结构非线性热弹耦合振动的理论分析与有限元计算
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摘要
本文首先推导了当温度场和应变场耦合时椭圆板的非线性热弹耦合振动的基本方程，其中包括横向热振动方程、协调方程和能量方程。为了求解方便，引入无量纲参数将其无量纲化。我们采用Galerkin法进行离散化后得到一耦合的非线性常微分方程组，利用Runge-Kutta法进行数值求解，得到的主要结论如下：影响椭圆板非线性热弹耦合振动的因素主要有耦合系数、温度幅值、组合参数τd~2、椭度以及初始位移参数；当给定的初始位移较小时，热弹耦合效应使椭圆板的振动频率加快，当给定的初始位移较大时，热弹耦合效应使椭圆板的振动频率减小；边界条件对耦合效应有较大的影响，较强的边界条件使椭圆板热弹耦合自由振动的频率变低，但振荡幅度增大。其次，我们用同样的方法研究了圆柱壳的非线性热弹耦合振动，主要结论如下：影响圆柱壳非线性热弹耦合振动的因素主要有耦合系数、温度幅值、长径比、径厚比以及初始位移参数；耦合系数越大，轴向应力、轴向力以及轴向弯矩越小。
最后利用有限元程序LS-DYNA对轴对称圆柱壳受高速热冲击作用进行了计算机仿真分析，得到了位移时间曲线和单元的应力时间曲线，同文献已有的结果吻合的较好。数值模拟表明：边界条件对位移和应力的影响非常小；位移随着线热膨胀系数、热传导系数的增大而增大，但是随着比热容的增大而减小；应力的最大值随着线热膨胀系数的增大而增大，而随着比热容的增大而减小，而热传导系数k对应力的影响比较小。
The basic equations of elliptic plate, when temperature and stress fields are coupled, are developed, which include transverse thermal vibration equation , the compatibility equation and energy equations. In order to solve them easily, we introduced the dimensionless quantities and non-dimensionlized the obtained equations. A nonlinear ordinary
differential equations are obtained by Galerkin's method. The numerical
results are presented by employing Runge-Kutta method. The important results are given as follows: The coupling coefficient the amplitude of temperature, the parameter, the ration of length of long axis to that of short axis and initial displacement are main factors which influence the thermoelastic vibration of elliptic plate; When the initial displacement is smaller, thermoelastic coupling makes the frequency of the elliptic plate larger and when the initial displacement is larger, thermoelastic coupling makes the frequency of the elliptic plate smaller; The boundary conditions has strong influence on the coupling effect, the stronger it is, the smaller frequency of free vibration is but the larger amplitude is. Second, the thermoelastic coupling vibration of cylindrical shell are discussed by means of the same method. The coupling coefficient, amplitude of temperature, initial displacement, ratio of length to radius and that of radius to thickness are main factors which influence the thermoelastic vibration of cylindrical shell; The higher the coupling coefficient is, the lower the axial force, the axial stress and the axial moment are.
The finite element code LS_DYNA is used to investigate the response of the axisymmetry cylindrical shell under the thermal impact, and the curves of displacement vs time and that of stress vs time are obtained. The results from simulation agree well with the conclusions of
conference [21], and indicate that the boundary conditions have no obvious effect on the displacement and stress; that the displacement and stress increase with linear heat expanding ratio, and those decrease with the specific heat of material; that the displacement increases with the heat transfer ratio, but the stress does not .
