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

International audienceIn the continuum structural mechanics framework, a unilateral contact condition between two flexible bodiesdoes not generate impulsive contact forces. However, finite-dimensional systems, derived from a Finite Elementsemi-discretization in space for instance, and undergoing a unilateral contact condition, require an additionalimpact law: unilateral contact occurrences then become impacts of zero duration unless (i) the impact law is purelyinelastic, or (ii) the pre-impact velocity is zero. This contribution explores autonomous periodic solutions withone contact phase per period and zero pre-impact velocity (case (ii)), for any n-dof mechanical systems involvinglinear free-flight dynamics together with a linear unilateral contact constraint. Such solutions seem to be limits ofperiodic trajectories with $k$ impacts per period as $k\to \infty$. Minimal analytic equations governing the existence ofsuch solutions are proposed and it is proven that, generically, they occur only for discrete values of the period. It isalso shown that the graphs of such periodic solutions have two axes of symmetry in time. Results are illustrated ona spring-mass system and on a 4-dof two-dimensional system made of 1D Finite Elements. Animation corresponding to Fig. 5 (left): Animation corresponding to Fig. 5 (right): Another 1 SPP for the same system (5 DOFs): 1 SPP with 10 DOFs 1 SPP with 20 DOFs 1 SPP with 30 DOFs Animation corresponding to Figure 7 (with rod Finite Elements and non-trivial geometry): Animation corresponding to Figure 8 (zero initial gap): Animation corresponding to Figure 9 (potential 1 SPP with punctual symmetry):