A collision between two bodies is a usual phenomenon in many engineering applications. The most important problem with the collision analysis is determining the hysteresis damping factor or the hysteresis damping ratio. The hysteresis damping ratio is related 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 unknown 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 completely 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 colliding bodies, the classic bouncing ball problem, the resilient impact damper, and a planar slider–crank mechanism, are presented and analyzed.

在许多工程应用中，两个物体之间的碰撞是一种常见现象。碰撞分析中最重要的问题是确定磁滞阻尼系数或磁滞阻尼比。磁滞阻尼比与恢复系数有关。在本文中，为此关系确定了一个显式表达式。为此，考虑变形和接触过程的速度之间的关系的参数表达式。该表达式由两个未知常数组成。利用能量平衡，推导了磁滞阻尼因子与恢复系数之间的新的显式参数表达式。为了确定未知常数，该新表达式相对于数值模型的磁滞阻尼比的均方根（RMS）最小。这种新模型完全适用于恢复系数的整个范围。因此，新模型可用于硬冲击和软冲击问题。最后，给出并分析了两个碰撞体的三个数值示例，即经典的弹跳球问题，弹性减震器和平面滑块-曲柄机构。