Nonsmooth modal analysis of a $N$-degree-of-freedom system undergoing a purely elastic impact law

Mathias Legrand 1 Stéphane Junca 2 Sokly Heng 2
2 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
Abstract : The dynamics of a $N$-degree-of-freedom autonomous oscillator undergoing an energy-preserving impact law on one of its masses is investigated in the light of nonlinear modal analysis. The impacted rigid foundation provides a natural Poincaré section of the investigated system from which is formulated a smooth First Return Map well-defined away from the grazing trajectory. In order to focus on the impact-induced nonlinearity, the oscillator is assumed linear. Continuous one-parameter families of $T$-periodic orbits featuring one impact per period and lying on two-dimensional invariant manifolds in the state-space are shown to exist. The geometry of these piecewise-smooth manifolds is such that a linear “flat” portion (on which contact is not activated) is continuously attached to a purely nonlinear portion (on which contact is activated once per period) exhibiting a velocity discontinuity through a grazing orbit. These features explain the newly introduced terminology “Nonsmooth modal analysis”. The stability of the periodic orbits lying on the invariant manifolds is also explored by calculating the eigenvalues of the linearized First Return Map. Internal resonances and multiple impacts per period are not addressed in this work. However, the pre-stressed case is succinctly described and extensions to multiple oscillators as well as self-contact are discussed.

A few animated nonlinear modes
  • Branch around $\omega_1$
  • Branch around $\omega_2$
  • Branch around $\omega_3/3$
  • Branch around $\omega_4/2$
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https://hal.archives-ouvertes.fr/hal-01185980
Contributeur : Mathias Legrand <>
Soumis le : mardi 7 février 2017 - 12:33:18
Dernière modification le : samedi 4 mars 2017 - 01:08:30

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Distributed under a Creative Commons Paternité 4.0 International License

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Mathias Legrand, Stéphane Junca, Sokly Heng. Nonsmooth modal analysis of a $N$-degree-of-freedom system undergoing a purely elastic impact law. Communications in Nonlinear Science and Numerical Simulation, Elsevier, 2017, 45, pp.190-219. <10.1016/j.cnsns.2016.08.022>. <hal-01185980v2>

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