GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE

Chen Xiangwei; Cao Qiupeng; Mei Fengxiang

doi:10.6052/0459-1879-15-268

Volume 48 Issue 3

Turn off MathJax

Article Contents

Abstract

References

Chinese Journal of Theoretical and Applied Mechanics > 2016 > 48(3): 684-691. > DOI: 10.6052/0459-1879-15-268 CSTR: 32045.14.0459-1879-15-268

Chen Xiangwei, Cao Qiupeng, Mei Fengxiang. GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 684-691. DOI: 10.6052/0459-1879-15-268

Citation:

PDF (129 KB)

GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE

1.
Department of Physics and Information Engineering, Shangqiu Normal University, Shangqiu 476000, China
2. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
3. School of Aerospace, Beijing Institute of Technology, Beijing 100081, China

Received Date: July 19, 2015
Revised Date: January 03, 2016

Graphical Abstract

Abstract

Abstract

It is an important and di cult problem to study the stability of the non-steady and nonholonomic mechanical systems, and it is di cult to construct the Lyapunov function directly from the di erential equation. This paper gives an indirect method. The ten kinds of generalized gradient systems are proposed and the di erential equations of the ten kinds of generalized gradient systems are given respectively. Furthermore, the generalized gradient representations of a nonholonomic system of Chetaev's type are studied. The condition under which a nonholonomic system can be considered as a generalized gradient system is obtained, so the nonholonomic system of Chetaev's type is transformed into each generalized gradient systems. The characteristic of the generalized gradient systems can be used to study the stability of the nonholonomic system. This method appears to be more e ective when it is di cult to construct the Lyapunov function directly. Some examples are given to illustrate the application of the result.
- nonholonomic system,
- generalized gradient system,
- stability

FullText(HTML)

References (30)

References

1 Hertz HR. Die Prinzipien der Mechanik. Leibzing: Gesammelte Werke, 1894

2 牛青萍. 经典力学基本微分原理与不完整力学组的运动方程. 力学学报, 1964, 7(2): 139-148 (Niu Qingping. The fundamental differential principle of classical mechanics and the equations of motion of nonholonomic systems. Acta Mechanica Sinica, 1964, 7(2):139-148 (in Chinese))

3 Mei FX. Nonholonomic mechanics. ASME Appl Mech Rev, 2000,53: 283-305

4 李子平. 经典和量子约束系统及其对称性质. 北京: 北京工业大学出版社, 1993 (Li Ziping. Classical and Quantum Constrained Systems and Their Symmetries. Beijing: Beijing University of Technology Press, 1993 (in Chinese))

5 Luo SK. Relativistic variational principles and equations of motion of high-order nonlinear nonholonomic systems. In: Proceedings of the International Conference on Dynamics, Vibration and Control, Beijing: Peking University Press, 1990, 645-652

6 Fu JL, Chen LQ, Luo Y, et al. Stability for the equilibrium state manifold of relativistic Birkhoff systems. Chinese Physics, 2003,12: 351-356

7 Zhang Y. Integrating factors and conservation laws for relativistic mechanical system. Communications in Theoretical Physics, 2005,44: 231-234

8 刘延柱. 航天器姿态动力学. 北京: 国防工业出版社, 1995 (Liu Yanzhu. Spacecraft Attitude Dynamics. Beijing: National Defense Industry Press, 1995 (in Chinese))

9 Ostrovskaya S, Angels J. Nonholonomic systems revisited within the frame work of analytical mechanics. ASME Appl Mech Rev,1998, 51: 415-433

10 Papastavridis JG. A panoramic overview of the principles and equations of motion of advanced engineering dynamics. ASME Appl Mech Rev, 1998, 51: 239-265

11 Han YL, Wang XX, Zhang ML, et al. Special Lie symmetry and Hojman conserved quantity of Appell equations for a Chetaev nonholonomic system. Nonlinear Dyn, 2013, 73: 357-361

12 杨新芳, 孙现亭, 王肖肖等. 变质量Chetaev 型非完整系统Appell 方程的Mei 对称性和Mei 守恒量. 物理学报, 2011, 60(11):111101 (Yang Xinfang, Sun Xianting, Wang Xiaoxiao, et al. Mei symmetry and Mei conserved quantity of Appell equations for nonholonomic systems of Chetaev's type with variable mass. Acta Phys Sin, 2011, 60(11): 111101 (in Chinese))

