Please wait a minute...
Chin. Phys. B, 2013, Vol. 22(11): 110205    DOI: 10.1088/1674-1056/22/11/110205
GENERAL Prev   Next  

Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

Li Chao (李超)a, Xu Wei (徐伟)a, Wang Liang (王亮)a, Li Dong-Xi (李东喜)b
a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China;
b School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
Abstract  A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function (PDF) of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.
Keywords:  vibro-impact system      colored noise      non-smooth transformation      stochastic bifurcation  
Received:  07 March 2013      Revised:  27 April 2013      Accepted manuscript online: 
PACS:  02.50.-r (Probability theory, stochastic processes, and statistics)  
  05.40.-a (Fluctuation phenomena, random processes, noise, and Brownian motion)  
  05.40.Ca (Noise)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11172233, 10932009, and 11202160) and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2012JQ1004).
Corresponding Authors:  Li Chao     E-mail:  xgdlc2011@163.com

Cite this article: 

Li Chao (李超), Xu Wei (徐伟), Wang Liang (王亮), Li Dong-Xi (李东喜) Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation 2013 Chin. Phys. B 22 110205

[1] Stratonovitch R L 1963 Topics in the Theory of Random Noise (Vol. 1) (New York: Gordon and Breach)
[2] Khasminskii R Z 1966 Theory of Probability & Its Applications 11 390
[3] Zhu W Q 1992 Random Vibration (Beijing: Science Press) (in Chinese)
[4] Lin Y K and Cai G Q 1995 Probabilistic Structural Dynamics (New York: McGraw-Hill)
[5] Roberts J B and Spanos P D 1986 Int. J. Non-Linear Mech. 21 111
[6] Zhu W Q and Cai G Q 2002 Acta Mech. Sin. 18 551
[7] Roberts J B 1983 IUTAM Symposium on Random Vibrations and Reliability October 31–November 6, 1982 Frankfurt, Germany, p. 285
[8] Dimentberg M, Cai G Q and Lin Y K 1995 Int. J. Non-Linear Mech. 30 677
[9] Zeng Y and Zhu W Q 2010 Int. J. Non-Linear Mech. 45 572
[10] Xu Y, Gu R C, Zhang H Q, Xu W and Duan J Q 2011 Phys. Rev. E 83 056215
[11] Brogliatoa B 1999 Nonsmooth Mechanics: Models, Dynamics and Control (London: Springer)
[12] Bernardo M di, Budd C J, Champneys A R and Kowalczyk P 2007 Piecewise-smooth Dynamical Systems: Theory and Applications (London: Springer)
[13] Chin W, Ott E, Nusse H E and Grebogi C 1994 Phys. Rev. E 50 4427
[14] Weger J D, Binks D and Molenaar J 1996 Phys. Rev. Lett. 76 3951
[15] Luo G W, Chu Y D, Zhang Y L and Zhang J G 2006 J. Sound Vib. 298 154
[16] Budd C and Dux F 1994 Phil. Trans. R. Soc. Lond. A 347 365
[17] Wagg D J and Bishop S R 2001 Int. J. Bifurcation Chaos 11 57
[18] Wagg D J 2004 Chaos Soliton. Fract. 22 541
[19] Feng J Q, Xu W and Niu Y J 2010 Acta Phys. Sin. 59 157 (in Chinese)
[20] Huang Z L, Liu Z H and Zhu W Q 2004 J. Sound Vib. 275 223
[21] Feng J Q, Xu W and Wang R 2008 J. Sound Vib. 309 730
[22] Feng J Q, Xu W, Rong H W and Wang R 2009 Int. J. Non-Linear Mech. 44 51
[23] Rong H W, Wang X D, Xu W and Fang T 2009 J. Sound Vib. 327 173
[24] Su M B and Rong H W 2011 Chin. Phys. B 20 060501
[25] Roberts J B 1978 J. Sound Vib. 60 177
[26] Cai G Q 1995 J. Eng. Mech. 121 633
[27] Xie W X, Xu W and Cai L 2006 Acta Phys. Sin. 55 1639 (in Chinese)
[28] Gan Z N, Ma J, Zhang G Y and Chen Y 2008 Chin. Phys. B 17 4047
[29] Wang B, Wu X Q and Shao J H 2009 Acta Phys. Sin. 58 1391 (in Chinese)
[30] Yang J H and Liu X B 2010 Chin. Phys. B 19 050504
[31] Zhang X Y, Xu W and Zhou B C 2011 Acta Phys. Sin. 60 060514 (in Chinese)
