf. Noise Vib. Eng. ISMA, pp. 2373–2401 (2004) 3. Allemang, R.J.: The modal assurance criterion—twenty years of use and abuse. Sound Vib. 37(8), 14–21 (2003) 4. Allemang, R., Brown, D.L.: Experimental Modal Analysis and Dynamic Component Sythesis - Volume 3 - Modal Parameter Estimation, p. 131 (1987) 5. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd Edition. Wiley (2006) 6. Noël, J.P., Kerschen, G.: 10 years of advances in nonlinear system identification in structural dynamics: a review. In: Proc. ISMA 2016 - Int. Conf. Noise Vib. Eng. USD2016 - Int. Conf. Uncertain. Struct. Dyn., pp. 2709–2745 (2016) 7. Kerschen, G., Worden, K., Vakakis, A.F., Golinval, J.C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006) 8. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill (1959) 9. Sathyamoorthy, M.: Nonlinear Analysis of Structures. Taylor & Francis (1997) 10. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley (2008) 11. Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modeling. CRC Press (2001) 12. Nagesh, M., Sharma, A., Allemang, R.J., Phillips, A.W.: Excitation techniques for nonlinear dynamic systems: a summary. In: Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 13. Scheel, M., Peter, S., Leine, R.I., Krack, M.: A phase resonance approach for modal testing of structures with nonlinear dissipation. J. Sound Vib. 435, 56–73 (2018) 14. Müller, F., et al.: Comparison Between Control-Based Continuation and Phase-Locked Loop Methods for the Identification of Backbone Curves and Nonlinear Frequency Responses - Nonlinear Structures & Systems, Volume 1, pp. 75–78 (2021) 15. Renson, L., Gonzalez-buelga, A., Barton, D.A.W., Neild, S.A.: Robust identification of backbone curves using control-based continuation. J. Sound Vib. 367, 145–158 (2016) 16. Peter, S., Leine, R.I.: Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation. Mech. Syst. Signal Process. 96, 139–158 (2017) 17. Peter, S., Riethmüller, R., Leine, R.I.: Tracking of backbone curves of nonlinear systems using phase-locked-loops. Nonlinear Dynamics. 1, 107–120 (2016) 18. Abramovitch, D.: Phase-Locked Loops: A Control Centric Tutorial, pp. 1–54. IEEE (2006)

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19. Best, R.E.: Phase-Locked Loops. McGraw-Hill (2003) 20. 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) 21. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.-C.: Nonlinear normal modes, Part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009) 22. MATLAB, 9.7.0.1190202 (R2019b). The MathWorks Inc., Natick, MA (2018) 23. Pandiya, N.: Design and validation of a MIMO nonlinear vibration test rig with hardening stiffness characteristics in multiple degrees of freedom. MS Thesis, University of Cincinnati, p. 133 (2017) 24. Nagesh, M., Allemang, R.J., Phillips, A.W.: Finite element (FE) model updating techniques for structural dynamics problems involving nonideal boundary conditions. In: Proc. ISMA 2020 - Int. Conf. Noise Vib. Eng. USD2020 - Int. Conf. Uncertain. Struct. Dyn., pp. 1937–1949 (2020)

Challenges of Characterizing Geometric Nonlinearity of a Double-Clamped Thin Beam Using Nonlinear Modal Testing Methods Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips

Abstract Phase-locked loop (PLL) controllers are increasingly employed for obtaining nonlinear frequency response curves (FRC) and backbone curves. Such controllers provide a specific phase lag between the response obtained and the excitation signal and isolate the nonlinear mode under consideration. The use of such feedback controllers provides many advantages against traditional sine-sweep methods and helps better characterize nonlinear behavior of dynamic systems. This paper briefly discusses about obtaining the nonlinear frequency response curves (FRC) and backbone curves of both symmetric and asymmetric modes of a double-clamped thin beam exhibiting geometric nonlinearity including a qualitative analysis such as stability and other prominent issues that arise during nonlinear modal testing particularly when the technique is applied to a thin and light structure. A comparison of nonlinear modal testing using PLL controllers as against other traditional methods such as sine-sweep methods is also demonstrated. Keywords Nonlinear modal testing · Phase-locked loop · Nonlinear frequency response · Geometric nonlinearity

1 Introduction Analysis of dynamic systems that exhibit nonlinear force-response relationships is very challenging. In particular, experimental characterization of such behavior is extremely challenging and requires adequate understanding of all aspects of experimentation while interpreting the experimental data. Nonlinearity is readily encountered in extremely thin and light structures, and widely available experimental modal analysis and modal parameter estimation techniques used for generally linear structures [1–5] cannot be applied to such thin and light structures. Current widespread use of such structures must be accompanied by adequate understanding of the complex phenomenon associated with them, such as variations of natural frequencies, deflection characteristics, damping, and other important parameters with variations in forcing levels and other relevant factors. In addition to the characteristics of the test structure, also of foremost importance during experimentation with thin and light structures is the use of traditional modal shakers to obtain such characteristics. While modal shakers are robust devices capable of various operations, their applications to thin and light structures are often limited and sometimes lead to incorrect characterization. Detailed theoretical analysis and modeling of dynamic behavior of nonlinear structures is widely available [6–8]. Experimental techniques for analysis of such nonlinear structures are widely available and well documented in [7–10]; common issues of applying linear modal analysis techniques to nonlinear structures are detailed in [11]. The variation of frequencies and other important parameters with variations in forcing levels applied to nonlinear structures is traditionally characterized using sine-sweep methods [7, 9, 10]. These methods provide frequency response curves (FRC) that show variations of characteristics at constant force level. A FRC is used for nonlinear analysis that provides actual response at a given force level instead of using a traditional frequency response function (FRF) approach used in linear systems that is generally force normalized. Lately, control-based phase resonance testing of nonlinear structures has gained widespread popularity. Two such techniques are widely available and in current use by researchers, viz. phase-locked loop (PLL) and control-based continuation [12–16].

M. Nagesh () · R. J. Allemang · A. W. Phillips Structural Dynamics Research Laboratory (SDRL), Department of Mechanical and Materials Engineering, College of Engineering and Applied Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected]; [email protected]; [email protected] © 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_20

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Phase-locked loop (PLL), a widely used technique in communication systems [17] and other control applications, was successfully adopted for nonlinear modal testing as described in [12–16]. This technique involves exciting a nonlinear mode, the nonlinear counterpart to a linear mode [18, 19], such that the excitation and response are in phase-lag quadrature (90◦ phase difference). At this condition, the damping and external forcing cancel each other, and the response is purely from the underlying conservative system. The backbone curve (BBC) is a curve representing the variations of this frequency at which the system is in phase quadrature at various response levels of the system. A double-clamped thin beam has been widely studied both analytically and experimentally to understand, characterize, and quantify nonlinear dynamic behavior of structures [6–8]. Such beams may exhibit more than one type of nonlinearity; they easily undergo buckling and can be used to demonstrate nonlinear phenomenon such as bifurcations and they also exhibit strong geometric nonlinearity where variation of natural frequencies, deflection shape, and response characteristics of the beam is observed with variations in forcing levels. Often, the dynamics of such beams is a combination of multiple nonlinearities. The thin double-clamped beam shows complex softening-hardening type behavior that requires adequate understanding and experimental study. This paper focuses on the geometric nonlinearity aspect of a thin double-clamped beam although it is implicit that isolation of only one type of nonlinear behavior in such structures is impractical. Some benchmark results for a thin double-clamped beam are available in [7] where several aspects of the beam loading are controlled and indirect excitation and measurement techniques are involved. However, in some situations direct measurement of structural response using lightweight accelerometers and force transducers attached to modal shakers is required. The characteristics of such experimentations may vary profoundly in comparison with results obtained from non-contact techniques. In this paper, a thin double-clamped beam with a slight unknown curvature due to an initial unknown buckling is considered, and excited using a shaker directly attached to the beam through a miniature force sensor. The responses are measured directly from the beam as well and variations in the softening-hardening type behavior of the beam for the first two modal frequencies, viz. first symmetric and first antisymmetric modes, are studied in this paper. A phase-locked loop is employed to obtain the backbone curve (BBC), and frequency response curves (FRC) are used in conjunction with the backbone curve (BBC) to better understand the nonlinear characteristics for the two modes under investigation.