13 Valery VK. On invariant manifolds of nonholonomic systems. Regular and Chaotic Dynamics, 2012, 17: 131-141

14 Hirsch MW, Smale S, Devaney RL. Differential Equations, Dynamical Systems, and An Introduction to Chaos. Singapore: Elsevier,2008

15 Mc Lachlan RI, Quispel GRW, Robidoux N. Geometric integration using discrete gradients. Phil Trans R Soc Lond A, 1999, 357: 1021-1045

16 梅凤翔. 关于梯度系统. 力学与实践, 2012, 34: 89-90 (Mei Fengxiang. On gradient system. Mechanics in Engineering, 2012, 34:89-90 (in Chinese))

17 梅凤翔. 分析力学下卷. 北京: 北京理工大学出版社, 2013 (Mei Fengxiang. Analytical Mechanics Ⅱ. Beijing: Beijing Institute of Technology Press, 2013 (in Chinese))

18 Chen XW, Zhao GL, Mei FX. A fractional gradient representation of the Poincaré equations. Nonlinear Dynamics, 2013, 73: 579-582

19 Tomáš B, Ralph C, Eva F. Every ordinary differential equation with a strict Lyapunov function is a gradient system. Monatsh Math, 2012,166: 57-72

20 Marin AM, Ortiz RD, Rodriguez JA. A generalization of a gradient system. International Mathematical Forum, 2013, 8: 803-806

21 陈向炜, 李彦敏, 梅凤翔. 双参数对广义Hamilton 系统稳定性的影响. 应用数学和力学, 2014, 35(12): 1392-1397 (Chen Xiangwei, Li Yanmin, Mei Fengxiang. Dependance of stability of equilibrium of generalized Hamilton system on two parameters. Applied Mathematics and Mechanics, 2014, 35(12): 1392-1397 (in Chinese))

22 梅凤翔, 吴惠彬. 广义Birkhoff 系统的梯度表示. 动力学与控制学报, 2012, 10(4): 289-292 (Mei Fengxiang, Wu Huibin. A gradient representation for generalized Birkhoff system. J of Dynam. and Control, 2012, 10(4): 289-292 (in Chinese))

23 梅凤翔, 吴惠彬. 广义Hamilton 系统与梯度系统. 中国科学: 物理学力学天文学, 2013, 43(4): 538-540 (Mei Fengxiang, Wu Huibin. Generalized Hamilton system and gradient system. Scientia Sinica Physica, Mechanica & Astronomica, 2013, 43(4): 538-540 (in Chinese))

24 高为炳. 运动稳定性基础. 北京: 高等教育出版社, 1989 (Gao Weibing. Foundations of Stability of Motion. Beijing: Higher Education Press, 1987 (in Chinese))

25王照林. 运动稳定性及其应用. 北京: 高等教育出版社, 1992 (Wang Zhaolin. Stability of Motion and Its Applications. Beijing: Higher Education Press, 1992 (in Chinese))

26 梅凤翔, 史荣昌, 张永发等. 约束力学系统的运动稳定性. 北京: 北京理工大学出版社, 1997 (Mei Fengxiang, Shi Rongchang, Zhang Yongfa, et al. Stability of Constrained Mechanical Systems. Beijing: Beijing Institute of Technology Press, 1997 (in Chinese))

27 Jiang WA, Luo SK. Stability for manifolds of equilibrium states of generalized Hamiltonian system. Meccanica, 2012, 47: 379-383

28 Luo SK, He JM, Xu YL. Fractional Birkhoffian method for equilibrium stability of dynamical systems. Inter J of Non-Linear Mech,2016, 78: 105-111

29 Novoselov VS. Variational Methods in Mechanics. Leningrad: LGU Press, 1966 (in Russian)

30 梅风翔. 非完整系统力学基础. 北京: 北京工业学院出版社, 1985 (Mei Fengxiang. Foundations of Mechanics of Nonholonomic Systems. Beijing: Beijing Institute of Technology Press, 1985 (in Chinese))

