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Stability and time-delayed feedback control of a relative-rotation nonlinear dynamical system under quasic-periodic parametric excitation

Shi Pei-Ming Li Ji-Zhao Liu Bin Han Dong-Ying

Stability and time-delayed feedback control of a relative-rotation nonlinear dynamical system under quasic-periodic parametric excitation

Shi Pei-Ming, Li Ji-Zhao, Liu Bin, Han Dong-Ying
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  • The dynamical equation of a relative-rotation nonlinear dynamic system, which contains quasi-periodic parametric excitation and time delays, is established. Bifurcation response equation of 1/2 subharmonic primary parametric resonance is obtained by the method of multiple scales, and the stability of the system is analyzed. By solving the steady state solutions of the uncontrolled system, the effect of quasi-periodic parametric excitation on system response is studied through discussing the dynamics of the system. Time-delay feedback control method is used to control the bifurcation and limit cycle(region). Numerical results show that the bifurcation and the stability of the limit cycle(region) are controlled effectively by changing the time-delay parameters.
    • Funds:
    [1]

    Carmeli M 1985 Found. Phys. 15 175

    [2]

    Carmeli M 1986 Inter. J. Theor. Phys. 15 89

    [3]

    Luo S K 1996 J. Beijing Inst. Technol. 16(S1) 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16(S1) 154]

    [4]

    Luo S K 1998 Appl. Math. Mech. 19 45

    [5]

    Fu J L, Chen X W, Luo S K 1999 Appl. Math. Mech. 20 1266

    [6]

    Fu J L, Chen X W, Luo S K 2000 Appl. Math. Mech. 21 549

    [7]

    Luo S K, Guo Y X, Chen X W 2001 Acta Phys. Sin. 50 2053 (in Chinese) [罗绍凯、 郭永新、 陈向炜 2001 物理学报 50 2053]

    [8]

    Luo S K 2002 Chin. Phys. Lett . 19 449

    [9]

    Luo S K, Chen X W, Guo Y X 2002 Chin. Phys. 11 523

    [10]

    Luo S K 2004 Acta Phys. Sin. 53 5 (in Chinese) [罗绍凯 2004物理学报 53 5]

    [11]

    Luo S K, Chen X W, Fu J L 2001 Chin. Phys. 10 271

    [12]

    Fang J H 2000 Acta Phys. Sin. 49 1028 (in Chinese) [方建会 2000 物理学报 49 1028]

    [13]

    Jia L Q 2003 Acta Phys. Sin. 52 1039 (in Chinese) [贾利群 2003 物理学报 52 1039]

    [14]

    Dong Q L, Liu B 2002 Acta Phys. Sin. 51 2191 (in Chinese) [董全林、 刘 彬 2002 物理学报 51 2191]

    [15]

    Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1321 (in Chinese) [时培明、 刘 彬、 侯东晓 2008 物理学报 57 1321]

    [16]

    Shi P M, Liu B,Hou D X 2009 Chinese Journal of Mechanical Engineering 22 132

    [17]

    Belhaq M, Guennoun K, Houssni M 2002 Int. J. Non-Lin Mech. 37 445

    [18]

    Guennoun K, Belhaq M, Houssni M 2002 Nonlin Dyn. 27 211

    [19]

    Maccari A 2003 Int. J. Non-Lin Mech. 38 123

    [20]

    Maccari A 2003 J. Sound Vib. 259 241

    [21]

    Qi W, Zhang Y, Wang Y H 2009 Chin. Phys. B 18 1404

    [22]

    Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 090306

    [23]

    Qian C Z, Tang J S 2006 Acta Phys. Sin. 55 617 (in Chinese) [钱长照、 唐驾时 2006 物理学报 55 617]

    [24]

    Nayfeh A H 1981 Introduction to Perturbation Techniques (New York:Academic) p46

    [25]

    Shi P M, Liu B, Jiang J S 2009 Acta Phys. Sin. 58 2147 (in Chinese) [时培明、 刘 彬、 蒋金水 2009 物理学报 58 2147]

  • [1]

    Carmeli M 1985 Found. Phys. 15 175

    [2]

    Carmeli M 1986 Inter. J. Theor. Phys. 15 89

    [3]

    Luo S K 1996 J. Beijing Inst. Technol. 16(S1) 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16(S1) 154]

    [4]

    Luo S K 1998 Appl. Math. Mech. 19 45

    [5]

    Fu J L, Chen X W, Luo S K 1999 Appl. Math. Mech. 20 1266

    [6]

    Fu J L, Chen X W, Luo S K 2000 Appl. Math. Mech. 21 549

    [7]

    Luo S K, Guo Y X, Chen X W 2001 Acta Phys. Sin. 50 2053 (in Chinese) [罗绍凯、 郭永新、 陈向炜 2001 物理学报 50 2053]

    [8]

    Luo S K 2002 Chin. Phys. Lett . 19 449

    [9]

