Nonlinear Dynamics, Volume 1

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Dynamics, Volume 1 Gaëtan Kerschen Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015 River Publishers

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

River Publishers Gaëtan Kerschen Editor Nonlinear Dynamics, Volume 1 Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-905-4 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2016 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 ten volumes of technical papers presented at the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 2–5, 2015. The full proceedings also include volumes on Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Sensors and Instrumentation; Special Topics in Structural Dynamics; Structural Health Monitoring & Damage Detection; Experimental Techniques, Rotating Machinery & Acoustics; Shock & Vibration Aircraft/Aerospace, Energy Harvesting; and Topics in Modal Analysis. 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. Liège, Belgium Gaëtan Kerschen Nashville, TN, USA D. Adams v

Contents 1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System...... 1 M. Di Bartolomeo, F. Massi, L. Baillet, A. Culla, and A. Fregolent 2 Non-linear Dynamics of Jointed Systems Under Dry Friction Forces ............................................. 11 Silvio Giuseppe Neglia, Antonio Culla, and Annalisa Fregolent 3 Prediction of Nonlinear Forced Response on Ancillary Subsystem Components Attached to Reduced Linear Systems............................................................................................. 23 Sergio E. Obando, Peter Avitabile, and Jason Foley 4 Numerical Round Robin for Prediction of Dissipation in Lap Joints.............................................. 53 L. Salles, C. Swacek, R.M. Lacayo, P. Reuss, M.R.W. Brake, and C.W. Schwingshackl 5 The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems .................. 65 T. Detroux, L. Renson, L. Masset, and G. Kerschen 6 Nonlinear Vibrations of a Beam with a Breathing Edge Crack .................................................... 83 Ali C. Batihan and Ender Cigeroglu 7 Stability Limitations in Simulation of Dynamical Systems with Multiple Time-Scales.......................... 93 Sadegh Rahrovani, Thomas Abrahamsson, and Klas Modin 8 Coupled Parametrically Driven Modes in Synchrotron Dynamics ................................................ 107 Alexander Bernstein and Richard Rand 9 Relating Backbone Curves to the Forced Responses of Nonlinear Systems....................................... 113 T.L. Hill, A. Cammarano, S.A. Neild, and D.J. Wagg 10 Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities .. 123 X. Liu, A. Cammarano, D.J. Wagg, S.A. Neild, and R.J. Barthorpe 11 Passive Flutter Suppression Using a Nonlinear Tuned Vibration Absorber ...................................... 133 Giuseppe Habib and G. Kerschen 12 Nonlinear Vibrations of a Flexible L-shaped Beam Using Differential Quadrature Method................... 145 Hamed Samandari and Ender Cigeroglu 13 Theoretical and Experimental Analysis of Bifurcation Induced Passive Bandgap Reconfiguration........... 155 Michael J. Mazzoleni, Brian P. Bernard, Nicolas Garraud, David P. Arnold, and Brian P. Mann 14 A Model of Evolutionary Dynamics with Quasiperiodic Forcing .................................................. 163 Elizabeth Wesson and Richard Rand 15 Experimental Demonstration of a 3D-Printed Nonlinear Tuned Vibration Absorber........................... 173 C. Grappasonni, G. Habib, T. Detroux, and G. Kerschen 16 The Effect of Gravity on a Slender Loop Structure ................................................................. 185 Lawrence N. Virgin, Raymond H. Plaut, and Elliot V. Cartee vii

viii Contents 17 Wave Propagation in a Materially Nonlinear Rod: Numerical and Experimental Investigations.............. 191 Yu Liu, Andrew J. Dick, Jacob Dodson, and Jason Foley 18 Experimental Nonlinear Dynamics and Chaos of Post-buckled Plates ............................................ 199 R. Wiebe and D. Ehrhardt 19 Control-Based Continuation of a Hybrid Numerical/Physical Substructured System.......................... 203 David A.W. Barton 20 Towards Finite Element Model Updating Based on Nonlinear Normal Modes .................................. 209 Simon Peter, Alexander Grundler, Pascal Reuss, Lothar Gaul, and Remco I. Leine 21 Experimental Modal Analysis of Nonlinear Structures Using Broadband Data ................................. 219 J.P. Noël, L. Renson, C. Grappasonni, and G. Kerschen 22 Measurement of Nonlinear Normal Modes Using Mono-harmonic Force Appropriation: Experimental Investigation............................................................................................. 241 David A. Ehrhardt, Matthew S. Allen, and Timothy J. Beberniss 23 Nonlinear System Identification Through Backbone Curves and Bayesian Inference........................... 255 A. Cammarano, P.L. Green, T.L. Hill, and S.A. Neild 24 Experimental Nonlinear Identification of an Aircraft with Bolted Connections ................................. 263 G. De Filippis, J.P. Noël, G. Kerschen, L. Soria, and C. Stephan 25 Non linear Finite Element Model Validation of a Lap-Joint........................................................ 279 A. delli Carri and D. Di Maio 26 Experimental Validation of Pseudo Receptance Difference (PRD) Method for Nonlinear Model Updating.. 293 Güvenç Canbalog˘lu and H. Nevzat Özgüven 27 Systems with Bilinear Stiffness: Extraction of Backbone Curves and Identification ............................ 307 Julian M. Londono, Simon A. Neild, and Jonathan E. Cooper 28 Simplifying Transformations for Nonlinear Systems: Part I, An Optimisation-Based Variant of Normal Form Analysis ............................................................................................... 315 N. Dervilis, K. Worden, D.J. Wagg, and S.A. Neild 29 Simplifying Transformations for Nonlinear Systems: Part II, Statistical Analysis of Harmonic Cancellation .............................................................................................. 321 N. Dervilis, K. Worden, D.J. Wagg, and S.A. Neild 30 Considerations for Indirect Parameter Estimation in Nonlinear Reduced Order Models ................................................................................................. 327 Lorraine C.M. Guerin, Robert J. Kuether, and Matthew S. Allen 31 Nonlinear Model Updating Methodology with Application to the IMAC XXXIII Round Robin Benchmark Problem.................................................................................................... 343 Mehmet Kurt, Keegan J. Moore, Melih Eriten, D. Michael McFarland, Lawrence A Bergman, and Alexander F Vakakis 32 Bridging the Gap Between Nonlinear Normal Modes and Modal Derivatives ................................... 349 Cees Sombroek, Ludovic Renson, Paolo Tiso, and Gaetan Kerschen 33 Validation of Nonlinear Reduced Order Models with Time Integration Targeted at Nonlinear Normal Modes ........................................................................................................... 363 Robert J. Kuether and Mathew S. Allen 34 Model Order Reduction of Nonlinear Euler-Bernoulli Beam...................................................... 377 Shahab Ilbeigi and David Chelidze 35 Identification of Dynamic Nonlinearities of Bolted Structures Using Strain Analysis........................... 387 D. DiMaio

