变截面Timoshenko简支梁动力特性的半解析解
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摘要
基于直接模态摄动法基本原理,建立一种求解变截面Timoshenko梁动力特性的近似分析方法。该方法在等效均匀Euler-Bernoulli梁的模态子空间内实施,将变截面Timoshenko梁特征方程的微分方程组转化为代数方程组求解。对于等截面Timoshenko简支梁情况,该方法可以得到精确的特征值和特征向量。对于变截面情况,数值分析结果表明这一方法简单适用,且具有良好的精度。
Based on modal perturbation method, a novel approach is presented to analyze the free vibration of non-prismatic Timoshenko beams.In the method, the dynamic characteristics of non-prismatic Timoshenko beams are calculated approximately in the modal subspace spanned by several lower models of equivalent prismatic Euler-Bernoulli beams. As a result, the set of characteristic equations of non- prismatic Timoshenko beams is transformed into a set of algebraic equations. For prismatic Timoshenko simply supported beams, the exact eigenvalues and eigenfunctions can be obtained by using the method. For non- prismatic ones, the results of numerical examples show that the method is simple, practicable and accurate.
Based on modal perturbation method, a novel approach is presented to analyze the free vibration of non-prismatic Timoshenko beams.In the method, the dynamic characteristics of non-prismatic Timoshenko beams are calculated approximately in the modal subspace spanned by several lower models of equivalent prismatic Euler-Bernoulli beams. As a result, the set of characteristic equations of non- prismatic Timoshenko beams is transformed into a set of algebraic equations. For prismatic Timoshenko simply supported beams, the exact eigenvalues and eigenfunctions can be obtained by using the method. For non- prismatic ones, the results of numerical examples show that the method is simple, practicable and accurate.
引文
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[4]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 Menglin,Shi Shuzhong.An approach for solving the eigenvalue problem of Timoshenko clamped beam[J].Chinese Journal of Applied Mechanics,2003,20(1):140-143.(in Chinese)
[5]段秋华,楼梦麟.Timoshenko悬臂梁自由振动特性的近似分析方法[J].结构工程师,2004,21(5):20-23.Duan Qiuhua,Lou Menglin.An approach for analyzing free vibration characteristics of Timoshenko cantilever beams[J].Structural Engineers,2004,21(5):20-23.(in Chinese)
[6]Gutierrez R H,Laura P A A,Rossi R E.Fundamental frequency of vibration of a Timoshenko beam of non-uniform thickness[J].Journal of Sound and Vibration,1991,145:341-344.
[7]Leung A Y T,Zhou W E.Dynamic stiffness analysis of axially loaded non-uniform Timoshenko columns[J].Computers&Structures,1995,56:577-588.
[8]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.
[9]Auciello N M,Ercolano A.A general solution for dynamic response of axially loaded non-uniform Timoshenko beams[J].International Journal of Solids and Structures,2004,41:4861-4874.
[10]Eisenberger M.Derivation of shape functions for an exact4-D.O.F.Timoshenko beam element[J].Communications in Numerical Methods in Engineering,1994,10:673-681.
[11]Eisenberger M.Dynamic stiffness matrix for variable cross-section Timoshenko beams[J].Communications in Numerical Methods in Engineering,1995,11:507-513.
[12]Rossi R E,Laura P A A,Gutierrez R H.A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other[J].Journal of Sound and Vibration,1990,143:491-502.
[13]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 Menglin.Dynamic analysis for modal characteristics and seismic response of soil layer with variable properties[J].Journal of Tongji University,1997,25(2):155-160.(in Chinese)
[14]楼梦麟,吴京宁.复杂梁动力问题的近似分析方法[J].上海力学,1997,18(3):234-240.Lou Menglin,Wu Jingning.An approach to solve dynamic problems of complicated beams[J].Shanghai Journal of Mechanics,1997,18(3):234-240.(in Chinese)
[15]潘旦光,董聪.局部损伤梁动力问题的近似计算方法[J].应用力学学报,2005,22(1):119-122.Pan Danguang,Dong Cong.An approach to solve dynamic problems of cracked beams[J].Chinese Journal of Applied Mechanics,2005,22(1):119-122.(in Chinese)
[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 Menglin.Perturbation method for the linear generalized eigenvalue problem in modal subspace[J].Journal of Tongji University,1994,22(3):268-273.(in Chinese)
[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]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 Menglin,Ren Zhigang.Precise solution to modal characteristics of Timoshenko pin-ended beams[J].Journal of Tongji University,2002,30(8):911―915.(in Chinese)
[4]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 Menglin,Shi Shuzhong.An approach for solving the eigenvalue problem of Timoshenko clamped beam[J].Chinese Journal of Applied Mechanics,2003,20(1):140-143.(in Chinese)
[5]段秋华,楼梦麟.Timoshenko悬臂梁自由振动特性的近似分析方法[J].结构工程师,2004,21(5):20-23.Duan Qiuhua,Lou Menglin.An approach for analyzing free vibration characteristics of Timoshenko cantilever beams[J].Structural Engineers,2004,21(5):20-23.(in Chinese)
[6]Gutierrez R H,Laura P A A,Rossi R E.Fundamental frequency of vibration of a Timoshenko beam of non-uniform thickness[J].Journal of Sound and Vibration,1991,145:341-344.
[7]Leung A Y T,Zhou W E.Dynamic stiffness analysis of axially loaded non-uniform Timoshenko columns[J].Computers&Structures,1995,56:577-588.
[8]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.
[9]Auciello N M,Ercolano A.A general solution for dynamic response of axially loaded non-uniform Timoshenko beams[J].International Journal of Solids and Structures,2004,41:4861-4874.
[10]Eisenberger M.Derivation of shape functions for an exact4-D.O.F.Timoshenko beam element[J].Communications in Numerical Methods in Engineering,1994,10:673-681.
[11]Eisenberger M.Dynamic stiffness matrix for variable cross-section Timoshenko beams[J].Communications in Numerical Methods in Engineering,1995,11:507-513.
[12]Rossi R E,Laura P A A,Gutierrez R H.A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other[J].Journal of Sound and Vibration,1990,143:491-502.
[13]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 Menglin.Dynamic analysis for modal characteristics and seismic response of soil layer with variable properties[J].Journal of Tongji University,1997,25(2):155-160.(in Chinese)
[14]楼梦麟,吴京宁.复杂梁动力问题的近似分析方法[J].上海力学,1997,18(3):234-240.Lou Menglin,Wu Jingning.An approach to solve dynamic problems of complicated beams[J].Shanghai Journal of Mechanics,1997,18(3):234-240.(in Chinese)
[15]潘旦光,董聪.局部损伤梁动力问题的近似计算方法[J].应用力学学报,2005,22(1):119-122.Pan Danguang,Dong Cong.An approach to solve dynamic problems of cracked beams[J].Chinese Journal of Applied Mechanics,2005,22(1):119-122.(in Chinese)
[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 Menglin.Perturbation method for the linear generalized eigenvalue problem in modal subspace[J].Journal of Tongji University,1994,22(3):268-273.(in Chinese)
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