Nonlinear Vibrations Analysis of Hyperelastic Cylindrical Shells Utilizing the Method of Multiple Scales

In this study, the primary resonance of a hyperelastic thin-walled cylindrical shell subjected to external excitation is examined. The incompressible Moony–Rivlin model, nonlinear Donnell’s shell theory, and Hamilton principle are employed to extract the fundamental equations in three directions. The Galerkin approach is employed to reduce the governing equation into ordinary differential equations form. Then, by neglecting the membrane inertia, three equations are reduced to one equation to analyze the transverse vibrations of the hyperelastic shells. Natural frequencies of different asymmetric modes are used to define the fundamental mode. Then, the multiple scales method is applied to evaluate the frequency–amplitude curves for various parameters. In addition, the model’s accuracy is validated with those presented in the literature for a cylindrical shell and the comparison shows good arrangement results. In the simulation part, the effects of the amplitude of excitation, the ratio of thickness–radius, and the ratio of radius–length on the primary resonances of shells are evaluated. The simulation plots show that for small values of radius-to-length ratios, increasing its value causes to decrease in the stiffness of the thin hyperelastic shell. Also, by increasing the radius–length ratios from 0.3 to 1.18 the hardening of the shell is decreased, and by increasing the radius–length ratios from 1.18 to 2 the hardening behavior is increased.