Contents ix 36 The Effects of Boundary Conditions, Measurement Techniques, and Excitation Type on Measurements of the Properties of Mechanical Joints.......................................................... 415 S. Smith, J.C. Bilbao-Ludena, S. Catalfamo, M.R.W. Brake, P. Reuß, and C.W. Schwingshackl 37 Numerical Model for Elastic Contact Simulation.................................................................... 433 D. Botto and M. Lavella 38 Efficient and Accurate Consideration of Nonlinear Joint Contact Within Multibody Simulation ............ 441 Florian Pichler, Wolfgang Witteveen, and Peter Fischer 39 Model Reduction for Nonlinear Multibody Systems Based on Proper Orthogonal- and Smooth Orthogonal Decomposition............................................................................................. 449 Daniel Stadlmayr and Wolfgang Witteveen 40 Cam Geometry Generation and Optimization for Torsion Bar Systems .......................................... 459 Ergin Kurtulmus and M.A. Sahir Arikan 41 Dynamics Modeling and Accuracy Evaluation of a 6-DoF Hexaslide Robot ..................................... 473 Enrico Fiore, Hermes Giberti, and Davide Ferrari 42 A Belt-Driven 6-DoF Parallel Kinematic Machine................................................................... 481 Navid Negahbani, Hermes Giberti, and Davide Ferrari 43 Bearing Cage Dynamics: Cage Failure and Bearing Life Estimation ............................................. 491 Silvio Giuseppe Neglia, Antonio Culla, and Annalisa Fregolent 44 Bias Errors of Different Simulation Methods for Linear and Nonlinear Systems ............................... 505 Yousheng Chen, Kjell Ahlin, and Andreas Linderholt 45 Internal Resonance and Stall Flutter Interactions in a Pitch-Flap Wing in the Wind-Tunnel ................. 521 E. Verstraelen, G. Kerschen, and G. Dimitriadis

Chapter 1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System M. Di Bartolomeo, F. Massi, L. Baillet, A. Culla, and A. Fregolent Abstract Friction affects almost the entirety of the mechanical systems in relative motion. In spite of intense and long-time research activities many aspects of this phenomenon still lack of a meaningful interpretation. Some of them could be explained by not focusing only on the interface properties. In fact recent literature confirms the picture of a macroscopic frictional behaviour of a mechanical system as the outcome of a complex interaction between the local dynamics at the frictional interface (wave and rupture nucleation and propagation) and the global dynamics of the system. This paper presents the results of a 2D non-linear finite element analysis under large transformations of the onset and evolution of sliding between two isotropic elastic bodies separated by a frictional interface. The aim is to investigate the trigger of the dynamic rupture at the interface, which preludes and goes with the sliding and its interaction with the global dynamics to determine the observed macroscopic frictional behaviour (stick-slip, continuous sliding). The analysis is focused on the observed phenomena during the onset of the sliding (micro-slips, precursors, macro-slips), accounting for the frictional properties and the inertial and elastic properties of the system. Keywords Dry friction • Stick-slip • Contact instabilities • Dynamic rupture • Body waves 1.1 Introduction When two bodies in dry frictional contact start to move relative to each other different types of behaviour could be observed: (1) the whole interface undergoes sliding gradually, without relevant fluctuations in the frictional force, this is the case of the continuous or “smooth” sliding; (2) the system presents an unstable behaviour, characterized by strong variations of the frictional force that result in powerful vibrations transmitted to the whole system (friction-induced vibrations). This behaviour, termed as stick-slip, is related to the temporal succession of zones of the interface that start to slide and return in stick alternately; (3) the system also exhibits severe vibrations, whose origin is to be found in different types of instabilities (sprag-slip, negative friction slope, mode coupling instability, parametric resonance [1, 2]). Friction instabilities are at the origin of very common acoustic phenomena as the creaking of door hinges, the rubbing of a shoe on the floor, the squeaking of a chalk on a board, squeal in the automotive braking system, the excitation of the violin strings by the bow (see [3] for an extensive review). Since the paper of Brace and Byerlee [4], stick-slip is also proposed as the mechanism at the origin of earthquakes, stimulating a considerable number of papers on this topic [5]. The presence of these instabilities is generally undesired. In fact the strong vibrations that they induce could bring to severe stresses acting on the system, fatigue phenomena, lack of functionality, surface damages, annoying noise etc. Issues that could affect every system/mechanism with parts in frictional contact in particular in the machine tools industry. However there are situations where, on the contrary, the stick-slip is necessary as in stringed instruments. M. Di Bartolomeo ( ) • F.Massi DIMA, Department of Mechanical and Aerospace Engineering,“La Sapienza” University of Rome, via Eudossiana 18, Rome 00184, Italy LaMCoS, Contacts and Structural Mechanics Laboratory, Université de Lyon, CNRS, INSA Lyon, UMR 5259 20 rue des Sciences, F-69621 Villeurbanne, France e-mail: mariano.dibartolomeo@uniroma1.it L. Baillet ISTerre, Institut des Sciences de la Terre, CNRS, Joseph Fourier University, Grenoble 1381 rue de la Piscine Domaine universitaire, St. Martin D’Heres 38400, France A. Culla • A. Fregolent DIMA, Department of Mechanical and Aerospace Engineering, “La Sapienz” University of Rome, via Eudossiana 18, Rome 00184, Italy © The Society for Experimental Mechanics, Inc. 2016 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15221-9_1 1