2 Theoretical Background Dynamic systems can be represented by a second-order differential equation as shown in Eq. (1), where M denotes the mass matrix, C denotes damping matrix (assumed viscous), K denotes the linear stiffness matrix, FNL denotes the nonlinear restoring forces (conservative), and Fext denotes external forces acting on the system comprising m degrees of freedom. M x(t) ¨ + C x(t) ˙ + Kx(t) + fNL (x(t)) = Fext (t)

(1)

A detailed mathematical explanation about phase resonance testing and corresponding implementation of a phase-locked loop is available in [15]. When a nonlinear normal mode type motion is enforced on the system represented in Eq. (1) and the underlying conservative system is isolated as represented by Eq. (2), the external forces and damping cancel each other out as represented by Eq. (3). M x(t) ¨ + Kx(t) + fNL (x(t)) = 0

(2)

C x(t) ˙ = Fext (t)

(3)

Similar to the approach followed in [15], the motion of the dynamic system can be represented by a Fourier series as shown in Eq. (4), where Xn is a complex vector of dimensions (m X 1) that represents the complex Fourier coefficients for any harmonic n. The complex notation enables correct phase representation for the motion of individual degrees of freedom of the system which is important for isolating symmetric and antisymmetric modes of the system. Thus, the damping forces fC can be represented by Eq. (5) using Eq. (1), where Vn = inωXn . This indicates that all harmonics of the damping forces are at 90◦ phase difference with reference to the displacement. x(t) =

∞ 

Xn einωt

(4)

n=−∞

fC = C

∞  n=−∞

inωXn einωt = C

∞  n=−∞

Vn einωt

(5)

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If the external excitation is also considered periodic and described by a Fourier series as shown in Eq. (6), then it can be said that for a perfect nonlinear mode isolation, the displacement compulsorily has a phase difference of 90◦ with the external excitation, i.e., phase lag quadrature exists between excitation and motion of the system at all harmonics across all degrees of freedom as shown in Eq. (7). The backbone curve (BBC) obtained using a phase-locked loop (PLL) is essentially a curve whose constituent points all satisfy this phase lag quadrature between the excitation and response at all degrees of freedom. Fext (t) =

∞ 

Fn einωt

(6)

Fn = CVn = CinωXn

(7)

n=−∞

3 Phase-Locked Loop: Theory and Implementation A phase-locked loop (PLL) is a control system that maintains specific phase relationship between the input and output signal. It follows from the corollary that two signals starting from the same reference will have same frequency only when the phase between them is zero. Conversely for dynamic systems, the same principle is used to obtain a phase quadrature relationship between the excitation and displacement; the frequency where the quadrature is observed can be assumed to satisfy one or more conditions for a normal mode (linear or nonlinear) as described in the previous section. Traditional sine-sweep type approaches to nonlinear testing lack any such control loops that provide monitoring and control of input and output relationships. The schematic for a simple PLL circuit is given in Fig. 1. The circuit comprises a phase detector, a loop filter, and a voltage-controlled-oscillator (VCO). The phase detector compares the input p(t) and output s(t) signals and provides the comparison information to a signal q(t). The DC component of this signal q(t) typically contains the phase information and hence a loop filter is used to extract this information and provided to a VCO that generates a signal that is proportional to the phase information obtained and the target phase required. Detailed descriptions for each of these components are provided in [15]. For nonlinear system responses and nonlinear modal testing, the fundamental harmonic of the frequency under consideration is tuned to obtain phase lag quadrature and effects of other higher harmonics are disregarded for the PLL operation. Additional filtering may be required to ensure this assumption is valid throughout the PLL operation. The operations of the above-described schematic can be implemented using a software platform [17] as followed in this paper. A software phase-locked loop (sPLL) has several advantages particularly for experimental analysis of vibration and dynamic systems. Traditional PLL circuits require experience for adjusting stability of the loop using additional PID controllers. When a sPLL is used, since various parameters are controllable, stability of the PLL can be easily achieved with simpler operations. In addition, the use of sPLL enables averaging of various parameters. Most experimental vibration analysis requires robust averaging to ensure confidence in the data obtained. Moreover, most modern vibration testing equipment already have ready software packages that are easily adaptable for implementation of a sPLL and various other data acquisition and processing techniques can be easily adapted for nonlinear modal testing. The sPLL method used in this paper uses MATLAB [20] and its available features as described in the further sections. A VXI mainframe is used for data acquisition and control of the modal shaker through a power amplifier. The complete schematic for operation of the sPLL is given in Fig. 2. Consistent with [15], two assumptions are made for nonlinear modal testing using sPLL, viz. (1) The fundamental harmonic is used for phase computations and effect of higher harmonics are ignored and (2) The RMS of input and output quantities is mostly dominated by the signal at the fundamental frequency and hence is used for force and response computations. A linear modal test and modal parameter estimation is initially performed on the system to establish the linear modes, mode shapes, and other relevant information. Once a mode has been selected for nonlinear analysis, a steady sine wave is

Fig. 1 Phase-locked loop schematic

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Fig. 2 Software phase-locked loop schematic

generated at a frequency very close to this linear modal frequency. At extremely low amplitudes of vibration, the steady sine wave excitation and response will be at quadrature at almost the same frequency as the linear modal frequency of the chosen mode. To obtain a backbone curve, this response and frequency of quadrature are essential. Excitation levels are steadily increased which disturbs the phase quadrature between the excitation and response. Depending on the phase difference of this new configuration, the frequency of excitation is slowly adjusted to re-establish the phase lag quadrature. Once this condition has been reached, the sPLL does not change the excitation characteristics any further and is hence in a locked state. Since the test structure considered in this paper is a thin and light double-clamped beam, sudden changes in excitation parameters may cause inadvertent jumps that are undesired particularly at very high excitation levels. Hence, a frequency step size of 0.1 Hz is used for coarse adjustments (0 <  φ  < 80 or 100 <  φ  < 180) and 0.01 Hz is used for fine adjustments (80 1, wl sin 2πτ l s − R, if m = 1,

(4)

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Fig. 1 illustration of the energy-based prediction of frequency responses from conservative backbone curves. Plot (a) shows the phase condition between the l-th harmonic of Qf (t), indicated with the notation [Qf ]l (t), and the forcing such that Ml:1 (s) has the form in Eq. (4). This latter function is shown in plot (b) for three different conservative motions, also highlighted in plot (c) with corresponding colors. Plot (c) shows the conservative backbone curve (blue line) and the frequency response (red line, the solid part depicts asymptotically stable motions, while the dashed part unstable ones) in the classic amplitude A and frequency Ω plane

˙ f (t) = q˙  (t)F , under the where wl is the amplitude of the l-th harmonic in the Fourier series of the periodic function Q 0  assumption that the l-th harmonic of the Fourier series of Qf (t) = q0 (t)F has the opposite phase of forcing, as shown in Fig. 1(a). First, we note that only primary resonances m = l = 1, so that T = τ , and subharmonic resonances, m = 1, l > 1, are possible, whenever the forcing direction is not orthogonal to the conservative motion. The term subharmonic refers to the fact that the forcing frequency is higher than that of the response. Other types of resonances occurring when m > 1, i.e., superharmonic, ultrasubharmonic, or ultrasuperharmonic ones, are always destroyed by the damping for small monoharmonic forcing and weak positive definite damping. When m = 1, the forcing contribution wl must overcome the resistance R of the damping in order to generate a response; otherwise the Melnikov function shapes as the yellow line in Fig. 1(b). We also note that, for subharmonic resonances l > 1, the motion amplitude hence needs to have a contribution on the l-th harmonic and this may only occur for sufficiently large amplitudes or nonlinear terms, since the linearized resonant response only features a single harmonic. Therefore, subharmonic resonances typically appear as isolated response curves from the main branch of the frequency response. If wl > R, then the energy balance in (3) has two zeros s+ , s− in a forcing period where M1:l (s+ ) = M1:l (s− ) = 0, M1:l (s+ ) > 0 and M1:l (s− ) < 0, as shown with the green line in Fig. 1(b). This means that two forced-damped frequency responses bifurcate from the conservative limit q0 (t). If the damping is acting positively, these  two solutions have opposite stability type as discussed in [4]. For hardening-type behaviors ω (h0 ) > 0, the solution occurring from the phase shift s+ is unstable, while the other one from s− is asymptotically stable, cf. Fig. 1(c). The converse is true  for locally softening conservative backbone curves at which ω (h0 ) < 0. This analytical conclusion matches with numerical and experimental studies, thus explaining for multi-degree-of-freedom systems the hysteretic behavior occurring when the frequency response is in the vicinity of conservative backbone curves [1, 2, 9, 10]. The case wl = R indicates that a single, forced-damped solution persists from q0 (t), having a phase shift equal to T/4, which is depicted in Fig. 1(b). Specifically, the displacement response projected onto the forcing direction, i.e., the product q0 (t)F , is in phase lag quadrature with respect to the forcing signal. When this occurs, then the conservative periodic solution q0 (t) is very close to either maximal or minimal responses of the frequency response curve, as highlighted in Fig. 1(c). In other words, this analytical evidence extends the phase lag quadrature criterion to more complex damping shapes with respect to previous studies [9]. We also remark that the phase lag needs to be evaluated looking at the same displacement location where the forcing excites the system and with reference to the same harmonic of the forcing, namely the l-th harmonic. This is particularly important for systems that exhibit non-synchronous NNM motions.

3 Conclusion We have shown some among the conclusions that can be derived from applying exact energy arguments for evaluating the survival of conservative backbone curves as forced-damped motions. These are justified from a mathematically rigorous Melnikov analysis, which can be also exploited for investigating existence and stability of motions arising from other nonconservative forces (e.g., parametric forcing) with respect to those considered in this paper.