[1]	Zhang Yi. THE GRADIENTIZATION OF CONSTRAINED HERGLOTZ EQUATIONS AND THE STABILITY OF THEIR SOLUTIONS FOR NONHOLONOMIC SYSTEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2025, 57(2): 516-522. DOI: 10.6052/0459-1879-24-539
[2]	Chen Xiangwei, Zhang Yey, Mei Fengxiang. STABLE GENERALIZED BIRKHOFF SYSTEMS CONSTRUCTED BY USING A GRADIENT SYSTEM WITH NON-SYMMETRICAL NEGATIVE-DEFINITE MATRIX[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 149-153. DOI: 10.6052/0459-1879-16-280
[3]	Wang Zaihua, Hu Haiyan. STABILITY OF A FORCE CONTROL SYSTEM WITH SAMPLED-DATA FEEDBACK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1372-1381. DOI: 10.6052/0459-1879-16-102
[4]	Qing Li, Tianshu Wang, Xingrui Ma. Parametric stability of a translational rigid-liquid coupled system subject to vertical excitations[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(3): 529-534. DOI: 10.6052/0459-1879-2010-3-2008-795
[5]	A METHOD FOR DETERMINING THE PERIODIC SOLUTION AND ITS STABILITY OF A DYNAMIC SYSTEM WITH LOCAL NONLINEARITIES 1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 1998, 30(5): 572-579. DOI: 10.6052/0459-1879-1998-5-1995-163
[6]	PERIODIC MOTIONS AND ROBUST STABILITY OF THE MULTI-DEGREE-OF-FREEDOM SYSTEMS WITH CLEARANCES[J]. Chinese Journal of Theoretical and Applied Mechanics, 1997, 29(1): 74-83. DOI: 10.6052/0459-1879-1997-1-1995-198
[7]	ROBUST STABILITY ANALYSIS OF IMPULSIVE DYNAMICAL SYSTEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1995, 27(2): 213-221. DOI: 10.6052/0459-1879-1995-2-1995-424
[8]	THE FREE MOT1ON OF NONHOLONOMIC SYSTEM AND DISAPPEARANCE OF THE NONHOLONOMIC PROPERTY[J]. Chinese Journal of Theoretical and Applied Mechanics, 1994, 26(4): 470-476. DOI: 10.6052/0459-1879-1994-4-1995-569
[9]	THE DYNAMIC STABILITY ANALYSIS FOR THE BULLET BY THE NONAUTONOMOUS SYSTEMS MODEL[J]. Chinese Journal of Theoretical and Applied Mechanics, 1993, 25(4): 500-504. DOI: 10.6052/0459-1879-1993-4-1995-672

Cited By

Cited by

Periodical cited type(12)

1.	张毅. 非完整系统的约束Herglotz方程的梯度化及其解的稳定性. 力学学报. 2025(02): 516-522 . 本站查看
2.	齐皓晖，王鹏. 力学系统的任意阶分数维梯度表示. 力学学报. 2024(08): 2408-2414 . 本站查看
3.	尹明旭，陈向炜. Nielsen方程的三重组合梯度表示及其稳定性分析. 动力学与控制学报. 2023(02): 24-32 .
4.	尹明旭，陈向炜. Lorentz-Dirac模型的梯度表示与稳定性研究. 云南大学学报(自然科学版). 2023(03): 621-627 .
5.	尹明旭，陈向炜. Nielsen方程的半负定矩阵的广义梯度系统表示. 商丘师范学院学报. 2022(06): 43-48 .
6.	尹明旭，谢煜，陈向炜. Nielsen方程的广义梯度表示及其稳定性分析. 力学季刊. 2022(02): 340-354 .
7.	王嘉航，鲍四元. 事件空间中Birkhoff系统的两类广义梯度表示. 华中师范大学学报(自然科学版). 2020(02): 174-177 .
8.	章婷婷，张毅，张成璞，陈向炜. 判定定常Chetaev型非完整系统稳定性的三重组合梯度方法. 动力学与控制学报. 2019(04): 306-312 .
9.	董孟峰，陈向炜. 判定广义Birkhoff系统稳定性的三重组合梯度方法. 力学季刊. 2019(03): 543-548 .
10.	李彦敏，章婷婷，梅凤翔. 用具有负定矩阵的梯度系统构造稳定的变质量力学系统. 力学学报. 2018(01): 109-113 . 本站查看
11.	章婷婷，陈向炜. 判定非自治Birkhoff系统稳定性的广义组合梯度方法. 力学季刊. 2018(04): 771-777 .
12.	陈向炜，张晔，梅凤翔. 用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统. 力学学报. 2017(01): 149-153 . 本站查看

Other cited types(3)

Get Citation

PDF

XML

Article Metrics

Article views (1295) PDF downloads (568) Cited by(15)

GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE