热机耦合粘弹性矩形板混沌与分岔研究
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摘要
随着各种粘弹性材料和结构的广泛应用及其工作温度范围的进一步拓宽,而粘弹性结构动力学行为表现出与温度的极大相关性,所以在研究粘弹性结构的非线性动力学行为时有必要考虑热效应的影响,即必须考虑热学和力学的耦合作用。
本文主要由以下三部分组成:
1)基于薄板大挠度Karman理论和Boltzmann叠加原理描述的变温粘弹性材料本构方程,结合变形几何方程、动力学平衡方程和热传导方程建立了受面内均布作用力和横向周期激励作用下,同时具有热膨胀、热传导和边界对流效应的热机耦合粘弹性矩形薄板的非线性动力学模型。使用Galerkin方法,简化为非线性积分—微分动力系统。且从实用性出发,讨论了针对不同使用条件的简化模型。
2)研究了热机耦合粘弹性板非线性振动模型的数值计算方法。首先引入差分,将控制方程转化为简单易求的差分微分型动力学模型。进一步,通过引入Newmark积分格式、中心差分格式或新的变量,分别得到基于Newmark法的迭代型数值计算方法、基于中心差分法的完全差分数值计算方法和一类特殊热机耦合粘弹性板数值计算方法。
3)综合利用研究非线性动力学系统中的经典方法,如时程曲线、相平面图、Poincare映射、功率谱、Lvapunov指数和关于横向荷载的分岔图,揭示了热机耦合粘弹性板、粘弹性板和热机耦合弹性板的动力学行为,并考察了热膨胀、热传导效应对热机耦合粘弹性板动力学行为的影响。研究表明:热机耦合粘弹性矩形板在横向周期激励和面内均布力作用下具有十分丰富的动力学行为,比不考虑温度效应的粘弹性板的混沌性更强,还出现了超混沌现象;热膨胀系数的增大和热传导系数的减小都会引起热机耦合系数增大,进而引起热机耦合系统振幅变大。
With the extensive application of viscoelastic materials and structures and the widening of the working temperature, the dynamic behaviors of the viscoelastic structures are closely related to thermal effect. So, it is necessary to consider the thermal effect in the study of nonlinear dynamic behaviors of the viscoelastic structures. Namely, the coupling with thermal and mechanical effect should be considered.
Three main parts are included in the paper:
1) The nonlinear dynamic model of a thermo-mechanical coupling viscoelastic rectangular plate is established. The processes base on the Karman theory for thin plates with large deflection, thermo-viscoelastic constitutive equations which are described by Boltzman's superposition principle, deformation geometry equations, dynamic equilibrium equation, and heat conduction equation. The rectangular plate has thermal expansion, heat conduction, convection effect, and is subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force. The model may be converted to a nonlinear differential-integral dynamical system by using Galerkin's method. And the simplified models are discussed according to the application conditions.
2) The numerical models of nonlinear vibration of thermo-mechanical coupling viscoelastic plate are studied. First, the difference-differential dynamic model is obtained from control equation by difference method. Then, the iteration numerical model is obtained by Newmark method. And difference numerical model is obtained by central difference method. Especially, a special kind of numerical model are obtained by introducing new variables.
3) The dynamic behaviors of the thermo-mechanical coupling viscoelastic plate are revealed. The effects of thermal expansion and heat conduction on the nonlinear dynamic behaviors are investigated. The processes use several methods in dynamic systems. The results show that the dynamic properties of the thermo-mechanical coupling viscoelastic plate are abundant, while the plate is subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force. With the influence of temperature, the model's chaotic property also is stronger. Especially, the motion state of hyperchaos appears. Coefficient of thermo-mechanical coupling and the amplitude of thermo-mechanical coupling system are enlarged by the increasing coefficient of thermal conduction and the decreasing coefficient of heat expansion.
