First return time to the contact hyperplane for n-degree-of-freedom vibro-impact systems - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Discrete and Continuous Dynamical Systems - Series B Year : 2022

First return time to the contact hyperplane for n-degree-of-freedom vibro-impact systems

Abstract

The paper deals with the dynamics of conservative $N$-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.
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Dates and versions

hal-01957546 , version 1 (17-12-2018)
hal-01957546 , version 2 (11-05-2020)

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Huong Le Thi, Stéphane Junca, Mathias Legrand. First return time to the contact hyperplane for n-degree-of-freedom vibro-impact systems. Discrete and Continuous Dynamical Systems - Series B, 2022, 27 (1), pp.1-44. ⟨10.3934/dcdsb.2021031⟩. ⟨hal-01957546v2⟩
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