This work presents a new numerical integration method for determining dynamics of a class of multibody systems involving impact and friction. Specifically, these systems are subject to a set of equality constraints and can exhibit single frictional impact events. Such events are associated to significant numerical stiffness, appearing in the equations of motion. The new method is a time-stepping scheme, involving proper incorporation of a novel return mapping into an augmented Lagrangian formulation, developed recently for systems with bilateral constraints only. Namely, when an impact is detected during a time step, this map is applied at the end of the step in order to bring the system position back to the configuration manifold with the allowable motions. The construction of this map is based on the concept of Jacobi fields on non-flat manifolds. Moreover, once an impact event is detected, the post-impact state is determined by employing a combination of analytical and numerical tools. First, a proper coordinate transformation is performed, bringing the system into a new set of coordinates, which are suitable for describing the impact dynamics. In these coordinates, the dominant dynamics is described by a system of three equations of motion only, which are valid during the short contact interval. In addition, these equations are geometrically discretized by using appropriate cubic splines on the configuration manifold. In this way, the inherent numerical stiffness of the class of systems examined is properly addressed, since it is restricted to a space with a much smaller dimension and a much shorter time scale. Finally, the accuracy and efficiency of the new method is demonstrated by applying it to a selected set of mechanical examples.

这项工作提出了一种新的数值积分方法，用于确定涉及冲击和摩擦的一类多体系统的动力学。具体而言，这些系统受到一组相等性约束，并且可能表现出单个摩擦冲击事件。这样的事件与出现在运动方程中的显着的数值刚度有关。新方法是一种时间步长方案，涉及将新的返回映射正确地合并到增强的拉格朗日公式中，该方法最近仅针对具有双边约束的系统开发。即，当在一个时间步骤中检测到撞击时，在该步骤的末尾应用此图，以便以允许的运动将系统位置带回到配置歧管。该图的构建基于非平面流形上的Jacobi场的概念。此外，一旦检测到撞击事件，就可以通过结合使用分析工具和数值工具来确定撞击后的状态。首先，执行适当的坐标转换，使系统进入一组新的坐标，这些坐标适合描述冲击动力学。在这些坐标中，主要动力学仅由三个运动方程式的系统描述，这在较短的接触间隔内有效。此外，通过在配置歧管上使用适当的三次样条，可将这些方程式进行几何离散。这样，就可以正确解决所检查系统类别的固有数值刚度，因为它被限制在一个尺寸更小，时间尺度更短的空间中。最后，通过将新方法应用于一组选定的机械示例，证明了该新方法的准确性和效率。