Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback

doi:10.1088/1674-1056/24/1/014501

Abstract The stability and the Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback are studied. By considering the energy in the air-gap field of the AC motor, the dynamical equation of the electromechanical coupling transmission system is deduced and a time delay feedback is introduced to control the dynamic behaviors of the system. The characteristic roots and the stable regions of time delay are determined by the direct method, and the relationship between the feedback gain and the length summation of stable regions is analyzed. Choosing the time delay as a bifurcation parameter, we find that the Hopf bifurcation occurs when the time delay passes through a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given by using the normal form method and the center manifold theorem. Numerical simulations are also performed, which confirm the analytical results.

Keywords: electromechanical coupling time delay Hopf bifurcation stability

Received: 18 June 2014
Revised: 06 August 2014
Accepted manuscript online:

PACS:	45.20.dc	(Rotational dynamics)
	05.45.-a	(Nonlinear dynamics and chaos)
	02.30.Ks	(Delay and functional equations)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 61104040), the Natural Science Foundation of Hebei Province, China (Grant No. E2012203090), and the University Innovation Team of Hebei Province Leading Talent Cultivation Project, China (Grant No. LJRC013).

Corresponding Authors: Li Hai-Bin
E-mail: hbli@ysu.edu.cn

Cite this article:

Liu Shuang (刘爽), Zhao Shuang-Shuang (赵双双), Wang Zhao-Long (王兆龙), Li Hai-Bin (李海滨) Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback 2015 Chin. Phys. B 24 014501

[1]	Kim T C, Rook T E and Singh R 2005 J. Sound Vib. 281 965
[2]	Kim T C, Rook T E and Singh R 2005 J. Sound Vib. 281 995
[3]	El-Bassiouny A F 2006 Physica A 366 167
[4]	Saigo M, Tanaka N and Nam D H 2004 J. Sound Vib. 270 657
[5]	Wang X Y and Wang M J 2008 Physica A 387 3751
[6]	Ma C and Wang X Y 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 721
[7]	Wang X Y and Luo C 2006 Appl. Math. Comput. 183 30
[8]	Wang X Y and Wu X J 2006 Acta Phys. Sin. 55 5083 (in Chinese)
[9]	Liu S, Liu B and Shi P M 2009 Acta Phys. Sin. 58 4384 (in Chinese)
[10]	Shi P M, Han D Y and Liu B 2010 Chin. Phys. B 9 090306
[11]	Meng Z, Fu L Y and Song M H 2013 Acta Phys. Sin. 62 054501 (in Chinese)
[12]	Huber D and Tsimring L S 2003 Phys. Rev. Lett. 91 260601
[13]	Zhang L S, Cai L and Feng C W 2011 Acta Phys. Sin. 60 060306 (in Chinese)
[14]	Park J H and Kwon O M 2005 Chaos Soliton. Fract. 23 495
[15]	Balasubramaniam P, Kalpana M and Rakkiyappan R 2012 Chin. Phys. B 21 048402
[16]	Wang X Y and Zhao Q 2008 Chin. Phys. B 22 048902
[17]	Wu R C 2009 Acta Phys. Sin. 58 139 (in Chinese)
[18]	Zhang L P, Wang H N and Xu M 2011 Acta Phys. Sin. 60 010506 (in Chinese)
[19]	Wang X Y and Zhang H 2013 Chin. Phys. B 22 4383
[20]	Ren H P, Li W C and Liu D 2010 Chin. Phys. B 19 030511
[21]	Jin S X and Zhang Y 2014 Chin. Phys. B 23 054501
[22]	Olgac N and Holm-Hansen B T 1994 J. Sound Vib. 176 93
[23]	Olgac N and Sipahi R 2002 IEEE T. Automa. Cont. 47 793
[24]	Hosek M, Elmali H and Olgac N 1997 J. Sound Vib. 205 151
[25]	Zhang W M, Li X, Liu S and Li Y Q 2013 Acta Phys. Sin. 62 94502 (in Chinese)
[26]	Liu S, Liu B, Zhang Y K and Wen Y 2010 Acta Phys. Sin. 59 38 (in Chinese)
[27]	Zhao Y Y and Li C A 2011 Acta Phys. Sin. 60 409 (in Chinese)
[28]	Zhao Y Y and Xu J 2007 J. Sound Vib. 308 212
[29]	Ruan S G and Wei J J 2003 Dyn. Conti. Discrete Impuls. Syst. Ser. A Math. Anal. 10 863
[30]	Hassard B D, Kazarinoff N D and Wan Y H 1981 Theory and Applications of Hopf bifurcation (London: Cambridge University Press)

