考虑奇异性的平面多刚体系统冲击问题理论解

Theoretical dynamical impact solution on planar multi-rigid-body systems considering singularity

当处理多刚体系统的冲击问题时,经典的分析方法是通过在无限小的时间间隔上直接积分得到冲击前后的广义速度之间的关系。当库仑干摩擦的因素考虑在内时,经典的方法遇到了困难。首先,冲击的过程中切向模式的变化导致了摩擦力表达式的变化;其次,双边约束的特性会带来库仑摩擦定律中未知法向约束反力的绝对值的引入;最后,系统刚性与库仑摩擦的耦合会导致求解广义速度过程中的奇异性问题等。作者采取了分段分析的方法来处理冲击中复杂的切向模式,引入符号函数来处理未知法向约束反力的绝对值,采用"无碰冲击"来解决奇异性问题。且针对一个摩擦接触点的情况,给出了冲击前后广义速度关系的理论表达式,通过该表达式,冲击后的系统状态可以通过代数的方式获得,从而避免进入繁琐的微分过程,且在该方法中能够预测刚体冲击过程中的奇异位置以及切向滑动模式的变化条件,并因此得出结论:即使在冲量/速度水平上来研究非理想的双面约束系统的动力学问题,依然不能避免解的奇异性问题。采用文中例子验证了方法的合理性。

Impact problems on planar multi-rigid-body systems with friction are considered. When friction cannot be ignored, there are the following problems to be solved. First, the phenomenon of slip-stick and slip-reversal at the frictional applied point can cause the change of direction of the frictional force and make it difficult to integrate the frictional force over the impact interval according to the traditional impulsive dynamics of multi-rigid-body systems. Second, absolute values of unknown constraint reactions due to Coulomb frictional law are included in the dynamical equations. Furthermore, the singularity due to the coupling between tangential frictional forces and normal reactions may arise in calculating unknown reactions. Piecewise analysis over the period of different tangential motion is used to overcome the energy inconsistence and the sign function is introduced to solve the second problem. The singularity problem is solved by the method of ＂tangential impact＂. The analytical impact solution for multi-rigid-body systems with friction at one joint is obtained, i.e. , the dynamic generalized velocities after impact can be obtained according to initial conditions and geometry properties of the system before impact by using the provided formulation. This method can avoid the difficulties in solving differential equations with variable scale and the resuits can avoid the energy inconsistency before and after impact due to considering the complex of the tangential sliding mode. The singularity can be predicted by this method and draw a conclusion that the work at the impulse/velocity level cannot avoid the singularity for this sort of system. The example implies that the scheme provided in the paper is valid.