本文主要由以下三部分组成:
1)基于薄板大挠度Karman理论和Boltzmann叠加原理描述的变温粘弹性材料本构方程,结合变形几何方程、动力学平衡方程和热传导方程建立了受面内均布作用力和横向周期激励作用下,同时具有热膨胀、热传导和边界对流效应的热机耦合粘弹性矩形薄板的非线性动力学模型。使用Galerkin方法,简化为非线性积分—微分动力系统。且从实用性出发,讨论了针对不同使用条件的简化模型。
2)研究了热机耦合粘弹性板非线性振动模型的数值计算方法。首先引入差分,将控制方程转化为简单易求的差分微分型动力学模型。进一步,通过引入Newmark积分格式、中心差分格式或新的变量,分别得到基于Newmark法的迭代型数值计算方法、基于中心差分法的完全差分数值计算方法和一类特殊热机耦合粘弹性板数值计算方法。
3)综合利用研究非线性动力学系统中的经典方法,如时程曲线、相平面图、Poincare映射、功率谱、Lvapunov指数和关于横向荷载的分岔图,揭示了热机耦合粘弹性板、粘弹性板和热机耦合弹性板的动力学行为,并考察了热膨胀、热传导效应对热机耦合粘弹性板动力学行为的影响。研究表明:热机耦合粘弹性矩形板在横向周期激励和面内均布力作用下具有十分丰富的动力学行为,比不考虑温度效应的粘弹性板的混沌性更强,还出现了超混沌现象;热膨胀系数的增大和热传导系数的减小都会引起热机耦合系数增大,进而引起热机耦合系统振幅变大。
With the extensive application of viscoelastic materials and structures and the widening of the working temperature, the dynamic behaviors of the viscoelastic structures are closely related to thermal effect. So, it is necessary to consider the thermal effect in the study of nonlinear dynamic behaviors of the viscoelastic structures. Namely, the coupling with thermal and mechanical effect should be considered.
Three main parts are included in the paper:
1) The nonlinear dynamic model of a thermo-mechanical coupling viscoelastic rectangular plate is established. The processes base on the Karman theory for thin plates with large deflection, thermo-viscoelastic constitutive equations which are described by Boltzman's superposition principle, deformation geometry equations, dynamic equilibrium equation, and heat conduction equation. The rectangular plate has thermal expansion, heat conduction, convection effect, and is subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force. The model may be converted to a nonlinear differential-integral dynamical system by using Galerkin's method. And the simplified models are discussed according to the application conditions.
2) The numerical models of nonlinear vibration of thermo-mechanical coupling viscoelastic plate are studied. First, the difference-differential dynamic model is obtained from control equation by difference method. Then, the iteration numerical model is obtained by Newmark method. And difference numerical model is obtained by central difference method. Especially, a special kind of numerical model are obtained by introducing new variables.
3) The dynamic behaviors of the thermo-mechanical coupling viscoelastic plate are revealed. The effects of thermal expansion and heat conduction on the nonlinear dynamic behaviors are investigated. The processes use several methods in dynamic systems. The results show that the dynamic properties of the thermo-mechanical coupling viscoelastic plate are abundant, while the plate is subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force. With the influence of temperature, the model's chaotic property also is stronger. Especially, the motion state of hyperchaos appears. Coefficient of thermo-mechanical coupling and the amplitude of thermo-mechanical coupling system are enlarged by the increasing coefficient of thermal conduction and the decreasing coefficient of heat expansion.
引文
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[8]王礼立,施绍裘.ZWT非线性热粘弹性本构关系的研究与应用.宁波大学学报(理工版).2000,13(B12):141-149
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[11]Akhtar S.Khan,Oscar Lopez-Pamies,Rehan Kazmi.Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures.International Journal of Plasticity.2006,22:581-601
[12]Giorgi C.,Naso M.G.Mathematical models of Reissner-Mindlin thermoviscoelastic plates.Journal of Thermal Stresses.2006,29:699-716
