Nonlinear Dynamics, Volume 2

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Dynamics, Volume 2 Gaetan Kerschen Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc., Bethel, CT, USA

River Publishers Gaetan Kerschen Editor Nonlinear Dynamics, Volume 2 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-890-3 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2014 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Nonlinear Dynamics represents one of the eight volumes of technical papers presented at the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 3–6, 2014. The full proceedings also include volumes on Dynamics of Coupled Structures; Model Validation and Uncertainty Quantification; Dynamics of Civil Structures; Structural Health Monitoring; Special Topics in Structural Dynamics; Topics in Modal Analysis I; and Topics in Modal Analysis 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, 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. Additionally, there are papers which discuss the results obtained by different research groups on a Round Robin exercise on nonlinear system identification. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Liege, Belgium Gaetan Kerschen Nashville, TN, USA D. Adams v

Contents 1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System................................ 1 Lawrence N. Virgin, Joshua J. Waite, and Richard Wiebe 2 Complex Behavior of a Buckled Beam Under Combined Harmonic and Random Loading.................... 11 R. Wiebe and S.M. Spottswood 3 The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems ...... 19 T. Detroux, L. Renson, and G. Kerschen 4 Frequency Response Calculations of a Nonlinear Structure a Comparison of Numerical Methods ........... 35 Yousheng Chen, Andreas Linderholt, and Thomas Abrahamsson 5 A Framework for the Computational Dynamic Analysis of Coupled Structures Using Nonlinear Modes.... 45 Malte Krack, Lars Panning-von Scheidt, and Jörg Wallaschek 6 Subspace and Nonlinear-Normal-Modes-Based Identification of a Beam with Softening-Hardening Behaviour ....................................................................................... 55 C. Grappasonni, J.P. Noël, and G. Kerschen 7 Model Updating of Nonlinear Structures ............................................................................. 69 Güvenç Canbalog˘lu and H. Nevzat Özgüven 8 Detection of Nonlinear Behaviour of Composite Components Before and After Endurance Trials ........... 83 D. Di Maio, A. delli Carri, F. Magi, and I.A. Sever 9 Model Calibration of a Locally Non-linear Structure Utilizing Multi Harmonic Response Data.............. 97 Yousheng Chen, Vahid Yaghoubi, Andreas Linderholt, and Thomas Abrahamsson 10 Nonlinear Time Series Analysis Using Bayesian Mixture of Experts.............................................. 111 Tara Baldacchino, Jennifer Rowson, and Keith Worden 11 Identifying Robust Subspaces for Dynamically Consistent Reduced-Order Models ............................ 123 David Chelidze 12 Identification of Sub- and Higher Harmonic Vibrations in Vibro-Impact Systems.............................. 131 Simon Peter, Pascal Reuss, and Lothar Gaul 13 An Efficient Simulation Method for Structures with Local Nonlinearity ......................................... 141 V. Yaghoubi and T. Abrahamsson 14 Parametric Nonlinearity Identification of a Gearbox from Measured Frequency Response Data............. 151 Murat Aykan and Elif Altuntop 15 Nonlinear Gear Transmission System Numerical Dynamic Analysis and Experimental Validation........... 159 Dimitrios Giagopoulos, Costas Papadimitriou, and Sotirios Natsiavas 16 A Stochastic Framework for Subspace Identification of a Strongly Nonlinear Aerospace Structure.......... 169 J.P. Noël, J. Schoukens, and G. Kerschen vii

viii Contents 17 Composite Non-Linearity in High Cycle Fatigue Experimentation................................................ 183 A.M.J. Pickard 18 A Procedure to Identify the Handling Characteristics of Agricultural Tyre Through Full-Scale Experimental Tests ...................................................................................................... 191 F. Cheli, E. Sabbioni, and A. Zorzutti 19 Nonparametric Analysis and Nonlinear State-Space Identification: A Benchmark Example.................. 203 A. Van Mulders, J. Schoukens, and L. Vanbeylen 20 Nonlinear Black-Box Identification of a Mechanical Benchmark System........................................ 215 L. Vanbeylen and A. Van Mulders 21 Suppression of Multiple Order Friction Torque Fluctuations with Modulated Actuation Pressure........... 223 Osman Taha Sen, Jason T. Dreyer, and Rajendra Singh 22 Two-Dimensional Nonlinear Dynamics of Axially Accelerating Beam Based on DQM......................... 231 Dongmei Wang, Wei Zhang, Minghui Yao, and Wenhua Hu 23 Nonlinear Structural Coupling: Experimental Application........................................................ 241 Taner Kalayc{og˘lu and H. Nevzat Özgüven 24 State Estimation in Nonlinear Structural Systems................................................................... 249 Kalil Erazo and Eric M. Hernandez 25 An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 1: Theoretical Investigation ... 259 Sadegh Rahrovani, Thomas Abrahamsson, and Klas Modin 26 An Efficient Exponential Integrator for Large Nonlinear Stiff Systems Part 2: Symplecticity and Global Error Analysis.............................................................................................. 269 Sadegh Rahrovani, Thomas Abrahamsson, and Klas Modin 27 Vibration Suppression of a Flexible Parallel Kinematic Manipulator............................................. 281 Hermes Giberti and Cristiano Marinelli 28 Analysis of Nonlinear System Response to an Impulse Excitation................................................. 297 G. Manson, K. Worden, and P.I. Reed 29 Experimental Evaluation of Veering Crossing and Lock-In Occurring in Parameter Varying Systems ...... 309 O. Giannini, A. Sestieri, and C. Cannarella