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References 1. Vakakis, A., Manevitch, L., Mikhlin, Y., Pilipchuk, V., Zevin, A.: Normal Modes and Localization in Nonlinear Systems. Wiley Blackwell, New York, NY (2008) 2. Kerschen, G., Peeters, M., Golinval, J., Vakakis, A.: Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23, 170–194 (2009) 3. Cenedese, M., Haller, G.: How do conservative backbone curves perturb into forced responses? A Melnikov function analysis. Proc. R. Soc. London A. 476, 20190494 (2020) 4. Cenedese, M., Haller, G.: Stability of forced–damped response in mechanical systems from a Melnikov analysis. Chaos. 30, 083103 (2020) 5. Nayfeh, A., Mook, D.: Nonlinear Oscillations. Wiley, New York, NY (2004) 6. Sanders, J., Verhulst, F., Murdock, J.: Averaging methods in nonlinear dynamical systems. In: Applied Mathematical Sciences, vol. 59, 2nd edn. Springer, New York, NY (2007) 7. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273, 77–101 (2004) 8. Hill, T., Cammarano, A., Neild, S., Wagg, D.: Interpreting the forced responses of a two- degree-of-freedom nonlinear oscillator using backbone curves. J. Sound Vib. 349, 276–288 (2015) 9. Peeters, M., Kerschen, G., Golinval, J.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330, 486–509 (2011) 10. Renson, L., Gonzalez-Buelga, A., Barton, D., Neild, S.: Robust identification of backbone curves using control-based continuation. J. Sound Vib. 367, 145–158 (2016) 11. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. In: Applied Mathematical Sciences, vol. 42. Springer, New York, NY (1983)

Joint Interface Contact Area Predictions Using Surface Strain Measurements Aryan Singh and Keegan J. Moore

Abstract Bolted joints are vital constituents of almost every small- to large-scale built-up structure. An especially important aspect of bolted joint modeling is the prediction of the contact area inside the interface of the joint based on the torque applied to the joint and surface characteristics. This is because the joint interface contact area effectively determines the stiffness of the joint as well as the dissipative qualities resulting from slip-stick friction interactions. This research introduces a novel method of estimating actual contact areas inside the interfaces of bolted joints by measuring the strains on the external surfaces induced by the tightening of a bolt. This method can determine contact areas without directly interfering with or altering the interface like existing methods. The experimental specimen used to demonstrate the technique consists of two flat plates with a carriage bolt on the top plate. A carriage bolt instead of a conventional bolt has been used to facilitate surface strain measurements, measured using the digital image correlation measurement technique. Contact areas inside the interface are measured using pressure-sensitive films, and the torque is measured using a digital torque wrench. Keywords Joints · Interface contact area · Data-driven modeling · Digital image correlation · Surface strains

1 Introduction Mechanical joints have a significant impact on the response of built-up structures. A critical amount of nonlinearity is introduced into the dynamics because of friction, wear, and non-idealized boundary conditions [1]. In fact, the stiffness and the dissipative characteristics resulting from slip-stick friction interactions inside the joint effectively depend upon the contact area. Thus, the prediction of the contact area inside the interface of the joint is an important aspect of modeling bolted joints. However, since the contact area inside the surface cannot be measured directly without altering the joints, the prediction of the contact area remains a challenge. Also, standard theory and the finite element models predict that perfectly circular contact areas arise in the interface, while the experimental measurements obtained using pressure-sensitive films rarely display a circular contact area. These contact areas vary substantially depending on bolt torque and the three-dimensional shape of the surface.

2 Experimental Specimen In this work, we consider a set of two flat steel plates of equal thickness. One of the plates is fixed to the optical table using six aluminum bars, as shown in Fig. 1. To facilitate the movement of the torque wrench, we did not put the aluminum bar on one side of the plate. We measure the surface displacement of the top plate using high-speed, three-dimensional digital image correlation at 1000 frames per second (FPS) for 30 s using two Photron Mini-AX100 540K-M-32 GB high-speed digital cameras and VicSnap software (Correlated Solutions Inc., Irmo, SC, United States). We crop the image to 800 by 764

A. Singh () · K. J. Moore Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 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_22

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(a)

(b)

Experimental system

(c)

Surface Strains

Contact Pressure

Fig. 1 (a) Experimental setup, (b) surface strain on top plate measured using high-speed digital cameras and DIC, (c) non-circular contact area found using pressure indicating film in the joint interface

pixels to maximize the recording time, and each measurement resulted in 60,000 pictures for a total of 20 GB of data. The full-field, three-dimensional displacements are obtained using Vic3D analysis software (Correlated Solutions Inc.) with a subset size of 23 and a step size of 11. This combination of subset and step size resulted in three-dimensional displacements at 72 points along the length of the plate and 64 points along the width of the beam for a total of 4608 measurement points. A carriage bolt was used instead of a regular hex bolt to avoid having to use a wrench to hold the bolt head still while tightening or loosening. The use of a carriage bolt maximizes the area around the bolt where strains can be measured (i.e., the presence of a wrench at the top blocks part of the plate during tightening or loosening). A square hole was machine in the top plate to accommodate the carriage bolt and to prevent it from rotating during assembly.

3 Results To further investigate the surface measurement technique being used, we measured the resulting strains when a halfwasher was placed inside the interface. We also measured the strains while tightening and loosening the carriage bolt. Our expectation with these sets of the experiment was to observe a clear distinction in the strain field because of the presence of the half-washer on one side of the interface. We also expected to see the exact opposite strain fields while loosening the carriage bolt. The result of such a set of experiments is presented in Fig. 2. It can be seen in Fig. 2(c) that we have a positive strain on one side of the interface and a negative strain on the other side while tightening the carriage bolt. The opposite behavior is observed when the bolt is loosened as can be seen in Fig. 2(d).

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Half -washer

Surface Strains – Tightening

(d)

Contact Pressure

Surface Strains – Loosening

Fig. 2 (a) Half-washer on top of an unused pressure film, (b) contact pressure due to half-washer in the interface, (c) corresponding residual surface strains after tightening, (d) surface strains after loosening

Reference 1. Kosova, G., et al.: Nonlinear system identification of a jointed structure using full-field data: part II analysis. In: Kerschen, G., Brake, M.R., Renson, L. (eds.) Nonlinear Structures & Systems, vol. 1, pp. 185–188. Springer, Cham (2021)

Towards Compact Structural Bases for Coupled Structural-Thermal Nonlinear Reduced Order Modeling X. Q. Wang and Marc P. Mignolet

Abstract Great progress has been made in the last two decades on the construction of non-intrusive reduced order models (ROMs) for the prediction of the response of structures with nonlinear geometric effects subjected to mechanical loading. Nevertheless, some challenges remain when the technique is extended to coupled structural-thermal problems. One such challenge is the construction of basis functions to account for the thermal effects on the structural deformations, especially when the temperature field is local and varies with time. The basis construction considered here starts with the basis relevant to the structure without temperature effects and then adds “enrichment modes” that capture the specificities of the thermal response. A systematic analysis of such possible enrichments and their potential benefits was recently performed (Wang and Mignolet, Proceedings of the 38th IMAC, conference and exposition on structural dynamics, Houston, TX, 2020). In the present study, the curved panel studied in that investigation is considered again but the optimal enrichment strategy established there is extended to a two-temperature-field local heating scenario, heating near the quarter of the panel in one case and near its middle in the second. The established enrichment strategy is firstly used to construct the enriched structural basis that captures the response of the panel under any linear combination of the two temperature fields which serve as two thermal modes. Two approaches are then followed to reduce the number of nonlinear enrichment modes and construct compact ROM bases. The first approach invokes the recently developed “progressive POD” method which was originally used for the reduction of the CFD data stored in multidimensional arrays (Wang et al., J. Aircr., 56:2248–2259, 2019). A notable reduction in the size of the basis is observed with this method. The second approach is using the static condensation to incorporate the in-plane components of the enrichment modes. The three ROMs constructed were found to lead to predictions of the structural responses that closely matched their counterparts determined from the underlying full finite element model. Keywords Heated structure · Reduced order modeling · Nonlinear geometric vibration · Basis enrichment for thermal effects

1 Introduction The construction of non-intrusive reduced order models (ROMs) for the prediction of the response of structures undergoing large displacements, i.e., with nonlinear geometric effects, due to mechanical loading has received significant attention and great progresses have been made in the last two decades. Nevertheless, some challenges remain when the technique is extended to coupled structural-thermal or structural-thermal-aerodynamic problems. One such challenge is the construction of basis functions to account for the thermal effects on structural deformations, especially when the temperature field is local and varies with time, e.g., see [1–5]. Appropriate basis functions have been devised as enrichment modes added to the isothermal basis used for mechanical loads only, but the form of these enrichments has varied with the particular application. Recently a systematic analysis of possible enrichments and their potential benefits was performed [6], showing that dominant modes extracted from structural responses to temperature loads related to thermal modes are most efficient as the enrichments. This analysis was carried out using temperature fields generated experimentally by a local heating, see [7] for discussion, on the panel of Fig. 1 which is a part of a cylindrical shell with radius of curvature of 100 in. curved along the x-axis while straight along the z-axis. When projected on the x-z plane, its dimensions are 9.75 in. by 15.75 in., and its

