AMPLITUDE-MODULATED DYNAMICS OF A RESONANTLY EXCITED AUTOPARAMETRIC 2-DEGREE-OF-FREEDOM SYSTEM

Forced, weakly nonlinear oscillations of a two degree-of-freedom autop arametric vibration absorber system are studied for resonant excitatio ns, The method of averaging is used to obtain first-order approximatio ns to the response of the system. A complete bifurcation analysis of t he averaged equations is undertaken in the subharmonic case of interna l and external resonance. The ''locked pendulum'' mode of response is found to bifurcate to coupled-mode motion for some excitation frequenc ies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mis tuned away from the exact internal resonance condition. The software p ackages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super- and s ub-critical Hopf bifurcations arise and the limit cycles quickly under go period-doubling bifurcations to chaos. These imply chaotic amplitud e modulated motions for the system.