A Model for Contact of a Rod with an Obstacle Using the Damped Normal Compliance Condition

Abstract

This work models, analyses and simulates the contact process between a linear elastic rod and a reactive obstacle, using the new `damped normal compliance' (DNC) contact condition. This condition allows for the dissipation of energy during the contact process, replacing the notion of `restitution coefficient' which is shown to be not well defined, since it depends on the applied force and the initial conditions. We establish the energy balance equation and study the possible steady states of the system. It is found that with sufficient damping once in contact, the rod never leaves the obstacle. On the other hand, when the damping is below a critical value, the rod leaves the obstacle and then returns. The existence of  the unique solution when viscosity is added to the rod is shown using the Banach fixed point theorem, and in the limit when the viscosity vanishes, the existence is proved, but the uniqueness is an unresolved question. The various theoretical results are depicted using computer simulations, which are based on a simple explicit finite difference scheme. The simulations show that as the stiffness of the obstacle increases and the damping decreases, the penetration decreases.

Keywords: dynamic contact, damped normal compliance, coefficient of restitution, existence, simulations