Dissipative gravitational bouncer on a vibrating surface

Abstract

We study the dynamical behavior of a particle flying under the influence of a gravitational field, with dissipation constant λ (Stokes-like), colliding successive times against a rigid surface vibrating harmonically with restitution coefficient α. We define re-scaled dimensionless dynamical variables, such as the relative particle velocity Ω with respect to the surface's velocity; and the real parameter τ accounting for the temporal evolution of the system. At the particle-surface contact point and for the k′th collision, we construct the mapping described by (τk ; Ωk ) in order to analyze the system's nonlinear dynamical behavior. From the dynamical mapping, the fixed point trajectory is computed and its stability is analyzed. We find the dynamical behavior of the fixed point trajectory to be stable or unstable, depending on the values of the re-scaled vibrating surface amplitude Γ, the restitution coefficient α and the damping constant λ. Other important dynamical aspects such as the phase space volume and the one cycle vibrating surface (decomposed into absorbing and transmitting regions) are also discussed. Furthermore, the model rescues well known results in the limit λ = 0.