X. Q. Wang · M. P. Mignolet () Faculties of Mechanical and Aerospace Engineering, School for Engineering of Matter, Transport, and Energy, Arizona State University, Tempe, AZ, USA e-mail: [email protected]; [email protected] © 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_23

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Fig. 1 Finite element model of the curved panel

thickness is 0.048 in. The panel material is stainless steel with elastic modulus of 2.85 × 106 psi, Poisson’s ratio of 0.3, and density of 7.48 × 10−4 lb-s2 /in4 . The panel is clamped along all its edges. A finite element model of the panel was constructed using 2457 CQUAD4 shell elements (a 64-by-40 nodes mesh) in Nastran. In [6], the temperature fields considered were those induced only by a local heating near the quarter of the panel as in Fig. 2(a). In the present scenario, there are two basic temperature fields: the first one, as in [6], due to heating near the quarter of the panel and the second one induced by heating near its center, see Fig. 2(b). The focus of the present investigation is on the structural response of the panel to temperature distributions which are linear combinations of these two basic ones, considered as two thermal modes. One such a linear combination is shown on Fig. 2(c). In the present study, the enrichment procedure developed in [6] is firstly followed to construct the enriched structural basis for the two-temperature-field scenario, and the ROM coefficients are identified. A linear random combination of the two temperature fields is generated and the responses of the curved panel to a set of scaled versions of this temperature distribution are computed using both ROM and Nastran. As will be shown, the agreement between these Nastran and ROM predictions is very good but the number of enrichment modes is relatively large. In order to obtain a compact basis, two methods are next developed to reduce the number of enrichment modes. One is the progressive POD method which was originally used for the reduction of the CFD data stored in multidimensional arrays [8]. The other method relies on a static condensation of the enrichment modes into the other modes to de facto remove them from the basis. The performance of the two methods is finally assessed using the same randomly combined temperature field as compared to the original enriched basis.

2 Coupled Structural-Thermal NLROM In the coupled structural-thermal nonlinear reduced order models (NLROM) considered here, the structural displacement field u and the temperature field T are expressed as u (t) =

NS 

qn (t) φ (n) ,

(1)

τn (t) T (n) ,

(2)

n=1

and T (t) =

NT  n=1

where the functions φ (n) and T (n) are structural and thermal basis functions (modes) defined in the undeformed configuration, respectively. They satisfy the corresponding boundary conditions.

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(b)

(c) Fig. 2 Three temperature fields in the two-temperature-field scenario. (a) Quarter heating, (b) center heating, and (c) a linear combination of the quarter and the center heating

Using a Galerkin approach, the governing equations of the coupled structural-thermal NLROM have been derived based on thermoelasticity theory, see [1] for details. In particular, when the structural properties are independent of the temperature, the governing equations for the structural generalized coordinates qn (t) are   (1) (th) (2) (3) (th) Mij q¨j + Dij q˙j + Kij − Kij l τl qj + Kij l qj ql + Kij lp qj ql qp = Fi + Fil τl

(3)

The thermal effects on the structural deformation are twofolds: one of them is the change of the linear stiffness of the structure through the term Kij(th) l τl on the left hand side (LHS) of Eq. (3), usually inducing a softening effect responsible for (th)

thermal buckling. The other effect is the appearance of the pseudo force Fil τl on the right hand side (RHS) of Eq. (3), which typically induces notable in-plane deformations at the contrary of mechanical loads, e.g., pressure, which are typically transverse dominated. The in-plane deformations induced by the pseudo force are different from the membrane stretching effects due to the geometric nonlinear effect of large deformation, as will be shown later. The current investigation focuses on the construction of the structural basis φ (n) which must be able to capture the LHS and the RHS thermal effects, in addition to the nonlinear geometric effects due to large structural deformations. The strategy employed here for this construction is an “enrichment” approach, i.e., starting from the isothermal (cold) structural basis, additional modes (enrichments) are sought to capture the thermal effects. Various enrichment options have been defined in

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[6] and are classified through a triplet of 1’s or 0’s depending on the type of finite element computations they are derived from. The first number in the triplet refers to whether the RHS thermal effect is included in these computations, 1 if it is, 0 otherwise. The second number refers to whether the LHS thermal effect is included, 1 if it is, 0 otherwise. Finally, the third number of the triplet refers to the presence of geometric nonlinearity in the computations, it is a 1 if the enrichment is computed with nonlinearity, 0 otherwise. To assess the quality of the enriched basis, the representation error is employed. It is defined as: εre =

8 8u

8 8u

basis

8 − uNastran 8 8 × 100%, 8

(4)

Nastran

where uNastran is the vector of structural response to be modeled (i.e., Nastran displacement), and ubasis is its best approximation for a given basis , expressed as ubasis =  · q proj ,

(5)

where q proj is the vector of projection coefficients, which is obtained by a least squares fit of uNastran by ubasis . The displacement vector u could be the vector of displacements of all six degrees of freedoms or of a single one (usually a translation degree of freedom, either transverse or in-plane).

3 Enrichment for Two-Temperature-Field Scenario In [6], the procedure for constructing the enriched structural basis for one thermal mode was as follows: 1. A “cold” basis, i.e., a basis corresponding to the isothermal scenario is first constructed. 2. Linear enrichments: the linear static response to the single temperature field (the single thermal mode) is taken as one enrichment. 3. Nonlinear enrichments: the single thermal mode is scaled at a set of levels (different from the levels used in the validation), to create a set of temperature distributions to which the corresponding nonlinear static responses are determined with Nastran. A POD analysis of these nonlinear static displacements is carried out and the dominant POD eigenvectors are taken as nonlinear enrichment modes. When the strategy is extended to the current two-temperature-field scenario, step (1) is not changed, and the cold basis of 19 modes remains the same. In step (2), however, two linear enrichment modes are determined rather than a single one since there are two thermal modes. They are still given by the linear structural responses to the two temperature fields of the two thermal modes, each in turn. Similarly, in step (3), the temperature loads are not from a single thermal mode, but from the two thermal modes and their combinations. To this end, denote the two thermal modes as T n , n = 1, 2. The temperature distributions to generate the nonlinear enrichments are then constructed as   T ij,s = αi,s T i + αj,s T j /2,

(6)

where i and j are the indices of the thermal modes, s is the index of the chosen level and α i, s and α j, s are the corresponding scaling factors. In the current scenario, there are three combinations: two of them, T 11,s and T 22,s , represent the cases where the temperatures are constructed from a single thermal mode only, and the third, T 12,s , represents the case where the temperatures are constructed from the linear combination of the two thermal modes. In this latter case, it was specified that α i, s = α j, s = α s . For each combination, 11 levels (s = 1, 2, . . . , 11) are selected so that the nonlinear response data computed by Nastran covers the range from nearly linear (about 0.02 thickness) to strong nonlinear (about 4–5 thicknesses), then a POD analysis of the nonlinear static displacements is carried out, and the dominant POD eigenvectors are taken as the nonlinear enrichment modes. For the three combinations, a set of 16 nonlinear enrichment modes were taken and added to the isothermal basis, giving rise to a 37-mode basis. Of these 16 modes, 6 originated from the temperatures T 11,s , 5 from T 22,s , and 5 also from T 12,s . This latter number of modes suggests that the displacements induced by the combined temperature field T 12,s are essentially different from those induced by each temperature mode alone, even scaled.

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Representation error - Tz (%)

Representation error - Tx (%)

(a)

(b)

(c)

Fig. 3 Means and standard deviations of the representation errors of the nonlinear static responses to the set of random temperature fields vs. the number of modes in the 37-mode basis. (a) Transverse, Ty; (b) in-plane, Tx; and (c) in-plane, Tz

To check the quality of the enriched 37-mode basis, a set of 10 linear random combinations of the two temperature fields is constructed as T rand,i = ηi T 1 + (1 − ηi ) T 2 ,

i = 1, 2, . . . , 10,

(7)

where ηi is a random number from the uniform distribution in the interval (0,1). The temperature field shown in Fig. 2(c) is one of the random fields. Each random temperature field was then scaled at 11 levels and applied to Nastran to determine the corresponding nonlinear static response. Thus, a total of 110 displacement fields were obtained and the corresponding representation errors were then computed. Their means and standard deviations are shown in Fig. 3 as a function of the number of modes retained in the basis, starting from 19 corresponding to the cold basis to 37, the full basis constructed above. For this basis, the error in the transverse (Ty) direction is very small and the errors in the two in-plane directions (Tx and Tz) are about 1% and 3%, respectively, suggesting that the fully enriched basis is appropriate for these random temperature fields.

4 Methods for Construction of Compact Enriched Bases While the above enriched structural basis appears fully appropriate, a potential concern is that the number of nonlinear enrichment modes is relatively large for this 2 thermal mode scenario and it may increase rapidly if more thermal modes are necessary to represent the temperature fields well. Accordingly, it was of interest to investigate strategies to reduce the number of enrichment modes to obtain a compact basis and two such methods were developed as described below.

