含分数阶微分项的Van Del Pol振子的动力学分析

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振动与冲击 ›› 2020, Vol. 39 ›› Issue (20) : 91-96.

论文

王晓娜1,2，申永军1，张娜2，石亚朋2，王军1

作者信息 +

Dynamical analysis of a Van Del Pol oscillator with fractional-order derivative

WANG Xiaona1,2，SHEN Yongjun1，ZHANG Na2，SHI Yapeng2，WANG Jun1

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文章历史 +

摘要

研究了含分数阶微分项的Van Del Pol振子。利用平均法得出系统的近似解析解，发现在近似解中分数阶微分项的系数和阶次以等效线性阻尼和线性刚度的形式影响着系统的振幅、共振频率以及稳定性，这与现有文献中直接把分数阶微分项单纯看做阻尼处理的方式不同。随后把解析解和数值解进行了比较，发现二者吻合较好，证明了本文解析解的正确性。通过数值仿真进一步验证了理论推导的正确性。

Abstract

A Van Del Pol oscillator with fractional-order derivative was studied by the averaging method, and the approximately analytical solution was obtained. The effects of the system parameters on response amplitude, frequency and stability, including the fractional coefficient and the fractional order, were characterized by the equivalent linear damping and the equivalent stiffness. The conclusion is entirely different from the published results which directly treat the fractional-order derivative as simple damping. The comparison of the analytical solution with the numerical results verifies the correctness of the approximately analytical results. The following analysis about the effects of the fractional parameters on the amplitude frequency was fulfilled. The correctness of the theoretical derivation was verified by numerical simulation.

导出引用

王晓娜1,2，申永军1，张娜2，石亚朋2，王军1. 含分数阶微分项的Van Del Pol振子的动力学分析[J]. 振动与冲击, 2020, 39(20): 91-96

WANG Xiaona1,2，SHEN Yongjun1，ZHANG Na2，SHI Yapeng2，WANG Jun1. Dynamical analysis of a Van Del Pol oscillator with fractional-order derivative[J]. Journal of Vibration and Shock, 2020, 39(20): 91-96

