Dependence of the coefficient of restitution on the shape of an impacting body

Abstract

The effect of the shape possessed by an impacting body on the coefficient of restitution is addressed using the case of low velocity axial impact of axisymmetric bars. We consider bars of a fixed mass but varying shapes that are parameterized by certain non-dimensional parameters. The eigenvalue problems in all such cases are solved analytically, and the obtained eigenfunctions are used to address the problem of impact with a rigid wall. This transient response problem is solved exactly using the method of Laplace transform. The analytical solutions are compared with fully numerical solutions obtained from the impact of three-dimensional axisymmetric bars using the Finite Element Method (FEM), and a very good match is obtained in all cases considered. In each case, the deformation and restitution times are determined for calculating the coefficient of restitution (COR) using Poisson’s definition, which is compared with Newton’s definition of COR. It is observed for variable cross-section bars that the end of deformation phase is not marked by the bar being completely at rest with the energy purely as internal strain energy, as is conventionally thought of. The redistribution of the initial kinetic energy of the bar after impact in the form of gross motion of the bar and its internal motion is studied. It is found that the longitudinal cross-section variation of the bar decides its rebound characteristic, which is observed by the velocity of gross motion of the bar post impact.