4.1 Method of Progressive POD One of the methods is the progressive POD (pPOD) method which was originally used for the reduction of the CFD data stored in multidimensional arrays [8]. For example, if a set of data U is written in the form of a three-dimensional matrix of dimensions n1 × n2 × n3 , there are different arrangements to reshape it into a two-dimensional matrix of dimensions

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M × N for a POD analysis. One arrangement is to select M1 = n1 × n2 and N1 = n3 , then the corresponding matrices of POD eigenvectors and POD coefficients will be M1 ×N1P OD and AN1P OD ×N1 so that UM1 ×N1 = M1 ×N1P OD · AN1P OD ×N1 . Another arrangement is to select M2 = n1 and N2 = n2 × n3 which leads to another set of POD matrices be M2 ×N2P OD and AN2P OD ×N2 . When the number of dimensions is larger, it is possible to reshuffle the matrix of POD coefficients and carry out further POD operation. Two conclusions from the pPOD experience in [8] are: (1) a set of successive POD operations can be carried out with the POD matrix of coefficients to further reduce the number of POD eigenvectors which is needed to have a good representation of the data; and (2) different arrangements lead to different levels of reduction, and an optimal arrangement can be found by going through various arrangements. Considering the nonlinear enrichments, the displacement data is naturally a three-dimensional matrix of dimensions equal to the number of structural degrees of freedom, Ndof , of levels, Nload , and of combinations, Ncombo , denoted as UNdof ×[Nload ×Ncombo ] . The POD enrichment procedure was carried out by grouping the two-dimensional data blocks, Ndof × Nload , for each combination of Ncombo , and the POD operation was carried out one combination at a time. The dominant POD eigenvectors, selected as the enrichment modes, were then stacked together as the final set of nonlinear enrichment modes generating a matrix Ndof ×NENL where NENL is the total number (=16) of the nonlinear enrichment modes. In the framework of the progressive POD, it is possible to further reduce the number of enrichment modes by doing a second POD operation, this time on the matrix of POD coefficients resulting from the projection of the displacement data on the final enrichment modes. These projection coefficients are grouped in the matrix ANENL ×[Nload ×Ncombo ] with ANENL ×[Nload ×Ncombo ] = #Ndof ×NENL UNdof ×[Nload ×Ncombo ] ,

(8)

where # denotes the pseudo inverse of a matrix. The second POD is carried out with the matrix ANENL ×[Nload ×Ncombo ] , so that ANENL ×[Nload ×Ncombo ] = NENL ×NP OD2 ANP OD2 ×[Nload ×Ncombo ]

(9)

Once this POD has been accomplished, the NENL enrichment modes have been reduced to a smaller set of NPOD2 such modes, referred to as the pPOD enrichment modes hereafter, which are the columns of the modal matrix Ndof ×NP OD2 = Ndof ×NENL NENL ×NP OD2 .

(10)

The above process was implemented with the 37-mode enriched basis which has 16 nonlinear enrichment modes and shown in Fig. 4 are the corresponding POD eigenvalues in decreasing order. It can be seen that the first three POD eigenvectors are dominant. To assess the performance of these pPOD enrichment modes, the means and standard deviations of the representation errors of the 110 displacement fields were computed and are shown in Fig. 5 as a function of the number of pPOD enrichment modes retained. It is seen from this figure that the representation errors in both transverse and in-plane directions are reduced significantly when taking only the three pPOD enrichment modes corresponding to the three dominant POD eigenvalues of Fig. 4. When taking more pPOD enrichments, the reduction of error is very slow. These three pPOD enrichment modes are thus taken to replace the original 16 enrichment modes, resulting in a 24-mode enriched basis.

Fig. 4 Eigenvalues of the secondary POD

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Fig. 5 Means and standard deviations of the representation errors of the nonlinear static responses to the set of random temperature fields vs. the number of modes in the 37-mode basis with pPOD enrichments. (a) Transverse, Ty; (b) in-plane, Tx; and (c) in-plane, Tz

4.2 Method of Static Condensation The other strategy investigated to reduce the size of the basis is to perform an approximate static condensation of the enrichments. There are significant assumptions associated with this process. Specifically, the enrichments must be very strongly in-plane dominant so that (1) the natural frequencies to which they are associated are much larger than those of the transverse modes. It is also required that (2) the nonlinear restoring forces present in the enrichment equations be essentially linear with respect to the enrichment generalized coordinates and involve a coupling with the transverse modes that is only through the quadratic terms of their generalized coordinates. Effectively, the equations for the enrichments generalized coordinates must be approximated in the form (th)

(1)

(th)

(2)

(th)

(1) Kee qe − Keel qe τl + Ket qt − Ketl qt τl + Kett qt2 = Fe + Fel τl ,

(11)

where the subscript e and t refer to the enrichments and to the rest of the basis, respectively. To maintain a cubic stiffness for the generalized coordinates of the remaining modes, it is further assumed that these have the following form before condensation (1)

(th)

(1)

(th)

(2)

(2)

(3)

(th)

Ktt qt − Kttl qt τl + Kte qe − Ktel qe τl + Kttt qt2 + Ktte qt qe + Ktttt qt3 = Ft + Ftl τl ,

(12)

Even then, for the structural-thermal coupled ROM, the static condensation is more complicated since the structuralthermal coupling coefficients also need to be considered. The process is exemplified below with a 2 mode model with 1 such enrichment (1 e mode and 1 t mode). Note that the linear coupling stiffness coefficients and the in-plane static forces, which are usually ignored in the static condensation, are considered in the present study. From Eq. (11),

qe =

    (th) (1) (th) (2) Fe + Fel τl − Ket qt − Ketl qt τl + Kett qt2 (1) (th) Kee − Keel τl

.

(13)

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Substituting Eq. (13) into Eq. (12), we have 

    (th) (2)  Fe + Fel(th) τl − Ket(1) qt − Ketl  qt τl + Kett qt2 (1) (th) (1) (th) (2) (2) (3) Ktt − Kttl τl qt + Kte − Ktel τl + Ktte qt + Kttt qt2 + Ktttt qt3 (1) Kˆ ee 

(th)

= Ft + Ftl τl

(14)

where (th) (1) (1) = Kee − Keel τl . Kˆ ee

(15)

After some manipulations, Eq. (14) can be written as       (1) (th) (th) (2) (th) (3) (th) (th) (1) (1) Kˆ tt − Kˆ ttl τl − Kˆ ttlm τl τm qt + Kˆ ttt − Kˆ tttl τl qt2 + Kˆ tttt qt3 = Ft + Fˆtl τl + Fˆtlm τl τm − Kte Fe /Kˆ ee , (16) where     (1) (2) (1) (1) Kˆ tt(1) = Ktt(1) − Kte Ket(1) /Kˆ ee + Ktte Fe /Kˆ ee

(17a)

      (th) (th) (1) (th) (th) (1) (2) (th) (1) (1) (1) − Ktel Ket /Kˆ ee − Ktte Fel /Kˆ ee Kˆ ttl = Kttl − Kte Ketl /Kˆ ee

(17b)

  (th) (th) (th) (1) Kˆ ttlm = Ktel Ketm /Kˆ ee

(17c)

    (2) (2) (1) (2) ˆ (1) (2) (1) Kett Ket(1) /Kˆ ee Kˆ ttt = Kttt − Kte /Kee − Ktte

(17d)

    (th) (th) (2) (2) (th) (1) (1) − Ktte Ketl /Kˆ ee Kˆ tttl = −Ktel Kett /Kˆ ee

(17e)

  (3) (3) (2) (2) (1) Kˆ tttt = Ktttt − Ktte Kett /Kˆ ee

(17f)

    (th) (th) (1) (th) (th) (1) (1) + Ktel Fe /Kˆ ee Fˆtl = Ftl − Kte Fel /Kˆ ee

(17g)

  (th) (th) (th) ˆ (1) Fˆtlm = −Ktel Fem /Kee

(17h)

From the governing equation of the condensed structural-thermal coupled ROM, Eq.(16), it can be seen that the linear stiffness matrix and the modal force are now quadratic functions of the temperature generalized coordinates, and the quadratic stiffness matrix is a linear function of them, even though the material properties are independent of temperature. Owing to the assumptions described above, the static condensation was carried out through a splitting of the transverse (Ty component) and in-plane (Tx and Tz components) enrichment modes. Specifically, the original 16 enrichment modes were first split into 16 transverse (Ty component) and 16 in-plane (Tx and Tz components) enrichment modes. A POD analysis

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POD eigenvalue

POD eigenvalue

Towards Compact Structural Bases for Coupled Structural-Thermal Nonlinear Reduced Order Modeling

(a)

(b)

Fig. 6 Eigenvalues of POD eigenvectors of the transverse and the in-plane enrichment modes. (a) Transverse; and (b) in-plane

Maximum inplane displacement Tx / Thickness

Maximum inplane displacement Tz / Thickness

(a)

(b)

(c)

Fig. 7 Predictions of the nonlinear static responses to the temperature field of Fig. 2(c) scaled by a load factor. Computations from Nastran, the 37-mode enriched ROM and its reduced counterparts, the 24-mode ROM by the pPOD method and the 27-mode ROM by the static condensation. (a) Transverse, Ty; (b) in-plane, Tx; and (c) in-plane, Tz

was next performed for these two sets of modes separately, and shown in Fig. 6 are the corresponding eigenvalues. Among the 16 POD eigenvectors of the transverse enrichment modes, 6 of them have relatively small eigenvalues and are taken as the dominant enrichment modes to be added to the 19 + 2 cold modes and linear enrichments to form a 27-mode basis. Then, the 16 in-plane enrichment modes were condensed into those 27 modes.

