Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator

The numerical study of a 10 degree of freedom impact oscillator is presented. The mathematical model consists of a linear array of masses in which each mass is connected to its nearest neighbor by identical springs and dashpots. One end mass is attached to a rigid support while the other is free to impact a sinusoidally vibrating rigid table. Bifurcation diagrams based on the impact Poincaré map are obtained over the entire range of natural frequencies, taking the table frequency as the bifurcation parameter. The diagrams reveal many chaotic bands as well as a wide variety of period-(n, m) (P_{n, m}) orbits, where n is the period with respect to the impact Poincaré map and m is the number of table periods. Perhaps the most interesting solutions are P_{∞, m} solutions (where m is finite). These arise from sticking events in which the time between impacts approaches zero in finite time. The convergence properties of the impact time sequence during a sticking event is shown to be controlled by the coefficient of restitution of the impact law. The spatial complexity of the motions is studied by finding proper orthogonal modes for various steady states, and the number of excited degrees of freedom is estimated by finding the number of modes needed to exceed 99% of the signal power. These results are compared to the predictions of fractal dimension theory.