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The dimensionality reduction and bifurcation of some high-dimensional relative-rotation nonlinear dynamical system are studied. Considering the nonlinear influence factor of a relative-rotation nonlinear dynamic system, the high-dimensional relative-rotation torsional vibration global dynamical equation is established based on Lagrange equation. The equivalent low-dimensional bifurcation equation, which can reveal the low-dimensional equivalent bifurcation equation between the nonlinear dynamics and parameters, can be obtained by reducing the dimensionality system using the method of Lyapunov-Schmidt reduction. On this basis, the bifurcation characteristic is analyzed by taking universal unfolding on the bifurcation equation through using the singularity theory. The simulation is carried out with actual parameters. The parameter region of torsional vibration and the effect of the parameters on the vibration are discussed.
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Keywords:
- relatively rotation /
- high-dimensional system /
- L-S reduction /
- singularity
[1] Carmeli M 1985 Found. Phys. 15 175
[2] Carmeli M 1986Inter. J. Theor. Phys. 15 89
[3] Luo S K 1998 Appl. Math. Mech. 19 45
[4] Fu J L, Chen X W, Luo S K 2000 Appl. Math. Mech. 21 549
[5] 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]
[6] Luo S K, Chen X W, Guo Y X 2002 Chin. Phys. 11 523
[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] 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]
[9] Shi P M, Liu B, Hou D X 2009 Chinese Journal of Mechanical Engineering 22 132
[10] Shi P M, Han D Y, Liu B 2009 Chin. Phys. B 19 090306
[11] Redkar S, Sina S C 2008 ASME J. Comput Nonlinear Dyn. 3 1
[12] Cao D Q, Wang J L, Huang W H 2010 Sci. China Tech. Sci. 53 684
[13] AL-Shudeifat M A, Butcher E A 2010 Nonlinear Dyn. 62 821
[14] Wang F Z, Qi G Y, Chen Z Q, Zhang Y H, Yuan Z Z 2006 Acta Phys. Sin. 55 4005 (in Chinese) [王繁珍, 齐国元, 陈增强, 张宇辉, 袁著社 2006 物理学报 55 4005]
[15] Zha X M, Zhang Y, Sun J J, Fan Y P 2012 Acta Phys. Sin. 61 020505 (in Chinese) [查晓明, 张扬, 孙建军, 樊友平 2012 物理学报 61 020505]
[16] Dong J X, Hua Y, Wei C H. 2001 Automation of Electric Power Systems 28 24 (in Chinese) [邓集祥, 华瑶, 韦春华 2001 电力系统自动化 28 24]
[17] Wang Y X, Wang Y M, Liu X S 2003 Science in China (Series E) 33 56 (in Chinese) [王玉新, 王仪明, 刘学深 2003 中国科学(E辑) 33 56]
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[1] Carmeli M 1985 Found. Phys. 15 175
[2] Carmeli M 1986Inter. J. Theor. Phys. 15 89
[3] Luo S K 1998 Appl. Math. Mech. 19 45
[4] Fu J L, Chen X W, Luo S K 2000 Appl. Math. Mech. 21 549
[5] 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]
[6] Luo S K, Chen X W, Guo Y X 2002 Chin. Phys. 11 523
[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] 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]
[9] Shi P M, Liu B, Hou D X 2009 Chinese Journal of Mechanical Engineering 22 132
[10] Shi P M, Han D Y, Liu B 2009 Chin. Phys. B 19 090306
[11] Redkar S, Sina S C 2008 ASME J. Comput Nonlinear Dyn. 3 1
[12] Cao D Q, Wang J L, Huang W H 2010 Sci. China Tech. Sci. 53 684
[13] AL-Shudeifat M A, Butcher E A 2010 Nonlinear Dyn. 62 821
[14] Wang F Z, Qi G Y, Chen Z Q, Zhang Y H, Yuan Z Z 2006 Acta Phys. Sin. 55 4005 (in Chinese) [王繁珍, 齐国元, 陈增强, 张宇辉, 袁著社 2006 物理学报 55 4005]
[15] Zha X M, Zhang Y, Sun J J, Fan Y P 2012 Acta Phys. Sin. 61 020505 (in Chinese) [查晓明, 张扬, 孙建军, 樊友平 2012 物理学报 61 020505]
[16] Dong J X, Hua Y, Wei C H. 2001 Automation of Electric Power Systems 28 24 (in Chinese) [邓集祥, 华瑶, 韦春华 2001 电力系统自动化 28 24]
[17] Wang Y X, Wang Y M, Liu X S 2003 Science in China (Series E) 33 56 (in Chinese) [王玉新, 王仪明, 刘学深 2003 中国科学(E辑) 33 56]
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