On the secondary resonance of a spinning disk under space-fixed excitations

We investigate the possibility of secondary resonance of a spinning disk under space-fixed excitations. Von Karmans plate model is employed in formulating the equations of motion of the spinning disk. Galerkin's procedure is used to discretize the equations of motion, and the multiple scale method is used to predict the steady state solutions. Attention is focused on the nonlinear coupling between a pair of forward (with frequency omega(mn)) and backward (with frequency omega(mn)) traveling waves. It is found that combination resonance may occur when the excitation frequency is close to 2 omega(mn) + omega(mn), omega(mn) + 2omega(mn), or 1/2(omega(mn) + omega(mn)). When the combination resonance does occur the frequencies of the free oscillation components are shifted slightly from the respective natural frequencies omega(mn) and omega(mn). The final response is therefore quasiperiodic. However in the case when the excitation frequency is close to 1/2(omega(mn) - omega(mn)), no combination resonance is possible. In the case when the excitation frequency is close to 1/3omega(mn) and 1/2(omega(mn) - omega(mn)) simultaneously, internal resonance between the forward and backward modes can occur The frequencies of the free oscillation components are exactly three times and five times that of the excitation frequency. In this special case both saddle-node and Hopf bifurcations are observed.