Bifurcation Analysis of a Double Pendulum With Internal Resonance
-
摘要: 应用normal form理论,首先分析了复摆自治系统在1:1内共振临界点附近的Hopf分岔解及其在参数平面上的分岔转迁集的解析表达式,并与数值解进行了比较;然后,应用数值方法,得到了复摆非自治系统通向混沌的过程。Abstract: By employing the normal form theory, the Hopf bifurcation and the transition boundary of an autonomous double pendulum with 1:1 internal resonance at the critical point is studied. The results are compared with numerical solutions. Further, by numerical methods, the road to chaos of a non-autonomous system is presented in the end.
-
Key words:
- internal resonance /
- Hopf bifurcation /
- transition boundary /
- chaos
-
[1] Huseyin K.Multiple-Parameter Stability Theory and Its Applications[M].Oxford:Oxford University Press,1986. [2] Yu P, Huseyin K.A perturbation analysis of interactive static and dynamic bifurcations[J].IEEE Transactions of Automatic Control,1988,33:28~41. [3] Guckenheimer J, Holmes P.Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[M].New York:Springer-Verlag,1983. [4] 毕勤胜,陈予恕.Duffing系统解的转迁集的解析表达式[J].力学学报,1997,8(12):573~581. [5] Yu P, Huseyin K.Static and dynamic bifurcations associated with a double zero eigenvalues[J].Dynamics and Stability of Systems,1986,1:73~86. [6] 陈予恕.非线性振动系统的分叉和混沌理论[M].北京:高等教育出版社,1993. [7] 毕勤胜,陈予恕.周期激励浅拱的内共振分岔研究[J].力学学报,1998,30(1):83~88. [8] Mandadi V, Huseyin K.Nonlinear bifurcation analysis of non-gradient systems[J].International Journal of Non-Linear Mechanics,1980,15(2):159~172. [9] Huseyin K, Yu P.On bifurcations into nonresonant quasi-periodic motions[J].Applied Mathematical Modelling,1988,12:189~201. [10] Scheidl R, Troger H, Zaman K.Coupled flutter and divergence bifurcation of a double pendulum[J].International Journal of Non-Linear Mechanics,1984,19(2):163~176. [11] Samaranayake S, Bajaj A K.Bifurcation in the dynamics of an orthogonal double pendulum[J].Nonlinear Dynamics,1993,4:605~633. [12] Pei Yu, Bi Qinsheng.Analysis of nonlinear dynamics and bifurcations of a double pendulum[J].Journal of Sound and Vibration,1998,217(4):691~736. [13] Bi Qinsheng and Pei Yu, Symbolic computation of normal forms for semi-simple cases[J].Journal of Computational and Applied Mathematics,1999,102(2):195~220.
点击查看大图
计量
- 文章访问数: 2453
- HTML全文浏览量: 58
- PDF下载量: 827
- 被引次数: 0