最后利用有限元程序LS-DYNA对轴对称圆柱壳受高速热冲击作用进行了计算机仿真分析，得到了位移时间曲线和单元的应力时间曲线，同文献已有的结果吻合的较好。数值模拟表明：边界条件对位移和应力的影响非常小；位移随着线热膨胀系数、热传导系数的增大而增大，但是随着比热容的增大而减小；应力的最大值随着线热膨胀系数的增大而增大，而随着比热容的增大而减小，而热传导系数k对应力的影响比较小。
The basic equations of elliptic plate, when temperature and stress fields are coupled, are developed, which include transverse thermal vibration equation , the compatibility equation and energy equations. In order to solve them easily, we introduced the dimensionless quantities and non-dimensionlized the obtained equations. A nonlinear ordinary
differential equations are obtained by Galerkin's method. The numerical
results are presented by employing Runge-Kutta method. The important results are given as follows: The coupling coefficient the amplitude of temperature, the parameter, the ration of length of long axis to that of short axis and initial displacement are main factors which influence the thermoelastic vibration of elliptic plate; When the initial displacement is smaller, thermoelastic coupling makes the frequency of the elliptic plate larger and when the initial displacement is larger, thermoelastic coupling makes the frequency of the elliptic plate smaller; The boundary conditions has strong influence on the coupling effect, the stronger it is, the smaller frequency of free vibration is but the larger amplitude is. Second, the thermoelastic coupling vibration of cylindrical shell are discussed by means of the same method. The coupling coefficient, amplitude of temperature, initial displacement, ratio of length to radius and that of radius to thickness are main factors which influence the thermoelastic vibration of cylindrical shell; The higher the coupling coefficient is, the lower the axial force, the axial stress and the axial moment are.
The finite element code LS_DYNA is used to investigate the response of the axisymmetry cylindrical shell under the thermal impact, and the curves of displacement vs time and that of stress vs time are obtained. The results from simulation agree well with the conclusions of
conference [21], and indicate that the boundary conditions have no obvious effect on the displacement and stress; that the displacement and stress increase with linear heat expanding ratio, and those decrease with the specific heat of material; that the displacement increases with the heat transfer ratio, but the stress does not .
引文
1．王洪纲，《热弹性力学概论》，清华大学出版社，1998．
2．曹志远，《板壳振动理论》，中国铁道出版社，1989．
3．严宗达，王洪礼，《热应力》，高等教育出版社，1993．
4．徐芝伦，《弹性力学》，高等教育出版社，1996．
5．杨桂通，《弹塑性力学》，人民教育出版社，1981．
6．杨桂通，张善元，《弹性动力学》，中国铁道出版社，1988．
7．刘延柱，陈文良，陈立群，《振动力学》，高等教育出版社，1998．
8. Theodore R Tacchert, Thermally induced flexure, buckling, and vibration of plates, Appl. Mech. Rev, Vol. 44, NO. 8, pp. 347-360, Aug. 1991.
9. Chang W.P., Wan S.M., Thermomechanically coupled nonlinear vibration of plates, Int. J.Non-linear Mechanics, Vol.21, No.5, pp.375-389, 1986.
10. Chang W.P., Jen S.C., Nonlinear free vibration of heated orthotropic rectangular plate, Int. J. Solid Structures, Vol.22, No.3, pp.267-281, 1986.
11．戴德成，任勇生，矩形板的非线性热弹耦合振动，《振动工程学报》，Vol．3，No．2，pp．65-72，Jun．1990．
12．戴德成，板的非线性热弹耦合振动—近似解析解，《应用力学学报》，Vol．7，No．4，pp．63-70，Dec．1990．
13．树学锋，张晓晴，张晋香，周边固支圆板非线性热弹耦合振动分析，《应用数学和力学》，Vol．21，No．6，pp．647-654，Jun．2000．
14．树学锋，张晓晴，简支圆板非线性热弹耦合振动问题的研究，《工程力学》，Vol．17，No．2，pp．97-101，Apr．2000．
15．树学锋，陈贻平，张晓晴，热弹耦合圆板非线性振动的研究，《固体力学学报》，Vol．20，No．3，pp．245-250，Sep．1999．
16. Boley, B. A., and Barber, A. D., "Dynamic response of beam and
plates to rapid heating", Journal of applied mechanics, Vol. 24, Trans. ASME, Vol. 79, pp.413, Sept. 1957.
17. E.J.Mcquillen, M.A.Brull, Dynamic thermoelastic response of cylindrical shells, Journal of Applied Mechanics, pp. 661-670, Sept. 1970.
18. A. H. Ghson and M. Sabbaghian, Quasi-static coupled problems of thermoelasticity for cylindrical regions, Journal of thermal stresses, Vol. 5, pp. 299-313, 1982.
19. Y. Y. Li, H. Ghoneim, and Y. Chen, and J. Davis, A numerical method in solving a coupled thermoelasticity equation and some results, Journal of thermal stresses, Vol. 6, pp. 253-280, 1983.
