Multibody Syst Dyn https://doi.org/10.1007/s11044-020-09732-2 A new model of the contact force for the collision between two solid bodies Hossein Safaeifar 1 · Anooshirvan Farshidianfar 1 Received: 1 December 2019 / Accepted: 4 March 2020 © Springer Nature B.V. 2020 Abstract A collision between two bodies is a usual phenomenon in many engineering ap- plications. The most important problem with the collision analysis is determining the hys- teresis damping factor or the hysteresis damping ratio. The hysteresis damping ratio is re- lated to the coefficient of restitution. In this paper, an explicit expression is determined for this relation. For this reason, a parametric expression is considered for the relation between the deformation and its velocity of the contact process. This expression consists of two un- known constants. Using the energy balance, a new explicit parametric expression between the hysteresis damping factor and the coefficient of restitution is derived. For determining the unknown constants, the root mean square (RMS) of the hysteresis damping ratio of this new expression with respect to the numerical model is minimized. This new model is com- pletely suitable for the whole range of the coefficient of restitution. So, the new model can be used in the hard and soft impact problems. Finally, three numerical examples of two col- liding bodies, the classic bouncing ball problem, the resilient impact damper, and a planar slider–crank mechanism, are presented and analyzed. Keywords Collision · Damping · Contact force model · Hysteresis damping factor · Hysteresis damping ratio Nomenclature RMS Root of mean square F Contact force K Generalized stiffness parameter  Relative normal deformation R 1 and R 2 Radii of spheres σ 1 and σ 2 Material parameters v 1 and v 2 Poisson’s ratios B A. Farshidianfar farshid@um.ac.ir H. Safaeifar safaeifar@mail.um.ac.ir 1 Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran