基础水平运动的弹簧摆的周期解与稳定性

张利娟; 张华彪; 李欣业

doi:10.7498/aps.67.20181676

摘要

针对基础水平运动的弹簧摆的非线性动力学响应进行研究，利用拉格朗日方程建立了系统的动力学方程.将离散傅里叶变换、谐波平衡法以及同伦延拓方法相结合，对系统的周期响应进行求解，避免了传统方法计算中使用泰勒展开引起的小振幅的限制，与数值计算结果的对比表明该求解方法具有较高的精确度.利用Floquet理论分析了周期响应的稳定性，给出了基础运动振幅和频率对系统周期响应的影响.研究发现：对应某些基础频率和振幅，系统的周期响应可能发生Hopf分岔；利用数值计算研究了Hopf分岔后系统响应随基础频率和振幅的变化，发现系统出现了倍周期运动、拟周期运动和混沌等复杂的动力学行为.研究表明系统进入混沌的主要路径是拟周期环面破裂和阵发性.

关键词:

Abstract

In this paper, the nonlinear dynamic response of spring pendulum with horizontal base motion is studied. The dynamical equations of the system are established by using Lagrange equation. The discrete Fourier transform, harmonic balance method and homotopy continuation method are combined to solve the periodic response of the system, which avoids the limitation of the small amplitude caused by the Taylor expansion in the traditional analytical method. The comparison with the numerical results shows that the proposed method in this paper can not only be used to solve the large amplitude vibration of spring pendulum, but also has a high accuracy. The stability of periodic response is studied by using Floquet theory. The effects of amplitude and frequency of base motion on the periodic response of the system are given, and the bifurcation characteristics of the periodic solution are analyzed. The results show that the influence curve of the base frequency on the periodic response has two peaks, and with the increase of the amplitude of the base motion, the two peaks will shift to the different sides respectively. When the base amplitude is large, the periodic response amplitude changes with the frequency of the foundation motion, and there will be two jumps. The amplitude of the periodic solution increases with the base amplitude. For some base frequencies, the amplitude of the periodic solution will jump with the change of the base amplitude. When the amplitude and frequency of the system are large, the periodic response of the system may be unstable. After the instability, the spring pendulum enters the continuous rotation state, and the amplitude in the breathing direction is great, the system will be destroyed. It is found that Hopf bifurcation may occur in the periodic response of the system corresponding to some base frequencies and amplitudes. The variation of the system response with the base frequency and amplitude after the Hopf bifurcation is studied numerically by the Runge-Kutta method. Complex dynamical behaviors such as periodic motion, almost periodic motion, torus doubling and chaos are found. It is shown that the main path of the system entering chaos is almost periodic torus rupture and paroxysmal. Finally, the influence analysis of the base frequency and amplitude is synthesized, and the transition of the response form on the plane of the basic motion parameters is given. The results of this paper provide a theoretical reference for the analysis and design of spring pendulum in engineering.

Keywords:

作者及机构信息

1. 河北工业大学机械工程学院, 天津 300130;