2 M. Di Bartolomeo et al. Focusing our attention on the stick-slip it is often assumed that it is related to a variation of frictional resistance during sliding [6–10], resulting in a temporal succession of phases of elastic energy storage (stick phase) followed by phases of energy release (slip phase) with an associated drop of the macroscopic frictional force. In the literature, the onset of this instability was usually related to the difference between the static and the dynamic coefficient of friction, to a rate-weakening feature of the friction coefficient or to a variation of friction coefficient along the interface [11, 12]. In [13–16] it has been observed that stick-slip motion also arises in systems without such assumption, i.e. with a constant friction coefficient and no distinction between static and kinetic friction. Stick-slip in this case arises as a consequence of a dynamic instability involving the coupling between normal and tangential motion [14] or to a destabilization of the interface waves [17]. Recent works showed that the variation of frictional resistance is rather the result of both the frictional properties at the interface and the elastic and inertial properties of the system [18, 19], assuming that stick-slip configures as an interchange between the elastic energy stored into the system and its kinetic energy by means of the friction force. This last statement places itself from the viewpoint of the recent literature which tends to consider the frictional behaviour as the result of a complex mutual interaction between the local dynamics at the frictional interface and the global dynamics of the systems rather than to be only related to the frictional characteristics of the interface [20–26]. In this new perspective the study of the local contact dynamics (wave and rupture generation and propagation) assumes a primary role. In this context in this paper the evolution of the contact forces during stick-slip instability is investigated in function of the variations of the local contact dynamics and of its relative energy flows occurring during the different phases characterizing the ruptures propagation [27]. 1.2 Finite Element Model The model (2-D, plain strain) l (Fig. 1.1) consists of two bodies kept in contact along a frictional interface by a normal force Nand subjected to a relative velocityV applied at the lower boundary of the lower body. The frictional interface is governed by a classic Coulomb friction law. Table 1.1 shows the material parameters used for the simulations. The degree of material contrast 1C DcS1/cS2 [28] (cS Dshear wave speed) falls in the range of existence of the generalized Rayleigh wave (GRW) in order to account for the so called “bimaterial” effect [17, 29–33] (Table 1.2). The simulations have been performed by the explicit dynamic finite element code PLASTD whose contact algorithm allows to calculate the mechanical quantities at the contact interface such as local contact forces, slip, velocity, contact Fig. 1.1 Geometry of the 2-D bimaterial model (HD3mmLD10mm) Table 1.1 Input data Body Young modulus E[GPa] Density [Kg/m3] Poisson ratio Longitudinal wave speed cP [m/s] Shearwave speed cS [m/s] Generalized Rayleigh w. cGRW [m/s] Material contrast Mass Matrix damping coeff. ˛ [s-1] Stiffness Matrix damping coeff. ˇ [ms] 1. 3.9 1,300 0.33 2,100 1,062 866 0.22 40 0.5 e-6 2. 2.5 1,200 0.38 1,973 868

