基于波传播法的椭圆柱壳自由振动特性研究
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摘要
基于Flügge壳体理论推导出椭圆柱壳的自由振动方程。由于椭圆柱壳周向曲率的变化,造成振动方程在周向模态阶数域内不解耦,采用波传播法将壳体位移以双Fourier级数形式展开,椭圆截面曲率半径以单Fourier级数形式展开,通过级数变换,将变系数的偏微分方程组转换为关于周向模态阶数相互耦合的有限阶常系数线性方程组,通过求解耦合振动方程得到椭圆柱壳的固有频率。随后对影响壳体固有频率的主要参数进行分析,得到椭圆柱壳对称和反对称模态频率随椭圆度、壳长比等参数的变化规律。
The free vibration equations of an elliptic cylindrical shell were derived based on the Flügge shell theory.Vibration equations about the circumferential wave number are not decoupled due to the varied circumferential curvature.The shell's displacements were expanded in double Fourier series in the wave propagation method and the circumferential curvature was expanded in single Fourier series.The partial differential equations with variable coefficients were converted into a set of linear equations which couple with each other about circumferential wave numbers.The natural frequencies of the elliptic cylindrical shell were obtained by solving the coupled equations.The influences of main parameters of the elliptic cylindrical shell,such as ellipticity parameter and shell length ratio,on the vibration characteristics were discussed.The symmetric and anti-symmetric modes of the elliptic cylindrical shell were both considered.
The free vibration equations of an elliptic cylindrical shell were derived based on the Flügge shell theory.Vibration equations about the circumferential wave number are not decoupled due to the varied circumferential curvature.The shell's displacements were expanded in double Fourier series in the wave propagation method and the circumferential curvature was expanded in single Fourier series.The partial differential equations with variable coefficients were converted into a set of linear equations which couple with each other about circumferential wave numbers.The natural frequencies of the elliptic cylindrical shell were obtained by solving the coupled equations.The influences of main parameters of the elliptic cylindrical shell,such as ellipticity parameter and shell length ratio,on the vibration characteristics were discussed.The symmetric and anti-symmetric modes of the elliptic cylindrical shell were both considered.
引文
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[13]NAMAZI M M,AGHANAJAFI C.The analyze of vibration of composite elliptical shell[J].Australian Journal of Basic and Applied Sciences,2010(7):1542-1554.
[14]FLGGE W.Stress in shells[M].2nd ed.Berlin and New York:Spring-Verlag,1973.
[15]MARGUERRE K.Stability of the cylindrical shells of variable curvature[R].NASA TM 1302,1951.
[16]ROMANO F,KEMPNER J.Stresses in short noncircular cylindrical shells under lateral pressure[J].Journal of Applied Mechanics,1962,29(4):669-674.
[17]ZHANG X M,LIU G R,LAM K Y.Vibration analysis of thin cylindrical shells using wave propagation approach[J].Journal of Sound and Vibration,2001,239(3):397-403.
[2]张盛,金翔,周桦.加肋圆柱壳制造误差对声学性能的影响研究[J].中国舰船研究,2011,6(4):43-50.ZHANG Sheng,JIN Xiang,ZHOU Hua.Influence of construction error on sound radiation for ring-stiffened cylindrical shell[J].Chinese Journal of Ship Research,2011,6(4):43-50.
[3]KLOSNER J M,POHLE F V.Natural frequencies of an infinitely long noncircular cylindrical shell[R].PIBAL Rept,476,1958.
[4]KLOSNER J M.Frequencies of an infinitely long noncircular cylindrical shell[R].PIBAL Rept.552,1959.
[5]CULBERSON L D,BOYD D E.Free vibrations of freely supported oval cylinders[J].AIAA Journal,1971,9(8):1474-1480.
[6]CARL E K,BOYD D E.Free vibrations of noncircular cylindrical shell segments[J].AIAA Journal,1971,9(2):239-244.
[7]ELSBERND G F,LEISSA A W.The vibrations of noncircular cylindrical shells with initial stresses[J].Journal of Sound and Vibration,1973,29(3):309-329.
[8]朱建雄,曹志远,李国豪.非圆柱壳在各种边界条件下的自由振动分析[J].力学学报.1992,24(2):171-179.ZHU Jianxiong,CAO Zhiyuan,LI Guohao.Free vibration of noncircular cylindrical shells with arbitrary boundary conditions[J].Journal of Theoretical and Applied Mechanics,1992,24(2):171-179.
[9]KHALIFA M.Effects of non-uniform Winkler foundation and non-homogeneity on the free vibration of an orthotropic elliptical cylindrical shell[J].European Journal of Mechanics-A/Solids,2015,49:570-581.
[10]AHMED,KHALIFA M.Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness[J].Mathematical Methods in the Applied Sciences,2011,34(14):1789-1800.
[11]曹雷,马运义,黄玉盈.环肋加强变厚度圆柱壳的自由振动[J].华中科技大学学报(城市科学版),2007,24(2):63-66.CAO Lei,MA Yunyi,HUANG Yuying.Free vibration of ring-stiffened circular cylindrical shell with variable thickness[J].Journal of Huazhong University of Science and Technology(Urban Science Edition),2007,24(2):63-66.
[12]TORNABENE F,FANTUZZI N,BACCIOCCHI M,et al.Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method[J].Thin-Walled Structures,2015,97:114-129.
[13]NAMAZI M M,AGHANAJAFI C.The analyze of vibration of composite elliptical shell[J].Australian Journal of Basic and Applied Sciences,2010(7):1542-1554.
[14]FLGGE W.Stress in shells[M].2nd ed.Berlin and New York:Spring-Verlag,1973.
[15]MARGUERRE K.Stability of the cylindrical shells of variable curvature[R].NASA TM 1302,1951.
[16]ROMANO F,KEMPNER J.Stresses in short noncircular cylindrical shells under lateral pressure[J].Journal of Applied Mechanics,1962,29(4):669-674.
[17]ZHANG X M,LIU G R,LAM K Y.Vibration analysis of thin cylindrical shells using wave propagation approach[J].Journal of Sound and Vibration,2001,239(3):397-403.