关于线性和非线性系统内在的本质联系——多自由度非线性系统的定量和定性分析  

INTRINSIC AND SIMPLE CONNECTION OF LINEAR SYSTEMS WITH NON-LINEAR ONES——QUANTITATIVE AND QUALITATIVE ANALYSIS OF LARGE SCALE MULTIPLE DOF NONLINEAR SYSTEMS

作　　者：郑兆昌[1] 

机构地区：[1]清华大学航天航空学院工程动力学研究所,北京100084

出　　处：《振动与冲击》2008年第1期4-8,共5页Journal of Vibration and Shock

摘　　要：多自由度非线性振动问题是历史性国际难题,其求解方法有数值解和渐近解析解或两者结合。基于近代有限元和子结构模态综合法的动力学建模方法,获得非线性系统动力学微分方程,其自由度几乎没有限制,对左端首次近似齐次方程进行模态分析,选取对响应有贡献的部分本征对,同样对右端激励和非线性伪力作模态变换,得到减缩后非线性系统耦合动力学微分方程。用数值方法求出系统非线性响应进行定量分析,也可获得在指定参数的变动中可能发生的主谐、超谐、亚谐和组合共振,极限环和分岔、混沌等各种非线性振动现象,但其缺点是不能作一般性定性分析。渐近解析解可进行分岔混沌现象的定性分析,但迄今只限于单、两自由度系统。若系统进入共振状态,系统响应相应急剧增加到大振幅振动,振动从微幅线性振动过渡到大幅非线性振动,因之系统运动主要由所涉及的各阶单一主模态所控制。这可称为"单模态共振理论"。当发生共振时,单模态理论可把多自由度系统变换为解耦的多个单自由度系统,因之可采用渐近解析法逐个进行分岔混沌等定性分析,这就克服了高自由度非线性系统定性分析的困难。为了剖析线性和非线性系统内在的本质联系,论证了首次近似表征线性系统特性的主模态是沟通线性振动和非线性振动之间的桥梁,揭示了高自由度线性振动和非线性振动都是以线性主模态呈现其运动规律。Nonlinear vibration of a large scale multi-DOF system is a historical difficult problem in the world.There are two methods to get its solution,the numerical method and the asymptotic analytic method or both of them combined.Based on the modern FEM with the dynamic substructure method,itd dynamic equation is established with almost no limits to DOFs of the system,in which the left side is the first approach of the nonlinear system and the right side is the excitation and coupling nonlinear pseudo forces.The modal analysis method is used to get the eigen solution to the reduced linear homogeneous dynamic differential equation.When the system is excited by the harmonic excitation at the right side of the equation,the nonlinear dynamic response can be calculated by the numerical method to give quantitative analysis.If the system is in the case of some selected parameters,its main resonances,suppr-resonances,sub-resonances,and combined ones as well as phenomena of limit cycles,bifurcation and chaos can be found in the further analysis,but the qualitative analysis can not be performed.The perturbation method is used to get the asymptotic analytic solution,it gives the qualitative analysis,but only for one or two-DOF systems.When the system enters the resonance state,the system vibration response increases to large amplitude and the vibration is turned from small amplitude linear vibration to large amplitude nonlinear vibration,the system motion is controlled by a main mode only.This phenomenon may be called 'single mode resonance theory'.When the resonance happens,the mode theory can be used to decompose the system as some one-DOF systems.Thus,the asymptotic analytic method can be applied to give the qualitative analysis for each one-DOF system to have bifurcation and chaos analysis.The intrinsic and simple connection of linear systems with nonlinear ones is established with the single main mode theory being a bridge communicating linear vibrations and non-linear ones.With amalgamation of the numerical method and the a

关 键 词：动力学建模方法 非线性振动响应数值解法 渐近奇异摄动法 主模态 分岔 

分 类 号：O322[理学—一般力学与力学基础]