Chapter 1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System Lawrence N. Virgin, Joshua J. Waite, and Richard Wiebe Abstract A key feature of many nonlinear dynamical systems is the presence of co-existing solutions, i.e, nonlinear systems are often sensitive to initial conditions. While there have been many studies to explore this behavior from a numerical perspective, in which case it is trivial to prescribe initial conditions (for example using a regular grid), this is more challenging from an experimental perspective. This paper will discuss the basins of attraction in a simple mechanical experiment. By applying both small and large stochastic perturbations to steady-state behavior, it is possible to interrogate the initial condition space and map-out basins of attraction as system parameters are changed. This tends to provide a more complete picture of possible behavior than conventional bifurcation diagrams with their focus on local steady-state behavior. Keywords Nonlinear dynamics • Basins of attraction • Snap-through • Competing solutions • Experimental mechanics 1.1 Introduction Nonlinear systems frequently exhibit a dependence on initial conditions, i.e., in contrast to linear dynamical systems, the ultimate destination of transients generated from a perturbation (say) may lead to different outcomes, just depending on where the trajectory originates (effectively its initial conditions). The study of co-existing attractors and their basins of attraction is well established, however, even in the simplest systems these boundaries may be highly complex, even fractal. Although it is not difficult to prescribe initial condition in a numerical setting (usually as a regular grid) [1, 2], it is much more challenging to access a full range of initial conditions experimentally, especially in a mechanical context. We focus attention here on a simple mechanical oscillator (essentially an arrangement of springs and masses) which was configured to have a double-well potential energy, i.e., two nominally symmetric equilibria separated by an unstable equilibrium (at the origin): the classic bi-stable situation. Depending on the system and forcing parameters the motion in such systems may either present only single well response, only cross-well (snap-through) response [3], or either type of response depending on initial conditions. Thus, taking the control parameter to be the harmonic forcing frequency (as is done in the work herein) and holding all other system and forcing parameters constant, we can anticipate entering and exiting a regime of cross-well behavior as the frequency is swept through a target range (including the underlying resonance effect). It is shown in this paper that even sweeping “up” and “down” through a control parameter range is no guarantee of capturing all the available attractors. In order to obtain the full spectrum of behavior, the system is subjected to a wide variety of perturbations (small and large) at essentially every value of the control parameter. The position and velocity of the mass together with the instantaneous phase under harmonic forcing provide a threedimensional phase space, requiringthreeinitial conditions to prescribe a uniquely evolving trajectory. For such a periodically excited system (generating a cyclic phase dimension) a stroboscopic sampling technique based on a Poincaré section can be conveniently utilized to reduce the system to a two-dimensional mapping (in Cartesian coordinates) [4]. For this system L.N. Virgin ( ) • J.J. Waite School of Engineering, Duke University, Durham, NC 27708-0300, USA e-mail: l.virgin@duke.edu R.Wiebe Structural Sciences Center, Air Force Research Laboratory, WPAFB, Dayton, OH 45433, USA Universal Technology Corporation, Dayton, OH, USA G. Kerschen (ed.), Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04522-1__1, © The Society for Experimental Mechanics, Inc. 2014 1

2 L.N. Virgin et al. we seek to determine what types of steady-state responses are possible, how distinct these responses are (e.g. snap-through vs single-well response), and in a relative sense how dominant these competing attractors are in capturing random initial conditions. A thorough numerical study has also been accomplished to model this specific system [5], but the focus of the current paper is purely on experimental results. 1.2 The Experimental System A photograph of the experimental system under investigation is shown in Fig. 1.1. Two 12 inch rigid links are hinged at their ends and at their common central point. This combination of springs and masses is configured such that there are two nominally symmetric stable equilibria (point attractors) separated by an unstable equilibrium. The system is also subject to an external harmonic excitation that is transmitted to the central mass from a harmonic motion of the remote end of the spring as indicated in part (b). This is a single-degree-of-freedom (SDOF) system with three variables: angle .t/, measured from a straight configuration, angular velocity P .t/, and the forcing phase defining the state space. Even in the absence of forcing we see competing attractors as the system (under the influence of a little viscous damping) will come to rest at one of the two stable equilibrium points; whether the system comes to rest at the positive or negative equilibrium depends entirely on the starting conditions. When the forcing is switched on (and going from a 2D to a 3D phase space) the dominant behavior consists of periodic and possibly chaotic attractors. We shall keep the discussion to the essential details of co-existing responses in experimental systems. The reader is referred to [6] for a complete description of the experimental setup, along with a derivation of the equation of motion and a Fig. 1.1 (a) Photographic image of the experimental system of the link model system, (b) the Scotch-yoke forcing mechanism, and (c) the low-friction fixed-end pivot

