Regular Nonlinear Dynamics and Bifurcations of an Impacting System under General Periodic Excitation

A class of periodic motions of an inverted pendulum with rigid lateralconstraints is analyzed under the hypothesis that the system is forcedby an arbitrary periodic excitation. The piecewise linear natureof the problem permits to obtain analytical results. The periodicsolutions are determined as fixed points of the stroboscopic Poincarémap, and it is shown that the stability is lost through classicalsaddle-node or period-doubling bifurcations. It is shown that theexistence paths can be determined, both geometrically and analytically,on the basis of a function which can easily be derived from theexcitation f(t). The main qualitative properties of these paths arediscussed, and attention is paid to the detection of the bifurcationsdetermining the stability of the solutions. None of the obtained resultsdepend on the specific properties of the excitation, and all can beemployed in the analysis of various cases with both symmetric andunsymmetric excitations. Some illustrative examples are reported at theend of the paper.