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The dynamic equation of a relative rotation time-delay nonlinear dynamic system is established, which contains time-varying stiffness, nonlinear damping and harmonic excitation. The bifurcation equation of time-delay dynamic system is deduced by the method of multiple scales. The structure stability of the system is studied by singularity theory, the transfer concourse of primary resonance equation and topological structure of bifurcation function are obtained. The dynamic stability of the system is discussed by the Hopf bifurcation theory and the condition for the limit cycle occurrance is given. Moreover, it is indicated by numerical method that parameters of time delay affect the limit cycle amplitude.
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Keywords:
- relative rotation /
- time-delay dynamic system /
- stability /
- limit cycle
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[2] Carmeli M 1986 Inter. J. Theor. Phys. 15 89
[3] Luo S K 1996 J. Beijing Inst. Technol. 16( S 1) 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16( 154] 〖4] Luo S K 1998 Appl. Math. Mech. 19 45
[4] Luo S K , Fu J L , Chen X W 2001 Acta Phys. Sin. 50 383 (in Chinese) [罗绍凯、傅景礼、陈向炜 2001 物理学报 50 383]
[5] Fang J H 2000 Acta Phys. Sin. 49 1028 (in Chinese) [方建会 2000 物理学报 49 1028]
[6] Jia L Q 2003 Acta Phys. Sin. 52 1039 (in Chinese) [贾利群 2003 物理学报 52 1039]
[7] 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]
[8] Dong Q L, Liu B , Wang K , Zhang C X 2004 Acta Phys. Sin. 53 337 (in Chinese) [董全林、刘 彬、王 坤、张春熹 2004 物理学报 53 337]
[9] Zhao W, Liu B 2005 Acta Phys. Sin. 54 4543 (in Chinese) [赵 武、刘 彬 2005 物理学报 54 4543]
[10] Wang K 2005 Acta Phys. Sin. 54 3987 (in Chinese) [王 坤 2005 物理学报 54 3987]
[11] Shi P M, Liu B 2007 Acta Phys. Sin. 56 3678 (in Chinese) [时培明、刘 彬 2007 物理学报 56 3678]
[12] Meng Z, Liu B 2007 Acta Phys. Sin. 56 6194 (in Chinese) [孟 宗、刘 彬 2007 物理学报 56 6194]
[13] Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1312 (in Chinese) [时培明、刘 彬、侯东晓 2008 物理学报 57 1312]
[14] Lu H T, He Z Y 1996 IEEE Trans. Circuit. Sys. I 43 700
[15] Fischer I, Kühne H, Richter H 1994 Phys. Rev. Lett. 73 2188
[16] Belair J, Campbell S, van der Driessche P 1996 J. Appl. Math. 56 245
[17] Mo J Q, Lin W T, Zhu J 2004 Acta Phys. Sin.53 3245 (in Chinese)[莫嘉琪、林万涛、朱 江 2004 物理学报 53 3245]
[18] Ji C J, Leung A Y T 2002 Int. J. Sound Vib. 253 985
[19] 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]
[20] Nbendjo B N , Tchoukuengno R , Woafo P 2003 Chaos Soliton. Fract. 18 345
[21] Fu W B, Tang J S 2004 Acta Phys. Sin. 53 2889 (in Chinese) [符文彬、唐驾时 2004 物理学报 53 2889]
[22] Shi P M, Liu B, Liu S 2008 Acta Phys. Sin. 57 3321 (in Chinese) [时培明、刘 彬、刘 爽 2008 物理学报 57 3321]
[23] EI-Bassiouny A F 2003 Appl. Math. Comput. 134 217
[24] EI-Bassiouny A F 2006 Physica A 366 167
[25] Nayfeh A H 1981 Introduction to Perturbation Techniques (New York:Academic) p46
[26] Zhang W, Zu J W 2008 Chaos Soliton. Fract. 38 1152
[27] 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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[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( S 1) 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16( 154] 〖4] Luo S K 1998 Appl. Math. Mech. 19 45
[4] Luo S K , Fu J L , Chen X W 2001 Acta Phys. Sin. 50 383 (in Chinese) [罗绍凯、傅景礼、陈向炜 2001 物理学报 50 383]
[5] Fang J H 2000 Acta Phys. Sin. 49 1028 (in Chinese) [方建会 2000 物理学报 49 1028]
[6] Jia L Q 2003 Acta Phys. Sin. 52 1039 (in Chinese) [贾利群 2003 物理学报 52 1039]
[7] 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]
[8] Dong Q L, Liu B , Wang K , Zhang C X 2004 Acta Phys. Sin. 53 337 (in Chinese) [董全林、刘 彬、王 坤、张春熹 2004 物理学报 53 337]
[9] Zhao W, Liu B 2005 Acta Phys. Sin. 54 4543 (in Chinese) [赵 武、刘 彬 2005 物理学报 54 4543]
[10] Wang K 2005 Acta Phys. Sin. 54 3987 (in Chinese) [王 坤 2005 物理学报 54 3987]
[11] Shi P M, Liu B 2007 Acta Phys. Sin. 56 3678 (in Chinese) [时培明、刘 彬 2007 物理学报 56 3678]
[12] Meng Z, Liu B 2007 Acta Phys. Sin. 56 6194 (in Chinese) [孟 宗、刘 彬 2007 物理学报 56 6194]
[13] Shi P M, Liu B, Hou D X 2008 Acta Phys. Sin. 57 1312 (in Chinese) [时培明、刘 彬、侯东晓 2008 物理学报 57 1312]
[14] Lu H T, He Z Y 1996 IEEE Trans. Circuit. Sys. I 43 700
[15] Fischer I, Kühne H, Richter H 1994 Phys. Rev. Lett. 73 2188
[16] Belair J, Campbell S, van der Driessche P 1996 J. Appl. Math. 56 245
[17] Mo J Q, Lin W T, Zhu J 2004 Acta Phys. Sin.53 3245 (in Chinese)[莫嘉琪、林万涛、朱 江 2004 物理学报 53 3245]
[18] Ji C J, Leung A Y T 2002 Int. J. Sound Vib. 253 985
[19] 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]
[20] Nbendjo B N , Tchoukuengno R , Woafo P 2003 Chaos Soliton. Fract. 18 345
[21] Fu W B, Tang J S 2004 Acta Phys. Sin. 53 2889 (in Chinese) [符文彬、唐驾时 2004 物理学报 53 2889]
[22] Shi P M, Liu B, Liu S 2008 Acta Phys. Sin. 57 3321 (in Chinese) [时培明、刘 彬、刘 爽 2008 物理学报 57 3321]
[23] EI-Bassiouny A F 2003 Appl. Math. Comput. 134 217
[24] EI-Bassiouny A F 2006 Physica A 366 167
[25] Nayfeh A H 1981 Introduction to Perturbation Techniques (New York:Academic) p46
[26] Zhang W, Zu J W 2008 Chaos Soliton. Fract. 38 1152
[27] 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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