1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System 3 numerical study of the system dynamics. The experimental measurement used to study the system was the “response” angle .t/, with a positive direction as indicated in Fig. 1.1. In addition to the acquisition of continuous time series data, a laser tachometer, triggered by a reflective tab on a Scotch-yoke flywheel (a device for converting rotational motion to unidirectional harmonic motion), was used to stroboscopically sample the data at intervals of the forcing period: the Poincaré section [4]. 1.3 Techniques of Investigation 1.3.1 Time Lag Embedding The forced system is completely described at every point in time by its position, velocity, and forcing phase. As is often the case, it is difficult to obtain these three quantities simultaneously in an experiment, leaving gaps in the knowledge of the system state. Fortunately we are able to reconstruct a structure that is topologically equivalent to the true phase portrait using time lag embedding [7]. Proper application of time lag embedding typically requires some manual tuning of the time lag that is used to reconstruct the phase portrait. For introductory purposes we will use the lateral shaking mechanism (the Scotch-yoke) as a means of illustrating the role of the time-lag. The intent is to produce a harmonic input to the structure, which in this instance is prescribed to be x.t/ D8:1sin.1:65.2 /t C / in which the magnitude is given in cm and the frequency is in Hz. The time series can be seen in Fig. 1.2a, along with a frequency transform (b), and reconstructed phase portraits at various time lags. As is visible in parts (a) and (b), the time series is a relatively clean harmonic motion, which in the phase space of position and velocity should yield an elliptical trace. As a consequence, it is a quarter period lag (part (d)) that qualitatively yields the best reconstruction of the true phase portrait. Shorter (longer) time lags tend to yield overly constricted in phase (out of phase) portraits, as the position and its time lagged counterpart are strongly positively (negatively) correlated (parts (c) and (e)). The distortions from a perfect circle in Fig. 1.2d also indicates that the forcing is not a perfect harmonic signal, however, departures from the ideal case is not out of the ordinary for experimental studies. 0 1 2 3 4 −10 0 10 0 1 2 3 4 0 2 4 6 8 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 τ = T/8 τ = T/4 τ = T/2 τ = 1/1.65 −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 x(t) (cm) x(t) x(t) x(t) x(t + τ) x(t + τ) x(t + τ) time (seconds) f = 1.65 Hz a b e d c Fig. 1.2 (a) A typical measured time series, (b) frequency content, (c)–(e) reconstructed phase projections using time-lag embedding. The superimposed red points are representative Poincaré data extracted at a given forcing phase

4 L.N. Virgin et al. −40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40 0 5 10 15 20 25 30 θ(deg) θ(t) time (s) θ(t + τ/4) a b f = 1.33 Hz Fig. 1.3 A typical period-7 response, (a) time series, (b) phase projection in time-lag coordinates. The time is the forcing period of the system, which in this case is 1:33Hz While it provides a good demonstration of time lag reconstruction and selection of the time lag, harmonic motions are not especially interesting. When faced with a time series with an unknown structure that has been produced by a nonlinear system, it is not usually obvious which time lag should be chosen. Motivated by the harmonic forcing itself, however, it is usually best to select a quarter forcing period as a starting point from which to tune the time lag. When reconstructing the response of the system we shall typically use one quarter of the forcing period as a default time-lag for embedding purposes. 1.3.2 Poincaré Section Another useful technique popularly used in nonlinear dynamics is to stroboscopically sample a trajectory using a Poincaré section. If a periodic signal is sampled once each period it, by definition, returns to where it started from. This is essentially the same as a strobe light in which the light flashes at the same frequency as the object under scrutiny, thus resulting in an apparently stationary image of the object. We can make use of a similar effect by sampling the response of our periodicallyforced system to assess the periodicity (or lack thereof) of the behavior. This also results in effectively reducing the three dimensional continuous phase space into a two dimensional discrete mapping, bringing with it other advantages in the general evaluation of nonlinear features. Here, we simply make use of Poincaré sectioning (and time-lag embedding) to identify distinct attractors in the overall response of the system under study. By way of introduction, consider a time series of a single measured variable .t/ as shown in Fig. 1.3a. If we extract instantaneous angles whenever the forcing pause reaches a certain value then we obtain the superimposed red points. Part (b) of this Fig. 1.3 then reconstructs the phase trajectory (using a quarter-cycle delay) to reveal the periodic nature of the response, perhaps more clearly than the time series, and with the red points (also based on a quarter-cycle delay) clearly indicating that this response is a “period-7”, i.e., a subharmonic oscillations that repeats itself every 7 forcing cycles. 1.3.3 Stochastic Interrogation As mentioned in the introduction, experimentally mapping basins of attraction requires some special techniques. In this paper we use a technique called stochastic interrogation [8]. Given a system undergoing steady-state motion and then subject to a (non-necessarily small) perturbation, the system will experience some transient motion before settling back down into its original behavior or perhaps be attracted to another co-existing behavior. Clearly the latter is more likely if the perturbations are relatively large. This approach also relates to the random (small) disturbance in the vicinity of an attractor to assess local stability [9]. If the process is repeated many times (with randomly varied perturbations), and careful book-keeping is used in conjunction with Poincaré sectioning and time-lag coordinates, it is possible to obtain basins of attraction. In the experimental system the means of achieving this sequence of varied perturbations is to switch the forcing to a different frequency for a short period of time, and after switching back to the baseline set of parameters track the subsequent transients

