简谐横向荷载作用下弹性地基上不可伸长梁的1/2次亚谐共振响应分析
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摘要
基于已建立的弹性地基上不可伸长梁的非线性动力学模型,本文针对简谐横向荷载作用下梁的1/2次亚谐共振响应进行研究。利用弹性地基梁的非线性运动方程和多尺度方法,求得梁1/2次亚谐共振响应的二次近似解。进而,运用幅频响应曲线对梁的亚谐共振响应进行研究,并分析了边界条件、地基模型、Winkler参数等对梁非线性共振响应的影响。结果表明:三参数地基模型中弹性层的引入对其支承梁的亚谐共振响应影响显著;临界激励幅值决定了弹性地基梁1/2次亚谐共振响应非平凡解的存在性;梁端约束条件在一定程上改变了Winkler参数对梁非线性动力响应的作用效应。
In this paper,the 1 /2 sub-harmonic resonance of the beam subjected to a harmonic lateral excitation is investigated. Based on the nonlinear dynamics model of an inextensional beam on elastic foundation,the frequencyresponse equation and the second-order approximate solution of the 1 /2 sub-harmonic resonance are obtained,employed the nonlinear motion equation of the beam and the method of multiple scales. Then,by means of the frequency-response curves,the effects of boundary conditions,foundation models and Winkler parameter on the 1 /2sub-harmonic resonances of the beam are analyzed. The results show that the effect of the spring layer included in Kerr model on the sub-harmonic resonance of the beam is significant. Obviously,the existence of the nontrivial solutions of the 1 /2 sub-harmonic resonance depended on the critical excitation amplitude. Moreover,the boundary condition changed the influence of Winkler parameter on the nonlinear response of the beam,in a sense.
In this paper,the 1 /2 sub-harmonic resonance of the beam subjected to a harmonic lateral excitation is investigated. Based on the nonlinear dynamics model of an inextensional beam on elastic foundation,the frequencyresponse equation and the second-order approximate solution of the 1 /2 sub-harmonic resonance are obtained,employed the nonlinear motion equation of the beam and the method of multiple scales. Then,by means of the frequency-response curves,the effects of boundary conditions,foundation models and Winkler parameter on the 1 /2sub-harmonic resonances of the beam are analyzed. The results show that the effect of the spring layer included in Kerr model on the sub-harmonic resonance of the beam is significant. Obviously,the existence of the nontrivial solutions of the 1 /2 sub-harmonic resonance depended on the critical excitation amplitude. Moreover,the boundary condition changed the influence of Winkler parameter on the nonlinear response of the beam,in a sense.
引文
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[3]Wang Y H,Tham L G,Cheung Y K.Beams and plates on elastic foundations:A review[J].Progress in Structural Engineering and Materials,2005,7:174-182.
[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.LOU Menglin,SHEN Xia.An approach for analyzing dynamic characteristic of reinforced concrete beam on elastic foundation[J].Chinese Journal of Applied Mechanics,2004,21(3):149-153.(in Chinese)
[5]Thambiratnam D,Zhuge Y.Free vibration analysis of beams on elastic foundation[J].Computers&Structures,1996,60(6):971-980.
[6]Pellicano F,Mastroddi F.Nonlinear dynamics of a beam on elastic foundation[J].Nonlinear Dynamics,1997,14(4):335-355.
[7]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 resonance of a beam externally excited on the Winkler foundation in temperature field[J].Earthquake Engineering and Engineering Dynamics,2006,26(3):91-93.(in Chinese)
[8]Lai Y C,Ting B Y,Lee W S,et al.Dynamic response of beams on elastic foundation[J].Journal of Structural Engineering-ASCE,1992,118(3):853-858.
[9]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.PENG Zhen,YANG Zhian.Analysis of 1/3 subharmonic resonance of a beam externally excited on Winkler foundation in temperature field[J].Earthquake Engineering and Engineering Dynamics,2006,26(4):132-135.(in Chinese)
[10]Zhu B,Leung A Y T.Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elememts[J].Computers and Geotechnics,2009,36(5):743-750.
[11]Wang L,Ma J,Zhao Y,et al.Refined modeling and free vibration of inextensional beams on the elastic foundation[J].Journal Applied Mechanics-ASME,2013,80:041026.
[12]马建军,王连华,赵跃宇.弹性地基有限长梁的动力学建模[J].中国科学:物理学力学天文学,2013,43:765-771.MA Jianjun,WANG Lianhua,ZHAO Yueyu.Dynamic modeling of the finite-length beam on the elastic foundation[J].Science China-Physics,Mechanics&Astronomy,2013,43:765-771.(in Chinese)
[13]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 vibration[M].Beijing:Higher Education Press,2001:83-95.(in Chinese)
[14]De Rosa M A,Maurizi M J.The influence of concentrated mass and Pasternak soil on the free vibration of Euler beams-exact solution[J].Journal of Sound and Vibration,1998,212(4):573-581.
[15]Ayvaz Y,Ozgan K.Application of modified Vlasov model to free vibration analysis of beams resting on elastic foundations[J].Journal of Sound and Vibration,2002,255(1):111-127.
[16]Nayfeh AH,Mook DT.Nonlinear Oscillations[M].New York:Wiley,1979:190-217.
[17]Nayfeh A H,Pai P F.Linear and Nonlinear Structural Mechanics[M].New York:Wiley-Interscience,2004:215-244.