【作者】 张志强；

【导师】 胡宇达；

【作者基本信息】 燕山大学 ， 一般力学与力学基础， 2010， 硕士

【摘要】 功能梯度材料是通过特定的材料制备工艺将不同性能的两种或几种材料按一定的规律组合起来的一种全新复合材料。它通常由抗高温的陶瓷和强度韧性较好的金属材料复合而成,具有强度高、耐高温、韧性好等优良的使用性能。与均质材料结构相比,功能梯度材料在机械载荷和热载荷共同作用下的动力行为更加复杂,已成为实际工程应用中极为重要的关键性技术问题,因此关于该类结构的动力学研究有着重要的理论和工程应用价值。本论文主要采用理论推导和数值计算相结合的方法对功能梯度圆形薄板进行分岔与混沌分析。基于板壳理论,本文首先给出了热环境中轴对称功能梯度圆板的基本振动方程。考虑周边固支无滑动边界条件,通过位移函数的设定,利用伽辽金方法导出了功能梯度圆板在横向简谐激励作用下的达芬型非线性微分方程。其次,针对功能梯度圆板的主共振问题和组合共振问题,应用多尺度法分别导出其分岔响应方程,并借助奇异性理论得到了对应的转迁集和开折参数域上的分岔图,分析了系统在主共振和组合共振不同情况下奇异点的分岔性态。通过数值模拟,分别给出了系统随相关参数变化的全局分岔图、波形图、相图、庞加莱映射图和幅频响应图。分析表明,材料体积分数指数、温度、激励幅值等参数对功能梯度圆板的振动都有影响,可使其发生周期、倍周期、混沌等运动。最后,应用梅利尼科夫方法对功能梯度圆板的混沌振动进行解析预测,求得了圆板在不同激励下可能出现混沌的参数阈,通过数值算例分析了功能梯度材料体积分数指数、温度等参数对系统混沌阈值的影响,并用分岔图、最大李雅普诺夫指数图加以验证。更多还原

【Abstract】 Functionally graded material(FGM) is a new type of composite consisting of two or more phases, which can be designed its composition varies in some spatial direction by special manufacturing process. Usually, functionally graded materials are made from a mixture metal and ceramics. It is known that ceramic constituent of the material has excellent characteristics in heat-resistance while the metal part keeps a certain extent of strength and toughness. Compared with the homogeneous materials, FGM can take a more complex nonlinear dynamics due to the affect of the mechanical and thermal loads. Research on the nonlinear dynamics of FGMs plates has an important theoretical and engineering application. The aim of the thesis focuses on the bifurcation and chaos behavior of functionally graded circular plate by theoretical analysis and numerical calculation.Based on the theory of plates and shells, the nonlinear forced axisymmetric vibration of a thin circular functionally graded plate in thermal environment is formulated firstly. For a circular plate with clamped immovable edge, the nondimensional Duffing vibration equation are derived by using Galerkin’s approach.Secondly, considering two cases of primary resonance and combination resonance, a multi-scale method is utilized to obtain the bifurcation equation, respectively. Using singularity theory, the transition sets in parameter and the bifurcation diagrams are plotted under some conditions for unfolding parameters, and then discuss the corresponding bifurcation behaviors of singularity. Numerical simulations including global bifurcation diagrams, wave forms, phase portraits, Poincare map and amplitude-frequency curves are plotted. The results show that the vibration of functionally graded circular plate should be influenced by the material volume fraction index, temperature and excitation. It is observed that periodic, multi-periodic and chaotic motions exist for the FGM circular plate under certain conditions.Finally, the Duffing system with different external forcing terms is investigated. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov method. Numerical simulations including homoclinic bifurcation curves, homoclinic bifurcation surfaces, global bifurcation diagrams and maximum Lyapunov exponents are given to illustrate the theoretical analysis, and to expose the influence of material volume fraction index and temperature etc.更多还原