Winkler地基上有限长梁的组合共振响应分析
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摘要
基于Winkler地基模型及Euler-Bernoulli梁理论,建立了两个横向激励共同作用下Winkler地基上有限长梁的组合共振模型,运用Galerkin方法和多尺度方法求得组合共振幅频响应方程,进而研究地基刚度、弹性模量、黏滞阻尼、外激励幅值等主要参数对其幅频响应曲线的影响。研究结果表明:增大地基刚度、弹性模量、黏滞阻尼或减小外激励幅值均可减小梁的响应幅值。黏滞阻尼的变化对影响的共振区域并无影响。
In this paper,based on Winkler foundation model and Euler-Bernoulli beam theory,the combination resonance model of the finite-length beam on Winkler foundation under two excitation forces is built.Then,amplitude-frequency response equation is derived by using Galerkin method and multi-scale method.Finally,various amplitude-frequency response curves are given to show the effect of kinds of parameters on response amplitude,such as stiffness coefficient of foundation,modulus of elasticity,damping coefficient,excitation amplitude and so on.The numerical results show that the amplitude of vibration decreases as the strength of stiffness coefficient of foundation,modulus of elasticity or damping coefficient increases.Nevertheless,decreasing the excitation amplitude will decrease the vibration amplitude of the beam.In addition,damping coefficient has little effect on the resonance field.
In this paper,based on Winkler foundation model and Euler-Bernoulli beam theory,the combination resonance model of the finite-length beam on Winkler foundation under two excitation forces is built.Then,amplitude-frequency response equation is derived by using Galerkin method and multi-scale method.Finally,various amplitude-frequency response curves are given to show the effect of kinds of parameters on response amplitude,such as stiffness coefficient of foundation,modulus of elasticity,damping coefficient,excitation amplitude and so on.The numerical results show that the amplitude of vibration decreases as the strength of stiffness coefficient of foundation,modulus of elasticity or damping coefficient increases.Nevertheless,decreasing the excitation amplitude will decrease the vibration amplitude of the beam.In addition,damping coefficient has little effect on the resonance field.
引文
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[3]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981.LONG Yuqiu.Calculation of elastic foundation beam[M].Beijing:People’s Education Press,1981.(in Chinese)
[4]Timoshenko S.Vibration problems in engineering,2nd Edition[M].New York:Wolfenden Press,2008:285-299.
[5]刘学山,冯紫良,胥兵.黏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 vibration analysis for elastic beam on viscous elastic foundation[J].Shanghai Journal of Mechanics,1999,20(4):470-476.(in Chinese)
[6]王赞芝,江林燕,辛立凤,等.与黏弹性地基相互作用的梁的自由振动分析[J].工业建筑,2009,39(10):57-61.WANG Zanzhi,JIANG Linyan,XIN Lifeng,et al.Free vibration analysis of the beam interaction with the viscoelastic foundation[J].IndustrialConstruction,2009,39(10):57-61.(in Chinese)
[7]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 approximate method of analyzing dynamic characteristic of beams on the Winkler foundation[J].Chinese Journal ofApplied Mechanics,2004,21(3):149-153.(in Chinese)
[8]彭震,杨志安.Winkler地基梁在温度场中受简谐荷载激励的1/3次亚谐共振进行了分析[J].地震工程与工程振动,2008,28(3):131-137.PENG Zhen,YANG Zhian.Analysis of 1/3 sub harmonic resonance of a beam externally excited on Winkler foundation in temperature field[J].Journal of Earthquake Engineering and Engineering Vibration[J].2008,28(3):131-137.(in Chinese)
[9]赵跃宇,马建军,刘齐建.简谐荷载作用下Winkler地基梁上有限长梁的1/3次亚谐共振进行了分析[J].地震工程与工程振动,2011,31:148-153.ZHAO Yueyu,MA Jianjun,LIU Qijian.1/3 sub-harmonic resonance of finite-length beams on the Winkler foundation subjected to harmonic later-al loads[J].Journal of Earthquake Engineering and Engineering Vibration,2011,31:148-153.(in Chinese)
[10]赵跃宇,赵珧冰,马建军,等.Winkler地基上有限长梁联合共振分析[J].地震工程与工程振动,2012,32(2):160-166.ZHAO Yueyu,ZHAO Yaobing,MA Jianjun,et al.Simultaneous resonance analysis of finite-length beams on Winkler foundation[J].Journal ofEarthquake Engineering and Engineering Vibration,2012,32(2):160-166.(in Chinese)
[11]王连华,马建军,刘齐建.横向荷载作用下弹性地基上梁的主共振分析[J].力学与实践,2012,34(1):57-61.WANG Lianhua,MA Jianjun,LIU Qijian.Primary resonance of beams on elastic foundations subjected to lateral loads[J].Mechanics in Engi-neering,2012,34(1):57-61.(in Chinese)
[2]Mook D T,Plaut R H,HaQuang N.The influence of an internal resonance on non-linear structural vibrations under subharmonic resonanceconditions[J].Journal of Sound and Vibration,1985,102:473-492.