1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System 3 Table 1.2 Simulation data Compressive ForceaN[N/mm] Initial normal stress ¢0 DN/L[MPa] Horizontal velocityV [mm/s] Coulomb friction coefficient 10 1 10 1 aper depth unit (2-D model) status. This software [34] is designed for large transformations and non-linear contact behaviour; it applies an incremental Lagrange multiplier method for the contact between deformable bodies. For the dynamic study, the formulation is discretized spatially by using the finite element method and temporally by using an explicit Newmark scheme. 1.3 Results 1.3.1 Onset of the Sliding Figure 1.2a shows the time evolution of the macroscopic (sum of the contact node components) normal and tangential contact forces at the frictional interface. At the first the normal contact force increases due to the application of the compressive force N; then the velocityVis imposed on the driving boundary (Fig. 1.1) and the tangential force increases linearly until the onset of the first sliding (time interval t1-t2). Afterwards the tangential force T shows irregular fluctuations in the form of a series of ramps followed by respective sudden drops, giving the typical macroscopic stick-slip pattern of Fig. 1.2 [35–38]. During the ramps the system stores elastic energy due to its deformation. When the shear contact stress locally reaches the critical value dictated by the friction law, zones of the interface switch in sliding and trigger the ruptures (in the sense of switch from local stick to slip) propagations. These ruptures can cross the whole interface, causing a global sliding and realising part of the elastic energy; the amount of released energy is proportional to the force drop at the end of every ramp. Figure 1.3a shows the seismic profile (local slip velocity at the interface along the time) and the respective contact status along the contact surface during the time period of the force ramp delimited by the two vertical dashed lines at t1 D0.725ms andt2 D0.755 ms in Fig. 1.2b. The colour on every seismic line represents the local status (sticking or sliding) of the contact at the interface. The area covered by every line shows the tangential local velocity of the upper side of the contact interface (that is the lower boundary of the body nı 2); the black (yellow) colour of the area indicates that the point is moving in x-negative (x-positive) direction. This graph can be considered as representative of what happens at the interface during each ramp shown in Fig. 1.2. In macroscopic stick-slip behaviour the interface phenomena occurring during the slipping (drop of the tangential force), can be divided into three characteristic distinct phases: 1. At approximately 0.7255 ms, micro-slips (slip confined to a reduced contact zone [25]) start to appear at the interface. Each micro-slip behaves as a single nucleated rupture and radiates different types of waves along the interface and inside the bodies [26]. For a longer time (0.7255 0.732 ms) the micro-slips coalesce with each other to form a sliding zone of about L/4 in length. Due to the energy continuously provided by the displacement of the lower body, this zone grows in width (0.732 0.735 ms) with a sliding propagation speed of about 100 mm/s toward the left and 400 mm/s toward the right. The behaviour of the rupture propagation recovered in this first phase is consistent with the phase of propagation of the “slow front” experimentally observed in [39] and with the “stable rupture front” of [40]. 2. When a sufficient number of micro-slips switches to sliding at nearly the same time they can release enough energy to sustain short range slip pulse rupture propagation (the so-called precursors); for example the supershear ruptures PR1 and PR2 in Fig. 1.3a (propagating rightwards at 1,800 m/s) [25]. As time increases, an increasing number of precursors are triggered and their distance of propagation increases, in agreement with what has been experimentally observed in [40]. 3. The propagation distance of the precursors increases until the main slip event the macro-slip is triggered in the form of an extensive crack-like supershear rupture propagating rightwards (SSR 3 in Fig. 1.3a) with speed crup 1,800 m/s. As macro-slip is intended a slip that involves a large part of the interface and a relative displacement between the two bodies. At the same time a leftwards rupture front (GRR 4), again crack-like, propagates at about the GRW speed (866 m/s). The passage of these two crack-like fronts releases most of the elastic energy and the whole interface switches to sliding state. The remaining elastic energy is discharged by subsequent pulse-like supershear ruptures (SSR 5, SSR 6). In this phase the ruptures cause local slipping with velocity an order of magnitude greater ( 103 mm/s) than the previous one, as can be inferred by the greater amplitude of the seismic profile oscillations. The waves generated at the interface propagate inside the two bodies as well, reaching the boundaries of the system, and are reflected exciting the dynamics of the whole system

4 M. Di Bartolomeo et al. 0.5 1 1.5 0.2 0.4 0.6 0.8 1 1.2 Time [ms] Normalized force 0.7 0.75 0.8 0.6 0.8 1 Tangential force Normal force t 1 t 2 a b Fig. 1.2 (a) Time evolution of the sum of the contact forces (normalized to the compressive force N); (b) detail around t1-t2 Fig. 1.3 (a) Seismic profile and contact status between 0.725 0.755ms (dashed lines of Fig. 1.2) blue spotsDadherence, green spotsDsliding, (at the top) directions of the wave speeds of the upper body, (on the right) stop times; (b) Acceleration of the point A (Fig. 1.1) during the same time interval (friction induced vibrations). In Fig. 1.3b is reported the time evolution of the acceleration of the point A during the same interval (Fig. 1.1), it can be noted the time delay in the signal due to the distance of the point A from the interface. After the main crack ruptures (>t D0.7455 ms in Fig. 1.3a), the whole interface returns to the sticking state. Waves still cross the interface due to the reflections of the rupture fronts at the interface boundaries (and to the reflections of the body waves at the system boundaries) as can be noted by the oscillations in the seismic profile. Nevertheless their magnitude cannot trigger sliding (there are no light-grey zones on the seismic lines) and the system returns to a quasi-static evolution during which it stores the elastic energy provided by the velocity imposed to the lower body.

1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System 5 Fig. 1.4 (a) Seismic profile for S II (b) Tangential force (c) Seismic energy The presented analysis shows how each phase of the stick-slip macroscopic friction behaviour can be associated with the local dynamics (wave and rupture generation) at the contact. 1.3.2 Phases Stability Analysis In order to investigate deeply the role of each phase (microslips, precursors) in triggering the macroslip, it should be possible to isolate and individually analyse each phase. To approach this condition, the relative motion of the two bodies has been “stopped” after each phase. This allows to investigate the energy flows during each phase, and to determine if that phase could be consider “stable” or “instable” with regard to the following macroslip activation. In particular as “stable” is intended a phase after which, if the system is stopped, the interface remains in quiet; conversely an “instable” phase means that after it the system undergoes macroslip, even if the system is stopped. A sequence of simulations have been performed and differ each other for the time, tstop, at which the motion has been stopped. Figure 1.3a reports, on the right, the different stop times (SI D0.7269e-3s, SIID0.7299e-3, SIII D0.7329e-3s, SIVD0.7339e-3s, SVD0.7350e-3s, SVI D0.7359e-3s, SVIID0.7389e-3) used for the different simulations (S1, S2, S3, etc.). In the following by S.A. is indicated the whole simulation, without stop. A particular attention has been placed on the modality to stop the motion between the two bodies. The interruption of the motion should take place in a time interval shorter than the time scale that characterize the different phases. Obviously the only action of reduce or stop the relative motion could affect the contact dynamics, due to the inertia forces that arise and that are inversely proportional to the applied deceleration. Reducing the braking interval tb, the inertial forces increase, increasing tb the inertial forces decrease but at the same time the breaking interval could cover different phases, making impossible to isolate the different effects. This fact make difficult, even impossible, to realize this type of experience in an experimental set-up, because the dynamics at the interface, during a single macroslip, is too fast compared to the possible rates of change of the macroscopic boundary conditions. The “braking” time used tb D1.67e-6s (represented in Figs. 1.4c, 1.5c, 1.6c, and 1.7c by the double arrow).

