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The dynamic equation of some nonlinear relative rotation system with two masses has been established, which contains a kind of nonlinear sick-slip friction. Chaotic motion and the control effect of multi-delay feedbacks of the nonlinear relative rotation system under external excitation are investigated. When the system is in chaos under external excitation, the multi-time delay is applied to control the chaotic motion. The chaotic critical conditions of parameters about Smale commutation are given in terms of Melnikov theory, while the influence of multi-delay feedbacks on the system movement state and chaotic critical conditions is studied. The control impacts of multi-time delay parameters on chaotic motion are studied by phase diagram, Poincare map and power spectrum, separately.
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Keywords:
- multi-time delay /
- relative rotation system /
- control /
- numerical simulation
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[2] Luo S K 1998 Appl. Math. Mech. 19 45
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[6] Luo S K 2003 Chin. Phys. 12 357
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[9] Liu S, Liu B, Zhang Y K, Wen Y 2010 Acta Phys.Sin. 59 38 (in Chinese) [刘爽, 刘彬, 张业宽, 闻岩 2010 59 38]
[10] Stepan G 1989 Retarded Dynamical Systems: Stability and Characteristic Functions (Essex: Longman Scientific & Technical) p80
[11] Kuang Y 1993 Delay Differential Equation with Application Dynamics (San Diego: Academic Press) p63
[12] Hu H Y, Wang Z H 1999 Advances in Mechanics 29 501 (in Chinese) [胡海岩, 王在华 1999 力学进展 29 501]
[13] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
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[15] Wang N, Ni M K 2011 Acta Phys. Sin. 60 050203 (in Chinese) [汪娜, 倪明康 2011 60 050203]
[16] Xu J, Pei L J 2006 Advances in Mechanics 36 17 (in Chinese) [徐鉴, 裴利军 2006 力学进展 36 17]
[17] Wang X Y, Zhao Q 2008 Acta Phys. Sin. 57 2812 (in Chinese) [王兴元, 赵群 2008 57 2812]
[18] Wang M J, Wang X Y 2009 Acta Phys. Sin. 58 1467 (in Chinese) [王明军, 王兴元 2009 58 1467]
[19] Feng W, Duan Y R, Yan J R 2000 Acta Mathematica Scientia 20 433 (in Chinese) [冯伟, 段永瑞, 燕居让 2000 数学 20 433]
[20] Chen N, Shen X Y, Gui W H, Yang C H, Wang L Y 2011 Control Theory {& Applications} 28 1099 (in Chinese) [陈宁, 沈晓瑜, 桂卫华, 阳春华, 王凌云 2011 控制理论与应用 28 1099]
[21] Zhang B L, Deng F F, Tang G Y, Cao F L 2010 Control and Decision 25 1723 (in Chinese) [张宝琳, 郑菲菲, 唐功友, 曹飞龙 2010 控制与决策 25 1723]
[22] Zhang X M, Wu M 2006 Control Theory {& Applications} 23 918 (in Chinese) [张先明, 吴敏 2006 控制理论与应用 23 918]
[23] Zhang L S, Cai L, Feng C W 2011 Acta Phys. Sin. 60 060306 (in Chinese) [张立森, 蔡里, 冯朝文 2011 60 060306]
[24] Zhang X M, Peng J H, Chen G R 2004 Acta Phys. Sin. 53 2864 (in Chinese) [张晓明, 彭建华, 陈关荣 2004 53 2864]
[25] Zhao Y Y, Yang R M 2011 Acta Phys. Sin. 60 104304 (in Chinese) [赵艳影, 杨如铭 2011 60 104304]
[26] Zhao Y Y, Li C A 2011 Acta Phys. Sin. 60 114305 (in Chinese) [赵艳影, 李昌爱 2011 60 114305]
