Periodic solutions of $n$-dof autonomous vibro-impact oscillators with one lasting contact phase

Anders Thorin (1) , Pierre Delezoide (2) , Mathias Legrand (1)

1 Structural Dynamics and Vibration Laboratory [Montréal]
2 Independent Researcher

In the continuum structural mechanics framework, a unilateral contact condition between two flexible bodies does not generate impulsive contact forces. However, finite-dimensional systems, derived from a finite element semi-discretization in space for instance, and undergoing a unilateral contact condition, require an additional impact law: Unilateral contact occurrences then become impacts of zero duration unless (i) the impact law is purely inelastic, or (ii) the pre-impact velocity is zero. This contribution explores autonomous periodic solutions with one contact phase per period and zero pre-impact velocity [case (ii)], for any n-dof mechanical systems involving linear free-flight dynamics together with a linear unilateral contact constraint. A recent work has shown that such solutions seem to be limits of periodic trajectories with k impacts per period as k increases. Minimal analytic equations governing the existence of such solutions are proposed, and it is proven that, generically, they occur only for discrete values of the period. It is also shown that the graphs of such periodic solutions have two axes of symmetry in time. Results are illustrated on a spring–mass system and on a 4-dof two-dimensional system made of 1D finite elements. Animations of SPPs with up to 30 dofs are provided.

With axial symmetry:

With zero initial gap:

With non-diagonal mass matrix:

Mots clés: Vibration analysis; Impact dynamics; Periodic solutions; Vibro-impact oscillators; Non-smooth modal Analysis

Domaines: Vibrations [physics.class-ph]; Génie mécanique [physics.class-ph]; Mécanique des structures [physics.class-ph]; Dynamique, vibrations; Structures

Anders Thorin, Pierre Delezoide, Mathias Legrand. Periodic solutions of $n$-dof autonomous vibro-impact oscillators with one lasting contact phase. Nonlinear Dynamics, 2017, 90 (3), pp.1771-1783. ⟨10.1007/s11071-017-3763-z⟩. ⟨hal-01505888⟩