参考文献

[1] Oldham K B, Spanier J. The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order.[M]. New York: Academic Press.1974.
[2] Podlubny I. Fractional Differential Equations.[M]. London: Academic Press.1999.
[3] Petras I. Fractional-Order Nonlinear System[M]. Beijing: Higher Education Press.2011.
[4] Rossikhin Y A, Shitikova M V. Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results[J]. Applied Mechanics Reviews. 2010.63(1):010801.
[5] Riewe F. Mechanics with fractional derivatives[J]. Physical Review E.1997.53(3):3581-3592.
[6] Wang Z H ,Hu H Y. Stability of linear vibration system with fractional derivative damping[J]. Science in China(Series G:Physics,Mechanics & Astronomy) ,2009,39(10):1495-1502.(in Chinese: 王在华,胡海岩.含分数阶导数阻尼的线性振动系统的稳定性[J].中国科学(G辑:物理学力学天文学),2009,39(10):1495-1502.
[7] Wang Z H, Du M L. Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system[J]. Shock and Vibration. 2011.18(1-2):257-268.
[8] Yu A. Rossikhin, Marina V. Shitikova. Application of fractional operators to the analysis of damped vibrations of viscoelastic single mass systems[J]. Acta Mechanica. 1997.199(4).567-586.
[9] Li Gen-guo, Zhu Zheng-you, Cheng Chang-jun. Dynamical stability of viscoelastic column with fractional derivative constitutive relation[J]. Applied Mathematics and Mechanics.2011.22(3):294-303.
[10]Cao Junyi, Xie Hang, Jiang Zhuangde. Nonlinear dynamics of Duffing system with fractional order damping[J]. Journal of Xi’an jiaotong University .2009.43(03):50-54.(in Chinese: 曹军义,谢航,蒋庄德.分数阶阻尼Duffing系统的非线性动力学特性[J].西安交通大学学报,2009,43(03):50-54.)
[11]Chen Juhn-horng, Chen Wei-Ching. Chaotic dynamics of the fractionally damped van der Pol equation[J]. Chaos. Solitons and Fractals,2006,35(1):188-198.
[12]Lu Jun Guo. Chaotic dynamics of the fractional-order Lü system and its synchronization[J]. Physics Letters A,2006,354(4).305-311.
[13]Pankaj Wahi, Anindya Chatterjee. Averaging oscillations with small fractional damping and delayed terms[J]. Nonlinear Dynamics, 2004, 38(1-2).3-22.
[14]Chen Lin-Cong, Zhu Wei-Qiu. Stationary response of Duffing oscillator with fractional derivative damping under combined exctation of harmonic and wide-band noise[J]. Journal of Applied Mechanics. 2010.27(03):517-521+643. (in Chinese:陈林聪,朱位秋.谐和与宽带噪声联合激励下含分数阶导数型阻尼的Duffing振子的平稳响应[J].应用力学学报.2010.27(03):517-521+643.)
[15]Huang Z L, Jin X L. Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative[J]. Journal of Sound and Vibration. 2008.319(3):1121-1135.
[16]Zhang Cheng-Fen, Gao Jin-Feng, Xu Lei. Chaos in fractional Liu system and fractional unified system and synchronization of their different structures[J]. Acta Physica Sinica. 2007.56(9):5124-5130.(in Chinese: 张成芬,高金峰,徐磊. 分数阶 Liu系统与分数阶统一系统中的混沌现象及二者的异结构同步[J].物理学报.2007.56(9):5124-5130.)
[17]Liu Chong-Xin. A hyperchaotic system and its fractional-order circuit simulation experiment[J]. Acta Physica Sinica. 2007.56(12):6865-6873.(in Chinese:刘崇新.一个超混沌系统及其分数阶电路仿真实验[J].物理学报.2007.56(12):6865-6873.)
[18]Chen Xiang-Rong, Liu Chong-Xin,Li Yong-Xun. Research and control of fractional Liu chaotic system and its circuit experiments[J]. Acta Physica Sinica. 2008.57(3):1416-1422.(in Chinese: 陈向荣,刘崇新,王发强,李永勋.分数阶Liu混沌系统及其电路实验的研究与控制[J].物理学报.2008.57(3):1416-1422.)
[19]Zhang Ruo-Xun,Yang Yang,Yang Shi-Ping. Adaptive synchronization of fractional unified Chaotic systems[J]. Acta Physica Sinica. 2009.58(9):6039-6044.(in Chinese: 张若洵,杨洋,杨世平.分数阶统一混沌系统的自适应同步[J].物理学报.2009.58(9):6039-6044.)
[20]Hu Jian-Bing, Zhang Guo-An,Zhao Ling-Dong, Zeng Jin-Quan. Fractional unified chaotic system with intermittent synchronization. Acta Physica Sinica. 2011.60(6):94-102.(in Chinese:胡建兵,章国安,赵灵冬,曾金全.间歇同步分数阶统一混沌系统[J].物理学报.2011.60(6):94-102.)
[21]LI Qing-Du. Chen Shu. Zhou Ping. Horseshoe and entropy in a fractional-order unified system[J]. Chinese Physics B.2011.20(1):010502(6pp).
[22] Zhang Ruo-Xun, Yang Shi-Ping. Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation [J].Chinese Physics B. 2009.18(8):3295-3303.
[23]Qi Dong-Lian, Wang Qiao, Yang Jie. Comparison between two different sliding mode controllers for a fractional-order unified chaotic system[J]. Chinese Physics B. 2011.20(10):100505(9pp).
[24] Wei Peng, Shen Yong-Jun, Yang Shao-Pu. Subharmonic resonance of fractional Duffing oscillator. Journal of Vibration Engineering. 2014, 27(6):811-818.(in Chinese:韦鹏,申永军,杨绍普.分数阶Duffing振子的亚谐共振[J].振动工程学报, 2014, 27(6):811-818.)
[25]Sanders J A, Verhulst F, Murdock J. Averaging methods in nonlinear dynamical systems[M]. New York: Springer-Verlag. 2007:150.
[26]Nei Zhen-Hua. Vibration Mechanics[M]. Xi’an: Xi’an Jiaotong University Press. 1988:79.(in Chinese:倪振华.振动力学[M].西安:西安交通大学出版社.1988:79.)
[27]Liu Yan-Zhu, Chen Wen-Liang,Chen Li-Qun. Vibration Mechanics[M]. Beijing: Higher Education Press. 1998:36. (in Chinese:刘延柱,陈文良,陈立群.振动力学[M].北京:高等教育出版社.1998:36.)
[28]Deng Wei hua, Li Chang Pin. The evolution of chaotic dynamics for fractional unified system. Physics Letters A. 2008.372(4):401-407.
[29]Deng Wei-Hua. Numerical algorithm for the time fractional Fokker–Planck equation [J]. Journal of Computational Physics. 2007.227(2):1510-1522.
[30]Shen Yong-Jun,Yang Shao-Pu, Xing Hai-Jun. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative[J]. Acta Physica Sinica. 2012.61(11):158-163.(in Chinese:申永军,杨绍普,邢海军.含分数阶微分的线性单自由度振子的动力学分析[J].物理学报.2012.61(11):158-163.)
[31] Shen Yong-Jun,Yang Shao-Pu, Xing Hai-Jun. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative[J]. Acta Physica Sinica. 2012,61(15):55-63.(in Chinese:申永军,杨绍普,邢海军.含分数阶微分的线性单自由度振子的动力学分析(Ⅱ)[J].物理学报,2012,61(15):55-63.)
[32]Shen Yong-Jun, Yang Shao-Pu, Xing Hai-Jun, Gao Guo-Sheng. Primary resonance of Duffing oscillator with fractional-order derivative[J]. Communications in Nonlinear Science and Numerical Simulation. 2012.17(7):3092-3100.