Nonlinear vibrations and dynamic instability analysis of a two-directional functionally graded conical shell under parametric and external excitations

Abstract

This paper analyzes the nonlinear vibrations and dynamic instability of two-directional functionally graded (2DFG) conical shells. To this end, the shell dynamic model is derived using the first-order shear deformation theory (FSDT) and nonlinear strain–displacement relations. The shell is subjected to a time-varying axial parametric force and harmonic external transverse excitation. In this article, a new hybrid solution method is used. Specifically, the generalized differential quadrature method (GDQM) is applied to solve the eigenvalue problem and obtain the mode shape vectors. These mode shapes are then used to discretize the equations using the Galerkin method. The multiple time scale method (MTSM) is used to analyze the nonlinear dynamics of the system. Initially, the shell’s nonlinear frequency response is obtained from steady-state motion, followed by an evaluation of the system’s nonlinear dynamic behavior in two resonance regions: primary and parametric. Finally, the effects of conical shell parameters such as the amplitude of the parametric and external excitation, detuning parameter, structural damping, FG power indices, semi-vertex angle, thickness-to-length ratio, and boundary conditions on the maximum amplitude and hardening behavior of the frequency response at primary resonance, as well as the dynamic instability of the shell, are investigated.