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[20]Altay G.A.,Dokmeci M.C.Thermo-viscoelastic analysis of high-frequency motions of thin plates.Acta Mechanicsa Sinica.2000,143:91-111
[21]Anastasia Muliana,Rami Haj-Ali.Multiscale Modeling for the Long-Term Behavior of Laminated Composite Structures.AIAA Journal.2005,43(8):1815-1822
[22]Yuming Qin,Jianan Fang.Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density.Nonlinear Analysis.2006,65:892-917
[23]Anastasia Muliana,Aravind Nair.Characterization of thermo-mechanical and long-term behaviors of multi-layered composite materials.Composites Science and Technology.2006,66:2907-2924
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[27]Argyris.Chaotic vibrations of a nonlinear viscoelastic beam.Chaos,Solitons and Fractals.1996,7(2):151-163
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[29]陈立群,程昌钧.非线性粘弹性柱的稳定性和混沌运动.应用数学和力学.2000,21(9):890-896
[30]Yen-Liang Yeh.Chaotic and bifurcation dynamic behavior of a simply supported rectangular orthotropic plate with thermo-mechanical coupling.Chaos,Solitons and Fractals.2005,24:1243-1255
[31]屈钧利,张义同.热粘弹性有限元法.河北煤炭建筑工程学院学报.1997,1:11-15
[32]陈予恕,唐云.非线性动力学中的现代分析方法.北京:科学出版社,2000
[33]龙运佳.混沌振动研究方法与实践.北京:清华大学出版社,1997
[34]闻邦椿,李以农,徐培民,韩清凯.工程非线性振动.北京:科学出版社,2007
[35]Giorgi C.,Naso M.G.Mathematical models of thin thermoviscoelastic plates.Journal of Mechanics and Applied Mathematics.2000,53:363-374
[36]王进廷,杜修力.有阻尼体系动力分析的一种显式差分法.工程力学.2002,19(3):109-112
[37]Yen-Liang Yeh,Chao-Kuang Chen,Hsin-Yi Lai.Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection.Chaos,Solitons and Fractals.2003,15:811-829
[38]Zhang Neng-hui,Cheng Chang-jun.Non-linear mathematical model of viscoelastic thin plates with its applications.Computer Methods in Applied Mechanics and Engineering.1998,165:307-319
[39]唐麟.分数算子描述的热粘弹性杆藕合问题研究.暨南大学硕士论文.2006
[40]陈立群,程昌钧.关于基于Galerkin截断的粘弹性结构动力学行为研究.自然杂志(专题综述).2000,21(1):1-4
[41]Johannes Zimmer.Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy.Journal of Mathematical Analysis and Applications.2004,292:589-604
[42]赵小红,李德生.一类组合的五阶非线性偏微分方程的一种显式差分格式.科学技术与工程.2008,8(3):751-752
[43]王进廷,杜修力.有阻尼体系动力分析的一种显式差分法.工程力学.2002,19(3):110-112
[44]朱加铭,欧贵宝,何蕴增.有限元和边界元法.哈尔滨:哈尔滨工程大学出版社,2002
[45]张燕,盛东发,程昌钧.在有限变形条件下损伤粘弹性梁的动力学行为.力学季刊.2004,25(2):230-238
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[2]Boit,M.A.General solutions of equations of elasticity and consolidation of a porous material.Journal of Applied Mechanics.1956,23:91-96.
[3]Lustig S.R.,Shay R.M.,Caruthers J.M.Thermodymamic constitutive equations for materials with memory on a material time scale.Journal of Rheology.1996,40(1):69-106.
[4]Coleman B.D.,Mizel V.A general theory of dissipation in materials with memory.Arch.Ration.Mechanics Analysis.1967,27:255.
[5]黄筑平,陈建康,王文标.有限变形热粘弹性本构关系的内变量理论.中国科学(A辑).1999,29(10):939-944
[6]罗文波,杨挺青,王霞瑜.高聚物自由体积与温度和应力水平的相关性.高分子材料科学与工程.2005,21:11-15
[7]张义同,严宗达.变温粘弹性的一般理论.力学学报.1993,25(6):685-696
[8]王礼立,施绍裘.ZWT非线性热粘弹性本构关系的研究与应用.宁波大学学报(理工版).2000,13(B12):141-149
[9]James Caruthersa M.,Douglas B.A thermodynamically consistent,nonlinear viscoelastic approach for modeling glassy polymers.Polymer.2004,45:4577-4597
[10]Drozdov A.D.,Agarwal S.,Gupta R.K.Linear thermo-viscoelasticity of isotactic polypropylene.Computational Materials Science.2004,29:195-213
[11]Akhtar S.Khan,Oscar Lopez-Pamies,Rehan Kazmi.Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures.International Journal of Plasticity.2006,22:581-601
