THE GENERALIZED GAUSS PRINCIPLE FOR MECHANICAL SYSTEM WITH VARIABLE MASS AND ITS GENERALIZATION TO HIGHER ORDER NONHOLONOMIC SYSTEMS

Zhang Yi; Chen Xinyu

doi:10.6052/0459-1879-22-202

Volume 54 Issue 10

Oct. 2022

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Chinese Journal of Theoretical and Applied Mechanics > 2022 > 54(10): 2883-2891. > DOI: 10.6052/0459-1879-22-202 CSTR: 32045.14.0459-1879-22-202

Zhang Yi, Chen Xinyu. The generalized Gauss principle for mechanical system with variable mass and its generalization to higher order nonholonomic systems. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(10): 2883-2891. DOI: 10.6052/0459-1879-22-202

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THE GENERALIZED GAUSS PRINCIPLE FOR MECHANICAL SYSTEM WITH VARIABLE MASS AND ITS GENERALIZATION TO HIGHER ORDER NONHOLONOMIC SYSTEMS

College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China

Funds: The project was supported by the (12345678)and (9876543)

Received Date: May 09, 2022
Accepted Date: June 30, 2022
Available Online: July 01, 2022

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Abstract

Abstract

Variational principle has great generality, which can be divided into differential and integral, Gauss principle is the variational principle with differential form. Among the existing differential variational principles, only the Gauss principle has the extreme value characteristics, which can be expressed as the Gauss variation of the compulsion function equals to zero. Gauss principle can be used to obtain the motion law of a particle system directly by finding the extreme value of function. Therefore, Gauss principle plays a unique role in the dynamics modeling and approximate calculation of complex systems, such as the design and analysis of robots, approximate solutions of nonlinear vibration equations and dynamics of multi-body systems. This paper deals with the generalized Gauss principle for mechanical systems with variable mass and its extension to higher order nonholonomic mechanics. Firstly, Gauss’s principle of least compulsion for mechanical system with variable mass is established, and extended to second order linear nonholonomic constrained systems by constructing modified compulsion function. Secondly, the generalized Gauss principle of mechanical system with variable mass for arbitrary order cases is proposed, and generalized Gauss’s principle of least compulsion is established, and the generalized compulsion function is constructed to extend the principle to high order nonholonomic constrained systems with variable mass. It is shown that for variable-mass mechanical system with bilateral ideal high-order nonholonomic constraints, the acceleration of real motion minimizes the generalized compulsion function under the $k{\text{-th}}$ Gauss variation in every instant among all the possible accelerations compatible with the constraints in the $k{\text{-th}}$ acceleration space. At the end of this paper, the differential equations of motion of a burning uniform sphere moving along a rough horizontal plane and the variable-mass Hamel problem are derived by applying the generalized Gauss’s principle of least compulsion.