1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System 5 time (s) θ(deg) 0 10 20 30 40 50 60 70 80 90 100 −40 −30 −20 −10 10 20 30 40 0 Fig. 1.4 Schematic illustration of random perturbations (stochastic interrogation) causing the steady-state to jump between co-existing attractors (and specifically as they pass through the Poincaré section) until they settle into steady-state behavior. This is best illustrated with an example. Figure 1.4 shows a time series of the measured angle. The superimposed red data points are again the instances when the angle passes through the Poincaré section. For a fixed set of parameters the motion initially consists of relatively small amplitude periodic motion about the negative equilibrium configuration. We label this motion as “green”. After about 8 s, the system is perturbed by switching to a randomly chosen disturbing frequency for a short but random duration (in this case a couple of seconds and indicated by the gray stripe in the figure). Once this is switched off and the frequency is returned to its baseline value, the system undergoes some large-amplitude transient motion and settles into a cross-well motion, which we label as “blue”. Upon subsequent disturbances the system can be jogged into small motion about the adjacent (positive) well, and this is labeled as “red”. Each Poincaré point (red points superimposed on the time series) “belongs” to the inset of a specific attractor; if we keep subjecting the system to such random perturbations the ensemble of Poincaré points associated with each attractor (its basin of attraction) can be mapped. It should also be pointed out that the cross-well motion may exhibit chaos (as well as higher-order subharmonics of the type shown in Fig. 1.3, but we shall simply use the blue designation to distinguish large cross-well motion from the red and green small amplitude single-well behavior, since this is an important practical distinction in most applications. 1.4 Results 1.4.1 A First Look A natural starting point in investigating possible behavior is to form a bifurcation diagram. Figure 1.5 shows the response of the system under a harmonic forcing a.t/ D Asin.2 ft C / for a frequency sweep up (a) and sweep down (b) for frequencies, f, between about 0:4 and 1:6Hz at a fixed forcing amplitude of AD8:1cm. This is the practical range of the aforementioned Scotch-yoke mechanism with the system tuned to exhibit broad resonance within this range. The period-7 response can be detected in the vicinity of 1.33 Hz, more clearly illustrated in the zoomed-in diagram in part (c), as well as period-6 and period-8 windows. The response is plotted in terms of the angle measured at the Poincaré section and again the color-coding system is used. This tends to be more convenient than the more conventional uses of response amplitude. Small-amplitude responses contained within each well (with only positive or negative angles) are labeled red and green. Large-amplitude (cross-well) behavior are labeled blue, and these contain mostly chaotic responses but occasionally higher-periodic behavior (within narrow windows) as just mentioned. By sweeping up and down, one is able to capture a hysteretic, i.e. a path-dependent, region. The ubiquity of a similar hysteresis in hardening and softening spring systems, has likely lulled analysts into a sense of security that these are the only solutions present in such systems. Hysteresis due to geometric nonlinearities, however, is a mere consequence of different solutions gaining and losing stability at different frequencies during the sweep, and there is no guarantee that they are the only steady-state solutions in existence. We note here that there is an inevitable bias or lack of perfect symmetry in this

6 L.N. Virgin et al. −40 −20 0 20 40 Frequency (Hz) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 −40 −20 0 20 40 a b 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 −40 −20 0 20 40 f (Hz) θp c 6 7 8 θp θp Fig. 1.5 Bifurcation diagrams based on a slow sweep in forcing frequency, (a) slowly increasing forcing frequency, (b) slowly decreasing forcing frequency, (c) a zoom within the apparently chaotic cross-well behavior. The angle is given in degrees experimental system. Thus, even the small-amplitude (red and green) responses in each are not necessarily equally likely to occur. We also printout here that the speed of the sweep can have an influence on the results, since a relatively fast sweep will not be as likely to eliminate the transients completely. 1.4.2 Basins of Attraction We next take a closer look at possible responses at specific (snapshot) values of the forcing frequency, partly in order to determine if any behavior is missed during the conventional bifurcation diagram based on frequency sweeps. In Fig. 1.6 we show basins of attraction of the Poincaré surface of section in time-lag reconstructed phase space during the transition into (left column) and out of (right column) cross-well behavior (see vertical dashed lines in Fig. 1.5 for relative locations) that were obtained using stochastic interrogation. For relatively low forcing frequencies, e.g., when f D 0:5Hz, we see that single-well (in each well) and cross-well behavior are both possible given an arbitrary initial condition. The swept bifurcation diagrams (Fig. 1.5) were effectively constrained to remain within whatever local basin they begin in, with minimal transients generated. Sample basins of attraction are shown in Fig. 1.6. The plots also have the attractors superimposed: red initial conditions are attracted to the yellow cluster, green to the black cluster, and blue to the purple cluster. For an intermediate range of forcing frequencies we have unavoidable cross-well behavior (and we have already seen that this may be chaotic or relatively high-order periodic). For higher frequencies we see a transition back to predominantly single-well behavior as shown in Fig. 1.6. There is a sudden transition from cross-well to single-well behavior betweenf D1:425andf D1:55Hz. At f D1:55Hz there are very few blue points indicating that cross-well behavior is shrinking in its prevalence. For the highest frequency shown we expect small amplitude behavior (a result anticipated even from linear oscillators forced well-above their resonant frequency), although using the Poincaré displacement is not necessarily the best representation of this effect.

