Singularity of nonideal rigid-body systems

The algorithm of linear complementary problem (LCP) has developed into a general method in modern dynamics of rigid-body systems with unilateral constraints. The nonideal bilateral constraint system can also be described into a linear complementary problem by some technique. Painleve's paradox is one of the basic difficulties when LCP is applied in a system with unilateral constraints, which shows singularity in systems subjected to unilateral constraints. The sort of singular problem also exists in systems subjected to bilateral constraints with friction. In modified Painleve's example, the singularity in systems subjected to bilateral constraints is showed. Comparison with unilateral system is made. The cause, conditions of the singularity and the method to break away from singularity are considered. A rule for the velocity's jump to break away from the singular states is given.