Winkler地基上有限长梁非线性自由振动
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摘要
基于经典Winkler地基模型及Euler-Bernoulli梁理论,考虑梁的几何非线性效应,运用Newton第二定律建立了弹性地基上有限长梁的非线性运动方程。采用Galerkin方法对运动方程进行一阶模态截断,进而利用多尺度法求得了该系统自由振动的一阶近似解。揭示了两端简支梁的非线性自由振动特性,分析了弹性模量、长细比及地基刚度系数等参数对系统固有频率的影响。并通过该系统的位移时程曲线,分析了阻尼对弹性地基上梁运动特性的影响。
The non-linear free vibration of a finite-length beam on the elastic foundation is investigated.Based on the Winkler foundation model and Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on an elastic foundation with geometric nonlinearity is deduced based on the Newton’s Second Law.The first-order mode truncation of the vibration function is obtained using the Galerkin method.The approximate solution of the free vibration of the finite-length beam is derived utilizing the multi-scale method to illustrate the behaviour of the non-linear free vibration.The effects of the slenderness ratio of beam,the modulus of elastic system and the stiffness of foundation on the natural frequency of the hinged-hinged beam on the Winkler foundation are analyzed.The influence of damping of the soil-beam system on the motion of the beam is also discussed.
The non-linear free vibration of a finite-length beam on the elastic foundation is investigated.Based on the Winkler foundation model and Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on an elastic foundation with geometric nonlinearity is deduced based on the Newton’s Second Law.The first-order mode truncation of the vibration function is obtained using the Galerkin method.The approximate solution of the free vibration of the finite-length beam is derived utilizing the multi-scale method to illustrate the behaviour of the non-linear free vibration.The effects of the slenderness ratio of beam,the modulus of elastic system and the stiffness of foundation on the natural frequency of the hinged-hinged beam on the Winkler foundation are analyzed.The influence of damping of the soil-beam system on the motion of the beam is also discussed.
引文
[1]Winkler E.Theory of elasticity and strength[M].Prague,Czechoslovakia:H.Domincus,1867.
[2]龙驭球.弹性地基梁的技计算[M].北京:人民教育出版社,1981:2―14.Long Yuqiu.Calculation of elastic foundation beam[M].Beijing:People’s Education Press,1981:2―14.(inChinese)
[3]Hetényi M.Beams on elastic foundation[M].Ann Arbor:University of Michigan Press,1946:50―63.
[4]Timoshenko S P,Gere J.Theory of elastic stability[M].New York:McGraw-Hill,1961:23―33.
[5]黎绍敏.稳定理论[M].北京:人民交通出版社,1989:12―15.Li Shaomin.Theory of stability[M].Beijing:ChinaCommunications Press,1989:12―15.(in Chinese)
[6]Timoshenko S.Vibration problems in engineering[M].2nd ed.New York:Wolfenden Press,2008:285―299.
[7]刘学山,冯紫良,胥兵.粘弹性地基上弹性梁的自由振动分析[J].上海力学,1999,20(4):470―476.Liu Xueshan,Feng Ziliang,Xu Bing.Free vibrationanalysis for elastic beam on viscoelastic foundation[J].Chinese Quarterly Mechanics,1999,20(4):470―476.(in Chinese)
[8]楼梦麟,沈霞.弹性地基梁振动特性的近似分析方法[J].应用力学学报,2004,21(3):149―153.Lou Menglin,Shen Xia.An approach for analyzingdynamic characteristic of reinforced concrete beam onelastic foundation[J].Chinese Journal of AppliedMechanics,2004,21(3):149―153.(in Chinese)
[9]彭震,杨志安.Winkler地基梁在温度场中受简谐激励的主共振分析[J].地震工程与工程振动,2006,26(3):91―93.Peng Zhen,Yang Zhian.Analysis of primary resonanceof a beam externally excited on the Winkler foundationin temperature field[J].Earthquake Engineering andEngineering Vibration,2006,26(3):91―93.(in Chinese)
[10]Ibrahim R A,Somnay R J.Nonlinear dynamic analysisof an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibration,2007,308:735―757.
