形状记忆合金薄板的分叉与激变

Bifurcations and Crises of a Shape Memory Alloy Plate

研究了受横向载荷作用的形状记忆合金矩形薄板的非线性动力学特性及混沌行为。基于形状记忆合金材料的热-机耦合行为和拟弹性行为的五次多项式本构关系及薄板的动力学平衡方程,建立了反映形状记忆合金薄板动力学行为的非线性动力学模型;用平衡态定性分析法讨论了矩形薄板的动力学稳定性与材料相转换间的关系。利用数值模拟的方法,研究了形状记忆合金薄板动力系统的分叉和激变。分叉分析研究了随外加载荷变化和温度变化这两种情况时,系统的分叉结构。通过所绘制的分叉图和局部区域放大的分叉图,发现系统呈现出倍周期分叉、倒置的倍周期分叉、激变和突减等分叉现象。

This paper is concerned with the dynamic characteristic and chaotic behaviors of a shape memory alloy plate subjected to transverse dynamic load. Based on a fifth order polynoiacl constitutive description of the thermo-mechanical and pseudo-elastic behaviors of the shape memory alloy and the dynamic equilibrium equation of plate, the nonlinear dynamic model that dominates the transverse vibration of the plate was established. The relations between the dynamic stability of the plate and the equilibrium of the material phase were investigated by using the qualitative analysis method of a differential dynamic system. Finally, the bifurcations and crises of the shape memory alloy plate were considered by using the numerical simulations. Bifurcation analysis was carried out by considering the structure of bifurcations varying with force amplitude parameter, g, as well as the parameter related to temperature, θ. The analyses of these bifurcations and the enlargement of bifurcations diagram allows us to identify interesting behaviors, such as period doubling cascades, direct and reverse crises and subductions.