变截面Timoshenko悬臂梁自由振动分析
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摘要
为考虑剪切变形和转动惯量的影响,基于模态摄动法基本原理,提出了一种求解变截面Timoshenko悬臂梁自由振动问题的近似解法。这一方法是利用等截面Euler梁的特征值和模态,将变截面Timoshenko梁特征方程的偏微分方程组转化为代数方程组进行求解,从而得到变截面Timoshenko梁的特征值和模态。该方法适用于求解任意复杂截面型式梁的动力特性,无论梁的截面变化是否连续。随后对截面阶跃变化和线性变化2类变截面梁进行算例分析,数值分析结果表明,这一方法简单、实用,具有良好的精度。
In order to consider the effect of shearing deformation and rotating inertia,an approximate method,which is based on modal perturbation method,is proposed for the free vibration analysis of non-uniform Timoshenko cantilever beams.With the eigenvalues and eigenvectors of uniform Euler beams,a set of partial differential characteristic equations of non-uniform Timoshenko beams is transformed into a set of algebraic equations for solutions of non-uniform Timoshenko beams.The method can solve the dynamic characteristics of beams with complicated cross-section,whether the cross-section variation is continuous or discontinuous.At the end,two types of beams,namely(a) discontinuous variation of thickness and(b) continuous and linear variation,are analyzed and shown that the method is simple,practicable and owns good precision.
In order to consider the effect of shearing deformation and rotating inertia,an approximate method,which is based on modal perturbation method,is proposed for the free vibration analysis of non-uniform Timoshenko cantilever beams.With the eigenvalues and eigenvectors of uniform Euler beams,a set of partial differential characteristic equations of non-uniform Timoshenko beams is transformed into a set of algebraic equations for solutions of non-uniform Timoshenko beams.The method can solve the dynamic characteristics of beams with complicated cross-section,whether the cross-section variation is continuous or discontinuous.At the end,two types of beams,namely(a) discontinuous variation of thickness and(b) continuous and linear variation,are analyzed and shown that the method is simple,practicable and owns good precision.
引文
[1]TIMOSHENKO S,YOUNG D H,WEAVER W.Vibration problems in engineering[M].4thedition,John Wiley&Sons,Inc.1974.
[2]VAN RENSBURG N F J,VAN DER MERWE A J.Natural frequencies and modes of a Timoshenko beam[J].Wave Motion,2006,44:58-69.
[3]ROSSI R E,LAURA P A A,GUTIERREZ R H.Anote on transverse vibrations of a Timoshenko beam ofnon-uniform thickness clamped at one end and carryinga concentrated mass at the other[J].Journal of Soundand Vibration,1990,143:491-502.
[4]EISENBERGER M.Derivation of shape functions foran exact 4-D.O.F.Timoshenko beam element[J].Communications in Numerical Methods in Engineering,1994,10:673-681.
[5]EISENBERGER M.Dynamic stiffness matrix for var-iable cross-section Timoshenko beams[J].Commun-ications in Numerical Methods in Engineering,1995,11:507-513.
[6]GUTIERREZ R H,LAURA P A A,ROSSI R E.Fundamental frequency of vibration of a Timoshenkobeam of non-uniform thickness[J].Journal of Soundand Vibration,1991,145:341-344.
[7]FARGHALY S H.Vibration and stability analysis ofTimoshenko beams with discontinuities in cross-section[J].Journal of Sound and Vibration,1994,174(5):591-605
[8]FARGHALY S H,GADELRAB R M.Free vibrationof a stepped composite Timoshenko cantilever beam[J].Journal of Sound and Vibration,1995,187(5):886-896.
[9]TONG X,TABARROK B,YEH K Y.Vibrationanalysis of Timoshenko beams with non-homogeneityand varying cross-section[J].Journal of Sound andVibration,1995,186(5):821-835.
[10]LEUNG A Y T,ZHOU W E.Dynamic stiffnessanalysis of axially loaded non-uniform Timoshenkocolumns[J].Computers&Structures,1995,56:577-588.
[11]AUCIELLO N M.Free vibration of a restrained shear-deformable tapered beam with a tip mass at its free end[J].Journal of Sound and Vibration,2000,237:542-549.
[12]AUCIELLO N M,ERCOLANO A.A general solutionfor dynamic response of axially loaded non-uniformTimoshenko beams[J].International Journal of Solidsand Structures,2004,41:4861-4874.
[13]楼梦麟,任志刚.Timoshenko简支梁的振动模态特性精确解[J].同济大学学报,2002,30(8):911-915.LOU MENG-LIN,REN ZHI-GANG.Precise solutionto modal characteristics of Timoshenko Pin-endedbeams[J].Journal of Tongji University,2002,30(8):911-915.
[14]楼梦麟,石树中.Timoshenko固端梁特征值问题近似计算方法[J].应用力学学报,2003,20(1):140-143.LOU MENG-LIN,SHI SHU-ZHONG.An Approachfor solving the eigenvalue problem of Timoshenkoclamped beam[J].Chinese Journal of AppliedMechanics,2003,20(1):140-143.