1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System 7 60 −60 −60 60 θp(t + τ/4) θp(t) 0 0 f = 0.5 f = 0.7 f = 0.9 f = 1.0 −60 θ p(t) 0 0 f = 1.375 f = 1.425 f = 1.55 f = 1.625 60 −60 60 θp(t + τ/4) Fig. 1.6 Initial condition plots for a frequency range during the transition into (left column) and out of (right column) cross-well motion. The reported frequencies, f, are in Hz, the angle in degrees, and the forcing amplitude, A, is 8.1cm It is noteworthy that despite the stochastic perturbations that are applied to the system, much of the initial condition space remains untouched. This occurs due to a rapid (and uneven) distortion and contraction of trajectories onto their respective attractors. Even in a relatively short period of time, that being the time required to reach the first Poincaré sampling point from the (random but bounded) time at which the random perturbation is completed plus the further quarter forcing period necessary to generate a data point in the time lag space, the phase space contracts very rapidly into the shapes visible in the plots. These persisting shapes are clearly indicative of the strength of the stable manifolds attached to each attractor. We are now in a position to revisit the “bifurcation diagram” but this time we include initial condition information as the control parameter is changed. In some ways this is better described as a “system response diagram”. Note, this is in marked contrast to a continuous sweep of the forcing frequency (based on very small increments), or for example, when the initial conditions are reset at the same value (say .0/ D P .0/ D0) at each increment of the forcing frequency. Figure 1.7 shows the end result of conducting long stochastic interrogation experiments at each value of the forcing frequency, where instead of plotting the response itself, we show (color coded) the percentage initial conditions observed

8 L.N. Virgin et al. 0.4 0.6 0.8 1 1.2 1.4 1.6 0 20 40 60 80 100 % Relative dominance Fig. 1.7 Bifurcation (response) diagrams including a more thorough sampling of initial conditions in terms of relative dominance of each attractor based on stochastic interrogation that lead to each particular type of response. In the midst of the resonant region (left blank in the figure) all preliminary experiments showed that the system presented only a single cross-well response, and hence there was no need to perform long time experiments throughout the entire region. At both the low and high frequency ends of the range we observe co-existing small amplitude motion about each well. Again, large amplitude cross-well motion occurs in the vicinity of resonance (the linearized natural frequency about each equilibrium is f D1.317 Hz). This small motion is actually characterized by softening spring effects and thus we would expect large amplitude responses to occur for frequencies slightly lower than the (undamped, linear) natural frequency. However, the important information is that we also uncover some additional (crosswell) behavior. This largely periodic behavior shows there are often three co-existing attractors, a result we saw for specific frequencies in the basin plots. We see that referring back to Fig. 1.5, the system was able to maintain its attraction to the small amplitude (green) response about the negative equilibrium until the local basin effectively disappeared close to a forcing frequency of a little over 1 Hz. However, over certain low-frequency ranges there are remote periodic cross-well responses (blue), but this is only revealed under quite careful interrogation of the initial condition space. It is interesting to view this behavior against the backdrop of “path-following” or continuation, i.e., primarily numerical algorithms that track the evolution of a solution. They effectively behave rather like the slow parameter sweeps in which only local behavior tends to be followed, with the distinct possibility of missing remote solutions perhaps not directly attached to the local solution path. Figure 1.7 seems to indicate an asymmetry in the structure, as the dominance of the two single-well responses is far from equal. This is not surprising as it was not easy to tune the multiple components of the structure to yield a perfectly symmetric underlying potential energy well. However, as discussed in the introduction the important practical distinction under study in this paper is between the small amplitude behavior (red and green) and the large amplitude behavior (blue). 1.5 Conclusions A bi-stable mechanical oscillator is shown to exhibit a variety of co-existing responses that occur over a range of forcing frequencies in the vicinity of resonance. In order to gain a more complete picture of possible behavior, the robustness of these responses in a global sense is investigated. A scheme of random disturbances is applied to an experimental system such that initial conditions are sampled, and basins of attraction are revealed. The responses are broadly classified as “single-well” for small-amplitude motion, and “cross-well” for large amplitude (the division is unambiguous in this system), and where coexisting responses do occur, their relative dominance is assessed using a measure based on the percentage of randomly generated initial conditions attracted to each long-term response. Using stochastic interrogation, the results discussed in this paper have addressed the questions raised in the introduction. For frequencies close to resonant conditions the behavior is dominated by snap-through events (blue); for frequencies higher than resonance we typically observe small amplitude non-snapping behavior (red and green); but for frequencies below resonance we see a rather complicated dependence on initial conditions leading to both small, and large, amplitude motion (red, green and blue). That is, for a specific harmonically-excited nonlinear mechanical oscillator we have detailed some interesting co-existing behavior, and thus, demonstrated the importance of thoroughly investigating initial conditions and parameter space in order to reveal the full range of behavior. Acknowledgements This work was partially supported by the US Air Force, AFOSR Grant no. FA9550-09-1-0201, and NSF grant 0927186 (Dynamical Systems).

