Nonsmooth Modal Analysis of Piecewise-Linear Impact Oscillators

Thorin, Anders; Delezoide, Pierre; Legrand, Mathias

doi:10.1137/16m1081506

Cited by 15 publications

(11 citation statements)

References 38 publications

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“…In addition, the study of more complex types of nonsmooth modes would be of great interest. In particular, one could allow particles to realize several impacts per period [38] or display sticking phases after a grazing contact [28]. The inclusion of dissipative impacts and forcing and the application of the method to more complex finite-element models of continuous impacting systems constitute additional challenging directions.…”

Section: Discussionmentioning

confidence: 99%

“…In addition, several analytical approaches have been used to obtain time-periodic solutions formally for different types of piecewiselinear dynamical systems with rigid impacts. One can mention Fourier and Green function methods [4,5,8,17,18,19,23,24,25,31,37], modal decomposition [27,38] and sawtooth time transformations [32]. Most of the results obtained for discrete systems concern impacts localized on a single particle, and different types of waves have been constructed.…”

mentioning

confidence: 99%

“…Most of the results obtained for discrete systems concern impacts localized on a single particle, and different types of waves have been constructed. In [27,32,38], nonsmooth normal modes have been obtained for general classes of conservative multiple degrees-of-freedom systems (the analysis of [32] is performed for a single or two impacting particles). Spatially-localized oscillations (breathers) with a single impacting node have been also studied for different classes of infinite or finite systems.…”

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confidence: 99%

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Periodic Motions of Coupled Impact Oscillators

James

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Pérignon

2018

Advanced Topics in Nonsmooth Dynamics

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We study the existence and stability of time-periodic oscillations in a chain of coupled impact oscillators, for rigid impacts without energy dissipation. We formulate the search of periodic solutions as a boundary value problem incorporating unilateral constraints. This problem is solved analytically in the vicinity of the uncoupled limit and numerically for larger coupling constants. Different solution branches corresponding to nonlinear localized modes (breathers) and normal modes are computed.

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

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Periodic Motions of Coupled Impact Oscillators

James

Acary

Pérignon

2018

Advanced Topics in Nonsmooth Dynamics

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“…The successive switches in boundary conditions at x D L, reflecting the unilateral contact constraint, are incorporated through appropriate extensions: Equation (10) for the free phase or Equation (11) for contact phase. The following section is concerned with analytical derivations for the computation of periodic solutions by employing the expressions (9), (10) and (11).…”

Section: Non-internally Resonant Elastic Barmentioning

confidence: 99%

“…Techniques traditionally employed for nonlinear modal analysis require a certain degree of smoothness in the nonlinearities [12] and thus fail for systems with nonsmooth nonlinearities such as unilateral contact constraints. Certainly, an accurate characterization of the vibratory response of these systems is essential to achieving enhanced and safer engineering applications [11]. Modal analysis of nonsmooth mechanical systems, also called nonsmooth modal analysis, has been recently proposed for a finite elastic bar of length L subject to a Dirichlet boundary condition at x D 0 and a unilateral contact constraint at x D L [13].…”

Section: Introductionmentioning

confidence: 99%

Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint

Yoong

Legrand

2019

Nonlinear Structures and Systems, Volume 1

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The present contribution describes a numerical technique devoted to the nonsmooth modal analysis (natural frequencies and mode shapes) of a non-internally resonant elastic bar of length L subject to a Robin condition at x D 0 and a frictionless unilateral contact condition at x D L. When contact is ignored, the system of interest exhibits non-commensurate linear natural frequencies, which is a critical feature in this study. The nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the boundary conditions. In order to find a few of the above families, the unknown displacement is first expressed using the well-known d'Alembert's solution incorporating the Robin boundary condition at x D 0. The unilateral contact constraint at x D L is reduced to a conditional switch between Neumann (open gap) and Dirichlet (closed gap) boundary conditions. Finally, T-periodicity is enforced. It is also assumed that only one contact switch occurs every period. The above system of equations is numerically solved for through a simultaneous discretization of the space and time domains, which yields a set of equations and inequations in terms of discrete displacements and velocities. The proposed approach is non-dispersive, non-dissipative and accurately captures the propagation of waves with discontinuous fronts, which is essential for the computation of periodic motions in this study. Results indicate that in contrast to its linear counterpart (bar without contact constraints) where modal motions are sinusoidal functions "uncoupled" in space and time, the system of interest features nonsmooth periodic displacements that are intricate piecewise sinusoidal functions in space and time. Moreover, the corresponding frequency-energy "nonlinear" spectrum shows backbone curves of the hardening type. It is also shown that nonsmooth modal analysis is capable of efficiently predicting vibratory resonances when the system is periodically forced. The pre-stressed and initially grazing bar configurations are also briefly discussed.

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Nonsmooth Modal Analysis: From the Discrete to the Continuous Settings

Thorin

Legrand

2018

Advanced Topics in Nonsmooth Dynamics

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This chapter addresses the prediction of vibratory resonances in nonsmooth structural systems via Nonsmooth Modal Analysis. Nonsmoothness in the trajectories is induced by unilateral contact conditions in the governing (in)equations. Semi-analytical and numerical state-of-the-art solution methods are detailed. The significance of nonsmooth modal analysis is illustrated in simplified one-dimensional space semi-discrete and continuous frameworks whose theoretical and numerical discrepancies are explained. This contribution establishes clear evidence of correlation between periodically forced and autonomous unilaterally constrained oscillators. It is also shown that strategies using semi-discretization in space are not suitable for nonsmooth modal analysis. The spectrum of vibration exhibits an intricate network of backbone curves with no parallel in nonlinear smooth systems. Terminology and acronyms The purpose of this chapter is to provide a general picture of the state-of-the-art vibratory analysis of nonsmooth systems. This topic lies at the interface between modal analysis of smooth nonlinear systems and nonsmooth contact dynamics dedicated to the time-evolution of nonsmooth systems, undergoing impact or dry friction, for instance. Some elementary concepts are succinctly recalled for the purpose of completeness. Unless otherwise stated, the epithet discrete (as in "discrete systems" or "discrete setting") designates semidiscretization in space, while continuous refers to everything else.

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Nonsmooth Modal Analysis of Piecewise-Linear Impact Oscillators

Cited by 15 publications

References 38 publications

Periodic Motions of Coupled Impact Oscillators

Periodic Motions of Coupled Impact Oscillators

Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint

Nonsmooth Modal Analysis: From the Discrete to the Continuous Settings

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