Collision between hard objects causes abrupt changes in the velocities of the system, which are characterized by very large contact forces over very small time durations. A common approach in the analysis of such collisions is to describe the system velocities using an impulse–momentum based relationship. The time duration of impact and the deformations at the contact points are usually assumed to be negligible for such an impact models, and the system velocities are evolved in terms of the impulses on the system. This type of impact models are usually relevant for hard (rigid) impacts, where deformations at the contact points can be considered negligible. However, these models cannot determine the forces during the impact process. The main objective of this work is to reformulate the impulse–momentum based model to determine the forces during an impact event, by relaxing the rigidity assumption to allow small deformations at the contact points.

硬物之间的碰撞会导致系统速度突然变化，其特征是在非常短的时间内会产生很大的接触力。分析此类碰撞的常用方法是使用基于冲量-动量的关系描述系统速度。对于这种冲击模型，通常假定冲击的持续时间和接触点处的变形可以忽略不计，并且系统速度根据对系统的脉冲而变化。此类冲击模型通常与硬（刚性）冲击有关，在这种情况下，接触点的变形可以忽略不计。但是，这些模型无法确定冲击过程中的作用力。