非保守热弹性圆形薄板的动力特性和稳定性
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摘要
本文主要研究了受面内切向均布随从力作用下的弹性非保守圆薄板的振动和稳定性问题。主要工作有：
(1)本文首先从微元体出发，通过分析圆板微元体的受力状况，建立了轴对称圆板在面内周边可移、不可移两种情况下的轴对称控制方程，这是一个变系数四阶偏微分方程。通过假设振动模态，将所得的混合初—边值问题转化为相应的变系数常微分方程两点边值问题，然后用打靶法直接导出求解该变系数常微分方程特征值问题数值解的计算式，通过数值计算，给出了周边可移、不可移的简支、固支圆板自振频率和临界载荷的特征曲线以及相应的临界发散载荷，并分析了泊松比对圆板自振频率和临界载荷的影响。
(2)在此基础上，本文采用Kantorovich平均法和打靶法，对于受切向均布随从力作用下的变厚度热弹性圆板和变温热弹性圆板的振动和稳定性问题分别进行了分析。通过数值计算，分别给出了在这两种情况下周边不可移的简支、固支热弹性圆板自振频率和临界载荷的特征曲线以及相应的临界发散载荷，并分别分析了厚度变化系数以及温度变化系数对圆板自振频率和临界载荷的影响。
(3)最后，基于圆板变形大挠度理论，导出了均匀加热非保守弹性圆板用中面位移表示的非线性振动的动力学控制方程，通过数值计算，获得了周边不可移简支及固支圆板的振动响应，绘出了在不同随从力下的幅—频响应曲线。
In this paper, dynamic Behavior and Stability of elastic non-conservative circular thin plate, subjected to tangentially uniformly distributed follower forces, is analyzed. The main research work is as follows:
(l)By the analysis of force-balance of elements, the axisymmetric governing equations of circular thin plate with movable and unmovable supports in the middle plane, subjected to tangentially uniformly distributed follower forces, are derived. The governing equation is a order four partial differential equation with variable coefficient. On the assumption of oscillation form, the mixed initial-boundary value problem is converted into corresponding two point boundary value problem of ordinary differential equation. Then numerical solutions for the eigenvalue problem of the ordinary differential equation with variable coefficients are obtained by shooting method. Moreover, the characteristic curves between self-excited frequencies and critical loads and their critical divergence load of the circular plate with movable and unmovable simple supports and clamped supports along edges are plotted, and the effect of Possion' s ratio on the self-excited frequencies and critical loads of the circular plate is analyzed.
(2) On the basis of the above study, by the Kantorovich averaging method and shooting method, the vibration and the stability of varying thickness thermal-elastic circular thin plate and varying thermal circular thin plate, subjected to non-conservative forces, are studied. Based on the numerical solutions for the eigenvalue problem of the ordinary differential equation, the characteristic curves, under those two conditions, between self-excited frequencies and critical loads and their critical divergence load of the circular plate with unmovable simple supports and clamped supports along edges are plotted. Besides, the effect of the variation of coefficient of thickness and the variation coefficient of thickness temperature on the self-excited frequencies and critical loads of the circular plate is analyzed.
(3)At last, based on large deflection theory of circular thin plate, the dynamic governing equation of nonlinear vibration of thermal-elastic circular plate subjected to distributed forces in terms of the displacement of the middle plate are derived. By numerical calculation, the vibration response of thermal-elastic thin plate with unmovable simple supports and clamped supports along edges are obtained. Furthermore, the characteristic curves of fundamental frequencies versus amplitudes for some specific values of follower force are plotted..