[12]Giorgi C.,Naso M.G.Mathematical models of Reissner-Mindlin thermoviscoelastic plates.Journal of Thermal Stresses.2006,29:699-716
[13]Aboudi J.,Cederbaum G.,Elishakoff I.General framework for analysis of buckling of viscoelastic structures.Impact and Buckling of Structures,Dallas,Texas,USA,1990:77-83
[14]Cederbaum,I.Analogy between VLSI floorplanning problems and realisation of a resistive network.IEE Proceedings.(Circuits,Devices and Systems).1992,139(1):99-103
[15]程昌均,范晓军.粘弹性环形板的临界载荷及动力稳定性.力学学报.2001,33(3):365-376
[16]彭凡,傅衣铭.湿热环境下粘弹性板的非线性动力稳定性分析,湘潭大学自然科学学报,2005,27:51-66
[17]Ilyasov M.H.Dynamic stability of viscoelastic plates.International Journal of Engineering Science.2007,45:111-122
[18]Brilla J.Solution of dynamic problems for thermoviscoelastic plates by the finite-element method.Journal of Thermal Stresses.1981,4:479-490
[19]Jaime E.Munoz Rivera,Rioco Kamei Barreto.Uniform Rates of Decay in Anisotropic Thermo-Viscoelasticity.Acta Applicandae Mathematicae.1998,50:207-224
[20]Altay G.A.,Dokmeci M.C.Thermo-viscoelastic analysis of high-frequency motions of thin plates.Acta Mechanicsa Sinica.2000,143:91-111
[21]Anastasia Muliana,Rami Haj-Ali.Multiscale Modeling for the Long-Term Behavior of Laminated Composite Structures.AIAA Journal.2005,43(8):1815-1822
[22]Yuming Qin,Jianan Fang.Global attractor for a nonlinear thermoviscoelastic model with a non-convex free energy density.Nonlinear Analysis.2006,65:892-917
[23]Anastasia Muliana,Aravind Nair.Characterization of thermo-mechanical and long-term behaviors of multi-layered composite materials.Composites Science and Technology.2006,66:2907-2924
[24]傅衣铭,汤可可,王永.变温场中具损伤粘弹性矩形板的非线性动力响应分析,固体力学学报.2006,27:243-248
[25]Zhang Neng-Hui,Xing Jing-Jing.Vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach.International Journal of Solids and Structures.2008,132:78-89
[26]Suire G,CederbaumG.Perfodic and chaotic behavior of viscoelastic nonlinear (elastic)bars under harmonic excitations.International Journal of Mechanics Science.1995,37:753-772
[27]Argyris.Chaotic vibrations of a nonlinear viscoelastic beam.Chaos,Solitons and Fractals.1996,7(2):151-163
[28]程昌钧,张能辉,粘弹性矩形板的混沌和超混沌行为.力学学报,1998,30(6):690-699
[29]陈立群,程昌钧.非线性粘弹性柱的稳定性和混沌运动.应用数学和力学.2000,21(9):890-896
[30]Yen-Liang Yeh.Chaotic and bifurcation dynamic behavior of a simply supported rectangular orthotropic plate with thermo-mechanical coupling.Chaos,Solitons and Fractals.2005,24:1243-1255
[31]屈钧利,张义同.热粘弹性有限元法.河北煤炭建筑工程学院学报.1997,1:11-15
[32]陈予恕,唐云.非线性动力学中的现代分析方法.北京:科学出版社,2000
[33]龙运佳.混沌振动研究方法与实践.北京:清华大学出版社,1997
[34]闻邦椿,李以农,徐培民,韩清凯.工程非线性振动.北京:科学出版社,2007
[35]Giorgi C.,Naso M.G.Mathematical models of thin thermoviscoelastic plates.Journal of Mechanics and Applied Mathematics.2000,53:363-374
[36]王进廷,杜修力.有阻尼体系动力分析的一种显式差分法.工程力学.2002,19(3):109-112
[37]Yen-Liang Yeh,Chao-Kuang Chen,Hsin-Yi Lai.Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection.Chaos,Solitons and Fractals.2003,15:811-829
[38]Zhang Neng-hui,Cheng Chang-jun.Non-linear mathematical model of viscoelastic thin plates with its applications.Computer Methods in Applied Mechanics and Engineering.1998,165:307-319
[39]唐麟.分数算子描述的热粘弹性杆藕合问题研究.暨南大学硕士论文.2006
[40]陈立群,程昌钧.关于基于Galerkin截断的粘弹性结构动力学行为研究.自然杂志(专题综述).2000,21(1):1-4
[41]Johannes Zimmer.Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy.Journal of Mathematical Analysis and Applications.2004,292:589-604
[42]赵小红,李德生.一类组合的五阶非线性偏微分方程的一种显式差分格式.科学技术与工程.2008,8(3):751-752
[43]王进廷,杜修力.有阻尼体系动力分析的一种显式差分法.工程力学.2002,19(3):110-112
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