[11]Nayfeh A H,Pai P Frank.Linear and nonlinear structuralmechanics[M].Hoboken:John Wiley&Sons,Inc,2004:215―224.
[12]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:83―95.Liu Yanzhu,Chen Liqun.Nonlinear vibrations[M].Beijing:Higher Education Press,2001:83―95.(inChinese)
[13]Shaw S W,Pierre C.Normal modes of vibration fornon-linear continuous systems[J].Journal of Sound andVibration,1994,163(3):319―347.
[14]Timoshenko S,Young D H,Weaver W.Vibrationproblems in engineering[M].4th ed.Wiley:New York,1974:309―312.
[15]Thambiratnam D,Zhuge Y.Free vibration analysis ofbeams on elastic foundation[J].Computers andStructures,1996,60:971―980.
[16]Friswell M I,Adhikari S,Lei Y.Vibration analysis ofbeams with non-local foundations using the finiteelement method[J].International Journal of NumericalMethods in Engineering,2007,71(11):1365―1386.
[2]龙驭球.弹性地基梁的技计算[M].北京:人民教育出版社,1981:2―14.Long Yuqiu.Calculation of elastic foundation beam[M].Beijing:People’s Education Press,1981:2―14.(inChinese)
[3]Hetényi M.Beams on elastic foundation[M].Ann Arbor:University of Michigan Press,1946:50―63.
[4]Timoshenko S P,Gere J.Theory of elastic stability[M].New York:McGraw-Hill,1961:23―33.
[5]黎绍敏.稳定理论[M].北京:人民交通出版社,1989:12―15.Li Shaomin.Theory of stability[M].Beijing:ChinaCommunications Press,1989:12―15.(in Chinese)
[6]Timoshenko S.Vibration problems in engineering[M].2nd ed.New York:Wolfenden Press,2008:285―299.
[7]刘学山,冯紫良,胥兵.粘弹性地基上弹性梁的自由振动分析[J].上海力学,1999,20(4):470―476.Liu Xueshan,Feng Ziliang,Xu Bing.Free vibrationanalysis for elastic beam on viscoelastic foundation[J].Chinese Quarterly Mechanics,1999,20(4):470―476.(in Chinese)
[8]楼梦麟,沈霞.弹性地基梁振动特性的近似分析方法[J].应用力学学报,2004,21(3):149―153.Lou Menglin,Shen Xia.An approach for analyzingdynamic characteristic of reinforced concrete beam onelastic foundation[J].Chinese Journal of AppliedMechanics,2004,21(3):149―153.(in Chinese)
[9]彭震,杨志安.Winkler地基梁在温度场中受简谐激励的主共振分析[J].地震工程与工程振动,2006,26(3):91―93.Peng Zhen,Yang Zhian.Analysis of primary resonanceof a beam externally excited on the Winkler foundationin temperature field[J].Earthquake Engineering andEngineering Vibration,2006,26(3):91―93.(in Chinese)
[10]Ibrahim R A,Somnay R J.Nonlinear dynamic analysisof an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibration,2007,308:735―757.
[11]Nayfeh A H,Pai P Frank.Linear and nonlinear structuralmechanics[M].Hoboken:John Wiley&Sons,Inc,2004:215―224.
[12]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:83―95.Liu Yanzhu,Chen Liqun.Nonlinear vibrations[M].Beijing:Higher Education Press,2001:83―95.(inChinese)
[13]Shaw S W,Pierre C.Normal modes of vibration fornon-linear continuous systems[J].Journal of Sound andVibration,1994,163(3):319―347.
[14]Timoshenko S,Young D H,Weaver W.Vibrationproblems in engineering[M].4th ed.Wiley:New York,1974:309―312.
[15]Thambiratnam D,Zhuge Y.Free vibration analysis ofbeams on elastic foundation[J].Computers andStructures,1996,60:971―980.
[16]Friswell M I,Adhikari S,Lei Y.Vibration analysis ofbeams with non-local foundations using the finiteelement method[J].International Journal of NumericalMethods in Engineering,2007,71(11):1365―1386.
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