(1)本文首先从微元体出发，通过分析圆板微元体的受力状况，建立了轴对称圆板在面内周边可移、不可移两种情况下的轴对称控制方程，这是一个变系数四阶偏微分方程。通过假设振动模态，将所得的混合初—边值问题转化为相应的变系数常微分方程两点边值问题，然后用打靶法直接导出求解该变系数常微分方程特征值问题数值解的计算式，通过数值计算，给出了周边可移、不可移的简支、固支圆板自振频率和临界载荷的特征曲线以及相应的临界发散载荷，并分析了泊松比对圆板自振频率和临界载荷的影响。
(2)在此基础上，本文采用Kantorovich平均法和打靶法，对于受切向均布随从力作用下的变厚度热弹性圆板和变温热弹性圆板的振动和稳定性问题分别进行了分析。通过数值计算，分别给出了在这两种情况下周边不可移的简支、固支热弹性圆板自振频率和临界载荷的特征曲线以及相应的临界发散载荷，并分别分析了厚度变化系数以及温度变化系数对圆板自振频率和临界载荷的影响。
(3)最后，基于圆板变形大挠度理论，导出了均匀加热非保守弹性圆板用中面位移表示的非线性振动的动力学控制方程，通过数值计算，获得了周边不可移简支及固支圆板的振动响应，绘出了在不同随从力下的幅—频响应曲线。
In this paper, dynamic Behavior and Stability of elastic non-conservative circular thin plate, subjected to tangentially uniformly distributed follower forces, is analyzed. The main research work is as follows:
(l)By the analysis of force-balance of elements, the axisymmetric governing equations of circular thin plate with movable and unmovable supports in the middle plane, subjected to tangentially uniformly distributed follower forces, are derived. The governing equation is a order four partial differential equation with variable coefficient. On the assumption of oscillation form, the mixed initial-boundary value problem is converted into corresponding two point boundary value problem of ordinary differential equation. Then numerical solutions for the eigenvalue problem of the ordinary differential equation with variable coefficients are obtained by shooting method. Moreover, the characteristic curves between self-excited frequencies and critical loads and their critical divergence load of the circular plate with movable and unmovable simple supports and clamped supports along edges are plotted, and the effect of Possion' s ratio on the self-excited frequencies and critical loads of the circular plate is analyzed.
(2) On the basis of the above study, by the Kantorovich averaging method and shooting method, the vibration and the stability of varying thickness thermal-elastic circular thin plate and varying thermal circular thin plate, subjected to non-conservative forces, are studied. Based on the numerical solutions for the eigenvalue problem of the ordinary differential equation, the characteristic curves, under those two conditions, between self-excited frequencies and critical loads and their critical divergence load of the circular plate with unmovable simple supports and clamped supports along edges are plotted. Besides, the effect of the variation of coefficient of thickness and the variation coefficient of thickness temperature on the self-excited frequencies and critical loads of the circular plate is analyzed.
(3)At last, based on large deflection theory of circular thin plate, the dynamic governing equation of nonlinear vibration of thermal-elastic circular plate subjected to distributed forces in terms of the displacement of the middle plate are derived. By numerical calculation, the vibration response of thermal-elastic thin plate with unmovable simple supports and clamped supports along edges are obtained. Furthermore, the characteristic curves of fundamental frequencies versus amplitudes for some specific values of follower force are plotted..
引文
[1] Leipholz H, Pfendt F. On the stability of rectangular,completely supported palates with uncoupling boundary conditions subjected to uniformly distributed follower forces.Computer Methods in Applied Mech. and Engng., 1992, 30:19～52.
[2] Leipholz H. and Pfendt F. Application of extended equations of Galerkin to stability problems of rectangular plates subjected to follower forces. Computer Methods in Applied Mech. and Engng., 1983, 37:341～365.
[3] Leipholz.H. H. E., Stability of Elastic Systems Computer Methods in Applied Mech. and Engng., 1980, 25:178～192
[4] Leipholz H., Variational Principle for Non-conservative Computer Methods. Applied Mech and Enging, 1979, 17: part 3.
[5] Leipholz H, Pfendt F. On the stability of rectangular,completely supported palates with uncoupling boundary conditions subjected to uniformly distributed follower forces.Computer Methods in Applied Mech. and Engng., 1992, 30:19～52.