6 M. Di Bartolomeo et al. Fig. 1.5 (a) Seismic profile for S III (b) Tangential force (c) Seismic energy Fig. 1.6 (a) Seismic profile for S V (b) Tangential force (c) Seismic energy

1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System 7 The results have shown that are simulations that do not undergo macroslip (SI, SII, SIII, SIV and SV) and other where the macroslip occurs (SVI, SVII). For example Fig. 1.4 shows the seismic profile (Fig. 1.4a), and the tangential force (Fig. 1.4b) for the simulation SII. It can be noted as the tangential force remains at about the same level reached at the instant of arrest. Figure 1.4c represents the seismic energy. From the seismic profile it can be observed as during the braking interval tb (black bracket) isolated microslips arise and extend their activity even after the body stops, until the interface return all in sticking state. This delay is related to the time necessary to the deceleration, applied to the lower boundary. Moreover, during the braking time the two bodies still undergo the effect of inertial forces, which active smooth oscillations following predominantly the first mode of vibration along the x direction. As can be seen in Fig. 1.4, they cause a further nucleation of microslips around 0.74 ms, i.e. after 0.06e-3s, which is consistent with the semi-period of the first mode of vibration of the system. The seismic energy is strongly related to energy released during every events (microslips, ruptures). A definition, used mainly in the geophysics field [41], is: Es;A D t2 Z t1 v.t/2 dt (1.1) The integral is referred to a generic point A of the solids in contact and measure the energy that arrives at the point A during the evolution of the sliding at the interface; similar to experimental geophysics where detectors measure the waves generated by earthquakes, this quantity could be measured experimentally on dedicated set-up to validate the numerical results. Even if this parameter is not an energy, missing a mass term, however it is directly related and proportional to the energy released and allows to easily examine the contribute of the different zones of the interface to the whole energy release. It can be observed as the microslips do not give detectable signals on point A (Fig. 1.4a). In the next SIII simulation (Fig. 1.5), the first precursors appear (PR1 in Fig. 1.3). Nevertheless, as can be observed in the figure, this rupture have not still enough magnitude to activate the macroslip. It can be also noted that also the precursor do not give detectable signals on point A (Fig. 1.5c); in fact their wave field propagating in the bulk material are quickly suppressed by the internal damping. In the simulation SV (Fig. 1.6), other precursors appear (e.g. PR 2 in Fig. 1.3), but also in this case, these rupture do not trigger the macroslip. To obtain the macroslip it is needed to keep the imposed relative motion until the nucleation of the main ruptures, i.e. the ruptures that cross the whole interface (Fig. 1.7), as indicated by the drop in the tangential force (Fig. 1.7b). The seismic energy (Fig. 1.7c) shows clearly the arrival of the wave fields generated by the different ruptures at the interface (ref. Fig. 1.3). The first wave field is that relative to the propagation of the rupture SSR 3, which is also the most energetic, followed by the one corresponding to the rupture SSR 5. After that, the wave field corresponding to the rupture (GRR 4) can be noted, which is lower in magnitude due to the strong localization of the rupture 4 on the left of the interface, farther from the point A. In Fig. 1.8 is reported the simulation unstopped (SA), it can be observed the greated magnitude of the seismic energy (Fig. 1.8c). It is interesting to note that once these ruptures nucleate, they evolve independently by the fact that the system is stopped as it can be inferred from the Fig. 1.8. That means that the energy necessary to feed them is already stored in the system and released independently from the system boundary conditions. This preliminary analysis suggests that the system, even after the appearance of the precursors, behaves as stable (in the meaning specified above), and the macroslip is not activated by the precursor propagation. Thus, they release only the amount of energy necessary for their own propagation, up to being extinguished, and seem to be not correlated with the subsequently macroslip. Comparing all the seismic profiles in Figs. 1.4–1.7, and 1.8, it can be observed that all the zones in sliding appear superimposable (where they exist), except for the replies of microslips due to system oscillation. The onset and evolution of the sliding seems to follow the same pattern, and seems to be not essentially affected by the stop operation. Of course, in reality the evolution of the rupture propagation will be necessarily affected by the macroscopic dynamic response due to the inertia forces, which could trigger further ruptures after the stop. As seen before these are strongly related to the magnitude of the deceleration. It can be inferred that the action of the imposed motion just around the onset of the sliding is limited to activate ruptures whose dynamics is led by previously stored energy. This is due to the fact that the dynamics of the ruptures evolves at time scales that are much shorter than the energy loading rate due to the imposed motion at the boundaries; consequently, the whole energy stored into the system during the time interval of Fig. 1.3 (0.725 0.755 ms) is at least one order of magnitude much lower that the energy released during the ruptures, which is stored during the whole loading ramp (Fig. 1.2).

