Open-Loop Centering of a Point Mass on a Horizontally Vibrating Frictional Table

In this paper we study a particle sliding on a horizontally vibrating frictional table. This dynamical system has applications in parts manipulation within robotics and manufacturing. We first show numerically that, upon vibrating the table in a specific open-loop way, the particle moves to a target location under Coulomb friction alone. We then turn to analytical treatment of the system. The governing equations have strong nonlinearities due to the friction. We apply the method of multiple scales (MMS) near a 1:2 resonance in two frequencies relevant to the forcing. Application of the MMS requires some difficult integrals, for which we develop asymptotic approximations. The MMS slow flow has logarithmic nonlinearities, is valid near the target location on the table, and is easy to integrate numerically since it retains parametric excitation only in slow time. The slow flow matches very well with full numerical solutions. This problem has a practical motivation, novel elements in the application of the MMS, a satisfactory slow flow, and potential for further analytical simplification.