5 Comparison of Three Enriched ROMs with Nastran The coefficients of the three enriched ROMs, i.e., the original 37-mode ROM, the pPOD 24-mode ROM, and the condensed 27-mode ROM, were identified and their predictions of the nonlinear structural responses to the random temperature field shown in Fig. 2(c) were computed. This temperature field was scaled by a factor ranging from nearly 0 to 2.5. At that final level, the maximum transverse displacement is approximately 4.5 thicknesses. The ROM predictions as compared to the Nastran results are shown in Fig. 7. For the transverse displacement (Ty), all three ROMs give very good prediction, and the 37-mode and the 24-mode ROMs are slightly better. For the in-plane displacements (Tx and Tz), the 37-mode ROM has very good predictions at lower load levels and they are not as good at higher levels, probably because it includes a larger number

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of coefficients and thus is more prone to be affected by identification errors\uncertainties. The 24-mode pPOD ROM and the 27-mode condensed ROM, on the contrary, have better predictions at higher load levels.

6 Conclusions The present investigation continues the work carried out earlier by the authors in [6] to construct a structural reduced order model basis that is applicable to nonlinear geometric responses induced by temperature. In this earlier investigation, a single temperature distribution (or mode) of varying peak temperature was considered. The present effort first extends the construction of the basis to 2 temperature modes but further investigates two possible strategies to reduce the number of enrichments added to the cold basis to account for the effects of temperature. The first such strategy is based on a secondary POD step which was found to reduce the number of enrichments from 16 to 3 in the example treated. The second strategy involves a static condensation of the part of the enrichments. This approach was also found efficient, reducing the number of enrichments from 16 to 6. The predictions of the displacements induced by a specific temperature distribution and obtained with the full set of enrichments and with the two reduced bases are found to match well to very well the corresponding Nastran predictions for peak transverse responses up to 4.5 thicknesses. Acknowledgments The authors gratefully acknowledge the support of this work by the AFRL-University Collaborative Center in Structural Sciences (Cooperative Agreement FA8650-13-2-2347) with Dr. Ben Smarslok as program manager.

References 1. Matney, A., Mignolet, M.P., Culler, A.J., McNamara, J.J., Spottswood, S.M.: Panel response prediction through reduced order models with application to hypersonic aircraft. In: Proceedings of the AIAA Science and Technology Forum and Exposition (SciTech2015), Orlando, FL, 5–9 Jan 2015, AIAA Paper AIAA 2015-1630 2. Matney, A., Perez, R., Mignolet, M.P.: Nonlinear unsteady thermoelastodynamic response of a panel subjected to an oscillating flux by reduced order models. In: Proceedings of the 52nd structures, structural dynamics and materials conference, 4–7 Apr 2011, Denver, Colorado, AIAA 2011-2016 3. Perez, R., Wang, X.Q., Mignolet, M.P.: Steady and unsteady nonlinear thermoelastodynamic response of panels by reduced order models. In: Proceedings of the 51st structures, structural dynamics, and materials conference, Orlando, FL, 12–15 Apr 2010, Paper AIAA-2010-2724 4. Gogulapati, A., Deshmukh, R., Crowell, A.R., McNamara, J.J., Vyas, V., Wang, X.Q., Mignolet, M., Beberniss, T., Spottswood, S.M., Eason, T.G.: Response of a panel to shock impingement: modeling and comparison with experiments. In: Proceedings of the AIAA Science and Technology Forum and Exposition (SciTech2014), National Harbor, MD, 13–17 Jan 2014, AIAA Paper AIAA 2014-0148 5. Gogulapati, A., Brouwer, K., Wang, X.Q., Murthy, R., McNamara, J.J., Mignolet, M.P.: Full and reduced order aerothermoelastic modeling of built-up aerospace panels in high-speed flows. In: Proceedings of the AIAA Science and Technology Forum and Exposition (SciTech2017), Dallas, TX, 9–13 Jan 2017, Paper AIAA 2017-0180 6. Wang, X.Q., Mignolet, M.P.: Enrichment of structural bases for the reduced order modeling of heated structures undergoing nonlinear geometric response. In: Proceedings of the 38th IMAC, conference and exposition on structural dynamics, Houston, TX, 10–13 Feb 2020. Paper 8472 7. Gordon, R.W., Hollkamp, J.J.: Reduced-order models for acoustic response prediction of a curved panel. In: Proceedings of the 52nd structures, structural dynamics and materials conference, Denver, CO, 4–7 Apr 2011, Paper AIAA 2011-2081 8. Wang, X.Q., Sarhaddi, D., Wang, Z., Mignolet, M.P., Chen, P.C.: Modeling-based hyper-reduction of multidimensional computational fluid dynamics data: application to ship airwake data. J. Aircr. 56, 2248–2259 (2019)

Ensemble of Multi-time Resolution Recurrent Neural Networks for Enhanced Feature Extraction in High-Rate Time Series Vahid Barzegar, Simon Laflamme, Chao Hu, and Jacob Dodson

Abstract Systems experiencing high-rate dynamic events, termed high-rate systems, typically undergo accelerations of amplitudes higher than 100 g in less than 10 ms. Examples include adaptive airbag deployment systems, hypersonic vehicles, and active blast mitigation systems. Given the critical functions of such systems, accurate and fast modeling tools are necessary for ensuring the target performance. However, the unique characteristics of these systems, which consist of (1) large uncertainties in the external loads, (2) high levels of nonstationarities and heavy disturbances, and (3) unmodeled dynamics generated from changes in system configurations, in combination with the fast-changing environment, limit the applicability of physical modeling tools. In this chapter, a neural network-based approach is proposed to model and predict high-rate systems. It consists of an ensemble of recurrent neural networks (RNNs) with short-sequence long short-term memory (LSTM) cells which are concurrently trained. To empower multi-step-ahead predictions, the input space for each RNN is selected individually using principal component analysis that extracts different resolutions on the dynamics. The proposed algorithm is validated on experimental data obtained from a high-rate system. Results showed that this algorithm significantly improves the quality of step-ahead predictions over a heuristic approach in constructing the input spaces. Keywords Recurrent neural network · Long short-term memory · Prediction · High rate · Nonstationary · Time series

1 Introduction High-rate systems are defined as those experiencing dynamic events of typical amplitudes higher than 100 g over durations less than 10 ms. Examples include adaptive airbag deployment systems, hypersonic vehicles, and active blast mitigation systems. Enabling closed-loop feedback capabilities for high-rate systems could empower their field deployments through enhanced operability and safety. However, this is a difficult task, as these systems are uniquely characterized by (1) large uncertainties in the external loads, (2) high levels of nonstationarities and heavy disturbances, and (3) unmodeled dynamics generated from changes in system configurations [1]. There have been recent research efforts in developing algorithms with real-time capabilities in the high-rate realm, including a sliding mode observer-based algorithm [2] and a frequency-based model updating strategy [3]. Others have studied algorithms enabling the online identification of highly nonstationary time series, without initial pre-training [4], but V. Barzegar () Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA, USA e-mail: [email protected] S. Laflamme Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA, USA Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, USA e-mail: [email protected] C. Hu Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, USA Department of Mechanical Engineering, Iowa State University, Ames, IA, USA e-mail: [email protected] J. Dodson Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, FL, USA e-mail: [email protected] © 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_24

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the algorithm was not applicable in real time. Inspired by this algorithm, the authors proposed an ensemble of recurrent neural networks (RNNs) constructed with short-sequence long short-term memory (LSTM) cells to learn nonstationary time series with minimal pre-training. The algorithm showed real-time capabilities with an average computation time of 25 μs, but its multi-step-ahead prediction was not evaluated [5]. In this chapter, we extend the work on the ensemble of RNN for step-ahead prediction. The algorithm is modified to select the input space of each RNN individually, such that different dynamic features are extracted from the time series. The extraction method is based on the embedding theorem, as recently used by others in [6] and principal components analysis (PCA) of the available time series data.