8 M. Di Bartolomeo et al. Fig. 1.7 (a) Seismic profile for S VI (b) Tangential force (c) Seismic energy Fig. 1.8 (a) Seismic profile for S A (b) Tangential force (c) Seismic energy

1 Interplay Between Local Frictional Contact Dynamics and Global Dynamics of a Mechanical System 9 1.4 Conclusions In this paper the interaction between the local contact rupture dynamics and macroscopic frictional is investigated. The macroscopic stick-slip evolution is described as a sequence of local contact phenomena: the occurrence of micro-slips is followed by the precursors, up to the macro-slip phase that brakes the whole surface and releases most of the elastic energy of the system. These results are in agreement with recent experimental observations [41, 42]. The microslips and the precursors seem to behave as stable phases with respect to the macroslip; in fact they provide only the amount of energy needed for their propagation but are not able to trigger the macroslip. The imposed motion just around the onset of the sliding is limited to activate ruptures whose dynamics is led by previously stored energy. This is due to the fact that the dynamics of the ruptures (micro-slips, rupture and wave propagation) evolves at time scales that are much shorter than the energy loading rate due to the imposed motion; References 1. Tonazzi D (2014) Macroscopic frictional contact scenarios and local contact dynamics: at the origins of “macroscopic stick-slip”, mode coupling instabilities and stable continuous sliding. Ph.D. thesis, University of Rome La Sapienza 2. Tonazzi D, Massi F, Culla A, Baillet L, Fregolent A, Berthier Y (2013) Instability scenarios between elastic media under frictional contact. Mech Syst Signal Process 40(2):754–766 3. Akay A (2002) Acoustics of friction. J Acoust Soc Am 111:1525 4. Brace WF, Byerlee JD (1966) Stick-slip as a mechanism for earthquakes. Science 153:990 5. Scholz CH (2002) The mechanics of earthquakes and faulting. Cambridge University Press, New York 6. Braun OM, Barel I, Urbakh M (2009) Dynamics of transition from static to kinetic friction. Phys Rev Lett 103:194301 7. Capozza R, Urbakh M (2012) Static friction and the dynamics of interfacial rupture. Phys Rev B 86:085430 8. Heslot F, Baumberger T, Perrin B (1994) Creep, stick-slip and dry-friction dynamics: experiments and a heuristic model. Phys Rev E 49:4973 9. Robbins MO, Thompson PA (1991) Critical velocity of stick-slip motion. Science 253:916 10. Trømborg J, Scheibert J, Amundsen DS, Thøgersen K, Malthe-Sørenssen A (2011) Transition from static to kinetic friction: insights from a 2D model. Phys Rev Lett 107:074301 11. Kammer DS, Yastrebov VA, Spijker P, Molinari JF (2012) On the propagation of slip fronts at frictional interfaces. Tribol Lett 48:27–32 12. Popp K, Shelter P (1990) Stick-slip vibrations and chaos. Philos Trans R Soc Lond A 332:89 13. Oden JT, Martins JAC (1985) Models and computational methods for dynamic friction phenomena. Comput Methods Appl Mech Eng 52:527 14. Martins JAC, Oden JT, Simoes FM (1990) A study of static and kinetic friction. Int J Eng Sci 28:29–92 15. Wikie B, Hill JM (2000) Stick-slip motion for two coupled masses with side friction. Int J Non-Linear Mech 35:953 16. Wensrich C (2006) Slip–stick motion in harmonic oscillator chains subject to Coulomb friction. Tribol Int 39:490 17. Adams GG (1995) Self-excited oscillations of two elastic half-spaces sliding with constant coefficient of friction. J Appl Mech 62:867 18. Persson BNJ, Popov VL (2000) On the origin of transition from slip to stick. Solid State Commun 114:261 19. Luan B, Robbins MO (2004) Effect of inertia and elasticity on stick-slip motion. Phys Rev Lett 93:036105 20. Baillet L, Link V, D’Errico S, Laulagnet B, Berthier Y (2005) Finite element simulation of dynamic instabilities in frictional sliding contact. J Tribol 127:652 21. Meziane A, D’Errico S, Baillet L, Laulangnet B (2007) Instabilities generated by friction in a pad-disc system during the braking process. Tribol Int 40:1127 22. Massi F, Berthier Y, Baillet L (2008) Contact surface topography and system dynamics of brake squeal. Wear 265:1784 23. Massi F, Rocchi J, Culla A, Berthier Y (2010) Coupling system dynamics and contact behaviour: modelling bearings subjected to environmental induced vibrations and ‘false Brinelling’ degradation. Mech Syst Signal Process 24:1068 24. Renouf M, Massi F, Saulot A, Fillot N (2011) Numerical tribology of dry contact. Tribol Int 44(7–8):834–844 25. Di Bartolomeo M, Massi F, Baillet L, Culla A, Fregolent A, Berthier Y (2012) Wave and rupture propagation at frictional bimaterial sliding interfaces: from local to global dynamics, from stick-slip to continuous sliding. Tribol Int 52:117 26. Di Bartolomeo M, Meziane A, Massi F, Baillet L, Fregolent A (2010) Dynamic rupture at a frictional interface between dissimilar materials with asperities. Tribol Int 43:1620 27. Bar-Sinai Y, Spatschek R, Brener EA, Bouchbinder E (2013) Instabilities at frictional interfaces: creep patches, nucleation, and rupture fronts. Phys Rev E 88:060403 28. Shi Z, Ben-Zion Y (2006) Dynamic rupture on a bimaterial interface governed by slip-weakening friction. Geophys J Int 165:469 29. Weertman J (1980) Unstable slippage across a fault separates elastic media of different elastic constant. J Geophys Res 85:1455 30. Ranjith K, Rice JR (2001) Slip dynamics at an interface between dissimilar materials. J Mech Phys Solids 49:341 31. Ben-Zion Y (2001) Dynamic ruptures in recent models of earthquake faults. J Mech Phys Solids 49:2209 32. Langer S, Weatherley D, Olsen-Kettle L, Finzi Y (2012) Stress heterogeneities in earthquake rupture experiments with material contrasts. J Mech Phys Solids. doi:10.1016/j.jmps.2012.11.002 33. Baillet L, Sassi T (2002) Finite element method with Lagrange multipliers for contact problems with friction. CR Mec 334:917 34. Ozaki S, Inanobe C, Nakano K (2014) Finite element analysis of precursors to macroscopic stick-slip motion in elastic materials: analysis of friction test as a boundary value problem. Tribol Lett 55:151–163

