Applicability of homotopy perturbation method to study the nonlinear vibration of doubly curved cross-ply shells

The objective of this paper is to study nonlinear vibrations of doubly curved cross-ply shells with simply supported boundary conditions. The governing equations of the system are constructed using the von-Karman geometric nonlinear theory with Donnell’s shell equations. The Galerkin procedure is used to reduce the nonlinear partial differential equations to a nonlinearly second-order ordinary differential equation. The homotopy perturbation method (HPM) is employed to analysis the present nonlinear ordinary equation. Based on the results of this study, the first order approximation of the HPM leads to highly accurate solutions for geometrically nonlinearity vibration of doubly curved cross-ply shells. Moreover, HPM in comparison with other traditional analytical methods (e.g. perturbation methods) has excellent accuracy for the whole range of oscillation amplitude and initial conditions.