Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

Yoong, Carlos; Thorin, Anders; Legrand, Mathias

doi:10.1007/s11071-017-4025-9

Cited by 15 publications

(64 citation statements)

References 35 publications

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“…For the initially grazing configuration, the NSMs emerge as vertical lines embedding periodic solutions with identical frequency and increasing energy. It also appears that every given collection of scattered points represents a purely independent subharmonic vibration, as opposed to the internally resonant bar which features subharmonic as well as internally resonant NSMs [13]. This observation coincides with the fact that the system of interest does not satisfy the full internal resonance condition.…”

Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 52%

“…2 as sets of scattered points supposedly belonging to NSMs. Contrary to main NSMs of the internally resonant bar studied in [13] where the energy continuously depends on the frequency, the depicted scattered points indicate more complicated backbone curves. Figure 2 suggests that a "nicely" connected continuum representing a possible "main NSM" does not exist.…”

Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 82%

“…NSMs computed for˛D 1 and˛D 1=2 are now investigated. In contrast to the results exposed in [13], the intricate dynamics caused by the spring complicates the identification of NSM branches and a highly fine discretization is required to discern how admissible periodic solutions organize to form a continuum. The backbone curves, emerging in the frequency range OE!…”

Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 99%

“…1 nor 1 , respectively [8, p. 245]. Accordingly, the complete internal resonance condition emerging when the bar is clamped at x D 0 no longer holds [13]. Non-trivial solutions of the unilateral contact problem described by Eqs.…”

Section: Non-internally Resonant Elastic Barmentioning

confidence: 99%

“…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]. This system satisfies a complete internal resonance condition, i.e.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 52%

Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 82%

Section: Spectrum Of Nonsmooth Vibrationmentioning

confidence: 99%

Section: Non-internally Resonant Elastic Barmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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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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Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

Cited by 15 publications

References 35 publications

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

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

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

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