Steady-state response of an axially moving circular cylindrical panel with internal resonance

Abstract

With the 3:1 internal resonance, the primary and secondary resonances of an axially moving thin circular cylindrical panel are investigated in the present work. The governing equation and the compatibility equation are established based on the Donnell's nonlinear shell theory and solved to obtain the nonlinear steady-state responses by combining the Galerkin method and the method of multiple scales. The analytical solutions are verified by numerical solutions based on the Runge-Kutta Method. The governing equation includes both the quadratic nonlinearity and the cubic nonlinearity, so the perturbation solutions need to consider three time scales. The quadratic nonlinearity causes the softening behavior of the system. Natural frequencies and the 3:1 internal resonance condition are obtained by the linear analysis. Under the primary resonance, the internal resonance causes the coupling of the first two modes to complicate the nonlinear dynamic response. The response for the second mode possesses an extra bulge or peak due to the internal resonance. The quadratic nonlinearity results in the zero frequency drift and the second-order harmonic. Under the secondary resonance, the exciting force only arouses the second mode. Results are shown to examine the effects of the internal resonance, the exciting force and viscous damping coefficients on the nonlinear dynamic response of an axially moving thin circular cylindrical panel.