10 M. Di Bartolomeo et al. 35. Prevost A, Scheibert J, Debrégeas G (2013) Probing the micromechanics of a multi-contact interface at the onset of frictional sliding. Eur Phys JE36, 17 36. Radiguet M, Kammer DS, Gillet P, Molinari JF (2013) Survival of heterogeneous stress distributions created by precursory slip at frictional interfaces. Phys Rev Lett 111:164302 37. Svetlizky I, Fineberg J (2014) Classical shear cracks drive the onset of dry frictional motion. Nature 509:205–208 38. Trømborg J, Sveinsson HA, Scheibert J, Thøgersen K, Amundsen DS, Malthe-Sørenssen A (2014) Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces. PNAS. doi: 10.1073/pnas.1321752111 39. Ben-David O, Fineberg J (2011) Static friction coefficient is not a material constant. Phys Rev Lett 106:254301 40. Nielsen S, Taddeucci J, Vinciguerra S (2010) Experimental observation of stick-slip instability fronts. Geophys J Int 180:697 41. Occhiena C, Coviello V, Arattano M, Chiarle M, Morra Di Cella U, Pirulli M, Pogliotti P, Scavia C (2012) Analysis of microseismic signals and temperature recordings for rock slope stability investigations in high mountain areas. Nat Hazards Earth Syst Sci 12:2283–2298 42. Rubinstein SM, Cohen G, Fineberg J (2007) Dynamics of precursors to frictional sliding. Phys Rev Lett 98:226103

Chapter 2 Non-linear Dynamics of Jointed Systems Under Dry Friction Forces Silvio Giuseppe Neglia, Antonio Culla, and Annalisa Fregolent Abstract Dry friction devices are usually adopted to reduce structure vibrations. Generally, an optimisation procedure is required to tune the system coefficients by performing the best effect. In this paper a continuous structure coupled with a lumped dissipative system is studied. This second system is alternatively loaded by a dry friction force, so that two states can be recognized: contact and not-contact. Aim of this work is the study of the system parameters influence on the structure response when the system passes through the different states before presented. In particular, during the contact state a stickslip motion of the lumped system is considered. The analysis is focused on the transition between chaotic and non-chaotic behaviour and on the correspondent power flows between continuous and lumped system during the motion. Keywords Non linear dynamics • Jointed systems • Stick-slip • Bifurcation diagrams • Maximum Lyapunov exponent 2.1 Introduction Aim of this paper is to present some results concerning the nonlinear coupling of a continuous mechanical system and a lumped one. The nonlinearity is due to a friction force between the mass of the lumped system and a moving belt. The motion of the system is composed of a sequence of stick and slip phases: during the stick phase the relative velocity of the mass in contact with the belt is null, on the contrary during the slip phase the relative velocity controls the friction force. In the past many works investigated this phenomenon, because in the field of applied sciences the phenomenon of friction and in particular stick-slip systems are very interesting [1–3]. In this paper a parametric analysis varying the belt velocity is performed by considering some indicators (bifurcation diagrams and Lyapunov exponents, see [3–5]). The nature of the system response is investigated in order to understand if the system exhibits a chaotic motion. The study is also focused on the power flows between the jointed systems. 2.2 Mechanical Model The system shown in Fig. 2.1 is made up of a continuous beam and an harmonic oscillator with two degrees of freedom driven by a belt, the belt has a constant speed vdr. The two systems are linked to each other via a springk1. During its motion, the mass mcan have two different behaviours: it can be attached to the belt (stick motion) and moves with it, or slide on the belt (slip motion). A non constant contact force is considered through a stiffness k3 between mass and belt along x.t/. In the following, the motion equations of the global system are derived. S.G. Neglia ( ) • A. Culla • A. Fregolent Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Rome, Italy e-mail: silvio.neglia@uniroma1.it © The Society for Experimental Mechanics, Inc. 2016 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15221-9_2 11

12 S.G. Neglia et al. Fig. 2.1 Mechanical system 2.2.1 Dimensional Motion Equations The continuous system is modelled using the Euler-Bernoulli theory of beam, its motion equation is expressed by the following equation: EI @4w.z; t/ @z4 C A @2w.z; t/ @t2 Dk1 Œy.t/ w.z; t/ ı.z zA/ (2.1) The beam has length L, Young modulus E, moment of inertia of area I, mass density and a cross section area A. The Dirac delta ı.z zA/ allows to consider the influence of the harmonic oscillator in the link point z DzA. The harmonic oscillator is modelled with two degrees of freedom: x.t/ and y.t/. Along the x.t/ direction the motion can be influenced of the non continuous contact between mass and belt. There is mass-belt contact if x.t/>0. In this case a contact force, N.t/, rises and it is equal to: N.t/ Dk3x.t/ (2.2) On the other hand, if there is no mass-belt contact, the normal force is N.t/ D0. The equation of motion along x.t/ is: mRx.t/ Ck2x.t/ CN.t/ D0 (2.3) Along the y.t/ direction the stick-slip condition must be taken into account. There is stick motion on the mass mif all the external forces acting on it are bounded in absolute value by the static friction force Ts.t/ D sN.t/, moreover the mass velocity Py.t/ must be equal to the belt speed vdr. In mathematical terms the stick condition is expressed by: jk1 Œy.t/ w.zA; t/ j < sN.t/ and Py.t/ Dvdr (2.4) The stick motion equation is the following: mRy.t/ D0 (2.5) The initial condition for Eq. (2.5) are: ( y.t/jtD0 D0 Py.t/jtD0 Dvdr (2.6)

