热环境中功能梯度圆形薄板的混沌运动

Chaos of thin circular functionally graded plate in thermal environment

针对陶瓷-金属功能梯度圆板,同时考虑几何非线性、材料物性参数随温度变化且材料组分沿厚度方向按幂律分布的情况,应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下,通过引入位移函数,利用伽辽金方法得到了达芬型非线性动力学方程。利用Melnikov方法,给出了热环境中功能梯度圆板可能发生混沌运动的临界条件。通过数值算例,给出了不同体积分数指数和温度的同宿分岔曲线,平面相图和庞加莱映射图,讨论其对临界条件的影响,证实了系统混沌运动的存在。通过分岔图和与其相对应的最大李雅普诺夫指数图,分析了激励频率和激励幅值对倍周期分岔的影响及变化规律,发现系统可出现周期、倍周期和混沌等复杂动力学响应。

A ceramic/metal functionally graded circular plate was considered in this study.The effect of geometric nonlinearity and temperature-dependent material properties are both taken into account.The material properties of the functionally graded plate are assumed to vary continuously through the thickness,according to a power law distribution of the volume fraction of the constituents.Using the principle of virtual work,the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation force and thermal load are derived.For the circular plate with clamped immovable edge,the Duffing nonlinear forced vibration equation is deduced by using Galerkin method.The criterion of existence of chaos is given with Melnikov method.Numerical simulation is carried out to plot the bifurcation curves for the homolinic orbits.Effect of the material volume fraction index and temperature to the criterion are discussed and the existence of chaos is validated by plotting phase portraits Poincare map.Also,the bifurcation diagram and the corresponding maximum Lyapunov exponent are plotted.It was found that periodic,multiplier periodic solutions and chaotic motions exist for the FGM plate under certain conditions.