20. Eslami, M. R., and Vahedi, H., A galerkin finite element displacement formulation of coupled thermoelasticity spherical problems, Journal of Pressure Vessel Technology, Vol. 114, No.3, pp.380-384, 1992.
21. M.R.Eslami, M.Shakeri, and R. Sedaghati, Coupled thermoelasticity of axially symmetric cylindrical shell, Journal of Thermal Stresses, Vol. 17, No. 1, pp. 115-135, 1994.
22. M.R. Eslami, M.Shakeri, A.R.Ohadi and B.Shiari, Coupled thermoelasticity of shells of revolution: Effect of normal stress and coupling, AIAA Journal, Vol. 37, No. 4, pp. 496-504, April 1999.
23．江晓禹，张相周，圆柱壳非线性振动的多重模式分析，《应用力学学报》，Vol．14，No．1，Mar l997，pp．5-10
24．吴晓，马建勋，圆柱壳热载荷作用下的非线性振动，《振动与冲击》，Vol．19，No．2，2000，pp．67-69
25．树学锋，韩强，杨桂通，大挠度板混沌运动的双模态分析，《应用数学和力学》，Vol．20，No．4 Apr．1999，pp．346-350
26．韩强，张年梅，杨桂通，非线性热弹耦合椭圆板的混沌运动，《应用数学和力学》，Vol．20，No．9，Sep．1999，pp．896-901
27．张年梅，韩强，杨桂通，非线性热弹耦合圆板中的混沌带现象，《固体力学学报》，Vol，19，No．3，Sep．1998，pp．265-268
28．江理平，圆柱壳扁壳的非线性动力响应的样条加权残值分析，《上海力学》，Vol．20，No．3，Sep．1999，pp．297-301
29．张年梅，弹塑性系统的混沌运动研究，博士论文，1997．3
30．程赫明，王洪纲，耦合热弹塑性问题的泛函及其变分原理，《应用数学和力学》，Vol．9，No．10，pp．805—810
31. H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, Vol. 15, pp. 299-309, 1967.
32. W. K. Liu and H. G. Chang, Uncondiyionally stable implicit-explicit algotithm for dyanmic coupled thermoelasticity, J. Appl. Mech., vol. 25, pp. 438-485, 1985.
33. M.R. Eslami and H.Vahedi ,Coupled Thermoelasticity Beam roblems, AIAA J., vol. 27, pp. 662-665, 1989.
34．兰姣霞，张晓晴，树学锋，轴对称圆柱壳非线性热弹耦合振动的基本方程，《华北工学院学报》，Vol．21，No．6，pp．513-516，2000．
35．戴德成，《非线性振动》，东南大学出版社，1993．
36．徐士良，《常用算法程序集》，清华大学出版社，1993．
2．曹志远，《板壳振动理论》，中国铁道出版社，1989．
3．严宗达，王洪礼，《热应力》，高等教育出版社，1993．
4．徐芝伦，《弹性力学》，高等教育出版社，1996．
5．杨桂通，《弹塑性力学》，人民教育出版社，1981．
6．杨桂通，张善元，《弹性动力学》，中国铁道出版社，1988．
7．刘延柱，陈文良，陈立群，《振动力学》，高等教育出版社，1998．
8. Theodore R Tacchert, Thermally induced flexure, buckling, and vibration of plates, Appl. Mech. Rev, Vol. 44, NO. 8, pp. 347-360, Aug. 1991.
9. Chang W.P., Wan S.M., Thermomechanically coupled nonlinear vibration of plates, Int. J.Non-linear Mechanics, Vol.21, No.5, pp.375-389, 1986.
10. Chang W.P., Jen S.C., Nonlinear free vibration of heated orthotropic rectangular plate, Int. J. Solid Structures, Vol.22, No.3, pp.267-281, 1986.
