Multiple scales analysis for the slowest flow of a particle on a horizontally vibrating frictional table

In recent work we examined frictionally driven motions of a point mass on a horizontally shaken table. Specifically, we examined an open-loop table motion that sends the sliding point mass approach a fixed point in space. Numerics followed by multiple scales analysis revealed three time scales: one fast time scale of the table oscillation, one slow time scale of a spiralling-in motion with superposed slow oscillations, and a slowest time scale of non-oscillatory approach to the fixed point. Of these, the motion at the intermediate or slow scale was captured using multiple scales analysis. In the present work, we further analyze the slow motions to remove the slow oscillations and extract the slowest spiralling-in motion which is of primary interest. The problem presents interesting analytical aspects. While in the first mutiple-scales analysis we used asymptotic approximations of some integrals, here we encounter logarithmic nonlinearities that cause difficulties. Fortunately, using a Fourier series expansion, we can extract the relevant integrals and the multiple scales analysis is successful again. While our results are based on assuming the table oscillation frequency is large and the friction is small, our demonstration of the doubly averaged motion towards the fixed point is more broadly convincing and presents interesting academic elements as well. This paper concludes the asymptotic analysis of a stable attractor for a frictionally driven point mass. We emphasize that such a stable attractor for purely horizontal open-loop motions of the table had not been reported in the literature before.