两自由度非对称三次系统非线性模态的奇异性质

徐鉴; 陆启韶; 黄克累

两自由度非对称三次系统非线性模态的奇异性质

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

基金项目: 国家自然科学基金资助项目(10072039);国家自然科学基金重大项目资助课题(19990510)

详细信息

作者简介:
徐鉴(1961),男,浙江余姚人,教授,博士.

中图分类号: O322
计量
- 文章访问数: 2352
- HTML全文浏览量: 148
- PDF下载量: 588
- 被引次数: 0
出版历程
- 收稿日期: 1999-09-27
- 修回日期: 2001-03-20
- 刊出日期: 2001-08-15

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

1.
Department of Engineering Mechanics and Technology, Tongji University, Shanghai 200092, P R China;
2.
Department of Applied Mathematics, Peking University of Aeronautics and Astronautics, Beijing 100083, P R China

摘要

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.
- 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
Abstract: 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 11 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.
- asymmetric system /
- nonlinear normal mode /
- localization of vibration /
- bifurcation of normal mode /
- nonlinear dynamics

HTML全文

参考文献(16)

[1]	Guchenheimer J,Holmes P.Nonlinear Oscillation,Dynam ical System and Bifurcation of VectorFields[M].New York:Springer-Verlag,1983.
[2]	陆启韶.分岔和奇异性[M].上海:上海科学技术出版社,1995.
[3]	Nayfeh A H,Mook D T.Nonlinear Oscillation s[M].New York:John Wiley & Sons Inc,1979.
[4]	Winggins S.In str oduction to Applied Nonlinear Dynamical Systems and Chaos[M].New York:Springer-Verlag,1990.
[5]	Chen Y S,Langford W F.The subharmonic bifurcation solution of nonlinear Mathieus equation and Euler dynamically buckling problem[J].Acta Mech Sinica,1998,19(3):522-532.
[6]	Iooss G,Joseph D D.Elementary Stability and Bifurcation Theory[M].New York:Springer-Ver-lag,1980.
[7]	Shaw S W,Pierre C.Normal modes for nonlinear vibration systems[J].Journal of Sound and Vibration,1993,164(1):85-124.
[8]	徐鉴,陆启韶,黄克累.两自由度非对称三次系统非奇异时的非线性模态及叠加性[J].应用数学和力学,1998,19(12):1077-1086.
[9]	Anderson P W.Absence of diffusion in certain random lattices[J].Phsical Review,1958,109(12):1492-1505.
[10]	Hodges C H.Confinement of vibration by structural chains[J].Journal of Sound and Vibration,1982,82(2):411-424.
[11]	Pierre C,Dowell E H.Localization of vibration by structural irregularity[J].Journal of Sound and Vibration,1987,114(3):549-564.
[12]	Pierre C.Model localization and eigenvalue lociveering phenomena in disordered structures[J].Journal of Sound and Vibration,1988,126(3):485-502.
[13]	Pierre C.Weak and strong vibration localization in disordered structures:a statistical investigation[J].Journal of Sound and Vibration,1990,139(1):111-132.
[14]	陈予恕.非线性振动[M].天津:天津科学技术出版社,1983.
[15]	Golubisky M,Stewart I,Schaeffer D G.Singularities and Groups in Bifurcation Theory[M].Voland Vol.New York:Springer-Verlag,1988.
[16]	Caughey T K,Vakakis A,Sivo J M.Analytical study of similar normal modes and their bifurcationsin class of strongly non-linear systems[J].Int J Non-Linear Mechanics,1990,25(5):521-533.