11．戴德成，任勇生，矩形板的非线性热弹耦合振动，《振动工程学报》，Vol．3，No．2，pp．65-72，Jun．1990．
12．戴德成，板的非线性热弹耦合振动—近似解析解，《应用力学学报》，Vol．7，No．4，pp．63-70，Dec．1990．
13．树学锋，张晓晴，张晋香，周边固支圆板非线性热弹耦合振动分析，《应用数学和力学》，Vol．21，No．6，pp．647-654，Jun．2000．
14．树学锋，张晓晴，简支圆板非线性热弹耦合振动问题的研究，《工程力学》，Vol．17，No．2，pp．97-101，Apr．2000．
15．树学锋，陈贻平，张晓晴，热弹耦合圆板非线性振动的研究，《固体力学学报》，Vol．20，No．3，pp．245-250，Sep．1999．
16. Boley, B. A., and Barber, A. D., "Dynamic response of beam and
plates to rapid heating", Journal of applied mechanics, Vol. 24, Trans. ASME, Vol. 79, pp.413, Sept. 1957.
17. E.J.Mcquillen, M.A.Brull, Dynamic thermoelastic response of cylindrical shells, Journal of Applied Mechanics, pp. 661-670, Sept. 1970.
18. A. H. Ghson and M. Sabbaghian, Quasi-static coupled problems of thermoelasticity for cylindrical regions, Journal of thermal stresses, Vol. 5, pp. 299-313, 1982.
19. Y. Y. Li, H. Ghoneim, and Y. Chen, and J. Davis, A numerical method in solving a coupled thermoelasticity equation and some results, Journal of thermal stresses, Vol. 6, pp. 253-280, 1983.
20. Eslami, M. R., and Vahedi, H., A galerkin finite element displacement formulation of coupled thermoelasticity spherical problems, Journal of Pressure Vessel Technology, Vol. 114, No.3, pp.380-384, 1992.
21. M.R.Eslami, M.Shakeri, and R. Sedaghati, Coupled thermoelasticity of axially symmetric cylindrical shell, Journal of Thermal Stresses, Vol. 17, No. 1, pp. 115-135, 1994.
22. M.R. Eslami, M.Shakeri, A.R.Ohadi and B.Shiari, Coupled thermoelasticity of shells of revolution: Effect of normal stress and coupling, AIAA Journal, Vol. 37, No. 4, pp. 496-504, April 1999.
23．江晓禹，张相周，圆柱壳非线性振动的多重模式分析，《应用力学学报》，Vol．14，No．1，Mar l997，pp．5-10
24．吴晓，马建勋，圆柱壳热载荷作用下的非线性振动，《振动与冲击》，Vol．19，No．2，2000，pp．67-69
25．树学锋，韩强，杨桂通，大挠度板混沌运动的双模态分析，《应用数学和力学》，Vol．20，No．4 Apr．1999，pp．346-350
26．韩强，张年梅，杨桂通，非线性热弹耦合椭圆板的混沌运动，《应用数学和力学》，Vol．20，No．9，Sep．1999，pp．896-901
27．张年梅，韩强，杨桂通，非线性热弹耦合圆板中的混沌带现象，《固体力学学报》，Vol，19，No．3，Sep．1998，pp．265-268
28．江理平，圆柱壳扁壳的非线性动力响应的样条加权残值分析，《上海力学》，Vol．20，No．3，Sep．1999，pp．297-301
29．张年梅，弹塑性系统的混沌运动研究，博士论文，1997．3
30．程赫明，王洪纲，耦合热弹塑性问题的泛函及其变分原理，《应用数学和力学》，Vol．9，No．10，pp．805—810
31. H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, Vol. 15, pp. 299-309, 1967.
32. W. K. Liu and H. G. Chang, Uncondiyionally stable implicit-explicit algotithm for dyanmic coupled thermoelasticity, J. Appl. Mech., vol. 25, pp. 438-485, 1985.
33. M.R. Eslami and H.Vahedi ,Coupled Thermoelasticity Beam roblems, AIAA J., vol. 27, pp. 662-665, 1989.
34．兰姣霞，张晓晴，树学锋，轴对称圆柱壳非线性热弹耦合振动的基本方程，《华北工学院学报》，Vol．21，No．6，pp．513-516，2000．
35．戴德成，《非线性振动》，东南大学出版社，1993．
36．徐士良，《常用算法程序集》，清华大学出版社，1993．