[15]段秋华,楼梦麟.Timoshenko悬臂梁自由振动特性的近似分析方法,结构工程师,2004,21(5):20-23.DUAN QIU-HUA,LOU MENG-LIN.An Approachfor analyzing free vibration characteristics ofTimoshenko cantilever beams[J].StructuralEngineers,2004,21(5):20-23.
[16]楼梦麟.线性广义特征值问题在模态子空间中的摄动解[J].同济大学学报,1994,22(3):268-273.LOU MENG-LIN.Perturbation method for the lineargeneralized eigenvalue problem in modal subspace[J].Journal of Tongji University,1994,22(3):268-273.
[17]楼梦麟.变参数土层的动力特性和地震反应分析[J].同济大学学报,1997,25(2):155-160.LOU MENG-LIN.Dynamic analysis for modalcharacteristics and seismic response of soil layer withvariable properties[J].Journal of Tongji University,1997,25(2):155-160.
[2]VAN RENSBURG N F J,VAN DER MERWE A J.Natural frequencies and modes of a Timoshenko beam[J].Wave Motion,2006,44:58-69.
[3]ROSSI R E,LAURA P A A,GUTIERREZ R H.Anote on transverse vibrations of a Timoshenko beam ofnon-uniform thickness clamped at one end and carryinga concentrated mass at the other[J].Journal of Soundand Vibration,1990,143:491-502.
[4]EISENBERGER M.Derivation of shape functions foran exact 4-D.O.F.Timoshenko beam element[J].Communications in Numerical Methods in Engineering,1994,10:673-681.
[5]EISENBERGER M.Dynamic stiffness matrix for var-iable cross-section Timoshenko beams[J].Commun-ications in Numerical Methods in Engineering,1995,11:507-513.
[6]GUTIERREZ R H,LAURA P A A,ROSSI R E.Fundamental frequency of vibration of a Timoshenkobeam of non-uniform thickness[J].Journal of Soundand Vibration,1991,145:341-344.
[7]FARGHALY S H.Vibration and stability analysis ofTimoshenko beams with discontinuities in cross-section[J].Journal of Sound and Vibration,1994,174(5):591-605
[8]FARGHALY S H,GADELRAB R M.Free vibrationof a stepped composite Timoshenko cantilever beam[J].Journal of Sound and Vibration,1995,187(5):886-896.
[9]TONG X,TABARROK B,YEH K Y.Vibrationanalysis of Timoshenko beams with non-homogeneityand varying cross-section[J].Journal of Sound andVibration,1995,186(5):821-835.
[10]LEUNG A Y T,ZHOU W E.Dynamic stiffnessanalysis of axially loaded non-uniform Timoshenkocolumns[J].Computers&Structures,1995,56:577-588.
[11]AUCIELLO N M.Free vibration of a restrained shear-deformable tapered beam with a tip mass at its free end[J].Journal of Sound and Vibration,2000,237:542-549.
[12]AUCIELLO N M,ERCOLANO A.A general solutionfor dynamic response of axially loaded non-uniformTimoshenko beams[J].International Journal of Solidsand Structures,2004,41:4861-4874.
[13]楼梦麟,任志刚.Timoshenko简支梁的振动模态特性精确解[J].同济大学学报,2002,30(8):911-915.LOU MENG-LIN,REN ZHI-GANG.Precise solutionto modal characteristics of Timoshenko Pin-endedbeams[J].Journal of Tongji University,2002,30(8):911-915.
[14]楼梦麟,石树中.Timoshenko固端梁特征值问题近似计算方法[J].应用力学学报,2003,20(1):140-143.LOU MENG-LIN,SHI SHU-ZHONG.An Approachfor solving the eigenvalue problem of Timoshenkoclamped beam[J].Chinese Journal of AppliedMechanics,2003,20(1):140-143.
[15]段秋华,楼梦麟.Timoshenko悬臂梁自由振动特性的近似分析方法,结构工程师,2004,21(5):20-23.DUAN QIU-HUA,LOU MENG-LIN.An Approachfor analyzing free vibration characteristics ofTimoshenko cantilever beams[J].StructuralEngineers,2004,21(5):20-23.
[16]楼梦麟.线性广义特征值问题在模态子空间中的摄动解[J].同济大学学报,1994,22(3):268-273.LOU MENG-LIN.Perturbation method for the lineargeneralized eigenvalue problem in modal subspace[J].Journal of Tongji University,1994,22(3):268-273.
[17]楼梦麟.变参数土层的动力特性和地震反应分析[J].同济大学学报,1997,25(2):155-160.LOU MENG-LIN.Dynamic analysis for modalcharacteristics and seismic response of soil layer withvariable properties[J].Journal of Tongji University,1997,25(2):155-160.
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