含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性. 

A LCP METHOD FOR THE DYNAMICS OF PLANAR MULTIBODY SYSTEMS WITH IMPACT AND FRICTION

This paper is presented to show the modeling and numerical method for the dynamics of the planar multi-rigid-body system with contact, impact and Coulomb's dry friction. The multibody system consists of the rigid bodies which are linked with ideal joints and driving motors, so the system constraint equations included two parts, scleronomic constraint equations and rheonomic constraint equations. Based on the theory of contact mechanics, the local deformations in contact bodies are taken into account and the normal forces of contact surfaces are expressed as nonlinear functions of relative penetration depth and its speed during impact between two bodies. The Coulomb dry friction model is used to describe the tangential frictional forces of contact surfaces. Using the concept of friction saturation and the relative acceleration of the contact point in the tangential direction, the complementarity conditions and formulations about the friction law are given. The problems of detecting stick-slip transitions of contact points and solving frictional forces in stick situation are formulated and solved as a linear complementarity problem (LCP) by the event-driven scheme. The dynamical equations of the system are obtained by Lagrange's equations of the first kind and Baumgarte stabilization method in order to reduce the constraint drift and solve the system motion, normal contact forces and tangential friction forces as well as ideal joint constraint forces and driven constraint forces in the system. Finally, the numerical example of a planar multi-rigid-body like flat beater is given to analyze its dynamical behavior and show the availability of the method in this paper.