Journal of Guangxi Normal University(Natural Science Edition) ›› 2021, Vol. 39 ›› Issue (4): 79-92.doi: 10.16088/j.issn.1001-6600.2020102901

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Co-dimension-two Grazing Bifurcation and Chaos Control of Three-degree-of-freedom Vibro-impact Systems

LI Songtao, LI Qunhong*, ZHANG Wen   

  1. College of Mathematics and Information Science, Guangxi University, Nanning Guangxi 530004, China
  • Revised:2020-12-16 Online:2021-07-25 Published:2021-07-23

Abstract: For a three-degree-of-freedom vibro-impact system, the dynamic behavior near the grazing periodic orbit is discussed by using discontinuous mapping method. The existence conditions of saddle-node bifurcation and period-doubling bifurcation in 1/n impact periodic motion are deduced theoretically. It is concluded that co-dimension-two bifurcations occur when saddle-node bifurcation, period-doubling bifurcation and grazing bifurcation are detected simultaneously and the numerical simulation results are consistent with the theoretical results above. The bifurcation and chaotic motions of the system near co-dimension-two bifurcation points are studied by combining Lyapunov exponent and the local bifurcation diagram and the periodic motion and chaotic motion of the system appear in turn within a certain range of parameters. Then, using pulse control of chaos, the chaotic motions of the system can be suppressed to the stable periodic orbit. The effectiveness of the control method is verified by comparing the control diagram.

Key words: discontinuous mapping, co-dimension-two bifurcation, Lyapunov exponent, chaotic motion, pulse control

CLC Number: 

  • O322
[1]BERNARDO M D L, CHAMPNEYS A R, BUDD C J, et al. Piecewise-smooth dynamical systems: theory and applications[M]. London: Springer-Verlag London Limited, 2008. DOI:10.1007/s00419-007-0125-1.
[2]NORDMARK A B. Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators[J]. Nonlinearity, 2001, 14(6): 1517-1542. DOI:10.1088/0951-7715/14/6/306.
[3]DANKOWICZ H, SVAHN F. On the stabilizability of near-grazing dynamics in impact oscillators[J]. International Journal of Robust and Nonlinear Control, 2007, 17(15): 1405-1429. DOI:10.1002/rnc.1252.
[4]LEINE R I. Non-smooth stability analysis of the parametrically excited impact oscillator[J]. International Journal of Non-Linear Mechanics, 2012, 47(9): 1020-1032. DOI:10.1016/j.ijnonlinmec.2012.06.010.
[5]赵益波, 罗晓曙, 唐国宁,等. 电流反馈型Buck-Boost变换器的非线性动力学研究[J]. 广西师范大学学报(自然科学版), 2005, 23(2): 9-12. DOI:10.3969/j.issn.1001-6600.2005.02.003.
[6]WEN G L, YIN S, XU H D, et al. Analysis of grazing bifurcation from periodic motion to quasi-periodic motion in impact-damper systems[J]. Chaos, Solitons and Fractals, 2016, 83: 112-118. DOI:10.1016/j.chaos.2015.11.039.
[7]NORDMARK A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration, 1991, 145(2): 279-297. DOI:10.1016/0022-460X(91)90592-8.
[8]XU H D, YIN S, WEN G L, et al. Discrete-in-time feedback control of near-grazing dynamics in the two-degree-of-freedom vibro-impact system with a clearance[J]. Nonlinear Dynamics, 2017, 87(2): 1127-1137. DOI:10.1007/s11071-016-3103-8.
[9]YIN S K, SHEN Y K, WEN G L, et al. Feedback control of grazing-induced chaos in the single-degree-of-freedom impact oscillator[J]. Journal of Computational and Nonlinear Dynamics, 2017, 13(1): 011012. DOI:10.1115/1.4037924.
[10]伍帅, 徐洁琼, 王子汉. 一类二自由度碰撞振动系统的余维二擦边分岔研究[J].振动与冲击, 2020, 39(20):113-120. DOI:10.13465/j.cnki.jvs.2020.20.015.
[11]PETERKA F. Bifurcations and transition phenomena in an impact oscillator[J]. Chaos, Solitons and Fractals, 1996, 7(10): 1635-1647. DOI:10.1016/S0960-0779(96)00028-8.
[12]DANKOWICZ H,ZHAO X P. Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators[J]. Physica D Nonlinear Phenomena, 2005, 202(3/4): 238-257. DOI:10.1016/j.physd.2005.02.008.
[13]LI Q H, XU J Q, CHEN P, Theoretical analysis of co-dimension-two grazing bifurcations in n-degree-of-freedom impact oscillator with symmetrical constrains[J]. Nonlinear Dynamics, 2015, 82(4): 1641-1657. DOI:10.1007/s11071-015-2266-z.
[14]金俐, 陆启韶. 非光滑动力系统Lyapunov指数谱的计算方法[J]. 力学学报,2005, 37(1): 40-46. DOI:10.3321/j.issn:0459-1879.2005.01.006.
[15]LI Q H, CHEN Y M. Analysis of the Lyapunov exponential spectrum in a vibro-impact system with two-sided rigid stops[J]. Journal of Vibration and Shock, 2012, 31(7): 148-152. DOI:10.13465/j.cnki.jvs.2012.07.017.
[16]苟向锋, 罗冠炜, 吕小红. 含双侧刚性约束碰撞振动系统的混沌控制[J]. 机械科学与技术, 2011, 30(8): 1262-1266.
[17]吕小红, 朱喜锋, 罗冠炜. 含双侧约束碰撞振动系统的OGY混沌控制[J]. 机械科学与技术, 2016, 35(4): 531-534. DOI:10.13433/j.cnki.1003-8728.2016.0406.
[18]WANG L, XU W , LI Y. Impulsive control of a class of vibro-impact systems[J]. Physics Letters A, 2008, 372(32): 5309-5313. DOI:10.1016/j.physleta.2008.06.027.
[19]丁旺才, 谢建华, 李国芳. 三自由度碰撞振动系统的周期运动稳定性与分岔[J].工程力学, 2004, 21(3): 123-128.DOI:10.3969/j.issn.1000-4750.2004.03.023.
[20]伍新,文桂林,徐慧东,等. 三自由度含间隙碰撞振动系统Neimark-Sacker 分岔的反控制[J]. 物理学报, 2015, 64(20): 200504. DOI:10.7498/aps.64.200504.
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