Subharmonics and chaos in loudspeakers with internal resonance

The Subharmonics and chaos in a loudspeaker shell are investigated when the axisymmetric mode is excited resonantly by the driven force,and the internal resonance of 2:1 exists between the axisymmetric mode and the asymmetric mode.The method of multiple scales is used to obtain the first-order approximations to the nonlinear modal equations,then to analyze their stabilities,and furthermore to determine the bifurcation sets on the driving frequency and force plane.The formula of the threshold voltage of the subharmonics is given in the case considered,which is lower than the one in the case without internal resonance.Besides the presence of the 1/2 subharmonics of the asymmetric mode,the amplitudes of the two modes undergo Hopf bifurcation to limit cycle motions,and then undergo perioddoubling bifurcations to chaos.The occurrence of the chaos is due to the strong energy interchange between the two modes.The experimental results agree with the theoretical ones.This agreement verifies the correctness in modeling the nonlinearity of the loudspeaker shell.