[6] 刘殿魁，张其浩．弹性理论中非保守问题的一般变分原理．力学学报，1981，6：562～570
[7] 黄玉盈，王武久．弹性非保守系统的拟固有频率变分原理及其应用．固体力学学报，1987，8(2)，：127～136
[8] 张其浩，单文秀．非保守稳定问题的有限元法．上海力学，1981，3：40～46
[9] 张其浩，单文秀．弹性非保守问题中全变分方程及其在有限元方法上的应用．上海力学，1984，3：22～29
[10] Higuchi K,Dowell E H.Dynamic stability of a completely free plate subjected to a controlled non-conservative follower forces. Journal of Sound & Vibration, 1989,132:115～128
[11] Higuchi K, Dowell E H. Effects of Poisson's ratio and negative thrust on the dynamic stability of free plate subjected to a follower force. Journal of Sound (?) Vibration, 1989,124:255～269
[12] 宋志远，万虹，梅占馨。非保守力作用下矩形薄板的稳定性问题。固体力学学报，1991，12(4)。
[13] Kim, Park. On the dynamic stability of rectangle plates subjected to intermediate follower forces. Journal of Sound and Vibration,1998: 109(5), 882-888
[14] 王忠民、计伊周，矩形薄板在随从力作用下的动力稳定性分析[J]，振动工程学报，1992，5(1)：78～83．
[15] 王忠民、计伊周，具有弹性约束的非保守矩形薄板的振动和稳定性分析，应用力学学报，1996，13：14～18
[16] 赵凤群、王忠民，点弹性支承的非保守矩形薄板的稳定性。西安理工大学学报，1998，14(4)：398～403。
[17] 王忠民，变厚度矩形薄板在随动了作用下的稳定性。固体力学学报，1997，18(12)：22～27。
[18] Celep Z., Axially symmetric stability of a completely free circular plate subjected to a non-conservative edge load.Journal of Sound and Vibration, 1979,65:549～556
[19] Celep Z., Vibration and stability of a free circular plate subject to non-conservative loading. Journal of Sound and Vibration, 1982,80:421～432
[20] Tauchert T.R. Thermally induced flexure, bucking and vibration of plates. Appl Mech, 1991,44(8):347～359
[21] Thornton E A.. Thermally bucking of plates and shells. Appl Mech, 1984,46(10):485～506
[22] Buckens F. Vibration in an thermally stressed thin plate. Journal of Thermal Stresses, 1979(2): 367～385
[23] Ganesan N, Dhotarad M. S., Hybrid method for analysis of thermally stressed plates. Journal of Sound & Vibration,1989,132:115～128
[24] Tani, J., Elastic instability of a heated annual plate under lateral pressure. Appl. Mech., 1981(48), 399-403
[25] Sarkar S K. Bucking of a heated circular plate. Indian Mech Math, 1967,5:39～42
[26] Grigolyuk E I, Magermova L A. Stability of homogeneous and Inhomogeneous plates. Mech. Solids(Engl. Tr), 1981, 16:93～115
[27] Ruju K K, Rao G V. Thermal-bucking of circular plates.Compute Struct, 1984, 18:1178～1182
[28] 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，1990，3(2)：65～72
[29] 李忠学、严宗达，周边固支的矩形板的动力耦合热弹性问题分析，工程力学，1998，15(3)：22～28
[30] 丁皓江、国凤林，耦合热弹性问题的一般解，应用数学和力学，2000：21(6)，573～577
[31] 程赫明、王洪纲，耦合热弹性问题的泛函及其变分原理，应用数学和力学，1988：9(10)，865～870
[32] 李世荣、程昌钧，复合载荷下环形薄板的热屈曲。应用数学和力学，1991，12(3)279～286。
[33] 李世荣，具有中心刚性质量的受热环板的非线性振动和热屈曲，应用数学和力学，1992，13(8)：817～824。
[34] 李世荣、宋曦，环板在任意轴对称面内载荷作用下自由振动和稳定性分析的一种差分方法，理论与应用力学，1993：4(1)：22～29
[35] 李世荣、宋曦、赵永刚，变厚度环板在面内变温下的固有频率，工程力学，1995，12(1)：58～65
[36] 李世荣，弹性圆板的热过屈曲，固体力学学报，1997，18(2)：179～182
[37] 李世荣，加热弹性圆板的大振幅自由振动，甘肃工业大学学报，1997，23(2)：103～107
[38] 张晓晴、树学锋，极坐标下板的非线性热弹耦合振动基本方程，太原理工大学学报，1997，30(7)：343～345
[39] 张晓晴、树学锋，几何非线性圆板的热弹性耦合分析，太原理工大学学报，2000，31(2)：116～119
[40] 树学锋、张晓晴，周边固支圆板非线性热弹耦合振动分析，应用数学和力学，2000，21(6)：647～654
[41] 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，2000，17(2)：97～101
[42] Gupta. For asymmetric response of linearly tapered circular plates. Journal of Sound and Vibration, 1999: 220(4), 641-657
[43] Singh, Saxena. Axisymmetric vibration of a circular plate with double linear variable thickness. Journal of Sound and Vibration,1995: 179(5), 879-897
[44] 徐芝纶，弹性力学，高等教育出版社，1987
[45] 何福保、沈亚鹏，板壳理论，西安交通大学出版社，1993
[46] William H. P., Brain P. F., Sau A T., Numerical recipes-the art of scientific computation. New York: Cambridge University Press,1986, 578～614.
