The Normal Impact on an Elastic Half Space of an Elastic Rod-Mass System

The normal impact problem of a one-dimensional elastic rod with a lumped mass on the trailing end onto an elastic half space is solved for the time-dependent interface displacement and stress. This problem is reduced to an integral equation, whose kernel is the solution of a simpler auxiliary problem, which is solved in closed form. After examining the graph of the kernel it is found that a simple linear expression adequately represents its half space contribution. This approximation allows the integral equation to be solved in closed form and provides insight into its solution. Numerical results are presented, which display the impact and rebound of the rod, and illustrate the presence of major effects from the Rayleigh wave in the half space and the reflected wave from the trailing end of the rod. Results are presented for various half-space materials, rod lengths, and masses. It is found that in the absence of the mass the maximum contact stress depends entirely on the rod material, but with the lumped mass added the contact stress can become much greater and depends on the rod, the half space, and the mass.