Winkler地基上有限长梁联合共振分析
|
摘要
为进一步完善弹性地基上梁的非线性振动理论以及满足实际工程的需要,本文对三种联合共振情况进行了研究。文中基于已建立的Winkler地基上有限长梁的非线性振动方程,运用Galerkin离散法以及多尺度法推导其稳态运动方程组,分析了主/次联合共振时,结构参数及初始条件对系统幅频响应曲线、调谐-相位曲线的影响,初步探讨了主/超联合共振、次/超联合共振的基本性质。研究结果表明:当且仅当两个激励频率是可公度关系时,联合共振才存在稳态响应;对于一个给定的调谐参数,系统最多存在七个解,而解的最终形式是由初始条件决定;系统参数及初始条件的微小变动,可能会激烈地改变系统的响应;工程中为了较好地抑制结构的振动,必须选择合适的方法提高系统的刚度。
In order to improve the nonlinear vibration theory for the finite-length beams on the elastic foundation and satisfy the requirements of practical engineering,this paper aims to study the primary,sub-harmonic and super-harmonic excitations occure simultaneously.Based on the derived nonlinear vibration equations of finite-length beams on the Winkler foundation,the method of Galerkin and multiple scales are adopted to formulate the equations of steady-state motions.We consider a case of double resonance in which primary and sub-harmonic resonances exist simultaneously and analyze the effect of the system parameters and the initial conditions on the amplitude-frequency response curves and the phase-tuning curves.Numerical results are presented that the steady state response is periodic,when two resonances occur simutaneously and the excitation frequencies are commensurable.There are as many as seven solutions for a given tuning parameter,in case of more than one stable solution,whose final forms will be determined by their initial conditions.It is also found that small changes in the system parameters or the initial conditions might drastically change the response of the system.To control the resonances of the beam in engineering,an appropriate way is demanded to enhance the stiffness of the system.
In order to improve the nonlinear vibration theory for the finite-length beams on the elastic foundation and satisfy the requirements of practical engineering,this paper aims to study the primary,sub-harmonic and super-harmonic excitations occure simultaneously.Based on the derived nonlinear vibration equations of finite-length beams on the Winkler foundation,the method of Galerkin and multiple scales are adopted to formulate the equations of steady-state motions.We consider a case of double resonance in which primary and sub-harmonic resonances exist simultaneously and analyze the effect of the system parameters and the initial conditions on the amplitude-frequency response curves and the phase-tuning curves.Numerical results are presented that the steady state response is periodic,when two resonances occur simutaneously and the excitation frequencies are commensurable.There are as many as seven solutions for a given tuning parameter,in case of more than one stable solution,whose final forms will be determined by their initial conditions.It is also found that small changes in the system parameters or the initial conditions might drastically change the response of the system.To control the resonances of the beam in engineering,an appropriate way is demanded to enhance the stiffness of the system.
引文
[1]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)
[2]Lee S Y,Kou Y H,Lin F Y.Stability of a Timoshenko beam resting on a Winkler elastic foundation.[J].Journal of Sound and Vibration,1992,153(2):193-202.
[3]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].Journalof Earthquake Engineering and Engineering Vibration,2006,26(3):91-93.(in Chinese)
[4]赵跃宇,马建军,刘齐建,等.简谐荷载作用下Winkler地基上有限长梁1/3次亚谐波共振分析[J].地震工程与工程振动,2011,31(3):148-153.ZHAO Yueyu,MA Jianjun,LIU Qijian,et al.1/3 sub-harmonic resonance of finite-length beams on the Winkler foundation subjected to harmoniclateral loads[J].Journal of Earthquake Engineering and Engineering Vibration,2011,31(3):148-153.(in Chinese)
[5]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.GU Yan,JIANG Xinliang.Research on superharmonic resonance and subharmonic resonance of soil-nonlinear structure interaction system[J].Journal of Earthquake Engineering and Engineering Vibration,2001,21(1):56-60.(in Chinese)
[6]冯志华,胡海岩.直线运动柔性梁非线性动力学—主参数共振与内共振联合激励[J].振动工程学报,2004,17(2):126-131.FENG Zhihua,HU Haiyan.Nonlinear dynamics of flexible beams undergoing a large linear motion of basement:principal parametric and internalresonances[J].Journal of Vibration Engineering,2004,17(2):126-131.(in Chinese)
[7]Nayfeh A H,Mook D T.Nonlinear Oscillations[M].New York:Wiley,1995:183-192.
