Chaotic dynamics and stability of functionally graded material doubly curved shallow shells

In this article, the nonlinear chaotic and periodic dynamic responses of doubly curved functionally graded shallow shells subjected to harmonic external excitation are numerically investigated. Material characteristics of the shell are defined according to a simple power law distribution through the thickness. Based on the first-order shear deformation shell theory and using the Donnell nonlinear kinematic relations the set of the governing equations are derived. The Galerkin method together with trigonometric mode shape functions is applied to solve the equations of motion. Also, the nonlinearly coupled time integration of the governing equation of plate is solved employing fourth-order Runge-Kutta method. The effects of amplitude and frequency of external force on the nonlinear dynamic response of shells are investigated. The bifurcation diagram and largest Lyapunov exponent are employed to detect the amplitude and frequency of external force critical parameter of periodic and chaotic response of shallow shells under periodic force. Having known the critical values, phase portrait, Poincare maps, time history and power spectrum are presented to observe the periodic and chaotic behavior of the system.