微分求积法在结构振动和稳定性分析中的应用
摘要
本文首先介绍了求解微分方程边值问题的一种数值方法—微分求积法,然后采用微分求积法分别对输流管道和非保守矩形薄板的稳定性问题进行了分析研究,具体有如下三方面的研究内容。
用微分求积法分析了具有弹性支承的弹性地基上输流管道的稳定性问题。对一端固定一端具有弹性支承的弹性地基上的输流管道,用D'Alembert原理建立了运动的微分方程,然后用微分求积法分析了管道颤振失稳和发散失稳情况,讨论了地基反力、地基剪切模量和端部支承的弹性常数对管道失稳形式的影响,并画出了稳定性区域图。
对变厚度矩形板在非保守力作用下的振动和稳定性问题,本文采用微分求积法进行研究。利用微分求积法处理边界条件的灵活性,对于三种不同边界条件(四边简支、四边固支以及一边固支三边简支)的非保守变厚度矩形薄板,分别计算了不同边长比下板的振动频率和临界载荷随板厚比的变化曲线,并讨论了不同边界约束下板的失稳形式。
对由金属和陶瓷两种材料制成的功能梯度材料(Functionally Graded Material,FGM)矩形板在非保守力作用下的稳定性问题用微分求积法进行研究。首先根据振动理论及FGM构成特性建立了FGM矩形板的振动微分方程,然后用微分求积法对其进行数值求解,分别讨论了FGM板和各单相材料板的振动频率和临界载荷之间的关系,并进一步讨论了材料组分对FGM板的振动频率和临界载荷的影响。
Differential Quadrature Method (DQM), one of numerical methods of solving boundary problem of differential equation is introduced in this paper. By using DQM, the dynamic stability of pipes conveying fluid and rectangular plates under the action of non-conservative force is studied. The main research work is as follows.The stability of pipes conveying fluid on the elastic foundation with elastic support is analyzed by using DQM. The governing differential equation of pipe conveying fluid on the elastic foundation with one end is fixed and the other end is subjected to elastic support is derived based on D'Alembert's principle. The divergence and flutter instability of the pipe is discussed, and the effect of foundation reaction, foundation shear modulus and elastic constant of the elastic support on shape of instability of elastic pipe conveying fluid are analyzed, and the stability regional graphs (divergence region and flutter region) are plotted.The stability of varying thickness rectangular plate under the action of non-conservative force is analyzed. Because DQM is easy to handle the boundary condition, For three different boundary conditions, variation curves between thickness ratio of plate and vibration frequencies and critical loads are plotted, and the instability shapes of non-conservative varying thickness rectangular plate under different boundary conditions are discussed by DQM.The stability of Functionally Graded Material (FGM) rectangular plate made of metal and ceramic under the action of non-conservative force is analyzed. Based on vibration theory and structure property of FGM, the governing differential equation is
derived, and the numerical solution is obtained by DQM. The vibration frequencies and critical loads of FGM plate and the two single-phase material plates (metal plate and ceramic plate) are compared, and the effect of volume fraction on vibration frequencies and critical loads are discussed.
