The 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 (dof) 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 attacked 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 stands as an open problem.
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Submitted on : Monday, December 17, 2018 - 2:07:25 PM
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  • HAL Id : hal-01957546, version 1

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Huong Le Thi, Stéphane Junca, Mathias Legrand. The first return time to the contact hyperplane for n-degree-of-freedom vibro-impact systems. 2018. ⟨hal-01957546⟩

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