[8]Mojaddidy Z.Nonlinear vibrations of structural elements subject to multi-frequency excitations[D].Virginia Polytechnic Institute and State Uni-versity,1977.
[9]Plaut R H,Gentry J J,Mook D T.Resonances in nonlinear structural vibrations involving two external periodic excitations[J].Journal of Soundand Vibration,1990,140(3):371-379.
[10]HaQuang N,Mook D T,Plaut R H.Non-linear structural vibrations under combined parametric and external excitations[J].Journal of Sound andVibration,1987,118(2):291-306.
[11]Yang S,Nayfeh A H,Mook D T.Combination resonances in the response of the duffing oscillator to a three-frequency excitation[J].Acta Me-chanica,1998,131:235-245.
[12]El-Bassiouny A F,Abd El-Latif G M.Resonances in nonlinear structure vibrations under multi-frequency excitations[J].Phys scr,2006,74:410-421.
[13]Nayfeh A H.Quenching of primary resonance by a super-harmonic resonance[J].Journal of Sound and Vibration,1984,92(3):363-377.
[14]Timoshenko S,Young D H,Weaver W.Vibration problems in engineering[M].New York:Wiley,1974.
[2]Lee S Y,Kou Y H,Lin F Y.Stability of a Timoshenko beam resting on a Winkler elastic foundation.[J].Journal of Sound and Vibration,1992,153(2):193-202.
[3]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].Journalof Earthquake Engineering and Engineering Vibration,2006,26(3):91-93.(in Chinese)
[4]赵跃宇,马建军,刘齐建,等.简谐荷载作用下Winkler地基上有限长梁1/3次亚谐波共振分析[J].地震工程与工程振动,2011,31(3):148-153.ZHAO Yueyu,MA Jianjun,LIU Qijian,et al.1/3 sub-harmonic resonance of finite-length beams on the Winkler foundation subjected to harmoniclateral loads[J].Journal of Earthquake Engineering and Engineering Vibration,2011,31(3):148-153.(in Chinese)
[5]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.GU Yan,JIANG Xinliang.Research on superharmonic resonance and subharmonic resonance of soil-nonlinear structure interaction system[J].Journal of Earthquake Engineering and Engineering Vibration,2001,21(1):56-60.(in Chinese)
[6]冯志华,胡海岩.直线运动柔性梁非线性动力学—主参数共振与内共振联合激励[J].振动工程学报,2004,17(2):126-131.FENG Zhihua,HU Haiyan.Nonlinear dynamics of flexible beams undergoing a large linear motion of basement:principal parametric and internalresonances[J].Journal of Vibration Engineering,2004,17(2):126-131.(in Chinese)
[7]Nayfeh A H,Mook D T.Nonlinear Oscillations[M].New York:Wiley,1995:183-192.
[8]Mojaddidy Z.Nonlinear vibrations of structural elements subject to multi-frequency excitations[D].Virginia Polytechnic Institute and State Uni-versity,1977.
[9]Plaut R H,Gentry J J,Mook D T.Resonances in nonlinear structural vibrations involving two external periodic excitations[J].Journal of Soundand Vibration,1990,140(3):371-379.
[10]HaQuang N,Mook D T,Plaut R H.Non-linear structural vibrations under combined parametric and external excitations[J].Journal of Sound andVibration,1987,118(2):291-306.
[11]Yang S,Nayfeh A H,Mook D T.Combination resonances in the response of the duffing oscillator to a three-frequency excitation[J].Acta Me-chanica,1998,131:235-245.
[12]El-Bassiouny A F,Abd El-Latif G M.Resonances in nonlinear structure vibrations under multi-frequency excitations[J].Phys scr,2006,74:410-421.
[13]Nayfeh A H.Quenching of primary resonance by a super-harmonic resonance[J].Journal of Sound and Vibration,1984,92(3):363-377.
[14]Timoshenko S,Young D H,Weaver W.Vibration problems in engineering[M].New York:Wiley,1974.
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心