Theory of Nonlinear Vibrations for Antisymmetric Cross-Ply Bistable Laminated Shells

Abstract

In this paper, on the basis of taking the von Karman nonlinear factors into consideration, the constitutive equation of the antisymmetric cross-ply laminated composite was used to calculate the internal force and internal moment of the bistable structure, and the dynamic equilibrium equation and the compatible equation were constructed, respectively. The two equations were combined to establish a nonlinear dynamic model for the antisymmetric cross-ply laminated glass fiber resin bistable shell. Then, the finite element numerical simulation software ABAQUS was adopted to perform simulation modeling and numerical analysis on a series of bistable specimens, so as to study the impact of different geometric parameters on the frequencies, mode shapes, and other vibration characteristics of the antisymmetric laminated fiber resin bistable shell. Galerkin discretization was conducted on the vibration partial differential equation. Since there are only even-order partial differential terms of deflection $w$ with respect to x and y in the vibration partial differential equation at this time, the form of series obtained by each term is the same, which simplifies the discretization of the dynamic equilibrium equation and the compatible equation. Finally, the two equations after discretization were merged to obtain the three-degree-of-freedom nonlinear ordinary differential equation of the antisymmetric cross-ply laminated glass fiber resin bistable shell. The system averaged equation was acquired by perturbation analysis through a multiscale method, and the periodic solution of the antisymmetric laminated bistable system was studied. Moreover, the system’s nonlinear dynamic behavior characteristics such as bifurcation and chaos were explored when the main resonance $\mathrm{\Omega }$ is close to ${\omega }_1$ and ${\omega }_2$ , respectively, and the internal resonance is 1 : 2 : 3.