[3]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981.LONG Yuqiu.Calculation of elastic foundation beam[M].Beijing:People’s Education Press,1981.(in Chinese)
[4]Timoshenko S.Vibration problems in engineering,2nd Edition[M].New York:Wolfenden Press,2008:285-299.
[5]刘学山,冯紫良,胥兵.黏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 vibration analysis for elastic beam on viscous elastic foundation[J].Shanghai Journal of Mechanics,1999,20(4):470-476.(in Chinese)
[6]王赞芝,江林燕,辛立凤,等.与黏弹性地基相互作用的梁的自由振动分析[J].工业建筑,2009,39(10):57-61.WANG Zanzhi,JIANG Linyan,XIN Lifeng,et al.Free vibration analysis of the beam interaction with the viscoelastic foundation[J].IndustrialConstruction,2009,39(10):57-61.(in Chinese)
[7]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 approximate method of analyzing dynamic characteristic of beams on the Winkler foundation[J].Chinese Journal ofApplied Mechanics,2004,21(3):149-153.(in Chinese)
[8]彭震,杨志安.Winkler地基梁在温度场中受简谐荷载激励的1/3次亚谐共振进行了分析[J].地震工程与工程振动,2008,28(3):131-137.PENG Zhen,YANG Zhian.Analysis of 1/3 sub harmonic resonance of a beam externally excited on Winkler foundation in temperature field[J].Journal of Earthquake Engineering and Engineering Vibration[J].2008,28(3):131-137.(in Chinese)
[9]赵跃宇,马建军,刘齐建.简谐荷载作用下Winkler地基梁上有限长梁的1/3次亚谐共振进行了分析[J].地震工程与工程振动,2011,31:148-153.ZHAO Yueyu,MA Jianjun,LIU Qijian.1/3 sub-harmonic resonance of finite-length beams on the Winkler foundation subjected to harmonic later-al loads[J].Journal of Earthquake Engineering and Engineering Vibration,2011,31:148-153.(in Chinese)
[10]赵跃宇,赵珧冰,马建军,等.Winkler地基上有限长梁联合共振分析[J].地震工程与工程振动,2012,32(2):160-166.ZHAO Yueyu,ZHAO Yaobing,MA Jianjun,et al.Simultaneous resonance analysis of finite-length beams on Winkler foundation[J].Journal ofEarthquake Engineering and Engineering Vibration,2012,32(2):160-166.(in Chinese)
[11]王连华,马建军,刘齐建.横向荷载作用下弹性地基上梁的主共振分析[J].力学与实践,2012,34(1):57-61.WANG Lianhua,MA Jianjun,LIU Qijian.Primary resonance of beams on elastic foundations subjected to lateral loads[J].Mechanics in Engi-neering,2012,34(1):57-61.(in Chinese)
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