功能梯度圆板的分岔与混沌
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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。
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.
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。
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.
引文
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20 M. M. Najfizadeh, M. R. Eslami. Buckling Analysis of Circular Plates of Functionally Graded Materials under Uniform Radial Compression. International Journal of Mechanical Sciences, 2002, 44(12):2479-2493
21 M. Bayat. Bending Analysis of Functionally Graded Rotating Disk Based on The First Order Shear Deformation Theory. Applied Mathematical Modeling, 2009, 33(11):4215-4230
22 S. Sahraee, A. R. Saidi. Axisymmetric Bending Analysis of Thick Functionally Graded Circular Plates Using Fourth-order Shear Deformation Theory. European Journal of Mechanics A/Solids, 2009, 42(4):997-1016
23 N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material. Transactions of Japan Society of Mechanical Engineers, Series A, 1991, 57(533):98-103
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33 Z. G. Zhou, L Z Wu. Non-local Theory Solution for the Anti-plane Shear of two Collinear Permeable Cracks in Functionally Graded Piezoelectric Materials. International Journal of Engineering Science, 2006, 44(18-19):1366-1379
34 G. N. Parveen, J. N. Reddy Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates. Journal of Solids Structures, 1998, 35(33):4457-4476
35 T. Y. Ng, K. Y. Lam, K. M. Liew. Effect of FGM Materials on the Parametric Resonance of Plate Structures. Computer Methods in Applied Mechanics and Engineering, 2000, 19(8-10):953-962
36 J. Yang, H. S. Shen. Dynamic Response of Initially Stressed Functionally Graded Rectangular Thin Plates. Composite Structures, 2001(4), 54:497-508
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2李进,田兴华.功能梯度材料的研究现状及应用.宁夏工程技术, 2007, 6(1):80-83
3赵军,艾兴,张建华.功能梯度材料的发展及展望.材料导报, 1997, 11(4):57-60
4田兴华.功能梯度材料及其强度分析. [浙江大学博士学位论文]. 2006:1-6
5黄旭涛,严密.功能梯度材料回顾与展望.材料科学与工程, 1997, 15(4):35-38
6王引真,孙永兴,阎国超.功能梯度材料的研究动态.机械工程材料, 1997, 21(4):35-37
7刘延柱,陈立群.非线性振动.北京:高等教育出版社, 2001
8张琪昌,王洪礼.分岔与混沌理论及运用.天津:天津大学出版社, 2005
9陆启韶.分岔与奇异性.上海:上海科技教育出版社, 1995
10刘秉正,彭建华.非线性动力学.北京:高等教育出版社, 2001
11高普云.非线性动力学—分叉、混沌与孤立子.长沙:国防科技大学出版社, 2005
12刘宗华.混沌动力学基础及其应用.北京:高等教育出版社, 2006
13边祖光.功能梯度材料板壳结构的耦合问题研究. [浙江大学博士学位论文]. 2005:2-10
14叶开沅.力学发展简史.北京:知识出版社, 1987
15 J. N. Reddy, C. M. Wang, S. Kitipornchai. Axisymmetric Bending of Functionally Grade Circular and Annular Plates. European Journal of Mechanics A/Solids, 1999, 18(2):185-195
16 Z. Q. Cheng, S. Kitipornchai. Exact Bending Solution of Inhomogeneous Plates from Homogeneous Thin-plate Deflection. AIAA Journal, 2000, 38(7):1289-1291
17 H. S. Shen. Nonlinear Bending Response of Functionally Graded Plates Subjected to Transverse Loads and in Thermal Environments. International Journal of Mechanical Sciences, 2002, 44(3):561-584
18刘进,武兰河,张晓炜.功能梯度材料板的弯曲问题.石家庄铁道学院学报, 2003, 16(2):1-5
19 Z. Q. Cheng. Nonlinear Bending of Inhomogeneous Plates. Engineering Structures, 2001, 23(10):1359-1363
20 M. M. Najfizadeh, M. R. Eslami. Buckling Analysis of Circular Plates of Functionally Graded Materials under Uniform Radial Compression. International Journal of Mechanical Sciences, 2002, 44(12):2479-2493
21 M. Bayat. Bending Analysis of Functionally Graded Rotating Disk Based on The First Order Shear Deformation Theory. Applied Mathematical Modeling, 2009, 33(11):4215-4230
22 S. Sahraee, A. R. Saidi. Axisymmetric Bending Analysis of Thick Functionally Graded Circular Plates Using Fourth-order Shear Deformation Theory. European Journal of Mechanics A/Solids, 2009, 42(4):997-1016
23 N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material. Transactions of Japan Society of Mechanical Engineers, Series A, 1991, 57(533):98-103
24 Y. Obata, N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material (Influence of Boundary Conditions). Transactions of Japan Society of Mechanical Engineers, Series A, 1992, 58(553):1689-1695
25杨杰,沈惠申.热/机械荷载下功能梯度材料矩形厚板的弯曲行为.固体力学学报, 2003, 24(1):119-124
26 G. N. Praveen, J. N. Reddy. Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates. International Journal of Solids and Structures, 1998, 35(33):4457-4476
27王铁军,马连生.功能梯度中厚圆/环板轴对称弯曲问题的解析解.力学学报, 2004, 36(3):348-353
28郑磊,仲政.弹性支承功能梯度弯曲问题的精确解.同济大学学报, 2009, 37(7):893-897
29李尧臣,亓峰.功能梯度压电材料圆板的简化理论和解析解.力学学报, 2008, 40(5):636-645
30孙建亮,周振功,王彪.功能梯度压电压磁材料中断裂问题的分析.力学学报, 2005, 37(1):9-14
31李春雨,邹振祝,段祝平.正交各向异性功能梯度材料反平面裂纹尖端力场.固体力学学报, 2001, 22(1):81-84
32李永东,贾斌,张男,戴耀等.功能梯度材料有限宽板的反平面断裂问题研究.应用数学和力学, 2006, 27(6):683-689
33 Z. G. Zhou, L Z Wu. Non-local Theory Solution for the Anti-plane Shear of two Collinear Permeable Cracks in Functionally Graded Piezoelectric Materials. International Journal of Engineering Science, 2006, 44(18-19):1366-1379
34 G. N. Parveen, J. N. Reddy Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates. Journal of Solids Structures, 1998, 35(33):4457-4476
35 T. Y. Ng, K. Y. Lam, K. M. Liew. Effect of FGM Materials on the Parametric Resonance of Plate Structures. Computer Methods in Applied Mechanics and Engineering, 2000, 19(8-10):953-962
36 J. Yang, H. S. Shen. Dynamic Response of Initially Stressed Functionally Graded Rectangular Thin Plates. Composite Structures, 2001(4), 54:497-508
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