Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Tom Proulx Society for Experimental Mechanics, Inc., Bethel, CT, USA For further volumes: http://www.springer.com/series/8922 D. Adams • G. Kerschen • A. Carrella Editors Topics in Nonlinear Dynamics, Volume 3 Proceedings of the 30th IMAC, A Conference on Structural Dynamics, 2012 Editors D. Adams G. Kerschen Purdue University University of Liege West Lafayette, IN, USA Belgium A. Carrella LMS International Leuven, Belgium ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4614-2415-4 e-ISBN 978-1-4614-2416-1 DOI 10.1007/978-1-4614-2416-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2012936657 # The Society for Experimental Mechanics, Inc. 2012 All rights reserved. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Topics in Nonlinear Dynamics represents one of six volumes of technical papers presented at the 30th IMAC, A Conference and Exposition on Structural Dynamics, 2012 organized by the Society for Experimental Mechanics, and held in Jacksonville, Florida, January 30–February 2, 2012. The full proceedings also include volumes on Dynamics of Civil Structures; Substructuring and Wind Turbine Dynamics; Model Validation and Uncertainty Quantification; and Modal Analysis, I & II. 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, in order to go From the Laboratory to the Real World 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. This volume addresses theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust computational algorithms) as well as experimental techniques and analysis methods. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. West Lafayette, IN, USA D. Adams Belgium G. Kerschen Leuven, Belgium A. Carrella Contents 1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure . . . . . . . . . . . . . . . . . . . . . . . 1 J.P. Noel, G. Kerschen, and A. Newerla 2 Nonlinear Dynamic Model and Simulation of Morphing Wing Profile Actuated by Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Cassio T. Faria, Carlos De Marqui Jr., Daniel J. Inman, and Vicente Lopes Jr. 3 Environmental Testing and Data Analysis for Non-linear Spacecraft Structures: Impact on Virtual Shaker Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Simone Manzato, Bart Peeters, Raphae¨l Van der Vorst, and Jan Debille 4 Using Impact Modulation to Detect Loose Bolts in a Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Janette Jaques and Douglas E. Adams 5 Nonlinear Modal Analysis of the Smallsat Spacecraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 L. Renson, G. Kerschen, and A. Newerla 6 Filter Response to High Frequency Shock Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Jason R. Foley, Jacob C. Dodson, and Alain L. Beliveau 7 Simplified Nonlinear Modeling Approach for a Bolted Interface Test Fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Charles Butner, Douglas Adams, and Jason R. Foley 8 Transmission of Guided Waves Across Prestressed Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Jacob C. Dodson, Janet Wolfson, Jason R. Foley, and Daniel J. Inman 9 Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Louis Thibault, Peter Avitabile, Jason R. Foley, and Janet Wolfson 10 Efficient Computational Nonlinear Dynamic Analysis Using Modal Modification Response Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Tim Marinone, Peter Avitabile, Jason R. Foley, and Janet Wolfson 11 Spectral Domain Force Identification of Impulsive Loading in Beam Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Pooya Ghaderi, Andrew J. Dick, Jason R. Foley, and Gregory Falbo 12 Free-Pendulum Vibration Absorber Experiment Using Digital Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Richard Landis, Atila Ertas, Emrah Gumus, and Faruk Gungor 13 Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 A. Nankali, Y.S. Lee, and T. Kalmar-Nagy 14 Force Displacement Curves of a Snapping Bistable Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Alexander D. Shaw and Alessandro Carrella 15 Characterization of a Strongly Nonlinear Vibration Absorber for Aerospace Applications. . . . . . . . . . . . . . . . . . . . 199 Sean A. Hubbard, Timothy J. Copeland, D. Michael McFarland, Lawrence A. Bergman, and Alexander F. Vakakis vii viii Contents 16 Identifying and Computing Nonlinear Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A. Cammarano, A. Carrella, L. Renson, and G. Kerschen 17 Nonlinear Identification Using a Frequency Response Function With the Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A. Carrella 18 Nonlinear Structural Modification and Nonlinear Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Taner KalaycIog˘lu and H. Nevzat O¨ zgu€ven 19 Nonlinear Dynamic Response of Two Bodies Across an Intermittent Contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Christopher Watson and Douglas Adams 20 Application of Continuation Methods to Nonlinear Post-buckled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 T.C. Lyman, L.N. Virgin, and R.B. Davis 21 Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Michael W. Sracic, Shifei Yang, and Matthew S. Allen 22 Identifying the Modal Properties of Nonlinear Structures Using Measured Free Response Time Histories from a Scanning Laser Doppler Vibrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Michael W. Sracic, Matthew S. Allen, and Hartono Sumali 23 Nonlinear System Identification of the Dynamics of a Vibro-Impact Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, and A.F. Vakakis 24 Modeling of Subsurface Damage in Sandwich Composites Using Measured Localized Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Sara S. Underwood and Douglas E. Adams 25 Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 ¨ Murat Aykan and H. Nevzat Ozgu€ven 26 Finding Local Non-linearities Using Error Localization from Model Updating Theory . . . . . . . . . . . . . . . . . . . . . . . . 