Third Order Superharmonic Resonance and Spatial Motion of a String
This paper theoretically and experimentally deals with the third order superharmonic resonance and the spatial motion occurring in a string. We consider the case when the upper end of the string is fixed and the lower end is harmonically excited in a direction. When the excitation frequency is in the neighborhood of a third the linear natural frequency of the string, the third order superharmonic resonance in the excitation direction occurs. Additionally, the superharmonic resonance(in-plane motion) can produce the resonance(out-of-plane motion) in the direction perpendicular to the excitation through the nonlinear coupling between the excitation direction and the perpendicular direction. First, the equations of motion of the string including the effect of the nonlinear restoring force are shown. Next, the analytical solutions of the equations and the amplitude equations are acquired by the method of multiple scales. Then, the amplitude equations give theoretical frequency response curve. Finally, the experimental results qualitatively confirm the theoretical characteristics.