Nonsmooth dynamics of a 3D rigid body on a vibrating plate

Abstract

In this work, we study a 3D rigid body, which is in a shape similar to a dumbbell, bouncing on a harmonically vibrating plate. The system involves two contact sets whose states vary with the relative motion between the plate and the dumbbell. These states are closely related to the physical properties of contact surfaces, and can be identified using the relationships of the relative kinematics. Under certain states, system-singular modes will likely occur due to the absence of the tangential compliance in Coulomb’s friction. Resolutions for these system singularities are given, and an integrated model, taking a hierarchical structure adaptable to the state variations, has been developed. Within an impact-free state, the contact forces to drive the motion of the dumbbell are obtained using an LCP (linear complementary problem) formulation. For the system in an impact state, the post-impact outputs to serve as the initial conditions for the subsequent motion can be calculated based on the new theory developed in impact dynamics. Specifying the dumbbell with initial states to bounce in a vertical plane, this model was justified by the comparison of our results and the experimental findings in other work. As certain periodic behaviors appear in the planar dynamics, the system in a 3D scenario also reveals intriguing patterns in the trajectories of the dumbbell’s mass center as projected onto the horizontal plane. They include a closed circular orbit, a dog-legged path, and a straight line.