用微分求积法分析了具有弹性支承的弹性地基上输流管道的稳定性问题。对一端固定一端具有弹性支承的弹性地基上的输流管道,用D'Alembert原理建立了运动的微分方程,然后用微分求积法分析了管道颤振失稳和发散失稳情况,讨论了地基反力、地基剪切模量和端部支承的弹性常数对管道失稳形式的影响,并画出了稳定性区域图。
对变厚度矩形板在非保守力作用下的振动和稳定性问题,本文采用微分求积法进行研究。利用微分求积法处理边界条件的灵活性,对于三种不同边界条件(四边简支、四边固支以及一边固支三边简支)的非保守变厚度矩形薄板,分别计算了不同边长比下板的振动频率和临界载荷随板厚比的变化曲线,并讨论了不同边界约束下板的失稳形式。
对由金属和陶瓷两种材料制成的功能梯度材料(Functionally Graded Material,FGM)矩形板在非保守力作用下的稳定性问题用微分求积法进行研究。首先根据振动理论及FGM构成特性建立了FGM矩形板的振动微分方程,然后用微分求积法对其进行数值求解,分别讨论了FGM板和各单相材料板的振动频率和临界载荷之间的关系,并进一步讨论了材料组分对FGM板的振动频率和临界载荷的影响。
Differential Quadrature Method (DQM), one of numerical methods of solving boundary problem of differential equation is introduced in this paper. By using DQM, the dynamic stability of pipes conveying fluid and rectangular plates under the action of non-conservative force is studied. The main research work is as follows.The stability of pipes conveying fluid on the elastic foundation with elastic support is analyzed by using DQM. The governing differential equation of pipe conveying fluid on the elastic foundation with one end is fixed and the other end is subjected to elastic support is derived based on D'Alembert's principle. The divergence and flutter instability of the pipe is discussed, and the effect of foundation reaction, foundation shear modulus and elastic constant of the elastic support on shape of instability of elastic pipe conveying fluid are analyzed, and the stability regional graphs (divergence region and flutter region) are plotted.The stability of varying thickness rectangular plate under the action of non-conservative force is analyzed. Because DQM is easy to handle the boundary condition, For three different boundary conditions, variation curves between thickness ratio of plate and vibration frequencies and critical loads are plotted, and the instability shapes of non-conservative varying thickness rectangular plate under different boundary conditions are discussed by DQM.The stability of Functionally Graded Material (FGM) rectangular plate made of metal and ceramic under the action of non-conservative force is analyzed. Based on vibration theory and structure property of FGM, the governing differential equation is
derived, and the numerical solution is obtained by DQM. The vibration frequencies and critical loads of FGM plate and the two single-phase material plates (metal plate and ceramic plate) are compared, and the effect of volume fraction on vibration frequencies and critical loads are discussed.
引文
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[1
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[2
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[3
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[43] 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。应用力学学报,13(增刊),1996
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[2] Bellman R E, Casti J. Differential quadrature and long-term integration. J. muth. Anal. Appl. , 34, 1971:235-238
[3] Bert C W, Jang SK, Striz A G. Two new approximate methods for analyzing free vibration of structural components. AIAAJ, 26(5), 1988: 612-618
[4] Bellman R E, Kashef B G. . Solution of the partial differential equation of the Hodgkins-Huxley model using differential quadrature. Math. Biosci, 19, 1974:1-8
[5] Mingle J O. Computational consideration in nonlinear diffusion. Int. Jnum er. Methods Engng. 7, 1973:103-116
[6] Mingle J O. The Method of differential quadrature for transient nonlinear diffusion. J. Math. Anal. Appl. , 60, 1977: 559-569
[7] Bellman R E, Roth R S. System identification with partial information. J. Math. Anal. Appl. , 68, 1979:321-333
[8] Civan F, Spiiepcevish C M. Solvinhg Integro-Differential Equations by the Quadrature Method. Intcgral Methods Sci & Engng. Hemisphe -re Publishing. Washingtcn. D. C. , 1986:106-113
[9] Civan F, Spiiepcevish C M. Application of differential quadrature to transport processes. J. Math. Anal. Appl. , 93, 1983:206-221
[10] Naadimuthu G, Bellman Re, Wang K M, Lee E S. Differential quadrature and partial differential equations:some numerical results. J. Math. Anal. Appl. , 98, 1984:220-235
[11] Malik M, Bert C W. Differential quadrature solutions for steady-state incompressible and compress sible lubrication problems. J. Tribology, 116, 1994:296-302
[1
[12] Shu C, Richards B E. Application of generalized differential quadrature to solve Two dimensional incompressible naviver-stokes equations. Int. J. Numer. Methods Fluids, 15, 1992:791-798
[13] Shu C, Richards B E. Parallel simulation of incompressible viscous flows by generalized differential quadrature. Computing Systems in Engng. 3, 1992:271-281
[14] Sherbourne A N, Pandey M D. Differential quadrature method in the buckling analysis of beams and composite plates. Comp. &Struct. 40(4), 1991: 903-913.
[15] Striz A G, Wang X, Bert C W. Harmonic differential quadrature method and applications to free vibration analysis of rectangular plates. Presented at the 2nd National Congress on Compatutional Mechanics. Washington, D C, USA, 1993:16-18
[16] Bert C W, Wang X, Striz A G. Differential quadrature for static and free vibration analysis of anisotropic plates. Int. J. Solids Struct. 30(13), 1993:1737-1744
[17] Shu c, Chen W. On optimal selection of interior points for applying discretized boundary conditions in DQ vibration of beams and plates. J. sound & Vibration, 222(2), 1999:239-257
[18] Bert C W, Jang S K, StrizA G. Two new approximate methods for analyzing free vibration of structural components. AIAAJ. , 26(5) 1988:612-618
[19] Wang X, Gu H. Static analysis of frame structures by the differential quadrature element method. Int. J. Num. Methods Eng. , 1997:759-772
[20] 王永亮,刘人怀,王鑫伟。弹性支承环形板的非线性弯曲。固体力 学学报,19,1998:107-110
[2
[21] Bert C W, Malik M. Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. , 49(1), 1996:1-28
[22] Malik M, Bert C W. Differential quadrature: a powerful new techni -que for analysis of composite structures. Compose. Struct. 39(1997): 179-189
[23] Chen,S.S.(美)。圆柱结构的流动诱发振动。冯振宇,张希农译,石油工业出版社,北京:1993:108-109。
[24] Pramila A, Laukkanen. Dynamics and stability of short fluid con -veying Timoshenko element pipes. Journal of Sound and Vivration, 144(3), 1991:421-425
[25] CUI Hong-wu, TANI Jun-ji. Effect of boundary conditions on the stability of a cantilever pipe discharging and aspirationg fluid. JSME International Journal, Series C, 39(1), 1996:20-22
[26] Lin H. C. , Chen S. S. Vibration and Stability of Fluid-conveying Pipe. Shock and Vibration, 46(2), 1976:267-283
[27] Noah S T. Hopking G R. Dynamic Stability of Elastically Supported Pipes Conveying Fluid. J. Sound and Vibration, 71(1), 1980:103-116
[28] 金基铎,林影,邹光胜。悬臂输流管道的颤振与混沌运动分析。振动工程学报,10(3),1997:314-320
[29] 倪樵,黄玉盈,陈贻平。微分求积法分析具有弹性支承输液管的临界流速。计算力学学报,18(2),2000:146-149
[30] 冯振宇,赵凤群。固—液耦合管道的稳定性分析。应用力学学报,16(3),1999:47-52
[31] 王忠民,冯振宇,赵风群,刘宏昭。弹性地基输流管道的耦合模态颤振分析。应用数学和力学,21(10),2000:1060-1068
[32] 黄玉盈编著。结构振动分析基础。武汉:华中理工大学出版社, 1988:225
[3
[33] 黄玉盈,邹时智等。输液管的非线性振动、分叉与混沌--现状与展望。力学进展,128(1),1998:30-32。
[34] Zhao Fengqun, Wang Zhongmin, Liu Hongzhao and Feng Zhenyu. Solid-liquid couple vibration of viscoelastic pipe conveying fluid with Maxwell model, Proceeding's of International Conference on Advanced Problems in Vibration Theory and Applications June, 2000:674-678.
[35] 赵凤群,王忠民,冯振宇。三参量固体模型粘弹性输流管道的动力特性分析。固体力学学报,23(4),2002:483-489
[36] 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,6,1981:563-570
[37] 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,2,1987:127-136
[38] 宋志远,万虹,梅占馨。非保守力作用下的矩形板的稳定性问题。固体力学学报,12(4),1991:359-363
[39] S. Y. Lee, H. Y. Kuo. Exact solutions for the non-conservative elastic stability of non-uniform columns. Journal of Sound and Vibration. 155(2) , 1992:291-301
[40] S. Y. Lee, C. C. Yang. Non-conservative instability of a Timoshenko beam resting on Winkler elastic foundation. Journal of Sound and Vibration. 162(1), 1993:177-184
[41] 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。振动工程学报,5(1),1992:78-83
[42] 许明田,程德林。用积分方程法解板的振动问题。应用数学和力学,17(7),1996:655-660
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