323 Andreas Linderholt and Thomas Abrahamsson Chapter 1 Application of the Restoring Force Surface Method to a Real-life Spacecraft Structure J.P. Noel, G. Kerschen, and A. Newerla Abstract Many nonlinear system identification methods have been introduced in the technical literature during the last 30 years. However, few of these methods were applied to real-life structures. In this context, the objective of the present paper is to demonstrate that the Restoring Force Surface (RFS) method can provide a reliable identification of a nonlinear spacecraft structure. The nonlinear component comprises an inertia wheel mounted on a support, the motion of which is constrained by eight elastomer plots and mechanical stops. Several adaptations to the RFS method are proposed, which include the elimination of kinematic constraints and the regularization of ill-conditioned inverse problems. The proposed methodology is demonstrated using numerical data. Keywords Nonlinear system identification • Space structure • Restoring force surface method 1.1 Introduction Nonlinear structural dynamics has been studied for a relatively long time, but the first contributions to the identification of nonlinear structural models date back to the 1970s. Since then, numerous methods have been developed because of the highly individualistic nature of nonlinear systems [1]. A large number of these methods were targeted to Single-Degree-Of- Freedom (SDOF) systems, but significant progress in the identification of Multi-Degree-Of-Freedom (MDOF) lumped parameter systems has been realized during the last 10 or 20 years. To date, simple continuous structures with localized nonlinearities are within reach. Among the well-established methods, there exist • Time-domain methods such as the Restoring Force Surface (RFS) and Nonlinear Auto-Regressive Moving Average with eXogeneous input (NARMAX) methods [2, 3]; • Frequency-domain methods such as the Conditioned Reverse Path (CRP) [4] and Nonlinear Identification through Feedback of the Output (NIFO) methods [5]; • Time-frequency analysis methods such as the Wavelet Transform (WT) [6]. The RFS method, introduced in 1979 by Masri and Caughey [7], constitutes the first attempt to identify nonlinear structures. Since then, many improvements of the RFS method were introduced in the technical literature. Without being comprehensive, we mention the replacement of Chebyschev expansions in favour of more intuitive ordinary polynomials [8], the design of optimized excitation signals [9] or the direct use of the state space representation of the restoring force as nonparametric estimate [10]. In theory [11], the RFS method can handle MDOF systems. However, a number of practical considerations diminish this capability and its scope is, in fact, bound to systems with a few degrees of freedom only. For example, Al-Hadid and Wright J.P. Noel (*) • G. Kerschen Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group Department of Aerospace and Mechanical Engineering, University of Lie`ge, Belgium e-mail: 2 J.P. Noel et al. [12] studied a T-beam structure with well-separated bending and torsion modes. Another extensively studied system of this kind is the automotive damper [13, 14]. The objective of the present paper is to demonstrate the usefulness of the RFS method in the particular case of a real-life nonlinear spacecraft structure: the SmallSat spacecraft from EADS-Astrium. Starting from a review of the required ingredients for a RFS-based identification, we will propose solutions to the inherent limits of the method. First, we will simplify the kinematics of the nonlinear device of the spacecraft, termed WEMS, in order to explicitely formulate its dynamic equations. This formulation will be based on the necessary use of the coordinates of its center of gravity. Eventually, we will discuss why our estimation of coefficients is ill-conditioned and how to circumvent this final issue. The whole procedure will then be demonstrated using numerical data. 1.2 The SmallSat Spacecraft and Its Finite Element Modelling The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [15]. It is a monocoque tube structure which is 1.2 m long and 1 m large. It incorporates eight flat faces for equipment mounting purposes, creating an octagon shape, as shown in Fig. 1.1a. The octagon is manufactured using carbon fibre reinforced plastic by means of a filament winding process. The structure thickness is 4.0 mm with an additional 0.25 mm thick skin of Kevlar applied to both the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the interface brackets is around 64 kg. The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various concentrated masses; its mass is around 140 kg. The telescope dummy plate is connected to the SmallSat top floor via three shock attenuators, termed SASSA (Shock Attenuation System for Spacecraft and Adaptator) [16], the behaviour of which is 2 considered as linear in the present study. The top floor is a 1 m sandwich aluminium panel, with 25 mm core and 1 mm skins. Finally, as shown in Fig. 1.1c a support bracket connects to one of the eight walls the so-called Wheel Elastomer Mounting System (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this device is to isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft interface between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMSmobile part during launch, which gives rise to nonlinear dynamic phenomena. Figure 1.1d depicts the WEMS overall geometry, but details are not disclosed for confidentiality reasons. The Finite Element (FE) model in Fig. 1.1b was created in Samcef software and is used in the present study to conduct numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this model. The WEMS mobile part (the inertia wheel and its cross-shaped support) was modeled as a flexible body, which is connected to the WEMS fixed part (the bracket and, by extension, the spacecraft itself) through four nonlinear connections, labeled NC 1–4 in Fig. 1.1d. Black squares signal such connections. Each nonlinear connection possesses • A nonlinear spring (elastomer in traction plus 2 stops) in the axial direction, • A nonlinear spring (elastomer in shear plus 1 stop) in the radial direction, • A linear spring (elastomer in shear) in the third direction. The spring characteristics (piecewise linear) are listed in Table 1.1 and are displayed in Fig. 1.1e. We stress the presence of two stops at each end of the cross in the axial direction. This explains the corresponding symmetric bilinear stiffness curve. In the radial direction, a single stop is enough to limit the motion of the wheel. For example, its +x motion is constrained by the lateral stop number 2 while the connection 1 x limits the opposite x motion. The corresponding stiffness curves are consequently asymmetric. Dissipation is introduced in the FEM through proportional damping and local dampers to model the elastomer plots. Sine-sweep excitation was applied locally at the bracket in different directions. The frequency band of interest spans the range from 5 to 50 Hz and the sweeping rate is chosen equal to four octaves per minute. This frequency range encompasses the local modes of the WEMS device and some elastic modes of the structure. More precisely, around 11 Hz, the WEMS vibrates according to two symmetric bending modes (around x and y axis). Around 30 Hz, two other symmetric modes appear combining bending (around x and y axis) and translation (along x and y axis). A mode involving the WEMS and the bracket is also present around 30 Hz. The first lateral bending modes and the first axial mode of the structure finally appear between 30 and 50 Hz.