Dynamic Instability and Nonlinear Vibration of Doubly-Curved Cross-Ply Shallow Shells

We consider the dynamic instability and nonlinear vibration of doubly-curved cross-ply laminated shallow shells with simply supported boundary conditions. We investigate their responses and stability to a primary resonance (i.e., Ω ≠ ω₁₁). The governing nonlinear partial-differential equations of motion are based on the von Karman-type geometric nonlinear theory and the first-order shear-deformation theory. We use the Galerkin procedure to reduce the governing nonlinear partial-differential equations to an infinite system of nonlinear coupled ordinary-differential equations. We use a combination of a shooting technique and Floquet theory to calculate the periodic responses of the shell and investigate their bifurcations. We show that for some shell parameters, a single-mode approximation misses some important dynamics, such as period-doubling, bifurcations.