2 Algorithm Architecture The machine learning algorithm is described in [5]. Briefly, it consists of an ensemble of RNNs constructed with long shortterm memory (LSTM) cells with transfer learning capabilities to cope with the highly limited availability of training data as is typical for high-rate systems. The use of an ensemble of RNNs empowers multi-rate sampling capability to capture multitemporal features of the time series, thus enabling modeling of nonstationarities. Also, because the RNNs use short-sequence LSTMs and are arranged in parallel, the computation time is substantially reduced to the sub-millisecond range. Here, the procedure to select the inputs of each RNN is altered to provide multi-step-ahead prediction capabilities. Figure 1a depicts the proposed procedure for extracting individual features in the source domain. At each discrete time step k, an RNN maps the input space xk = {xk−dτ , xk−(d−1)τ , . . . , xk } to the next discrete value xk+τ , where τ is the time delay and d the embedding length. The number of extracted features is taken as the number of principal components used in representing at least 90% of the source domain. For each RNN, variables τ and d are selected based on the embedding theorem to represent the essential dynamics of the associated principal components using the mutual information (MI) [7] and false nearest neighbors (FNN) [8] tests. Figure 1b depicts the algorithm for real-time one-step prediction in the target domain. The LSTM cells trained in the source domain are transferred to the target domain and run in parallel, each sampling the time series at different rates as data sequentially becomes available. A multi-resolution sampler at time step k extracts xik for the ith LSTM. Note that data are organized such that the target prediction value for all of the LSTMs is xk+1 . The features extracted in the LSTM layers are linearly scaled in an attention layer using a linear neuron. The squared error of the prediction is back-propagated to the network for updating its weights. Multi-step-ahead prediction is conducted by iterating the algorithm over time.

3 Simulations on Drop Tower Data The proposed algorithm was validated using an experimental high-rate dynamic dataset obtained from an accelerated drop tower test [4]. Briefly, the setup, illustrated in Fig. 2a, consists of an electronics package with four circuit boards mounted in a canister on an accelerated drop tower. At each time step k, four accelerometers measure the vibrations of the boards sampled at 1 MHz. In this study, a single time series from accelerometer TS1 is used as the source domain, and five different time series from accelerometer TS2 produced from five different tests are used for target domain prediction. A total of 5 RNNs are used. To investigate the performance of the proposed approach (labeled as “PCA inputs”), a comparison is made with the case where the input parameters were selected through a grid search (GS) for one-step-ahead prediction only, as done in [5]. Prediction performance was assessed over the range of 1–20 steps ahead using the mean absolute error (MAE) and root mean square error (RMSE) metrics. Results are plotted in Fig. 2b–c. As expected, GS over-

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performed PCA over small prediction ranges. However, PCA shows more stable performance over larger prediction horizons, yielding better performance after approximately 5 steps ahead, attributable to the extracted features enabling the modeling of multi-resolution dynamics. A typical prediction time history is presented in Fig. 3 for 14 steps ahead, conducted every 14 steps. Note that 1000 time steps is equivalent to 1 ms. A flat section indicates a naive prediction where the algorithm reports the previously predicted value as the current prediction. At the beginning of the prediction, both approaches exhibit a naive behavior, but PCA quickly improves its predictive performance as observable in the chaotic event around time step 500 and after the event passed 600 time steps.

4 Conclusion In this chapter, a new approach for selecting the input space for an ensemble of RNNs was proposed, with the objective of enabling multi-step-ahead prediction for high-rate systems. The selection was conducted based on the embedding theorem conducted on principal components representing the dynamics of the source domain. The performance of the proposed approach was evaluated on a set of experimental data and compared to that of a grid search approach to organizing inputs. Results showed that the proposed approach outperformed the grid search approach for long prediction horizons.

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Acknowledgments The work presented in this chapter is funded by the National Science Foundation under award number CISE-1937460. Their support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this work are those of the authors and do not necessarily reflect the views of the sponsor. The authors also acknowledge Dr. Janet Wolfson and Dr. Jonathan Hong for providing the experimental data.

References 1. Hong, J., Laflamme, S., Dodson, J., Joyce, B.: Introduction to state estimation of high-rate system dynamics. Sensors 18(2), 217 (2018) 2. Joyce, B., Dodson, J., Hong, J., Laflamme, S.: Practical considerations for sliding mode observers for high-rate structural health monitoring. In: ASME 2018 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, New York (2018) 3. Downey, A., Hong, J., Dodson, J., Carroll, M., Scheppegrell, J.: Millisecond model updating for structures experiencing unmodeled high-rate dynamic events. Mech. Syst. Signal Process. 138, 106551 (2020) 4. Hong, J., Laflamme, S., Cao, L., Dodson, J., Joyce, B.: Variable input observer for nonstationary high-rate dynamic systems. Neural Comput. Appl. 32(9), 5015–5026 (2018) 5. Barzegar, V., Laflamme, S., Hu, C., Dodson, J.: Multi-time resolution ensemble lstms for enhanced feature extraction in high-rate time series. Sensors. 21(6), 1954 (2021) 6. Kim, K., Kim, J., Rinaldo, A.: Time series featurization via topological data analysis (2018). Preprint, arXiv:1812.02987 7. Belghazi, M.I., Baratin, A., Rajeshwar, S., Ozair, S., Bengio, Y., Courville, A., Hjelm, D.: Mutual information neural estimation. In: International Conference on Machine Learning, pp. 531–540 (2018) 8. Laflamme, S., Slotine, J.J.E., Connor, J.J.: Self-organizing input space for control of structures. Smart Mater. Struct. 21(11), 115015 (2012)

Modelling the Effect of Preload in a Lap-Joint by Altering Thin-Layer Material Properties Nidhal Jamia, Hassan Jalali, Michael I. Friswell, Hamed Haddad Khodaparast, and Javad Taghipour

Abstract The joints in an assembled structure represent a significant source of energy dissipation and may lead to overall stiffness variation, which may affect high cycle fatigue failure. Many approaches have been developed to model and simulate the dynamics of bolted joint structures. However, the inherent dynamics of the contact interfaces still need further investigation in order to be able to generate accurate models to predict the behaviour in the contact interface. In this paper, the modelling of the contact interface of a bolted lap-joint and the prediction of its pressure distribution are considered using 2D and 3D FE models. A 3D finite element model with solid elements is developed to simulate the behaviour of the contact interface. The model is a modified thin-layer element where the material properties of a thin layer are distributed over the contact interface. Due to the high computational cost of the 3D model, a reduced-order model is proposed for the lap-joint in which beam elements are used. The material properties are introduced in these models to account for the variability in the contact parameters. Finally, experimental modal properties were used to identify the joint parameters. A good agreement is obtained between the detailed model and the reduced-order model in the prediction of the pressure distribution in the contact interface. Keywords Pressure distribution · Bolted lap-joint · Detailed model · Modified thin-layer element · Joint parameters

1 Introduction In almost all practical mechanical structures, there is at least one mechanical joint. After more than half century of efforts on modelling the hysteretic behaviour of jointed structures, it is still one of the most complicated tasks in structural dynamics. Modelling and investigating the behaviour of mechanical joints have received considerable attention. These efforts have been intensified during recent decades due to progress in computational tools and experimental equipment. Bograd et al. [1] reviewed various identification approaches developed to model the dynamics of mechanical joints. Iwan [2] presented a model to investigate the yielding behaviour of continuous and composite materials and structures by complementing and extending some of the earliest works [3–7]. Iwan models are capable of predicting both transient and steady-state yielding behaviour of jointed structures. The Iwan model is one of the most important mathematical formulations introduced to predict the dynamic behaviour of joint contact interfaces. Ref. [8] gives an overview of the Iwan model for mechanical jointed structures and presents a reduced-order model to investigate the qualitative properties of such systems using a small number of parameters. Many research works have been carried out based on this model. Segalman and Starr [9] compared various constitutive models of joints and discussed how the Iwan model can represent the properties of Massing models. Segalman [10] and Li and Hao [11] developed a four-parameter Iwan model and a six-parameter Iwan model, respectively, to investigate the dynamic response of lap-jointed structures. Quinn and Segalman [12] investigated the applicability and performance of the series-series Iwan model in predicting the dynamics of jointed structures. There are various identification methods introduced in the literature for modelling the dynamic behaviour of jointed structures [13–17]. Ahmadian and Jalali [13] suggested modelling the dynamics of bolted lap-joints by introducing a

N. Jamia () · M. I. Friswell · H. H. Khodaparast · J. Taghipour Faculty of Science and Engineering, Swansea University, Swansea, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] H. Jalali Department of Mechanical and Construction Engineering, Northumbria University, Newcastle upon Tyne, UK e-mail: [email protected] © 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_25

