简谐横荷载作用下Winkler地基上有限长梁1/3次亚谐共振分析
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摘要
基于Winkler地基模型及Euler-Bernoulli梁理论,建立了Winkler地基上有限长梁的非线性运动方程。运用Galerkin方法对运动方程进行一阶模态截断,得到离散的非线性振动方程,利用多尺度法求得该系统1/3次亚谐共振的一阶近似解。分析了长细比、弹性模量、地基刚度、阻尼、密度等参数对其亚谐共振幅频响应的影响,并通过与非共振硬激励情况对比分析1/3次亚谐共振对系统实际动力反应的影响。研究结果表明:1/3次亚谐共振区域对外激励幅值敏感;阻尼或地基刚度大于一定值后,系统将不出现1/3次亚谐共振响应;1/3次亚谐共振显著增大系统稳态动力响应位移。
The nonlinear vibration of finite-length beams on the Winkler foundation subjected to harmonic lateral loads is investigated in the present paper.Based on the Winkler foundation model and the Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on elastic foundation is derived.The approximate solution of the finite-length beam for the case of the 1/3 sub-harmonic resonance is obtained by using the Galerkin method and the multi-scale method.To illustrate the characteristics of the 1/3 sub-harmonic resonance,the effects of the slenderness ratio,modulus of elasticity,stiffness coefficient of foundation,damping coefficient and beam density on the frequency-response curves of the finite-length beam on the Winkler foundation are studied.The effect of the 1/3 sub-harmonic resonance on the actual dynamic response of this system is analyzed with the contrast to the non-resonant situation.The numerical results show that the excitation amplitude has significant impact on the 1/3 sub-harmonic resonance region;the resonance of the finite-length beam on elastic foundation will not appear when the damping coefficient or the stiffness coefficient of foundation is greater than a certain value;the 1/3 sub-harmonic resonance significantly increases the displacement of the stationary response.
The nonlinear vibration of finite-length beams on the Winkler foundation subjected to harmonic lateral loads is investigated in the present paper.Based on the Winkler foundation model and the Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on elastic foundation is derived.The approximate solution of the finite-length beam for the case of the 1/3 sub-harmonic resonance is obtained by using the Galerkin method and the multi-scale method.To illustrate the characteristics of the 1/3 sub-harmonic resonance,the effects of the slenderness ratio,modulus of elasticity,stiffness coefficient of foundation,damping coefficient and beam density on the frequency-response curves of the finite-length beam on the Winkler foundation are studied.The effect of the 1/3 sub-harmonic resonance on the actual dynamic response of this system is analyzed with the contrast to the non-resonant situation.The numerical results show that the excitation amplitude has significant impact on the 1/3 sub-harmonic resonance region;the resonance of the finite-length beam on elastic foundation will not appear when the damping coefficient or the stiffness coefficient of foundation is greater than a certain value;the 1/3 sub-harmonic resonance significantly increases the displacement of the stationary response.
引文
[1]A P S塞尔瓦杜雷.土与基础相互作用的弹性分析[M].范文田等译.北京:中国铁道出版社,1984:7-23.
[2]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981:2-14.
[3]Timoshenko S.Vibration problems in engineering,2nd Edition[M].New York:Wolfenden Press,2008:285-299.
[4]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.
[5]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.
[6]彭震,杨志安.Winkler地基梁在温度场中受简谐激励的1/3次亚谐共振分析[J].地震工程与工程振动,2006,26(4):132-135.
[7]侯兴民,马小燕,吴汉生,等.动力机器基础振动与设计若干问题的讨论[J].地震工程与工程振动,2008,28(3):131-137.
[8]谷岩,姜忻良.地基——非线性结构相互作用体系的超谐和亚谐振动[J].地震工程与工程振动,2001,21(1):56-60.
[9]Nayfeh A H,Mook D T.Nonlinear Oscillations[M].New York:Wiley,1979:174-175.
[10]Mcdonald P H.Nonlinear motion of a beam[J].Nonlinear Dynamics,1991,2:187-198.
[11]Nayfeh A H,Pai P Frank.Linear and nonlinear structural mechanics[M].Hoboken:John Wiley&Sons,Inc,2004:215-224.
[12]Xie W C,Lee HP,Lim S P.Normal modes of a non-linear clamped-clamped beam[J].Journal of Sound and Vibration,2002,250(2):339-349.
[13]Katsikadelis J T,Tsiatas G C.Non-linear dynamic analysis of beams with variable stiffness[J].Journal of Sound and Vibration,2004,270:847-863.
[14]Ibrahim R A,Somnay R J.Nonlinear dynamic analysis of an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibra-tion,2007,308:735-757.
[15]Wojciech S,Klosowicz M,Nadolsld W.Nonlinear vibrations of a simply supported viscoelastic iner-tensible beamand comparison of four methods[J].Acta Mechanic,1990,85(1):43-54.
[16]Shaw S W,Pierre C.Normal modes of vibration for non-linear continuous systems[J].Journal of Sound and Vibration,1994,163(3):319-347.
[2]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981:2-14.
[3]Timoshenko S.Vibration problems in engineering,2nd Edition[M].New York:Wolfenden Press,2008:285-299.
[4]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.
[5]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.
[6]彭震,杨志安.Winkler地基梁在温度场中受简谐激励的1/3次亚谐共振分析[J].地震工程与工程振动,2006,26(4):132-135.
[7]侯兴民,马小燕,吴汉生,等.动力机器基础振动与设计若干问题的讨论[J].地震工程与工程振动,2008,28(3):131-137.
[8]谷岩,姜忻良.地基——非线性结构相互作用体系的超谐和亚谐振动[J].地震工程与工程振动,2001,21(1):56-60.
[9]Nayfeh A H,Mook D T.Nonlinear Oscillations[M].New York:Wiley,1979:174-175.
[10]Mcdonald P H.Nonlinear motion of a beam[J].Nonlinear Dynamics,1991,2:187-198.
[11]Nayfeh A H,Pai P Frank.Linear and nonlinear structural mechanics[M].Hoboken:John Wiley&Sons,Inc,2004:215-224.
[12]Xie W C,Lee HP,Lim S P.Normal modes of a non-linear clamped-clamped beam[J].Journal of Sound and Vibration,2002,250(2):339-349.
[13]Katsikadelis J T,Tsiatas G C.Non-linear dynamic analysis of beams with variable stiffness[J].Journal of Sound and Vibration,2004,270:847-863.
[14]Ibrahim R A,Somnay R J.Nonlinear dynamic analysis of an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibra-tion,2007,308:735-757.
[15]Wojciech S,Klosowicz M,Nadolsld W.Nonlinear vibrations of a simply supported viscoelastic iner-tensible beamand comparison of four methods[J].Acta Mechanic,1990,85(1):43-54.
[16]Shaw S W,Pierre C.Normal modes of vibration for non-linear continuous systems[J].Journal of Sound and Vibration,1994,163(3):319-347.
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