横向荷载作用下Winkler地基上有限长梁3次超谐共振分析
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摘要
基于Winkler地基模型和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,得到了离散的非线性振动方程,然后利用多尺度法求得了该系统3次超谐共振的幅频响应方程及其位移的一阶近似解。为揭示弹性地基上有限长梁的3次超谐共振响应的特性,分别分析了长细比、弹性模量、基床系数、阻尼、密度等主要参数对该系统3次超谐共振幅频响应曲线的影响,并通过与非共振硬激励情况的对比分析了3次超谐共振对系统实际动力反应的影响。研究结果表明:3次超谐共振响应曲线有跳跃和滞后现象;增大阻尼和基床系数均对3次超谐共振的发生有抑制作用;增大外激励幅值,系统3次超谐共振区域增大;3次超谐共振将增大系统的稳态动力响应幅值和加速度。
The third super-harmonic resonance of finite-length beams on the Winkler foundation is investigated in this paper.Based on the Winkler foundation model and the Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on an elastic foundation is obtained.The approximate solution of the finite-length beam for the case of the third super-harmonic resonance is obtained by using the Galerkin method and the multi-scale method.To illustrate the characteristic of the third super-harmonic resonance,the effects of the slenderness ratio,modulus of elasticity,stiffness coefficient of foundation,damp coefficient and beam density on the frequency-response curves of the finite-length beam on the Winkler foundation are studied.The effect of the third super-harmonic resonance on the actual dynamic response of this system is analyzed by contrasting with the non-resonant situation.The numerical results show that the system's third super-harmonic resonance including delay and jumping phenomena;the increase of damping or stiffness coefficient of foundation may suppress the third super-harmonic resonance;the third super-harmonic resonance may enlarge the displacement and acceleration of the stationary response.
The third super-harmonic resonance of finite-length beams on the Winkler foundation is investigated in this paper.Based on the Winkler foundation model and the Euler-Bernoulli beam theory,the nonlinear motion equation of the finite-length beam on an elastic foundation is obtained.The approximate solution of the finite-length beam for the case of the third super-harmonic resonance is obtained by using the Galerkin method and the multi-scale method.To illustrate the characteristic of the third super-harmonic resonance,the effects of the slenderness ratio,modulus of elasticity,stiffness coefficient of foundation,damp coefficient and beam density on the frequency-response curves of the finite-length beam on the Winkler foundation are studied.The effect of the third super-harmonic resonance on the actual dynamic response of this system is analyzed by contrasting with the non-resonant situation.The numerical results show that the system's third super-harmonic resonance including delay and jumping phenomena;the increase of damping or stiffness coefficient of foundation may suppress the third super-harmonic resonance;the third super-harmonic resonance may enlarge the displacement and acceleration of the stationary response.
引文
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[4]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.
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[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]Xie W C,Lee H P,Lim S P.Normal modes of a non-linear clamped-clamped beam[J].Journal of Sound and Vibration,2002,250(2):339-349.
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[9]Ibrahim R A,Somnay R J.Nonlinear dynamic analysis of an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibration,2007,308:735-757.
[10]马建军,刘齐建,王连华,等.Winkler地基上有限长梁非线性自由振动[J].工程力学,2011(已录用).
[11]Nayfeh A H,Mook D T.Nonlinear oscillations[M].New York:Wiley,1979.
[12]Nayfeh A H,Pai P F.Linear and nonlinear structural mechanics[M].Hoboken:John Wiley&Sons Inc,2004.
[2]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981.
[3]Timoshenko S.Vibration problems in engineering[M].New York:Wolfenden Press,2008.
[4]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.
[5]Mcdonald P H.Nonlinear motion of a beam[J].Nonlinear Dynamics,1991(2):187-198.
[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]Xie W C,Lee H P,Lim S P.Normal modes of a non-linear clamped-clamped beam[J].Journal of Sound and Vibration,2002,250(2):339-349.
[8]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.
[9]Ibrahim R A,Somnay R J.Nonlinear dynamic analysis of an elastic beam isolator sliding on frictional supports[J].Journal of Sound and Vibration,2007,308:735-757.
[10]马建军,刘齐建,王连华,等.Winkler地基上有限长梁非线性自由振动[J].工程力学,2011(已录用).
[11]Nayfeh A H,Mook D T.Nonlinear oscillations[M].New York:Wiley,1979.
[12]Nayfeh A H,Pai P F.Linear and nonlinear structural mechanics[M].Hoboken:John Wiley&Sons Inc,2004.
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