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In this article, from the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems, three kinds of dynamics for high-order nonholonomic constrained systems are obtained, including the vakonomic dynamical model, Lagrange-d'Alembert model and a new one if utilizing respectively three kinds of conditional variation to them. And the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems is also fitted for the general nonholonomic systems when the constrained equation is reduced to a first-order one. Then, the vakonomic dynamic, Chetaev dynamics and a new model of general nonholonomic systems can also be obtained. Finally, two illustrated examples are used to verify the validity of the theory.
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Keywords:
- high-order nonholonomic constraint /
- variation identity /
- conditional variation /
- vakonomic dynamics
[1] Guo Y X, Luo S K, Mei F X 2004 Advance in Mechanics 34 3 (in Chinese) [郭永新, 罗绍凯, 梅凤翔 2004 力学进展 34 3]
[2] Neimark Ju I, Fufaev N A 1972 Dynamics of Nonholonomic Systems Providence: American Mathematical Society
[3] Mei F X, Liu D, Luo Y 1991 Advanced Analytical Mechanics (Beijing: Beijing Institute Technology Press) (in Chinese) [梅凤翔, 刘瑞, 罗勇 1991 高等分析力学 (北京: 北京理工大学出版社)]
[4] Shen H C 2005 Acta Phys. Sin. 54 6 (in Chinese) [沈惠川 2005 物理学报 54 6]
[5] Guo Y X, Zhao Z, Liu S X, Liu C 2008 Acta Phys. Sin. 57 3 (in Chinese) [郭永新, 赵喆, 刘世兴, 刘畅 2008 物理学报 57 3]
[6] Zhao Z, Guo Y X, Liu C, Liu S X 2008 Acta Phys. Sin. 57 4 (in Chinese) [赵喆, 郭永新, 刘畅, 刘世兴 2008 物理学报 57 4]
[7] Guo Z H, Gao P Y 1990 Chinese Journal of Theoretical and Applied Mechanics 22 2 (in Chinese) [郭仲衡, 高普云 1990 力学学报 22 2]
[8] Chen B 1991 Chinese Journal of Theoretical and Applied Mechanics 23 3 (in Chinese) [陈滨 1991 力学学报 23 3]
[9] Guo Z H, Gao P Y 1992 Chinese Journal of Theoretical and Applied Mechanics 24 2 (in Chinese) [郭仲衡, 高普云 1992 力学学报 24 2]
[10] Guo Y X, Zhao Z, Liu S X, Zhu N, Han X J 2006 Acta Phys. Sin. 55 8 (in Chinese) [郭永新, 赵喆, 刘世兴, 朱娜, 韩晓静 2006 物理学报 55 8]
[11] Guo Y X, Liu S X, Liu C, Chang P 2009 Phys. Lett. A 373
[12] Zheng X Y, Wu Y Q 2009 Int. J. Auto. Comp. 6 3
[13] Gao F Z, Yuan F S, Gao C C 2009 Fuzzy Systems and Mathematics 23 3 (in Chinese) [高芳征, 袁付顺, 高存臣 2009 模糊系统与数学 23 3]
[14] Zhang X W 2006 Journal of Longdong College 16 1 (in Chinese) [张相武 2006 陇东学院学报 16 1]
[15] Mu X W, Yu J M, Cheng G F 2006 Applied Mathematics and Mechanics 27 4 (in Chinese) [慕小武, 虞继敏, 程桂芳 2006 应用数学和力学 27 4]
[16] Gao F Z, Shang Y L, Yuan F S 2012 Journal of System Sciences and Mathematical Sciences 32 2 (in Chinese) [高芳征, 尚艳玲, 袁付顺 2012 系统科学与数学 32 2]
