动力学与控制论力学系统的自由度

胡海岩

doi:10.6052/0459-1879-18-219

动力学与控制论力学系统的自由度

doi: 10.6052/0459-1879-18-219

胡海岩^2), ,

北京理工大学宇航学院,飞行器动力学与控制教育部重点实验室, 北京 100081

基金项目: 1) 国家自然科学基金资助项目(11832005).

详细信息

作者简介:
2) 胡海岩,教授,主要研究方向：结构动力学与控制. E-mail: haiyan_hu@bit.edu.cn

通讯作者:
胡海岩

中图分类号: O313;
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出版历程
- 收稿日期: 2018-07-05
- 刊出日期: 2018-09-18

ON THE DEGREES OF FREEDOM OF A MECHANICAL SYSTEM

Hu Haiyan^{2)
, ,}

MOE Lab of Dynamics and Control of Flight Vehicles, School of Aerospace Engineering, Beijing Institute of Technology,Beijing 100081, China

摘要

摘要: 力学系统的自由度定义源自描述系统位形的独立坐标数.在分析力学发展过程中,人们通过对非完整约束的研究,将其拓展为独立的坐标变分数.本文指出,对于含非完整约束的力学系统,该定义存在不妥之处,给出的自由度会过度限制系统的力学行为.文中研究力学系统在状态空间中的可达流形,指出可达流形维数与描述系统动力学的一阶常微分方程组的最少未知函数个数一致,例如Gibbs-Appell方程与广义速度方程联立的未知函数个数,进而将可达流形维数的一半定义为系统自由度.通过含黏弹性支承的振动系统、在倾斜平面上运动的冰橇等案例,讨论了单个非完整约束导致的半自由度概念,指出其力学意义和与相邻整数自由度的关系.此外,文中还给出两个非完整约束导致系统减少一个自由度的案例,讨论了系统的切丛和余切丛维数.
- 自由度 /
- 半自由度 /
- 非完整约束 /
- 可达状态流形 /
- 切丛 /
- Gibbs-Appell方程 /
- 黏弹性
Abstract: The definition of degrees of freedom of a mechanical system originated from the number of independent coordinates to describe the system configuration. The definition turned to be the number of independent variations of generalized coordinates after the studies on non-hololomic constraints in the development of analytic mechanics. The paper points out that the above definition of degrees of freedom has some flaws for the mechanical system with non-holonomic constraints and may impose excessive limits on the system dynamics. The paper, hence, studies the accessible state manifold of a mechanical system with non-holonomic constraints in the state space and shows that the dimensions of the accessible state manifold is equal to the number of minimal unknown variables to describe the system dynamics, governed by a set of ordinary differential equations of the first order, such as the Gibbs-Appell equations together with the relation of generalized velocities and psudo-velocities. Then, the paper defines the degrees of freedom of a mechanical system as a half of the dimensions of the accessible state manifold. Afterwards, the paper demonstrates how to understand the concept of a half degree of freedom of a mechanical system with a single non-holonomic constraint via two case studies, that is, the vibration system having a viscoelastic mounting and the sleigh system moving on an inclined plane, presenting the relation between a half degree of freedom and the two neighboring integer degrees of freedom. Furthermore, the paper gives two examples of mechanical systems, each of which has two non-holonomic constraints and results in the reduction of a single degree of freedom, and addresses the dimensions of tangent and cotangent bundles of the two systems.
- degrees of freedom /
- half degree of freedom /
- non-holonomic constraint /
- accessible state manifold /
- tangent bundle /
- Gibbs-Appell equation /
- viscoelasticity

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参考文献(1)

