Nonlinear resonant responses of hyperelastic cylindrical shells with initial geometric imperfections

Abstract

The resonant responses of the hyperelastic cylindrical shell with the initial geometric imperfections are investigated by considering both geometric and material nonlinearities. The hyperelastic cylindrical shell is subjected a radial harmonic excitation. Based on Donnell's theory, hyperelastic constitutive relations and Lagrange equation, the differential governing equations of motion are derived for the hyperelastic cylindrical shell with the initial geometric imperfections. Using the multiple scale method, the perturbation analysis for the governing equations of motion is conducted under the 2:1 internal resonance condition. The amplitude-frequency and force-amplitude response curves are obtained for the hyperelastic cylindrical shell. The effect of different parameters on the linear frequencies, amplitude-frequency response curves, force-amplitude response curves and chaotic responses of the hyperelastic cylindrical shell are discussed. The numerical results indicate that the existence of the geometric imperfections leads to the increase of the linear frequencies of the hyperelastic cylindrical shell. The amplitude-frequency response curves have the typical double-jumping phenomena and the response amplitudes increase with the increase of the external excitation amplitudes. The imperfection amplitudes, imperfection forms and structure parameters have the obvious influences on the resonant peaks evolutions. The vibrations of the first-order and second-order modes have the synchronization phenomena. With the changes of the parameters, the dynamic responses of the hyperelastic cylindrical shell with the imperfections change the period to chaotic vibrations alternately.