摘要
针对横向简谐激励下弹性地基上不可伸长梁的1/3次亚谐共振响应进行研究.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,利用多尺度方法得到梁的1/3次亚谐共振幅频响应方程,进而作出幅频响应曲线,分析了弹性地基模型、Winkler参数、边界条件等对弹性地基梁非线性动力学特性的影响.结果表明:边界条件导致弹性地基剪切参数对1/3次亚谐共振响应的影响效应发生定性改变;在通常情况下,Winkler参数对弹性地基梁1/3次亚谐共振响应的特性起控制性作用,其增大将抑制幅频响应方程的非平凡解;若梁端约束为铰支-自由,应注意外激励幅值对系统动力学特性的影响.
In this study,the 1/3 sub-harmonic resonance of an inextensional beam on elastic foundation is investigated.Based on the nonlinear dynamics model of an inextensional beam on elastic foundation,the frequencyresponse equation of the 1/3sub-harmonic resonance of the beam is obtained via the method of multiple scales.To illustrate the characteristics of 1/3sub-harmonic resonance,by means of the frequency-response curves,the effects of the foundation models,boundary conditions and Winkler parameter on the dynamic response of the beam on elastic foundation are studied.The numerical results show that the boundary condition changed the influence of shear parameter on the 1/3sub-harmonic resonance response of the beam on elastic foundation,in qualitative level.Under normal circumstances,Winkler parameter dominated the 1/3sub-harmonic resonance response of the beam.The increase of Winkler parameter inhibited the amplitude of the non-trivial solution of the frequency-response response equation.When the constraint of beam is hinged-free,it is necessary to pay attention to the effect of the excitation amplitude on the dynamic nature of the beam on elastic foundation.
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 YQ.Calculation of elastic foundation beam[M].Beijing:People’s eduction press,1981.
[2]NAYFEH A H,NAYFEH S A.Nonlinear normal modes of a continuous system with quadratic nonlinearities[J].Journal of Vibration and Acoustics,1995,117(2):199-205.
[3]WANG Y H,THAM L G,CHEUNG Y K.Beams and plates on elastic foundations:A review[J].Progress in Structural Engineering Materials,2005,7:174-182.
[4]HETNYI M.Beams on Elastic Foundation[M].Ann Arbor:University of Michigan Press,1946.
[5]THAMBIRATNAM D,ZHUGE Y.Free vibration analysis of beams on elastic foundation[J].Computers and Structures,1996,60(6):971-980.
[6]孔德志,张庆贺.考虑土与结构共同作用的挡土结构内力和变形[J].河南大学学报(自然科学版),2003,33(3):75-78.KONG D Z,ZHANG Q H.Internal force and deformation of resisting structure with interaction between soil and structure[J].Journal of Henan University(Natural Science),2003,33(3):75-78.
[7]MORFIDIS K.Vibration of Timoshenko beams on three-parameter elastic foundation[J].Computers and Structures,2010,88(5-6):294-308.
[8]杨志安,赵雪娟,席晓燕.非线性弹性地基上矩形薄板的非线性振动与奇异性分析[J].振动与冲击,2006,25(5):69-73.YANG Z A,ZHAO X J,XI X Y.Nonlinear vibration and singularities analysis of a thin rectangular plate on nonlinear elastic foundation[J].Journal of Vibration and Shock,2006,25(5):69-73.
[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 Z,YANG Z A.Analysis of 1/3subharmonic resonance of a beam externally excited on Winkler foundation in temperature field[J].Earthquake Engineering and Engineering Vibration,2006,26(4):132-135.
[10]丁敏,李潇,马倩,等.基于Winkler弹性地基梁模型的半刚性轻钢柱脚底板受力分析[J].工程力学,2014,31(5):158-165.DING M,LI X,MA Q,et al.Mechanical analysis of the sole plate of semi-rigid light steel column footings based on winkler model of elastic foundation beams[J].Engineering Mechanics,2014,31(5):158-165.
[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 L H,MA J J,LIU Q J.Primary resonance of beams on elastic foundations subjected to lateral loads[J].Mechanics in Engineering,2012,34(1):57-61.
[12]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.
[13]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 J J,WANG L H,ZHAO Y Y.Dynamic modeling of the finite-length beam on the elastic foundation[J].Scientia Sinica:Physica,Mechanica&Astronomica,2013,43:765-771.
[14]NAYFEH A H,LACARBONARA W.On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities[J].Nonlinear Dynamics,1997,13(3):203-220.
[15]NAYFEH A H,MOOK D T.Nonlinear Oscillations[M].New York:Wiley,1979.
[16]NAYFEH A H,PAI P F.Linear and nonlinear structural mechanics[M].Hoboken:John Wiley&Sons,Inc,2004.