[1]	Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays Zhan-Hong Guo(郭展宏), Zhi-Jun Li(李志军), Meng-Jiao Wang(王梦蛟), and Ming-Lin Ma(马铭磷). Chin. Phys. B, 2023, 32(3): 038701.
[2]	Effect of autaptic delay signal on spike-timing precision of single neuron Xuan Ma(马璇), Yaya Zhao(赵鸭鸭), Yafeng Wang(王亚峰), Yueling Chen(陈月玲), and Hengtong Wang(王恒通). Chin. Phys. B, 2023, 32(3): 038703.
[3]	Modulational instability of a resonantly polariton condensate in discrete lattices Wei Qi(漆伟), Xiao-Gang Guo(郭晓刚), Liang-Wei Dong(董亮伟), and Xiao-Fei Zhang(张晓斐). Chin. Phys. B, 2023, 32(3): 030502.
[4]	Continuous-wave optical enhancement cavity with 30-kW average power Xing Liu(柳兴), Xin-Yi Lu(陆心怡), Huan Wang(王焕), Li-Xin Yan(颜立新), Ren-Kai Li(李任恺), Wen-Hui Huang(黄文会), Chuan-Xiang Tang(唐传祥), Ronic Chiche, and Fabian Zomer. Chin. Phys. B, 2023, 32(3): 034206.
[5]	Suppression of laser power error in a miniaturized atomic co-magnetometer based on split ratio optimization Wei-Jia Zhang(张伟佳), Wen-Feng Fan(范文峰), Shi-Miao Fan(范时秒), and Wei Quan(全伟). Chin. Phys. B, 2023, 32(3): 030701.
[6]	Improvement of coercivity thermal stability of sintered 2:17 SmCo permanent magnet by Nd doping Chao-Zhong Wang(王朝中), Lei Liu(刘雷), Ying-Li Sun(孙颖莉), Jiang-Tao Zhao(赵江涛), Bo Zhou (周波), Si-Si Tu(涂思思), Chun-Guo Wang(王春国), Yong Ding(丁勇), and A-Ru Yan(闫阿儒). Chin. Phys. B, 2023, 32(2): 020704.
[7]	Formation of nanobubbles generated by hydrate decomposition: A molecular dynamics study Zilin Wang(王梓霖), Liang Yang(杨亮), Changsheng Liu(刘长生), and Shiwei Lin(林仕伟). Chin. Phys. B, 2023, 32(2): 023101.
[8]	Formation of quaternary all-d-metal Heusler alloy by Co doping fcc type Ni₂MnV and mechanical grinding induced B2-fcc transformation Lu Peng(彭璐), Qiangqiang Zhang(张强强), Na Wang(王娜), Zhonghao Xia(夏中昊), Yajiu Zhang(张亚九),Zhigang Wu(吴志刚), Enke Liu(刘恩克), and Zhuhong Liu(柳祝红). Chin. Phys. B, 2023, 32(1): 017102.
[9]	Memristor hyperchaos in a generalized Kolmogorov-type system with extreme multistability Xiaodong Jiao(焦晓东), Mingfeng Yuan(袁明峰), Jin Tao(陶金), Hao Sun(孙昊), Qinglin Sun(孙青林), and Zengqiang Chen(陈增强). Chin. Phys. B, 2023, 32(1): 010507.
[10]	Ion migration in metal halide perovskite QLEDs and its inhibition Yuhui Dong(董宇辉), Danni Yan(严丹妮), Shuai Yang(杨帅), Naiwei Wei(魏乃炜),Yousheng Zou(邹友生), and Haibo Zeng(曾海波). Chin. Phys. B, 2023, 32(1): 018507.
[11]	Parametric decay instabilities of lower hybrid waves on CFETR Taotao Zhou(周涛涛), Nong Xiang(项农), Chunyun Gan(甘春芸), Guozhang Jia(贾国章), and Jiale Chen(陈佳乐). Chin. Phys. B, 2022, 31(9): 095201.
[12]	Kinetic theory of Jeans' gravitational instability in millicharged dark matter system Hui Chen(陈辉), Wei-Heng Yang(杨伟恒), Yu-Zhen Xiong(熊玉珍), and San-Qiu Liu(刘三秋). Chin. Phys. B, 2022, 31(7): 070401.
[13]	Propagation and modulational instability of Rossby waves in stratified fluids Xiao-Qian Yang(杨晓倩), En-Gui Fan(范恩贵), and Ning Zhang(张宁). Chin. Phys. B, 2022, 31(7): 070202.
[14]	The transition from conservative to dissipative flows in class-B laser model with fold-Hopf bifurcation and coexisting attractors Yue Li(李月), Zengqiang Chen(陈增强), Mingfeng Yuan(袁明峰), and Shijian Cang(仓诗建). Chin. Phys. B, 2022, 31(6): 060503.
[15]	All polarization-maintaining Er:fiber-based optical frequency comb for frequency comparison of optical clocks Pan Zhang(张攀), Yan-Yan Zhang(张颜艳), Ming-Kun Li(李铭坤), Bing-Jie Rao(饶冰洁), Lu-Lu Yan(闫露露), Fa-Xi Chen(陈法喜), Xiao-Fei Zhang(张晓斐), Qun-Feng Chen(陈群峰), Hai-Feng Jiang(姜海峰)^‡, and Shou-Gang Zhang(张首刚). Chin. Phys. B, 2022, 31(5): 054210.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback

Cite this article:

Online attention