On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations

Abstract

We investigate the possibility of secondary resonance of a spinning disk under space-fixed excitations. Von Karman’s 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 ωmn̄) and backward (with frequency ωmn) traveling waves. It is found that combination resonance may occur when the excitation frequency is close to 2ωmn+ωmn̄,ωmn+2ωmn̄, or 1/2(ωmn̄+ωmn). When the combination resonance does occur, the frequencies of the free oscillation components are shifted slightly from the respective natural frequencies ωmn̄ and ωmn. The final response is therefore quasiperiodic. However, in the case when the excitation frequency is close to 1/2(ωmn̄−ωmn), no combination resonance is possible. In the case when the excitation frequency is close to 1/3ωmn and 1/2(ωmn̄−ωmn) 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.