[17] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian Systems (Beijing: Beijing Institute Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff 系统动力学 (北京: 北京理工大学出版)]
[18] Mei F X 1991 Applied Mathematics and Mechanics 12 8 (in Chinese) [梅凤翔 1991 应用数学和力学 12 8]
[19] Chen L Q 1992 Journal of Anshan Institute I. & S. Technology 15 1 (in Chinese) [陈立群 1992 鞍山钢铁学院学报 15 1]
[20] Li Y, Fang J H, Zhang K J 2010 Journal of Dynamics and Control 8 4 (in Chinese) [李燕, 方建会, 张克军 2010 动力学与控制学报 8 4]
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[1] Guo Y X, Luo S K, Mei F X 2004 Advance in Mechanics 34 3 (in Chinese) [郭永新, 罗绍凯, 梅凤翔 2004 力学进展 34 3]
[2] Neimark Ju I, Fufaev N A 1972 Dynamics of Nonholonomic Systems Providence: American Mathematical Society
[3] Mei F X, Liu D, Luo Y 1991 Advanced Analytical Mechanics (Beijing: Beijing Institute Technology Press) (in Chinese) [梅凤翔, 刘瑞, 罗勇 1991 高等分析力学 (北京: 北京理工大学出版社)]
[4] Shen H C 2005 Acta Phys. Sin. 54 6 (in Chinese) [沈惠川 2005 物理学报 54 6]
[5] Guo Y X, Zhao Z, Liu S X, Liu C 2008 Acta Phys. Sin. 57 3 (in Chinese) [郭永新, 赵喆, 刘世兴, 刘畅 2008 物理学报 57 3]
[6] Zhao Z, Guo Y X, Liu C, Liu S X 2008 Acta Phys. Sin. 57 4 (in Chinese) [赵喆, 郭永新, 刘畅, 刘世兴 2008 物理学报 57 4]
[7] Guo Z H, Gao P Y 1990 Chinese Journal of Theoretical and Applied Mechanics 22 2 (in Chinese) [郭仲衡, 高普云 1990 力学学报 22 2]
[8] Chen B 1991 Chinese Journal of Theoretical and Applied Mechanics 23 3 (in Chinese) [陈滨 1991 力学学报 23 3]
[9] Guo Z H, Gao P Y 1992 Chinese Journal of Theoretical and Applied Mechanics 24 2 (in Chinese) [郭仲衡, 高普云 1992 力学学报 24 2]
[10] Guo Y X, Zhao Z, Liu S X, Zhu N, Han X J 2006 Acta Phys. Sin. 55 8 (in Chinese) [郭永新, 赵喆, 刘世兴, 朱娜, 韩晓静 2006 物理学报 55 8]
[11] Guo Y X, Liu S X, Liu C, Chang P 2009 Phys. Lett. A 373
[12] Zheng X Y, Wu Y Q 2009 Int. J. Auto. Comp. 6 3
[13] Gao F Z, Yuan F S, Gao C C 2009 Fuzzy Systems and Mathematics 23 3 (in Chinese) [高芳征, 袁付顺, 高存臣 2009 模糊系统与数学 23 3]
[14] Zhang X W 2006 Journal of Longdong College 16 1 (in Chinese) [张相武 2006 陇东学院学报 16 1]
[15] Mu X W, Yu J M, Cheng G F 2006 Applied Mathematics and Mechanics 27 4 (in Chinese) [慕小武, 虞继敏, 程桂芳 2006 应用数学和力学 27 4]
[16] Gao F Z, Shang Y L, Yuan F S 2012 Journal of System Sciences and Mathematical Sciences 32 2 (in Chinese) [高芳征, 尚艳玲, 袁付顺 2012 系统科学与数学 32 2]
[17] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian Systems (Beijing: Beijing Institute Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff 系统动力学 (北京: 北京理工大学出版)]
[18] Mei F X 1991 Applied Mathematics and Mechanics 12 8 (in Chinese) [梅凤翔 1991 应用数学和力学 12 8]
[19] Chen L Q 1992 Journal of Anshan Institute I. & S. Technology 15 1 (in Chinese) [陈立群 1992 鞍山钢铁学院学报 15 1]
[20] Li Y, Fang J H, Zhang K J 2010 Journal of Dynamics and Control 8 4 (in Chinese) [李燕, 方建会, 张克军 2010 动力学与控制学报 8 4]
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