Motion of a rigid body with a rounded base due to harmonic excitation

The asymptotic response of a rigid block to harmonic force is very rich and well worth studying, for both theoretical and technical reasons. The present study is motivated partly by the possibility of imperfections in the contact between the foundation and the block (therefore a rounded block is assumed), while the theoretical study of the dynamics of a particular softening oscillator provides a further motivation. The problem is mainly tackled numerically, assuming Poincaré sections. Frequency response and characteristic curves are evaluated by varying the relevant parameters. The stability boundaries of the relevant motions are evaluated on the basis of an analysis of the eigenvalues of the derivative of the Poincaré map. The evolution of the attracting basins of the various periodic orbits is followed using the cell-to-cell mapping technique. Comparisons with the one-well Duffing oscillator and the planar rigid block are made throughout. The study reveals new aspects of the wide class of softening oscillators.