[32] Xiao R, Wang C J and Zhang L 2012 Chin. Phys. B 21 110504
[33] Zhuravlev V F 1976 Mech. Solids 11 23
[34] Dimentberg M F and Iourtchenko D V 2004 Nonlinear Dyn. 36 229
[35] Gardiner C W 2004 Handbook of Stochastic Methods (New York: Springer)
[36] Zhu W Q 2003 Nonlinear Stochastic Dynamics and Control: Framework of Hamiltonian Theory (Beijing: Science Press) (in Chinese)
[37] Krenk S and Roberts J B 1999 J. Appl. Mech. 66 225
[38] Zhang H Q, Xu W and Xu Y 2009 Physica A 388 781
[39] Ludwig Arnold 1998 Random Dynamical Systems (New York: Springer)
[40] Xu W, He Q, Rong H W and Fang T 2003 Proceedings of the Fifth International Conference on stochastic Structural Dynamics-SSD03 May 26–28, 2003 Hangzhou, China, p. 509
[1] Nano-friction phenomenon of Frenkel—Kontorova model under Gaussian colored noise
Yi-Wei Li(李毅伟), Peng-Fei Xu(许鹏飞), and Yong-Ge Yang(杨勇歌). Chin. Phys. B, 2022, 31(5): 050501.
[2] Effects of colored noise on the dynamics of quantum entanglement of a one-parameter qubit—qutrit system
Odette Melachio Tiokang, Fridolin Nya Tchangnwa, Jaures Diffo Tchinda,Arthur Tsamouo Tsokeng, and Martin Tchoffo. Chin. Phys. B, 2022, 31(5): 050306.
[3] Stationary response of colored noise excited vibro-impact system
Jian-Long Wang(王剑龙), Xiao-Lei Leng(冷小磊), and Xian-Bin Liu(刘先斌). Chin. Phys. B, 2021, 30(6): 060501.
[4] Asymmetric stochastic resonance under non-Gaussian colored noise and time-delayed feedback
Ting-Ting Shi(石婷婷), Xue-Mei Xu(许雪梅), Ke-Hui Sun(孙克辉), Yi-Peng Ding(丁一鹏), Guo-Wei Huang(黄国伟). Chin. Phys. B, 2020, 29(5): 050501.
[5] Some new advance on the research of stochastic non-smooth systems
Wei Xu(徐伟), Liang Wang(王亮), Jinqian Feng(冯进钤), Yan Qiao(乔艳), Ping Han(韩平). Chin. Phys. B, 2018, 27(11): 110503.
[6] Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise
Wei Li(李伟), Mei-Ting Zhang(张美婷), Jun-Feng Zhao(赵俊锋). Chin. Phys. B, 2017, 26(9): 090501.
[7] Influence of the colored noise on determining the period of a torsion pendulum
Jie Luo(罗 杰), Wen-Ze Zhan(占文泽), Wei-Huang Wu(巫伟皇), Cheng-Gang Shao(邵成刚), Dian-Hong Wang(王典洪). Chin. Phys. B, 2016, 25(8): 080401.
[8] Multi-valued responses and dynamic stability of a nonlinear vibro-impact system with a unilateral non-zero offset barrier
Wei Xu(徐伟), Dong-Mei Huang(黄冬梅), Wen-Xian Xie(谢文贤). Chin. Phys. B, 2016, 25(3): 030502.
[9] Stochastic resonance for a metapopulation system driven by multiplicative and additive colored noises
Wang Kang-Kang (王康康), Liu Xian-Bin (刘先斌). Chin. Phys. B, 2014, 23(1): 010502.
[10] Enhancement of density divergence in an insect outbreak model driven by colored noise
Xiao Rong (肖荣), Wang Can-Jun (王参军), Zhang Lin (张林 ). Chin. Phys. B, 2012, 21(11): 110504.
[11] Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation
Su Min-Bang (苏敏邦), Rong Hai-Wu (戎海武). Chin. Phys. B, 2011, 20(6): 060501.
[12] Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops
Li Qun-Hong(李群宏) and Tan Jie-Yan(谭洁燕). Chin. Phys. B, 2011, 20(4): 040505.
[13] The Stochastic stability of a Logistic model with Poisson white noise
Duan Dong-Hai(段东海), Xu Wei(徐伟), Su Jun(苏军), and Zhou Bing-Chang(周丙常). Chin. Phys. B, 2011, 20(3): 030501.
[14] A stochastic epidemic model on homogeneous networks
Liu Mao-Xing(刘茂省) and Ruan Jiong(阮炯) . Chin. Phys. B, 2009, 18(12): 5111-5116.
[15] Dynamical behaviour of a controlled vibro-impact system
Wang Liang(王亮), Xu Wei(徐伟), and Li Ying(李颖) . Chin. Phys. B, 2008, 17(7): 2446-2450.
No Suggested Reading articles found!