1 Co-existing Responses in a Harmonically-Excited Nonlinear Structural System 9 References 1. Thompson JMT, Stewart HB (2002) Nonlinear dynamics and chaos: geometrical methods for engineers and scientists, 2nd edn. Wiley, Chichester 2. McDonald SW, Grebogi C, Ott E, Yorke JA (1985) Fractal basin boundaries. Physica D 17:125 3. Murphy KD, Virgin LN, Rizzi SA (1996) Experimental snap-through boundaries for acoustically excited, thermally buckled plates. J Exp Mech 36:312–317 4. Virgin LN (2000) Introduction to experimental nonlinear dynamics: a case study in mechanical vibration. Cambridge University Press, New York 5. Waite JJ, Virgin LN, Wiebe R (2013) Competing responses in a discrete mechanical system. Int J Bifurcat Chaos (to appear) 6. Wiebe R, Virgin LN, Stanciulescu I, Spottswood SM, Eason TG (2013) Characterizing dynamic transitions associated with snap-through: a discrete system. J Comput Nonlinear Dyn 8. doi:10.1115/1.4006201 7. Krantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, Cambridge 8. Cusumano JP, Kimble BW (1995) A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: application to a two-well oscillator. Nonlinear Dyn 8:213–235 9. Murphy KD, Bayly PV, Virgin LN, Gottwald JA (1994) Measuring the stability of periodic attractors using perturbation-induced transients: applications to two nonlinear oscillators. J Sound Vib 172:85–102

Chapter 2 Complex Behavior of a Buckled Beam Under Combined Harmonic and Random Loading R. Wiebe and S.M. Spottswood Abstract Nonlinearities have long been avoided in the design of structural systems. This was done to make problems tractable, to fit within current design paradigms, and often with the assumption that the resulting design would be conservative. Computational methods have made the investigation of nonlinear systems possible, which may yield more accurate and optimal designs. However, in venturing into the nonlinear regime, a designer must be aware of potential pitfalls, one of which is the possibility of unsafe responses “hiding in the weeds” of parameter or initial condition space. In this paper, an experimental study on a damped, post-buckled beam in the presence of noise is used to show that co-existing stationary solutions may be present in real-world scenarios. Stochastic resonance, a surprising phenomenon in which a small harmonic load interacts with, and magnifies the response to, an otherwise pure random load, is also studied and observed to occur in the beam. Keywords Nonlinear dynamics • Stochastic resonance • Experimental mechanics • Snap-through • Duffing 2.1 Introduction Co-existing solutions and their basins of attraction are well represented in the literature, however, despite decades of study, even in the simplest case of single-degree-of-freedom (SDOF) models such as the Duffing equation, one may see extremely complex fractal basin of attraction boundaries separating co-existing solutions in initial condition (IC) space [1–3]. An excellent review of the importance, and many of the applications of co-existing solutions in nonlinear dynamics can be found in [4]. Real engineering systems are always subjected to at least some noise, which provides another level of complexity. The response of the Duffing oscillator under noise is studied in [5] for white noise, and [6] for bounded noise under parametric excitation. In nonlinear systems, even when small relative to the random input, harmonic forces may provide a disproportionate amount of energy to the response. This can occur through stochastic resonance. Stochastic resonance typically occurs when the time scale of the mean snap-through rate due to the random loading is close to that of a small harmonic input. The harmonic loading works to alternatingly deepen and shallow the competing potential wells which, in an average sense, promotes (or makes it more likely that) the response due to the random load will show increased coherence with the harmonic input [7]. As a consequence of this, the harmonic load is also able to do more net work on the system [8]. Good reviews of stochastic resonance, both in two-state and continuous systems, with many other potential applications may be found in the works [9–11]. R.Wiebe ( ) Structural Sciences Center, Air Force Research Laboratory, WPAFB, Dayton, OH 45433, USA Universal Technology Corporation, Dayton, OH, USA e-mail: rwiebe@co.utcdayton.com S.M. Spottswood Structural Sciences Center, Air Force Research Laboratory, WPAFB, Dayton, OH 45433, USA G. Kerschen (ed.), Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04522-1__2, © The Society for Experimental Mechanics, Inc. 2014 11

12 R. Wiebe and S.M. Spottswood The studies of stochastic resonance that most closely resemble the work in this paper may be found in [12] and [13, 14]. In the former, a numerical SDOF analogue of a buckled beam is used to investigate the potential for stochastic resonance to improve the performance of vibrational energy harvesters. The latter two include experimental results showing the presence of stochastic resonance in a bi-stable nanomechanical oscillator. The experimental system studied in this paper is the clamped-clamped post-buckled beam. The nonlinear response of post-buckled beams has been studied for beams subjected to both harmonic loading [15], and random loading [16]. This system presents an ideal avenue for studying nonlinear phenomena in engineering structures as it is an intermediate step in complexity between qualitative discrete models and common structural systems such as curved panels and shells. 2.2 The Double-Well Duffing Equation It is well known that the damped harmonically forced Duffing equation given by Rx C2 Px x Cx3 DAsin!t ( D0:05 in all cases herein) may be used to generate co-existing responses for many different forcing parameters. Several sample time series are shown in Fig. 2.1, where Pn indicates a response that repeats once every n forcing cycles. The single-well responses shown in this figure also contain a mirrored twin (not shown) in the other well. Figure 2.2a shows the Poincaré section of the response, x, as a function of frequency, !, for a fixed forcing amplitude A D 0:3 in the absence of noise. This plot was obtained by running a simulation at a fixed frequency and plotting the 0 102030405060 −2 −1 0 1 2 x(t) t 0 102030405060 −2 −1 0 1 2 x(t) t a b 0 102030405060 −2 −1 0 1 2 x(t) t c Fig. 2.1 Sample time series for a damped harmonically forced double-well Duffing oscillator: (a) co-existing P1 cross-well and single-well responses (AD0:3, ! D0:75rad/s), (b) co-existing cross-well chaos and cross-well P1 responses (AD0:3, ! D1:20rad/s), and (c) co-existing single-well P1 and cross-well P3 responses (AD0:25, ! D1:80rad/s) 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 A (rad/s) 1,0,0,0 Single-Well Response 1,1,0,0 1,0,1,0 1,0,0,1 1,1,1,0 1,1,0,1 1,0,1,1 1,1,1,1 0,1,0,0 0,1,1,0 0,1,0,1 0,1,1,1 0,0,1,0 0,0,1,1 0,0,0,1 Unavoidable Cross-Well Response Potential Cross-Well Response b a 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (rad/s) 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 w w −2.5 x(nTP) Fig. 2.2 Co-existing solutions as a function of forcing parameters: (a) A bifurcation diagram in forcing frequency developed using Poincaré sampling at the forcing period. Four types of response are shown; (cyan) Single-well non-chaotic P1, (blue) chaotic, (red) cross-well Pn, (green) cross-well P1. (b) Response type chart in forcing parameter space, the four number sequences indicate the existence (1) or non-existence (0) of, in sequence, the cyan, blue, red, and green response types. The horizontal dashed line in part (b) corresponds to the forcing amplitude used in part (a) and the target markers denote forcing parameter pairs of interest (Color figure online)