[47] 徐士良，Fortran常用算法程序集，清华大学出版社，1995
[48] Li Shirong. Nonlinear vibration and thermal buckling of heated annular plate with variable thickness. Proceedings of 2nd International Conference on Nonlinear Mechanics. Beijing: Peking University Press, 1993. 535～538
[49] C.L.D. Huang and H. S. Walker, Nonlinear vibration of a hinged circular plate with a concentric rigid mass. Journal of Sound and Vibration, 1998,126: 9～17.
[50] 张志涌，精通MATLAB5.3，北京航空航天大学出版社，2000
[51] Mathworks,Matlab5.3 User Manual,1999
[52] 倪振华编著．振动力学．西安交通大学出版社，西安：1989．
[53] 陈铁云、沈慧中，结构的屈曲，上海科学技术文献出版社，1993
[54] 唐家祥、王仕统、裴若娟，结构稳定理论，中国铁道出版社，1989
[55] 武连元，板壳稳定性理论，华中科技大学出版社，1996
[56] Kanaka, R. K. and R. G. Venkateswara, Thermal-postbulking of circular plates, Comput. Struct., 18(1984), 1179-1182
[57] Bercin. Free vibration solution for clamped orthotropic plates using the Kantorovich method. Journal of Sound and Vibration,1996: 196(2), 243-247
[58] 王林祥、武际可，周边均布压力下圆薄板大范围非线性问题，计算力学学报(增刊)，1997，14：145～148
[2] Leipholz H. and Pfendt F. Application of extended equations of Galerkin to stability problems of rectangular plates subjected to follower forces. Computer Methods in Applied Mech. and Engng., 1983, 37:341～365.
[3] Leipholz.H. H. E., Stability of Elastic Systems Computer Methods in Applied Mech. and Engng., 1980, 25:178～192
[4] Leipholz H., Variational Principle for Non-conservative Computer Methods. Applied Mech and Enging, 1979, 17: part 3.
[5] Leipholz H, Pfendt F. On the stability of rectangular,completely supported palates with uncoupling boundary conditions subjected to uniformly distributed follower forces.Computer Methods in Applied Mech. and Engng., 1992, 30:19～52.
[6] 刘殿魁，张其浩．弹性理论中非保守问题的一般变分原理．力学学报，1981，6：562～570
[7] 黄玉盈，王武久．弹性非保守系统的拟固有频率变分原理及其应用．固体力学学报，1987，8(2)，：127～136
[8] 张其浩，单文秀．非保守稳定问题的有限元法．上海力学，1981，3：40～46
[9] 张其浩，单文秀．弹性非保守问题中全变分方程及其在有限元方法上的应用．上海力学，1984，3：22～29
[10] Higuchi K,Dowell E H.Dynamic stability of a completely free plate subjected to a controlled non-conservative follower forces. Journal of Sound & Vibration, 1989,132:115～128
[11] Higuchi K, Dowell E H. Effects of Poisson's ratio and negative thrust on the dynamic stability of free plate subjected to a follower force. Journal of Sound (?) Vibration, 1989,124:255～269
[12] 宋志远，万虹，梅占馨。非保守力作用下矩形薄板的稳定性问题。固体力学学报，1991，12(4)。
[13] Kim, Park. On the dynamic stability of rectangle plates subjected to intermediate follower forces. Journal of Sound and Vibration,1998: 109(5), 882-888
[14] 王忠民、计伊周，矩形薄板在随从力作用下的动力稳定性分析[J]，振动工程学报，1992，5(1)：78～83．
[15] 王忠民、计伊周，具有弹性约束的非保守矩形薄板的振动和稳定性分析，应用力学学报，1996，13：14～18
[16] 赵凤群、王忠民，点弹性支承的非保守矩形薄板的稳定性。西安理工大学学报，1998，14(4)：398～403。
[17] 王忠民，变厚度矩形薄板在随动了作用下的稳定性。固体力学学报，1997，18(12)：22～27。
[18] Celep Z., Axially symmetric stability of a completely free circular plate subjected to a non-conservative edge load.Journal of Sound and Vibration, 1979,65:549～556
[19] Celep Z., Vibration and stability of a free circular plate subject to non-conservative loading. Journal of Sound and Vibration, 1982,80:421～432
[20] Tauchert T.R. Thermally induced flexure, bucking and vibration of plates. Appl Mech, 1991,44(8):347～359
