参数振动系统亚谐共振理论的新结果:退化分叉解  

Some New Results about the Subharmonic Resonance of Nonlinear Parametrically Excited Systems Degenerate Bifurcation

作　　者：陈予恕[1] 詹凯君 Langford W.F. 

机构地区：[1]天津大学,Guelph大学,加拿大

出　　处：《振动工程学报》1990年第2期38-47,共10页Journal of Vibration Engineering

基　　金：国家自然科学基金;博士点基金

摘　　要：本文应用Liapunov-schmidt方法和奇异性理论,首次研究了参数激励非线性振动系统亚谐共振情况下的退化分叉解,这里的退化是指在分叉方程(10)中a_1=0时的情况,这种情况工程振动系统中有时会遇到.文中给出了用Liapunov-schmidt方法计算分叉方程系数的简有效的方法,使计算工作量大大减少.奇异性理论的应用,证明了系统在a_1=0的情况下为余三,这时系统呈现出十分复杂、有趣的动力学现象,如图2所示,从而使我们对该类非线性振动统动力学行为有了更深的了解.一般认为,参数振动系统只有当ε>δ时才可能有周期解,但本证明了由于Strutt图顶点的后移,在ε<δ时也可能存在周期解.数值计算表明,本文的理论果有足够的精度.The subharmonic degenerate bifurcation and its mechanics behaviour of a class of parametrically excited systems with Z_2-symmetry are studied first time in detail by use of Liapunov-Schmidt Method and singularity theory. Here the degenerate case is that a_1 = 0 in equation (10). This case we can meet in engineering vibration systems. The effective method for calculating coefficients of the bifarcation equation by Liapunov-Schmidt method is given. By use of the singularity theory we proved that the codimension of the degenerate system is three. The degenerate system presents some very complicated and interesting dynamical phenomena, as shown in Fig.2.usually, we consider that there is a periodic solution in nonlinear parametrically excited systems only when ε>δ.But we proved that there is a periodic solution whenε<δ. Numerical calculation shows that the result of theoretical analysis is accurate enough.

关 键 词：参数振动 亚谐共振 退化分叉 

分 类 号：O323[理学—一般力学与力学基础]