In this study, we develop an analytical formula to approximate the damping coefficient as a function of the coefficient of restitution for a class of continuous contact models. The contact force is generated by a logical point-to-point force element consisting of a linear damper connected in parallel to a spring with Hertz force–penetration characteristic, while the exponent of deformation of the Hertz spring can vary between one and two. In this nonlinear model, it is assumed that the bodies start to separate when the contact force becomes zero. After separation, either the restitution continues or a permanent penetration is achieved. Therefore, this model is capable of addressing a wide range of impact problems. Herein, we apply an optimization strategy on the solution of the equations governing the dynamics of the penetration, ensuring that the desired restitution is reproduced at the time of separation. Furthermore, based on the results of the optimization process along with analytical investigations, the resulting optimal damping coefficient is analytically expressed at the time of impact in terms of system properties such as the effective mass, penetration velocity just before the impact, coefficient of restitution, and the characteristics of the Hertz spring model.

在这项研究中，我们针对一类连续接触模型开发了一个解析公式，以将阻尼系数近似为恢复系数的函数。接触力由逻辑上的点对点力元件产生，该点对点力元件由线性阻尼器组成，该线性阻尼器与具有赫兹力-穿透特性的弹簧并联连接，而赫兹弹簧的变形指数可在一到两个之间变化。在该非线性模型中，假设当接触力变为零时，主体开始分离。分离后，要么恢复原状，要么实现永久渗透。因此，该模型能够解决各种冲击问题。在此，我们将优化策略应用于控制渗透动力学的方程式的解，确保在分离时再现所需的还原。此外，根据优化过程的结果以及分析研究，得出的最佳阻尼系数将在撞击时根据系统属性（例如有效质量，撞击前的穿透速度，恢复系数，以及赫兹弹簧模型的特性