2 Complex Behavior of a Buckled Beam Under Combined Harmonic and Random Loading 13 3 2 1 0 −1 −2 −3 −3 −4 −1 −2 0 1 2 3 4 x(0) x(0) a 3 2 1 0 −1 −2 −3 −3 −4 −1 −2 0 1 2 3 4 x(0) x(0) b t=6TP p(x) c x t=8TP t=10TP x (−2,2) ∋ Fig. 2.3 Basins of attraction under harmonic plus random loading: A D0:3, ! D0:75rad/s for (a) D0:0 and (b) D0:014; A D0:3, ! D1:20rad/s for (c) D0:0 and (d) D0:033. White (black) shading indicates ICs inevitably lead to cross-well P1 (single-well) response in parts (a) and (b) and chaotic (cross-well P1) response in parts (c) and (d). Gray shadingindicates ICs which may lead to either type of co-existing response under different realizations of the random forcing process position “x” each time the forcing reached a particular phase (after transients were allowed to decay). Hence, in the event of a Pn response these Poincaré samples repeatedly revisit the same npoints. All of the data points for a particular simulation were color-coded according to the labels in the caption. At each frequency, 60 simulations were performed at random ICs to search for co-existing solutions, which is why many frequencies contain multiple types of color-coded response. The complete procedure was then repeated over a grid of 200frequencies to yield the final plot. In order to include the effects of all the forcing parameters, in Fig. 2.2b the vertical axis is replaced with the forcing amplitude, A, (with a resolution of 200) and contours were plotted based on the number and types of responses co-existing at each parameter set. The determination of chaos was done by calculating the sign of the largest Lyapunov exponent ( 1) via the method discussed in [17]. The switching (snap-through) frequency, together with the Lyapunov exponent were then used to distinguish between the four types of response in the caption. The green (cross-well) and cyan (single-well) “paths” of Poincaré points in Fig. 2.2a appear to show more than just a single P1 response. This may indeed be the case, however, at least one of these is trivial as all responses exist with a 180ı phase-lagged mirror image counterpart due to the symmetry of the system. The level of complexity in Fig. 2.2 is quite surprising considering the simplicity of the equation of motion. The region of potential cross-well response is perhaps of most interest as it is mostly likely to lead to unsafe design in the event where a simulation or experiment, purely by happenstance, only captures the “safe” single-well response. It is therefore of interest whether this region persists under more realistic loading, i.e. in the presence of noise. A noisy forcing term (a normally distributed random process .t/ D u, where u is standard normal) was added to the right-hand-side of the Duffing equation. At low noise levels, the underlying deterministic attractors are not completely destroyed, instead the noise blurs the boundaries in the basins of attraction. This is shown in Fig. 2.3, where the basins of attraction yielding co-existing solutions are shown with and without the addition of the noise term .t/. The fractal basin boundaries observed in part (a) are a well-known feature of the Duffing system. For part (b), each IC pair was simulated 60 times and in the event that all responses ended up on the same attractor the IC pair was shaded black or white depending