    Luo S K, Chen X W, Guo Y X 2002 Chin. Phys. 11 523

    [10]

    Luo S K 2004 Acta Phys. Sin. 53 5 (in Chinese) [罗绍凯 2004物理学报 53 5]

    [11]

    Luo S K, Chen X W, Fu J L 2001 Chin. Phys. 10 271

    [12]

    Fang J H 2000 Acta Phys. Sin. 49 1028 (in Chinese) [方建会 2000 物理学报 49 1028]

    [13]

    Jia L Q 2003 Acta Phys. Sin. 52 1039 (in Chinese) [贾利群 2003 物理学报 52 1039]

    [14]

    Dong Q L, Liu B 2002 Acta Phys. Sin. 51 2191 (in Chinese) [董全林、 刘 彬 2002 物理学报 51 2191]

    [15]

    Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1321 (in Chinese) [时培明、 刘 彬、 侯东晓 2008 物理学报 57 1321]

    [16]

    Shi P M, Liu B,Hou D X 2009 Chinese Journal of Mechanical Engineering 22 132

    [17]

    Belhaq M, Guennoun K, Houssni M 2002 Int. J. Non-Lin Mech. 37 445

    [18]

    Guennoun K, Belhaq M, Houssni M 2002 Nonlin Dyn. 27 211

    [19]

    Maccari A 2003 Int. J. Non-Lin Mech. 38 123

    [20]

    Maccari A 2003 J. Sound Vib. 259 241

    [21]

    Qi W, Zhang Y, Wang Y H 2009 Chin. Phys. B 18 1404

    [22]

    Shi P M, Han D Y, Liu B 2010 Chin. Phys. B 19 090306

    [23]

    Qian C Z, Tang J S 2006 Acta Phys. Sin. 55 617 (in Chinese) [钱长照、 唐驾时 2006 物理学报 55 617]

    [24]

    Nayfeh A H 1981 Introduction to Perturbation Techniques (New York:Academic) p46

    [25]

    Shi P M, Liu B, Jiang J S 2009 Acta Phys. Sin. 58 2147 (in Chinese) [时培明、 刘 彬、 蒋金水 2009 物理学报 58 2147]

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    [5] Liu Bin, Zhang Ye-Kuan, Liu Shuang, Wen Yan. Hopf bifurcation and stability of periodic solutions in a nonlinear relative rotation dynamical system with time delay. Acta Physica Sinica, 2010, 59(1): 38-43. doi: 10.7498/aps.59.38
    [6] You Bo, Cen Li-Xiang. Phenomena of limit cycle oscillations for non-Markovian dissipative systems undergoing long-time evolution. Acta Physica Sinica, 2015, 64(21): 210302. doi: 10.7498/aps.64.210302
    [7] Zhang Wen-Ming, Li Xue, Liu Shuang, Li Ya-Qian, Wang Bo-Hua. Chaos and the control of multi-time delay feedback for some nonlinear relative rotation system. Acta Physica Sinica, 2013, 62(9): 094502. doi: 10.7498/aps.62.094502
    [8] Liu Bin, Zhao Hong-Xu, Hou Dong-Xiao, Liu Hao-Ran. Bifurcation and chaos of some strongly nonlinear relative rotation system with time-varying clearance. Acta Physica Sinica, 2014, 63(7): 074501. doi: 10.7498/aps.63.074501
    [9] Zhao Wu, Liu Bin, Shi Pei-Ming, Jiang Jin-Shui. Analysis of stability of the equilibrium state of periodic motion in a nonlinear relative-rotation system. Acta Physica Sinica, 2006, 55(8): 3852-3857. doi: 10.7498/aps.55.3852
    [10] Guan Xin-Ping, Qiao Jie-Min, Wang Kun. Precise periodic solutions and uniqueness of periodic solutions of some relative rotation nonlinear dynamic system. Acta Physica Sinica, 2010, 59(6): 3648-3653. doi: 10.7498/aps.59.3648
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  • Received Date:  19 September 2010
  • Accepted Date:  20 December 2010
  • Published Online:  15 September 2011

Stability and time-delayed feedback control of a relative-rotation nonlinear dynamical system under quasic-periodic parametric excitation

  • 1. (1)College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China; (2)College of Vehicles and Energy, Yanshan University, Qinhuangdao 066004, China

Abstract: The dynamical equation of a relative-rotation nonlinear dynamic system, which contains quasi-periodic parametric excitation and time delays, is established. Bifurcation response equation of 1/2 subharmonic primary parametric resonance is obtained by the method of multiple scales, and the stability of the system is analyzed. By solving the steady state solutions of the uncontrolled system, the effect of quasi-periodic parametric excitation on system response is studied through discussing the dynamics of the system. Time-delay feedback control method is used to control the bifurcation and limit cycle(region). Numerical results show that the bifurcation and the stability of the limit cycle(region) are controlled effectively by changing the time-delay parameters.

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