[27] Tang Q, Wang X Y 2012 Chinese Journal of Applied Mechanics 29 410 (in Chinese) [唐谦, 王兴元 2012 应用力学学报 29 410]
[28] Shang H L 2012 Acta Phys. Sin. 60 180506 (in Chinese) [尚慧琳 2012 60 180506]
[29] Thomsen J J, Fidlin A 2003 International Journal of Nonlinear Mechanics 38 389
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[1] Luo S K 1996 J. Beijing Inst. Technol 16 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16 (S1)154]
[2] Luo S K 1998 Appl. Math. Mech. 19 45
[3] 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 50 383]
[4] Luo S K, Chen X W, Guo Y X 2002 Chin. Phys. 11 429
[5] Luo S K 2002 Chin. Phys. Lett. 19 449
[6] Luo S K 2003 Chin. Phys. 12 357
[7] Liu S, Liu B, Shi P M 2010 Acta Phys. Sin. 58 4383 (in Chinese) [刘爽, 刘彬, 时培明 2010 58 4383]
[8] Liu S, Liu B, Shi P M 2010 Acta Phys. Sin. 58 4383 (in Chinese) [刘爽, 刘彬, 时培明 2010 58 4383]
[9] Liu S, Liu B, Zhang Y K, Wen Y 2010 Acta Phys.Sin. 59 38 (in Chinese) [刘爽, 刘彬, 张业宽, 闻岩 2010 59 38]
[10] Stepan G 1989 Retarded Dynamical Systems: Stability and Characteristic Functions (Essex: Longman Scientific & Technical) p80
[11] Kuang Y 1993 Delay Differential Equation with Application Dynamics (San Diego: Academic Press) p63
[12] Hu H Y, Wang Z H 1999 Advances in Mechanics 29 501 (in Chinese) [胡海岩, 王在华 1999 力学进展 29 501]
[13] Shang H L 2011 Acta Phys. Sin. 60 070501 (in Chinese) [尚慧琳 2011 60 070501]
[14] Meng J, Wang X Y 2008 Chinese Journal of Computaional Physics 25 247 (in Chinese) [孟娟, 王兴元 2008 计算物理 25 247]
[15] Wang N, Ni M K 2011 Acta Phys. Sin. 60 050203 (in Chinese) [汪娜, 倪明康 2011 60 050203]
[16] Xu J, Pei L J 2006 Advances in Mechanics 36 17 (in Chinese) [徐鉴, 裴利军 2006 力学进展 36 17]
[17] Wang X Y, Zhao Q 2008 Acta Phys. Sin. 57 2812 (in Chinese) [王兴元, 赵群 2008 57 2812]
[18] Wang M J, Wang X Y 2009 Acta Phys. Sin. 58 1467 (in Chinese) [王明军, 王兴元 2009 58 1467]
[19] Feng W, Duan Y R, Yan J R 2000 Acta Mathematica Scientia 20 433 (in Chinese) [冯伟, 段永瑞, 燕居让 2000 数学 20 433]
[20] Chen N, Shen X Y, Gui W H, Yang C H, Wang L Y 2011 Control Theory {& Applications} 28 1099 (in Chinese) [陈宁, 沈晓瑜, 桂卫华, 阳春华, 王凌云 2011 控制理论与应用 28 1099]
[21] Zhang B L, Deng F F, Tang G Y, Cao F L 2010 Control and Decision 25 1723 (in Chinese) [张宝琳, 郑菲菲, 唐功友, 曹飞龙 2010 控制与决策 25 1723]
[22] Zhang X M, Wu M 2006 Control Theory {& Applications} 23 918 (in Chinese) [张先明, 吴敏 2006 控制理论与应用 23 918]
[23] Zhang L S, Cai L, Feng C W 2011 Acta Phys. Sin. 60 060306 (in Chinese) [张立森, 蔡里, 冯朝文 2011 60 060306]
[24] Zhang X M, Peng J H, Chen G R 2004 Acta Phys. Sin. 53 2864 (in Chinese) [张晓明, 彭建华, 陈关荣 2004 53 2864]
[25] Zhao Y Y, Yang R M 2011 Acta Phys. Sin. 60 104304 (in Chinese) [赵艳影, 杨如铭 2011 60 104304]
[26] Zhao Y Y, Li C A 2011 Acta Phys. Sin. 60 114305 (in Chinese) [赵艳影, 李昌爱 2011 60 114305]
[27] Tang Q, Wang X Y 2012 Chinese Journal of Applied Mechanics 29 410 (in Chinese) [唐谦, 王兴元 2012 应用力学学报 29 410]
[28] Shang H L 2012 Acta Phys. Sin. 60 180506 (in Chinese) [尚慧琳 2012 60 180506]
[29] Thomsen J J, Fidlin A 2003 International Journal of Nonlinear Mechanics 38 389
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