Inelastic impact and the coefficient of restitution

A model for the impact between a structurally deformable sphere and a rigid flat surface is presented. It includes a nonlinear contact element that accounts for energy loss due to local plastic deformation or flattening of the sphere, a viscous element that accounts for energy loss due to wave propagation and/or damping, and a linear stiffness element that accounts for recoil effects during and after impact. A linearized version of the model facilitates normalization of the equations with helpful insights into the impact problem. The results show that the impact event can be characterized by two non-dimensional parameters, namely the relative stiffness., which accounts for recoil effects, plastic deformation and/or flattening of the sphere, and the damping factor zeta, which accounts for viscous and/or wave propagation effects. Model predictions compare well with experimental data for sports balls that are excellent examples of deformable spheres. The normalized impact force and the coefficient of restitution (COR) are dependent on both parameters. At low speeds and damping factors the normalized impact force and COR mainly depend on damping, whereas, at low speeds and large damping factors they depend on damping and flexibility. For a given damping factor, the normalized maximum impact force and COR decrease with higher flexibility. Depending on the deformation characteristics of the ball, at higher speeds the COR decreases further due to surface flattening effects.