2 Non-linear Dynamics of Jointed Systems Under Dry Friction Forces 13 Fig. 2.2 Dynamical friction force −2 0 2 −0.4 −0.2 0 0.2 0.4 dy(t)/dt−vdr[m/s] T d (t) [N] If the stick condition [Eq. (2.4)] is not verified, there is sliding between the mass and the belt and a dynamic friction force Td.t/ rises. The slip equation of motion is: mRy.t/ Ck1 Œy.t/ w.zA; t/ DTd.t/ (2.7) where the non linear dynamical friction force Td.t/ is expressed by: Td.t/ D sN.t/ 1C Œ Py.t/ vdr sgnŒPy.t/ vdr (2.8) In Fig. 2.2 the dynamic friction force plot is shown, the parameter represents the slope of the curve and its dimension is the inverse of the velocity. 2.2.2 Dimensionless Motion Equations In order to generalize the problem, the general motion equation are written in a dimensionless form. According to the Buckingham theorem, the dimensional variables are the following: 8 ˆ< ˆ: ˛1 DEI ˛2 DL ˛3 D A (2.9) The dimensionless constants . / are the following: t Dt˛ 1=2 1 ˛ 2 2 ˛ 1=2 3 y .t / Dy.t/˛ 1 2 k 1 Dk1˛ 1 1 ˛ 3 2 w .z ; t / Dw.z; t/˛ 1 2 k 2 Dk2˛ 1 1 ˛ 3 2 ı .z z A/ Dı.z zA/˛2 k 3 Dk3˛ 1 1 ˛ 3 2 T d .t / DTd.t/˛ 1 1 ˛ 2 2 m Dm˛ 1 2 ˛ 1 3 v dr Dvdr˛ 1=2 1 ˛2˛ 1=2 3 x .t / Dx.t/˛ 1 2 D ˛ 1=2 1 ˛ 1 2 ˛ 1=2 3

14 S.G. Neglia et al. Substituting the dimensionless constants in Eq. (2.1), the beam dimensionless equations is obtained: @4w .z ; t / @z 4 C @2w .z ; t / @t 2 Dk 2 y .t / w .z ; t / ı .z z A/ (2.10) The same procedure of the beam is applied to the harmonic oscillator. Along x .t / the motion equation is: m Rx .t / Ck 2 x .t / CN .t / D0 (2.11) where the mass-belt contact force is: (N .t / Dk 3 x .t / if x .t />0 N .t / D0 else (2.12) Along y .t /, the motion equation of the harmonic oscillator is: m Ry .t / DK.t / (2.13) where K.t / is the force due to the following stick-slip condition: ˇ ˇk 1 y .t / w .z A; t / ˇ ˇ < sN .t / and Py .t / Dv dr (2.14) the force K.t / is expressed by: (K.t / D0 (stick) K.t / D k 1 y .t / w .z A; t / CT d .t / (slip) (2.15) where the dimensionless dynamic friction force T d .t / is: T d .t / D sN .t / 1C Py .t / v dr sgn Py .t / v dr (2.16) 2.2.3 Solution Method Applying the Bubnov-Galerkin method and using the eigenfunction of the simply supported beam, it is possible to write: w .z ; t / D N Xi D1 i .z /q i .t / (2.17) where the eigenfunctions i .z / are expressed by: i .z / Dsin.i z / (2.18) and the beam eigenvalues are: ! i Di 2 2 (2.19)

2 Non-linear Dynamics of Jointed Systems Under Dry Friction Forces 15 Projecting the motion equation on a different eigenfunction basis j .z /, integrating on the domain and adopting the following orthonormalization conditions: (PN iD1 R 1 0 j .z / IV i .z /dz D! 2 i ıij PN iD1 R 1 0 j .z / i .z /dz Dıij (2.20) the following N ordinary differential equations are obtained: Rq j .t / Ch! 2 j Ck 2 2 j .z A/iq j .t / Dk 2 y .t / j .z A/ j D1;2; : : : ;N (2.21) With the modal basis expansion expressed in Eq. (2.17) the stick condition becomes: ˇ ˇ ˇ ˇ ˇ k 1 "y .t / N Xi D1 i .z A/q i .t /#ˇ ˇ ˇ ˇ ˇ < sN .t / and Py .t / Dv dr (2.22) and the force K.t /: 8 < : K.t / D0 (stick) K.t / D k 1 hy .t / P N iD1 i .z A/q i .t /iCT d .t / (slip) (2.23) 2.3 Results In this section the most relevant results for the system studied are shown. Different simulations were done in order to evaluate the sensitivity of the system solution to the variation of the belt velocity. In Table 2.1 the dimensionless parameters used for the simulation are shown. It is important to advance a consideration about the magnitude of the belt velocity. For small values of the velocity the system usually shows a continuous switching motion from stick to slip, while for big values of the velocity there will be an initial stick before a complete slip behavior for the whole simulation. Therefore, the velocity range is selected in order to have the system in stick-slip motion (see Table 2.1). 2.3.1 Bifurcation Diagrams In order to calculate the bifurcation diagram for every Lagrangian coordinates, the procedure based on the Poincaré map is used. The Lagrangian coordinates are: 8 ˆ< ˆ: L1 Dx .t / L2 Dy .t / L3 Dw .z A; t / (2.24) Table 2.1 Model parameters Variable Value k 1 1,000 k 2 50 k 3 800 m 10 v dr 10 4 3 10 2 z A 0:3 N 10 3

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