[1]	Hibbeler RC.Engineering Mechanics: Statics, 10th Edition. New Jersey: Prentice Hall, 2004: 555
[2]	Arnold VI.Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1978: 75-135
[3]	Ionescu TG.Terminology for the mechanism and machine science, Chapter 3: Dynamics. Mechanism and Machine Theory, 2003, 38(7-10): 787-812
[4]	Greenwood DT. Advanced Dynamics.Cambridge: Cambridge University Press, 2003: 34-37
[5]	汪家訸. 分析力学. 北京:高等教育出版社, 1982: 146-149
[5]	(Wang Jiahe. Analytic Mechanics.Beijing: Higher Education Publisher, 1982: 146-149 (in Chinese))
[6]	梅凤翔. 高等分析力学. 北京: 北京理工大学出版社, 1991: 25-28
[6]	(Mei Fengxiang. Advanced Analytic Mechanics.Beijing: Press of Beijing Institute of Technology, 1991: 25-28 (in Chinese))
[7]	陈滨. 分析动力学. 第二版. 北京: 北京大学出版社, 2012: 37-44
[7]	(Chen Bin. Analytic Dynamics, 2nd Edition. Beijing: Peking University Press, 2012: 37-44 (in Chinese))
[8]	陈滨. 关于经典非完整力学的一个争议. 力学学报, 1991, 23(3): 379-384
[8]	(Chen Bin.A contention to the classic nonholonomic dynamics. Acta Mechnica Sinica, 1991, 23(3): 379-384 (in Chinese))
[9]	Guo YX, Mei FX.Integrability for Pfaffian constrained systems: A connection theory. Acta Mechnica Sinica, 1998, 14(1): 85-91
[10]	傅宇. 分析力学中自由度的确定. 天中学刊, 1988, 13(5): 71-73
[10]	(Fu Yu.Determination of degrees of freedom in analytic mechanics. Journal of Tianzhong, 1988, 13(5): 71-73 (in Chinese))
[11]	江优良. 经典力学体系中广义坐标和自由度的数目. 株洲师范高等专科学校学报, 2003, 7(2): 32-35
[11]	(Jiang Youliang.Generalized coordinates and degrees of freedom in classical mechanics. Journal of Zhuzhou Teachers College, 2003, 7(2): 32-35 (in Chinese))
[12]	Lee JM.Introduction to Smooth Manifolds, 2nd Edition. New York: Springer-Verlag, 2013: 50-76
[13]	Lee T, Leok M, McClamroch NH. Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds. New York: Springer-Verlag, 2018: 1-388
[14]	Marsden JE, Ratiu TS.Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd Edition. New York: Springer-Verlag, 1999: 121-180
[15]	胡海岩. 结构阻尼模型及系统时域动响应. 应用力学学报, 1993, 10(1): 34-43
[15]	(Hu Haiyan.Structural damping model and dynamic system response in time domain. Chinese Journal of Applied Mechanics, 1993, 10(1): 34-43 (in Chinese))
[16]	Ziglin SL.Self-intersection of the complex separatrices and the nonexistence of the integrals in the Hamiltonian-systems with one-and-half degrees of freedom. PMM Journal of Applied Mathematics and Mechanics, 1981, 45(3): 411-413
[17]	Fung YC.Foundations of Solid Mechanics. Englewood Cliffs: Prentice-Hall, 1965: 20-29
[18]	Min CQ, Dahlmann M, Sattel T.A concept for semi-active vibration control with a serial-stiffness-switch system. Journal of Sound and Vibration, 2017, 405: 234-250
[19]	胡海岩, 李岳锋. 具有记忆特性的非线性减振器参数识别. 振动工程学报, 1989, 2(2): 17-27.
[19]	(Hu Haiyan, Li Yuefeng.Parametric identification of nonlinear vibration isolators with memory. Journal of Vibration Engineering, 1989, 2(2): 17-27 (in Chinese))
[20]	Wen YK.Method for random vibration of hysteretic systems. Proceedings of ASCE, Journal of Engineering Mechanics, 1976, 12(1): 249-263
[21]	Simionatto VGS, Miyasato HH, de Melo FM, et al. Singular mass matrices and half degrees of freedom: A general method for system reduction//Proceedings of the 21st International Congress of Mechanical Engineering. Natal, RN, Brazil, 2011
[22]	Liu C, Liu SX, Guo YX.Inverse problem for Chaplygin's nonholonomic systems. Science in China E, 2011, 54(8): 2100-2106