Abstract：A planar rigid rod-spring pendulum model was constructed and dimensionless dynamic equation was given. We numerically simulated and analyzed the dynamical behavior of the two-time-scale system while the frequency ratio and length ratio and initial conditions vary. The dynamic equation is strongly nonlinear as the fast and slow variables couple each other. A cubic interpolation precise integration method was applied to solve nonlinear dynamic equation. We also employ the Poincaré maps and the maximum Lyapunov exponent methods. Numerical simulation results demonstrate that the system presents complex chaotic motion in different parameters conditions. It is also found that the fast variable may transform to chaos via the quasi-periodic torus breakdown.
赵聪, 于洪洁. 一种刚性杆-弹簧摆模型的混沌动力学行为研究[J]. 复杂系统与复杂性科学, 2016, 13(3): 97-102.
ZHAO Cong, YU Hongjie. On the Chaotic Dynamic Behaviour of the Rigid Rod-Spring Pendulum Model[J]. Complex Systems and Complexity Science, 2016, 13(3): 97-102.
[1] Stachowiak T, Okada T. A numerical analysis of chaos in the double pendulum[J].Chaos, Solitons & Fractals, 2006, 29(2): 417-422. [2] Nikitina N V. Ultimate energy of a double pendulum undergoing quasiperiodic oscillations[J].International Applied Mechanics, 2007, 43(9): 1035-1042. [3] 司丽荣, 张竞夫. 弹簧摆内共振现象的实验研究[J].物理实验, 2002, 22(3): 9-12. Si Lirong, Zhang Jingfu. Study on the experiment of the autoparametric resonance of a spring fendulum[J].Physics Experimentation, 2002, 22(3): 9-12. [4] Lee W K, Park H D. Chaotic dynamics of a harmonically excited springpendulum system with internal resonance[J].Nonlinear Dynamics, 1997, 14(3): 211-229. [5] Zheng J L, Yu X W. Analysis of the autoparametric resonance of a spring pendulum[J].Physics and Engineering, 2010, 20(2): 13-16. [6] Van der Weele J P, De Kleine E. The Order-chaos-order sequence in the spring pendulum[J].Physica A: Statistical Mechanics and its Applications, 1996, 228(1-4): 245-272. [7] Li Y S, Shu X F. Internal resonance and chaotic motion of a spring pendulum[J].Journal of Taiyuan University of Technology, 1998, 29(6): 555-559. [8] 于洪洁, 张靖姝, 洪嘉振. 刚柔耦合弹簧摆的复杂动力学行为[J].科技导报, 2013, 31(18): 32-38. Yu Hongjie, Zhang Jingshu, Hong Jiazhen. Complex dynamical behavior of rigid-flexible coupling spring pendulum[J].Science and Technology Review, 2013, 31(18): 32-38. [9] 钟万勰. 结构动力方程的精细时程积分法[J].大连理工大学学报, 1994, 34(2): 131-136. Zhong Wanxie. On precise time-integration method for structural dynamics[J].Journal of Dalian University of Technology, 1994, 34(2): 131-136. [10] Zhong W X, Williams F W. A precise time step integration method[J].Journal of Mechanical Engineering Science, 1994, 208: 427-430. [11] 张洵安, 杨人风. 结构混沌运动的非线性精细积分计算方法[J].西安公路交通大学学报, 2000, 20(2): 99-101. Zhang Xunan, Yang Renfeng. The method of nonlinear precise integration for chaotic motion of structure[J].Journal of Xi’an Highway University, 2000, 20(2): 99-101. [12] Wang M F, Zhou X Y. Renewal frecise time step integration method of structural dynamic analysis[J].Acta Mechanica Sinica, 2004, 36(2): 191-195. [13] Gu Y X, Chen B S, Zhang H W. Precise time-integration with dimension expanding method[J].Acta Mechanica Sinica, 2000, 32(4): 447-456. [14] Zhang S Y, Deng Z C. Increment-dimensional precise integration method for nonlinear dynamic equation[J].Chinese Journal of Computational Mechanics, 2003, 20(4): 423-426. [15] Ge G, Wang H L, Tan G. Improved increment-dimensional precise integration method for the nonlinear dynamic equation with multi-degree-of-freedom[J].Journal of Tianjin University, 2009, 42(2): 114-117. [16] 吕和祥, 于洪洁, 裘春航. 精细积分的非线性动力学积分方程及其解法[J].固体力学学报, 2001, 22(3): 303-308. Lv Hexiang, Yu Hongjie, Qiu Chunhang. A integral equation of non-linear dynamics and Its solution method[J].Acta Mechanica Solida Sinaca, 2001, 22(3): 303-308.
