Tangential velocity during impact with friction in three-dimensional rigid multibody systems

This paper introduces an analysis of tangential velocity of the contact point during a single-point impact with friction in three-dimensional rigid multibody systems. Considering the normal component of impulse at the contact point as the independent variable, a system of nonlinear differential equations describing the progress of the tangential velocity is derived. Quartic algebraic equations for the invariant and the flow change directions are obtained. The system of differential equations could have a discontinuity problem if the tangential velocity instantaneously vanishes. Effectively, this problem is mathematically solved and the new sliding direction is specified. During impact, the sliding direction could continuously change or the sliding could stop and the non-sliding continues or the sliding could restart along a new direction. The stick-slip effect could also have its influence. Two friction models are used, and an appropriate numerical procedure is suggested that can handle all the possible sliding scenarios. To verify the proposed method, the numerical solution is compared to the analytical solution for a special case of a homogeneous ellipsoid that collides with a horizontal fixed rough surface. A case study of a four-degrees-of-freedom spatial robot that collides with its environment is considered. The bifurcation associated with the variation of the kinetic coefficient of friction is investigated, and a set of threshold values of it are determined. Some phase portraits are drawn, and the invariant directions and flow change directions are drawn for all portraits. Each portrait represents all the flow trajectories of the tangential velocity for different initial conditions with a specific kinetic coefficient of friction. This qualitative analysis not only helps in understanding the behavior of multibody systems in such situations but also it gives insight to the physical phenomenon of impact itself.