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 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 is close to and , respectively, and the internal resonance is 1 : 2 : 3.

中文翻译：

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反对称交叉层双稳态叠层壳非线性振动理论

本文在考虑von Karman非线性因素的基础上，采用反对称交叉层合复合材料的本构方程计算双稳态结构的内力和内力矩，动态平衡方程和分别构造了相容方程。结合这两个方程，建立了反对称交叉层合玻璃纤维树脂双稳态壳的非线性动力学模型。然后采用有限元数值模拟软件ABAQUS对一系列双稳态试件进行模拟建模和数值分析，研究不同几何参数对反对称叠层的频率、振型等振动特性的影响。纤维树脂双稳态外壳。对振动偏微分方程进行伽辽金离散化。由于偏转只有偶数阶偏微分项相对于此时振动偏微分方程中的x和y，各项得到的级数形式相同，简化了动力平衡方程和相容方程的离散化。最后，将离散化后的两个方程合并，得到反对称交叉层合玻璃纤维树脂双稳态壳的三自由度非线性常微分方程。采用多尺度方法通过微扰分析得到系统平均方程，研究了反对称叠层双稳态系统的周期解。此外，还探讨了系统在主谐振时的分岔和混沌等非线性动态行为特征。分别接近和，内部共振为 1 : 2 : 3。