Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation

A class of periodic motions of an inverted pendulum with rigid lateral barr iers is analysed under the hypothesis that the system is forced by impulsed periodic excitation. Due to the piece-wise linear nature of the problem, t he existence and the stability of the cycles are determined analytically. I t is found that they depend on both classical (saddle-node and period-doubl ing) and non-classical bifurcations, the latter involving a 'synchronizatio n' between impulses and impacts which leads to the sudden disappearing of t he orbits. Attention is paid to the physical interpretation of these bifurc ations, and to the determination of analytical criteria for their occurrenc e. We study how the relative position (with respect to the excitation ampli tude) of the local bifurcations determines the system response and the bifu rcation scenario. Symmetric and unsymmetric excitations are considered and the regions of stability of the periodic solutions are analytically determi ned. Finally, a comparison with the case of harmonic excitation is presente d showing both analogies and differences, and highlighting how the impulsed excitation allows to obtain stable periodic responses at higher values of the excitation amplitude. (C) 2000 Elsevier Science Ltd. All rights reserve d.