2. 天津商业大学机械工程学院, 天津 300134

Authors and contacts

1. Institute of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, China;

2. Institute of Mechanical Engineering, Tianjin University of Commerce, Tianjin 300134, China

参考文献

[1]	Nayfeh A H, Mook D T 1979 Nonlinear Oscillations (New York: Wiley) pp369-395
[2]	Eissa M, El-Serafi S A, El-Sheikh M, Sayed M 2003 Appl. Math. Comput. 145 421
[3]	Alasty A, Shabani R 2006 Nonlinear Anal.-Real. 7 81
[4]	Starosta R, Sypniewska-Kamińska G, Awrejcewicz J 2011 Int. J. Bifurcat. Chaos 21 3013
[5]	Awrejcewicz J, Starosta R, Sypniewska-Kamińska G 2014 Asymptotic Analysis and Limiting Phase Trajectories in the Dynamics of Spring Pendulum (Cham: Springer) pp161-173
[6]	Klimenko A A, Mikhlin Y V, Awrejcewicz J 2012 Nonlinear Dynam. 70 797
[7]	Sousa M C D, Marcus F A, Caldas I L 2018 Physica A 509 1110
[8]	Lee W K 1994 J. Sound Vib. 171 335
[9]	Lee W K, Park H D 1997 Nonlinear Dynam. 14 211
[10]	Lee W K, Park H D 1999 Int. J. Nonlin. Mech. 34 749
[11]	Zaki K, Noah S, Rajagopal K R 2002 Nonlinear Dynam. 27 1
[12]	Tian R L, Wu Q L, Xiong Y P 2014 Eur. Phys. J. Plus 129 85
[13]	Yang X W, Tian R L, Zhang Q 2013 Eur. Phys. J. Plus 128 159
[14]	Awrejcewicz J, Starosta R, Sypniewska-Kamińska G 2016 Procedia IUTAM 19 201
[15]	Digilov R M, Reiner M, Weizman Z 2005 Am. J. Phys. 73 901
[16]	Eissa M, Kamel M, El-Sayed A T 2010 Nonlinear Dynam. 61 109
[17]	Gitterman M 2010 Physica A 389 3101
[18]	Amer T S, Bek M A 2009 Nonlinear Anal.-Real. 10 3196
[19]	Amer T S, Bek M A, Hamada I S 2016 Adv. Math. Phys. 2016 8734360
[20]	Amer T S, Bek M A, Abouhmr M K 2018 Nonlinear Dynam. 91 2485

施引文献

[1]	Nayfeh A H, Mook D T 1979 Nonlinear Oscillations (New York: Wiley) pp369-395
[2]	Eissa M, El-Serafi S A, El-Sheikh M, Sayed M 2003 Appl. Math. Comput. 145 421
[3]	Alasty A, Shabani R 2006 Nonlinear Anal.-Real. 7 81
[4]	Starosta R, Sypniewska-Kamińska G, Awrejcewicz J 2011 Int. J. Bifurcat. Chaos 21 3013
[5]	Awrejcewicz J, Starosta R, Sypniewska-Kamińska G 2014 Asymptotic Analysis and Limiting Phase Trajectories in the Dynamics of Spring Pendulum (Cham: Springer) pp161-173
[6]	Klimenko A A, Mikhlin Y V, Awrejcewicz J 2012 Nonlinear Dynam. 70 797
[7]	Sousa M C D, Marcus F A, Caldas I L 2018 Physica A 509 1110
[8]	Lee W K 1994 J. Sound Vib. 171 335
[9]	Lee W K, Park H D 1997 Nonlinear Dynam. 14 211
[10]	Lee W K, Park H D 1999 Int. J. Nonlin. Mech. 34 749
[11]	Zaki K, Noah S, Rajagopal K R 2002 Nonlinear Dynam. 27 1
[12]	Tian R L, Wu Q L, Xiong Y P 2014 Eur. Phys. J. Plus 129 85
[13]	Yang X W, Tian R L, Zhang Q 2013 Eur. Phys. J. Plus 128 159
[14]	Awrejcewicz J, Starosta R, Sypniewska-Kamińska G 2016 Procedia IUTAM 19 201
[15]	Digilov R M, Reiner M, Weizman Z 2005 Am. J. Phys. 73 901
[16]	Eissa M, Kamel M, El-Sayed A T 2010 Nonlinear Dynam. 61 109
[17]	Gitterman M 2010 Physica A 389 3101
[18]	Amer T S, Bek M A 2009 Nonlinear Anal.-Real. 10 3196
[19]	Amer T S, Bek M A, Hamada I S 2016 Adv. Math. Phys. 2016 8734360
[20]	Amer T S, Bek M A, Abouhmr M K 2018 Nonlinear Dynam. 91 2485

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