介绍精确动力刚度法分析中厚椭球壳自由振动具体实施方法,据环向波数不同将中厚椭球壳自由振动分解为一系列确定环向波数的一维振动;利用控制方程Hamilton形式建立动力刚度关系,用常微分方程求解器COLSYS求解控制方程获得单元动力刚度,用Wittrick-Williams算法求得该环向波数下椭球壳自振频率。数值算例给出中厚圆球壳及椭球壳不同边界条件的自振频率,验证动力刚度法高效、可靠、精确。
Abstract
The application of exact dynamic stiffness method to the free vibration analysis of moderately thick elliptical shells is introduced. The free vibration of moderately thick elliptical shells is decomposed into a series of one-dimensional vibration problems with respect to different circumferential wave numbers. For any one-dimensional vibration problem, governing equations are written in Hamiltonian form from which the dynamic stiffness relationship of this one-dimensional problem is set up. Based on this dynamic stiffness relationship, the governing equations are solved by using the ordinary differential equations (ODE) solver COLSYS from which element dynamic stiffnesses can be obtained. By applying the Wittrick-Williams algorithm, natural frequencies under a specific circumferential wave number are found. Numerical examples on moderately thick spherical and elliptical shells with different boundary conditions are given, showing that the dynamic stiffness method is robust, reliable and accurate.
关键词
椭球壳 /
自由振动 /
动力刚度法 /
Wittrick-Williams算法 /
Hamilton形式
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Key words
elliptical shells /
free vibration /
dynamic stiffness method /
Wittrick-Williams algorithm /
Hamiltonian form
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参考文献
[1] Lamb H. On the vibration of a spherical shell[J]. Proceedings of the London Mathematical Society, 1883, 14: 50-56.
[2] DiMaggio F L, Silbiger A. Free extensional torsional vibrations of a prolate spheroidal shell[J]. Journal of Acoustical Society of America, 1961, 33(1): 56-58.
[3] Penzes L E, Burgin G. Free vibration of thin isotropic oblate-spheroidal shells[J]. Journal of Acoustical Society of America, 1966, 39(1): 8-13.
[4] Niordson F I. Free vibrations of thin elastic spherical shells [J].International Journal of Solids and Structures, 1984, 20(7): 667-687.
[5] Al-Jumaily A M, Najim F M. An approximation to the vibrations of oblate spheroidal shells[J]. Journal of Sound and Vibration, 1997, 204(4): 561-574.
[6] Sai Ram K S, Sreedhar Babu T. Free vibration of composite spherical shell cap with and without a cutout [J]. Computers and Structures, 2002, 80(23): 1749-1756.
[7] Hosseini-Hashemi S, Fadaee M. On the free vibration of moderately thick spherical shell panel-a new exact closed- form procedure[J]. Journal of Sound and Vibration, 2011, 330(17): 4352-4367.
[8] Su Z, Jin G, Ye T. Free vibration analysis of moderately thick functionally graded open shells with general boundary conditions [J]. Composite Structures, 2014, 117: 169-186.
[9] Kang J H, Leissa A W. Three-dimensional vibrations of thick spherical shell segments with variable thickness[J]. International Journal of Solids and Structures,2000,37: 4811- 4823.
[10] Kang J H, Leissa A W. Three-dimensional vibration analysis of solids and hollow hemispheres having varying thicknesses with and without axial conical holes[J]. Journal of Vibration and Control, 2004, 10: 199-214.
[11] Ascher U, Christiansen J, Russell R D. Collocation software for boundary value ODEs[J]. ACM Transactions on Mathematical Software, 1981, 7(2): 209-222.
[12] Ascher U, Christiansen J, Russell R D. Algorithm 569, COLSYS: collocation software for boundary value ODEs [D2] [J]. ACM Transactions on Mathematical Software, 1981, 7(2): 223-229.
[13] Williams F W, Wittrick W H. An automatic computational procedure for calculating natural frequencies of skeletal structures[J]. International Journal of Mechanical Sciences, 1970, 12(9): 781-791.
[14] Wittrick W H, Williams F W. A general algorithm for computing natural frequencies of elastic structures[J]. Quarterly Journal of Mechanics and Applied Mathematics, 1971, 24(3): 263-284.
[15] Yuan S, Ye K, Xiao C, et al. Exact dynamic stiffness method for non-uniform Timoshenko beam vibrations and Berboulli-Euler column buckling[J]. Journal of Sound and Vibration, 2007, 303(3/4/5): 526-537.
[16] 叶康生,赵雪健. 动力刚度法求解平面曲梁面外自由振动问题[J]. 工程力学,2012,29:1-8.
YE Kang-sheng, ZHAO Xue-jian. Dynamic stiffness method for out-of-plane free vibration analysis of planar curved beams [J]. Engineering Mechanics, 2012, 29: 1-8.
[17] Su H, Banerjee J R, Cheung C W. Dynamic stiffness formulation and free vibration analysis of functionally graded beams [J]. Composite Structures, 2013, 106: 854-862.
[18] El-Kaabazi N, Kennedy D. Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the Wittrick-Williams algorithm[J]. Computers and Structures, 2012, 104/105: 4-12.
[19] Gautham B P, Ganesan N. Free vibration analysis of thick spherical shells[J]. Computers and Structures,1992,45(2): 307-313.
[20] Shim H J, Kang J H. Free vibration of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory [J]. International Journal of Engineering Science, 2004, 42: 1793-1815.
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