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nonlinear generic element formulation. In [14], Ahmadian and Jalali proposed a nonlinear mathematical model for bolted lap-joints to predict the micro-slip/slap behaviour in the joint contact interface. Noel et al. [15] introduced a nonlinear statespace identification approach to estimate the hysteresis dynamics. Miller and Quinn [16] developed a two-sided interface model using a series-series Iwan model and an elastic chain to predict the dynamic behaviour of structures with frictional joints. They also developed a reduced-order model of the presented approach in order to reduce the computational costs of the modelling. Jalali et al. [18] showed the application of the force-state mapping technique in the identification of nonlinear lap-jointed structures. Thin-layer element theory is another method introduced by Desai et al. [19] to model the contact interface of jointed structures. By carrying out a parametric study, they illustrated that the ratio of the thickness of the thin-layer element to the mean dimension of the adjacent elements should be between 0.01 and 0.1. Jalali et al. [20] exploited thin-layer element theory to model the dynamics of bolted lap-joints experiencing micro-slip/slap behaviour. Ehrlich et al. [21] reduced the computational costs and calculation time required for model updating and uncertainty analysis using thin-layer element theory, by proposing a reduced-order model of the thin-layer element theory. Wang et al. [22] developed a parametric modelling approach, based on the finite element method and thin-layer element theory, to estimate the parameters of jointed structures. Iranzad and Ahmadian [23] modelled the dynamics of a lap-type mechanical joint using thin-layer element theory and utilizing experimentally measured data. Zhan et al. [24] focused on modelling and estimating the parameters of the dynamics of bolted joint structures. To this end, they utilized thin-layer elements to model the joint contact interface and exploited three-dimensional brick elements to model the substructures of the joint. This paper investigates modelling the dynamics of a bolted lap-joint utilizing a detailed three-dimensional model and a two-dimensional equivalent beam model. In both models, a modified thin-layer element approach is used to model the joint contact interface where the material properties of the thin layer are considered to be variable over the contact interface. The variable material properties resemble the distribution of the normal contact pressure in the contact interface. Impact modal testing was employed to obtain the measured natural frequencies of the lap-joint. A 3D detailed model and a 2D equivalent model are created using the thin-layer approach. An identification of the model parameters is employed using the experimental modal properties of the lap-joint obtained by modal testing.

2 Experimental Analysis of Lap-Joint In this section, modal testing is performed on a lap-joint structure in order to obtain the undamped natural frequencies, which will be used for the identification of the joint parameters in the next sections. Two different beams with two different lengths joined by two M8 bolts were designed and fabricated in order to compose a lap-joint structure. The design and dimension of the two beams are shown in Fig. 1. The beams are made of structural steel with the following material properties, E = 208 GPa, ρ = 7860 kg/m3 and υ = 0.3, where E is the Young’s modulus, ρ is the density and υ is Poisson’s ratio. In order to obtain the natural frequencies of the lap-joint, an impact hammer experiment was performed. In this experiment, a 4-channel (DAQ) system, an impact hammer and data acquisition software were used. In the modal testing procedure, care has been taken to keep the uncertainties to a minimum. Two identical M8 bolts and nuts are used to join the two beams and a KTC Digital Ratchet Torque Wrench was used to tighten the bolts to a specific level corresponding to a torque level equal to 23 Nm. Free-free boundary conditions were used for the lap-joint structure by suspending it using flexible strings as shown in Fig. 2.

Fig. 1 The lap-joint: (a) design, (b) dimensions

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In addition, three miniature PCB accelerometers with 0.5 g mass, 10 kHz frequency range and 10 mg/s2 sensitivity were bonded to the middle and the two extremities of the beams were used to measure the response of the structure. The details of the experimental setup are presented in Fig. 2. The measured natural frequencies are tabulated in Table 1 and they are used in the next sections to identify the joint parameters. In this study, the bending behaviour of the lap-joint structure is of interest. Therefore, the first eight bending modes and their corresponding natural frequencies are considered and will be used in the identification of the joint structure in the rest of this study.

3 The Modified Thin-Layer Approach The force-displacement behaviour of a contact interface in the normal and tangential directions is characterized by the condition of the contact surface and the level of the bolt preload. For instance, by increasing the bolt preload, the linear stiffness of the contact interface in the tangential direction increases. The other parameter which has a direct effect on the force-displacement behaviour is the size of asperities or the contact surface roughness characteristics. Both bolt preload and contact surface roughness are variable over the contact interface: the former has a deterministic nature and the latter has a stochastic nature. In the conventional thin-layer approach, equivalent constant material properties are considered for the layer representing the contact interface, as shown in Fig. 3(a). The constant material properties are the result of this assumption that both the contact pressure and surface roughness have constant distributions. In an attempt to consider the variation of the contact pressure on the thin-layer characteristics, in this paper, the distribution of material properties shown in Fig. 3(b) is considered for the thin layer. The applicability of this approach is examined using 3D and 2D models of the lap-joint structure shown in Fig. 2.

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4 3D Detailed Model In this section, a detailed 3D FE model of the lap-joint using the thin-layer approach is developed. The commercial software ANSYS Workbench is used to perform the FEM analysis. Two rectangular beams in contact with each other, as shown in Fig. 4(a), are considered. For simplicity, four cubes are employed to present the mass of the bolt heads and the nuts. The model geometry details are shown in Fig. 4(a). Structural steel with the same parameters as the real lap-joint was assigned as the material for the whole model. The dimensions of the model geometry are given in Fig. 1(b). The FE model was constructed using a fine mesh in the contact parts, as shown in Fig. 4(b). To provide a high accuracy in the contact interface, particular care was taken to ensure that the pairs of nodes on the two contact surfaces are coincident to ensure an accurate interface mesh. Tetrahedral solid elements were used in the mesh as shown in Fig. 4(b). The assembly has 57,000 elements and 65,070 nodes. The main approach in this paper is based on an identification process of the material distribution in the thin layer in order to simulate a bolted joint interface under the bolts’ preload. The difficulty raised here is the initial ‘guess’ of the material properties distribution in the contact interface in order to identify the final state of the material properties distribution. An assumption is made in this paper where the material properties in the contact interface behave similarly to the normal stress distribution in the thin layer. At the first stage, a thin layer with thickness equal to 2 mm is selected by selecting all the elements that are located in the contact interface. A constant Young’s modulus ETL is assigned to the thin layer. The bottom surfaces of the two cubes presenting the nuts are constrained as shown in Fig. 4(b). A force Fb is applied to the two upper cubes. This force presents the preload of the bolts (e.g. bolt pretensions). A static analysis is first performed to obtain the prestressed structure. This structure is then employed in a prestressed modal analysis to account for the effects of the applied force and the thin-layer stiffness on the natural frequencies of the structure. At the second stage, the natural frequencies are calculated at each set of the Young’s modulus ETL and the applied force Fb . The two values of ETL and Fb are identified by minimizing the difference between the measured and the simulated natural frequencies for the first eight bending modes. This operation is performed using the minimization function ‘fminsearch’ in MATLAB. The identified values are ETL = 1.262815928 GPa and Fb = −92.977 kN. The identified natural frequencies and the accuracy of the identified model parameters are reported in Table 2 for the experimental results corresponding to the first eight bending modes. After identifying ETL and Fb , a static analysis is performed using these two values to obtain the distribution of the normal stress at the contact interface, as shown in Fig. 5(a). Assuming that the material properties distribution in the thin layer is

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behaving as the distribution of the normal stress in the contact interface, the material properties distribution given in Fig. 5(b) is generated from the results shown in Fig. 5(a). Finally, to verify the assumption considered in this study, the obtained material properties distribution is employed in a modal analysis of the free-free case and the natural frequencies of the first eight bending modes are calculated. The accuracy of the obtained results is shown in Table 3. As shown in Table 3, the simulated natural frequencies are accurate for four modes. For the rest of the modes, the error is around 4% which can be reduced by using a more robust minimization algorithm, specifically that the identification in this study is performed over eight modes. Decreasing the number of modes identified might increase the accuracy.

5 2D Equivalent Model The same procedure as for the 3D case described in the previous section is followed for a 2D model of the beam structure shown in Fig. 2. Figure 6 shows the 2D FE model where the contact interface is modelled as a thin layer. The identified distribution of the material properties, i.e. E, in the contact interface is shown in Fig. 7 and Table 4 shows the comparison between the experimental and the identified natural frequencies. The results presented in these tables show that the method presented in this paper is able to model the effect of preload on contact interface.

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Fig. 6 The 2D FE model

Fig. 7 Identified distribution of the material properties in the thin layer using the 2D model

Table 4 Accuracy of the identified material properties distribution for the first eight bending modes (2D model) Mode No. Exp. (Hz) Simulated (Hz) Error (%)

f1 158.41 160.57 1.36

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f3 860.93 885.2 2.81

f4 1383.5 1424.3 2.80

f5 2045.7 2066.4 1.01

f6 2869.4 2980.4 3.87

f7 3438.9 3355.6 −2.42

f8 4279.7 4296.1 0.38

Modelling the Effect of Preload in a Lap-Joint by Altering Thin-Layer Material Properties

217

6 Conclusion In this paper 2D and 3D FE models were constructed to simulate a bolted lap-joint. The thin-layer approach was employed to simulate the contact interface of the joint structure. An identification of the joint model parameters was performed using the measured and simulated natural frequencies of the bending modes. The accuracy of the results showed that the material properties distribution in the thin layer can be generated from the normal stress distribution in the contact interface. This work showed a good agreement of the two models to predict the pressure distribution in the contact interface. This approach will be extended to stochastic modelling by considering a stochastic thin layer in the contact interface. Acknowledgements This research is funded by the Engineering and Physical Sciences Research Council through Grant no. EP/R006768/1. Javad Taghipour acknowledges financial support from the College of Engineering at Swansea University through the PhD scholarship in support of EPSRC project EP/P01271X/1.

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