14 R. Wiebe and S.M. Spottswood 0.0 0.1 0.2 0.3 0.4 0.5 0.6 W 0.0 0.2 0.4 0.6 0.8 W Wt fsw(Hz) 0.1 0.2 0.3 0.4 t 20,000 0 1 −1 s = 0.08 x 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 t 20,000 0 1 −1 s = 0.12 s x Fig. 2.4 Stochastic resonance in the Duffing oscillator under combined harmonic plus random forcing at AD0:1, ! D0:75rad/s on the type of response, however, if each response occurred at least once the IC pair was shaded gray. As the noise level is increased further, eventually the attractors are destroyed completely, at first by occasional jumping between the “ghosts” of the former attractors, and then at really high noise levels the response takes on a new, more random character. A response of the system in the gray shaded regions of Fig. 2.3b may be better understood by looking at part (c). In this figure the IC denoted by the red dot in part (b) was simulated 100,000 times, the results of which were used to create a time varying probability density plot. At time t D0 the probability density is a delta function at x D 2, Px D0. Only five forcing cycles later it produces a broad density plot. As the time progresses it begins to show three distinct peaks, and after ten forcing cycles it has reached a steady state. Beyond this point, the probability density becomes periodic returning to what is shown once every forcing cycle. It should be noted that the density plots shown are a projection, as the velocity is not shown. The two smaller peaks are in fact the solutions that were trapped in the two potential wells. These two peaks oscillate on either side, while the large peak oscillates across the both wells. After ten forcing cycles the three separate peaks no longer exchange any probability density, and in the full x and Px space they never come in contact. Under higher noise levels intermittency appears as “leakage” between the peaks. A more complete picture of the response of the Duffing oscillator to increasing noise levels is given in Fig. 2.4 for a grid of 100 values. The forcing parameters used to develop this plot were in a single-well response region just below potential snap-through (see Fig. 2.2b). This was selected to avoid confusion between stochastic resonance and primarily deterministic cross-well response. The top plot shows the average switching (or snap-through) frequency fSW, which is the frequency of crossings of the x D0 hilltop in either direction. The bottom plot in each panel contains the average work caused by the harmonic force component over a single forcing period (solid black curve, left axis), the maximum correlation envelope (shaded gray, left axis), and the ratio of the harmonic to total work (dashed black curve, right axis). Also shown in the insets of this figure is a time series before (showing intermittency) and near stochastic resonance. The value W is the average work done by the harmonic component of the total force (through the total displacement) over a single harmonic forcing period and presents a peak value at a nonzero noise level that is much higher than the work under the harmonic load alone. This phenomenon is known as stochastic resonance. Perhaps more meaningful than W, is the ratio of the average work done by the harmonic force to the total average work Wt done over a single forcing period. It is not obvious whether, for a given amount of average energy input, if it is better or worse, say in the context of fatigue, that the majority of the energy input be delivered by the harmonic or random force components. Another interesting result is obtained by plotting the cross-correlation for harmonics at different phases than the true harmonic input (this is no longer work but has the same units). The gray shaded region in Fig. 2.4 shows the envelope of the maximum cross-correlation at each frequency. The phase at which the maximum occurs is potentially different for each point on the envelope. Note that the actual work, W, can never exceed this envelope, and in this case it in fact does not make any contact.

2 Complex Behavior of a Buckled Beam Under Combined Harmonic and Random Loading 15 In the literature it is usually assumed that the resonant peaks in the workW should occur approximately where the average switching rate is approximately twice (a full response “cycle” requires two switching events) the harmonic forcing frequency (f D!=2 D0:12Hz). The dashed line construction at a frequency equal to twice the harmonic forcing frequency shows that this is not true for this system, although many results in the literature are for heavily overdamped systems. 2.3 The Post-Buckled Beam While useful to come to a phenomenological understanding, it is difficult to determine the relevance of the above results to real world structural systems. In this section, the same types of behavior are shown to occur, both experimentally and using a finite element (FE) model, in a post-buckled beam, which can be seen as an analogue for curved structural systems such as aircraft panels. The experimental system investigated in this section is a clamped-clamped stainless steel beam as shown in Fig. 2.5 with beam thickness t, width b, length L, modulus of elasticity E, density , and coefficient of thermal expansion ˛T. The beam was buckled under various axial loads that were induced by a heat lamp placed several feet in front of the beam (not seen in photograph). The beam was fastened to a shaker which provided harmonic and random loading through base excitation. The power spectral density that was used was a flat band between50 and 500Hz with no power at other frequencies. The various experimental results that follow were also obtained at two temperatures (T1 D34:7ıC, T2 D26:1ıC). This was done to promote snap-through at different load levels. The beam was also tested at two different realistic damping levels ( 1 D0:71%, 2 D1:70%) which was adjusted by attaching thin adhesive aluminum constrained layer damping foil strips to the back of the beam. Co-existing responses were quite easy to find at both temperatures and damping levels. The natural frequency of the post-buckled beam at T1 was found to be 163 Hz. In order to gain a sense of the robustness of experimental phenomena shown later, and to ensure that they are real physics as opposed to pathological effects of experimental imperfections, the results are compared with those of a numerical model. However, as the focus of this paper is not on developing accurate numerical models, a relatively simple FE approach is taken. Despite the simplicity of the model, the results will be shown to be surprisingly accurate. The finite element model was developed using the lumped mass co-rotational beam element formulation in [18] with 24 elements, and solved using an arc-length solution following routine. The damping ratios identified experimentally were used to select the coefficient for mass proportional damping. Slightly different temperatures were applied to the FE model (T1 D 35:3ıC, T2 D 27:1ıC) than were observed experimentally. This was done to best match all of experimentally measured static quantities, that being the natural frequencies, temperatures, and the y0 quantity (distance between the two post-buckled configurations) seen in Fig. 2.5. As opposed to independently generating harmonic and random loads to match that used in the experiments, the total experimental loading applied to the structure, measured with an accelerometer, was fed into the FE model. y Load Midspan Deflection t = 0.787mm b = 12.7 mm L = 228.6 mm E = 153.4 GPa r = 7567 kg/m3 a = 14.4 10−6/°C T 0 Fig. 2.5 Photograph of beam clamped to shaker with parameters inset top right, and illustrative load-deflection curve under transverse loadinset bottom left. Damping treatment was applied to the back of the beam

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