两自由度非对称三次系统非线性模态的奇异性质  

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric System With Cubic Nonlinearities

作　　者：徐鉴[1] 陆启韶[2] 黄克累[2] 

机构地区：[1]同济大学工程力学与技术系,上海200092 [2]北京航空航天大学应用数学系,北京100083

出　　处：《应用数学和力学》2001年第8期869-878,共10页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目 ( 10 0 72 0 39) ;国家自然科学基金重大项目资助课题 ( 19990 510 )

摘　　要：利用非线性模态子空间的不变性和摄动技术 ,研究两自由度非对称三次系统在奇异条件下系统的性质·　重点考虑子系统之间线性耦合退化时的奇异性质·　对于非共振情形 ,所得到的解析结果表明 ,系统出现单模态运动以及振动局部化现象 ,这种现象的强弱不但与非线性耦合刚度有关 ,而且与非对称参数有关·　并解析地得到了参数的门槛值 ;对于 1:1共振情形 ,模态随非线性耦合刚度和非对称参数的变化会出现分岔 ,得到了参数分岔集以及模态的分岔曲线·Nonlinear normal modes in a two degrees of freedom asymmetric system with cubic nonlinearities as singularity occurs in the system are studied, based on the invariant space in nonlinear normal modes and perturbation technique. Emphasis is placed on singular characteristics as the linear coupling between subsystems degenerates. For non_resonances, it is analytically presented that a single_mode motion and localization of vibrations occur in the system, and the degree of localization relates not only to the coupling stiffness between oscillators, but also to the asymmetric parameter. The parametric threshold value of localization is analytically given. For 1∶1 resonance, there exist bifurcations of normal modes with nonlinearly coupling stiffness and asymmetric parameter varying. The bifurcating set on the parameter and bifurcating curves of normal modes are obtained.

关 键 词：非对称系统 非线性模态 振动局部化 模态分岔 非线性动力学 两自由度 奇异性质 三次系统 非线性振动 

分 类 号：O322[理学—一般力学与力学基础]