[21] Thornton E A.. Thermally bucking of plates and shells. Appl Mech, 1984,46(10):485～506
[22] Buckens F. Vibration in an thermally stressed thin plate. Journal of Thermal Stresses, 1979(2): 367～385
[23] Ganesan N, Dhotarad M. S., Hybrid method for analysis of thermally stressed plates. Journal of Sound & Vibration,1989,132:115～128
[24] Tani, J., Elastic instability of a heated annual plate under lateral pressure. Appl. Mech., 1981(48), 399-403
[25] Sarkar S K. Bucking of a heated circular plate. Indian Mech Math, 1967,5:39～42
[26] Grigolyuk E I, Magermova L A. Stability of homogeneous and Inhomogeneous plates. Mech. Solids(Engl. Tr), 1981, 16:93～115
[27] Ruju K K, Rao G V. Thermal-bucking of circular plates.Compute Struct, 1984, 18:1178～1182
[28] 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，1990，3(2)：65～72
[29] 李忠学、严宗达，周边固支的矩形板的动力耦合热弹性问题分析，工程力学，1998，15(3)：22～28
[30] 丁皓江、国凤林，耦合热弹性问题的一般解，应用数学和力学，2000：21(6)，573～577
[31] 程赫明、王洪纲，耦合热弹性问题的泛函及其变分原理，应用数学和力学，1988：9(10)，865～870
[32] 李世荣、程昌钧，复合载荷下环形薄板的热屈曲。应用数学和力学，1991，12(3)279～286。
[33] 李世荣，具有中心刚性质量的受热环板的非线性振动和热屈曲，应用数学和力学，1992，13(8)：817～824。
[34] 李世荣、宋曦，环板在任意轴对称面内载荷作用下自由振动和稳定性分析的一种差分方法，理论与应用力学，1993：4(1)：22～29
[35] 李世荣、宋曦、赵永刚，变厚度环板在面内变温下的固有频率，工程力学，1995，12(1)：58～65
[36] 李世荣，弹性圆板的热过屈曲，固体力学学报，1997，18(2)：179～182
[37] 李世荣，加热弹性圆板的大振幅自由振动，甘肃工业大学学报，1997，23(2)：103～107
[38] 张晓晴、树学锋，极坐标下板的非线性热弹耦合振动基本方程，太原理工大学学报，1997，30(7)：343～345
[39] 张晓晴、树学锋，几何非线性圆板的热弹性耦合分析，太原理工大学学报，2000，31(2)：116～119
[40] 树学锋、张晓晴，周边固支圆板非线性热弹耦合振动分析，应用数学和力学，2000，21(6)：647～654
[41] 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，2000，17(2)：97～101
[42] Gupta. For asymmetric response of linearly tapered circular plates. Journal of Sound and Vibration, 1999: 220(4), 641-657
[43] Singh, Saxena. Axisymmetric vibration of a circular plate with double linear variable thickness. Journal of Sound and Vibration,1995: 179(5), 879-897
[44] 徐芝纶，弹性力学，高等教育出版社，1987
[45] 何福保、沈亚鹏，板壳理论，西安交通大学出版社，1993
[46] William H. P., Brain P. F., Sau A T., Numerical recipes-the art of scientific computation. New York: Cambridge University Press,1986, 578～614.
[47] 徐士良，Fortran常用算法程序集，清华大学出版社，1995
[48] Li Shirong. Nonlinear vibration and thermal buckling of heated annular plate with variable thickness. Proceedings of 2nd International Conference on Nonlinear Mechanics. Beijing: Peking University Press, 1993. 535～538
[49] C.L.D. Huang and H. S. Walker, Nonlinear vibration of a hinged circular plate with a concentric rigid mass. Journal of Sound and Vibration, 1998,126: 9～17.
[50] 张志涌，精通MATLAB5.3，北京航空航天大学出版社，2000
[51] Mathworks,Matlab5.3 User Manual,1999
[52] 倪振华编著．振动力学．西安交通大学出版社，西安：1989．
[53] 陈铁云、沈慧中，结构的屈曲，上海科学技术文献出版社，1993
[54] 唐家祥、王仕统、裴若娟，结构稳定理论，中国铁道出版社，1989
[55] 武连元，板壳稳定性理论，华中科技大学出版社，1996
[56] Kanaka, R. K. and R. G. Venkateswara, Thermal-postbulking of circular plates, Comput. Struct., 18(1984), 1179-1182
[57] Bercin. Free vibration solution for clamped orthotropic plates using the Kantorovich method. Journal of Sound and Vibration,1996: 196(2), 243-247
[58] 王林祥、武际可，周边均布压力下圆薄板大范围非线性问题，计算力学学报(增刊)，1997，14：145～148