高阶非完整约束系统嵌入变分恒等式的积分变分原理

宋端; 刘畅; 郭永新

doi:10.7498/aps.62.094501

摘要

本文从高阶非完整系统嵌入变分恒等式的积分变分原理出发, 根据三种不等价条件变分的选取, 得到了高阶非完整系统的三类不等价动力学模型, 即高阶非完整约束系统的vakonomic方程、Lagrange-d'Alembert 方程和一种新的动力学方程. 当高阶非完整约束方程退化为一阶非完整约束时, 利用此理论可以得到一般非完整系统的vakonomic模型、Chetaev模型和一种新的动力学模型. 最后借助于应用实例验证了结论的正确性.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
辽东学院影像物理教研室, 丹东 118001;

2.
辽宁大学物理学院, 沈阳 110036

基金项目: 国家自然科学基金(批准号:11202090,11172120,10932002)和辽宁省重点实验室建设项目(批准号:2008403009)资助的课题.

Authors and contacts

1.
Physics of medical imaging department, Eastern Liaoning University, Dandong 118001, China;

2.
College of Physics, Liaoning University, Shenyang 110036, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11202090, 11172120, 10932002) and the Program of Constructing Liaoning Provincial Key Laboratory, China (Grant No. 2008403009).

参考文献

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[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]
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[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
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[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]
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施引文献

[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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