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[1]	梅凤翔, 吴惠彬, 李彦敏. 分析力学史略. 北京: 科学出版社, 2019 Mei Fengxiang, Wu Huibin, Li Yanmin. A Brief History of Analytical Mechanics. Beijing: Science Press, 2019 (in Chinese)
[2]	陈滨. 分析动力学, 第2版. 北京: 北京大学出版社, 2012 Chen Bin. Analytical Dynamics. 2nd ed. Beijing: Peking University Press, 2012 (in Chinese)
[3]	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)
[4]	Udwadia FE, Kalaba RE. Analytical Dynamics-A New Approach. New York: Cambridge University Press, 2008
[5]	波波夫ЕП. 操作机器人动力学与算法. 遇立基, 陈循介译. 北京: 机械工业出版社, 1983 Попов ЕП. Operating Robot Dynamics and Algorithm. Yu Liji, Chen Xunjie, trans. Beijing: Mechanical Industry Press, 1983 (in Chinese)
[6]	刘延柱, 潘振宽, 戈新生. 多体系统动力学, 第2版. 北京: 高等教育出版社, 2014 Liu Yanzhu, Pan Zhenkuan, Ge Xinshen. Dynamics of Multibody Systems. 2nd ed. Beijing: Higher Education Press, 2014 (in Chinese)
[7]	刘延柱. 基于高斯原理的多体系统动力学建模. 力学学报, 2014, 46(6): 940-945 (Liu Yanzhu. Dynamic modeling of multi-body system based on Gauss’s principle. Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(6): 940-945 (in Chinese) doi: 10.6052/0459-1879-14-143
[8]	刘延柱. 杆网系统基于高斯原理的动力学建模. 动力学与控制学报, 2018, 16(4): 289-294 (Liu Yanzhu. Dynamical modeling of a net system of rods based on Gauss’s principle. Journal of Dynamics and Control, 2018, 16(4): 289-294 (in Chinese)
[9]	姚文莉, 刘彦平, 杨流松. 基于高斯原理的非理想系统动力学建模. 力学学报, 2020, 52(4): 945-953 (Yao Wenli, Liu Yanping, Yang Liusong. Dynamic modeling of nonideal system based on Gauss’s principle. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(4): 945-953 (in Chinese) doi: 10.6052/0459-1879-20-073
[10]	Yao WL, Yang LS, Song KW, et al. Optimization method for dynamics of non-holonomic system based on Gauss’s principle. Acta Mechanica Sinica, 2020, 36(5): 1133-1141 doi: 10.1007/s10409-020-00998-5
[11]	Yao WL, Yang LS, Guo MM. Gauss optimization method for the dynamics of unilateral contact of rigid multibody systems. Acta Mechanica Sinica, 2021, 37(3): 494-506 doi: 10.1007/s10409-020-01019-1
[12]	Orsino RMM. Extended constraint enforcement formulations for finite-DOF systems based on Gauss’s principle of least constraint. Nonlinear Dynamics, 2020, 101: 2577-2597 doi: 10.1007/s11071-020-05924-9
[13]	姚文莉, 戴葆青. 广义坐标形式的高斯最小拘束原理及其推广. 力学与实践, 2014, 36(6): 779-782, 785 (Yao Wenli, Dai Baoqing. Gauss principle of least constraint in generalized coordinates and its generalization. Mechanics in Engineering, 2014, 36(6): 779-782, 785 (in Chinese) doi: 10.6052/1000-0879-14-195
[14]	杨流松, 姚文莉, 薛世峰. 粒子群优化算法在具有奇异位置的多体系统动力学中的应用. 北京大学学报(自然科学版), 2021, 57(5): 795-803 (Yang Liusong, Yao Wenli, Xue Shifeng. Application of particle swarm optimization on the multi-body system dynamics with singular positions. Acta Scientiarum Naturalium Universitatis Pekinensis, 2021, 57(5): 795-803 (in Chinese)
[15]	薛纭, 刘延柱, 陈立群. 超细长弹性杆的分析力学问题. 力学学报, 2005, 37(4): 485-493 (Xue Yun, Liu Yanzhu, Chen Liqun. On analytical mechanics for a super-thin elastic rod. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(4): 485-493 (in Chinese) doi: 10.3321/j.issn:0459-1879.2005.04.014
[16]	薛纭, 翁德玮. 超细长弹性杆动力学的Gauss原理. 物理学报, 2009, 58(1): 34-39 (Xue Yun, Weng Dewei. Gauss principle for a super-thin elastic rod dynamics. Acta Physica Sinica, 2009, 58(1): 34-39 (in Chinese) doi: 10.3321/j.issn:1000-3290.2009.01.006
[17]	刘延柱, 薛纭. 基于高斯原理的Cosserat弹性杆动力学模型. 物理学报, 2015, 64(4): 044601 (Liu Yanzhu, Xue Yun. Dynamical model of Cosserat elastic rod based on Gauss principle. Acta Physica Sinica, 2015, 64(4): 044601 (in Chinese) doi: 10.7498/aps.64.044601
[18]	薛纭, 曲佳乐, 陈立群. Cosserat生长弹性杆动力学的Gauss最小拘束原理. 应用数学和力学, 2015, 36(7): 700-709 (Xue Yun, Qu Jiale, Chen Liqun. Gauss principle of least constraint for Cosserat growing elastic rod dynamics. Applied Mathematics and Mechanics, 2015, 36(7): 700-709 (in Chinese) doi: 10.3879/j.issn.1000-0887.2015.07.003
[19]	杰格日达 СА, 索尔塔哈诺夫 ШХ, 尤士科夫 МП. 非完整系统的运动方程和力学的变分原理: 新一类控制问题. 梅凤翔译. 北京: 北京理工大学出版社, 2007 Зегжда СА, Солтаханов ШХ, Юшков МП. Equations of Motion for Nonholonomic Systems and Variational Principles of Mechanics: A New Class of Control Problems. Mei Fengxiang trans. Beijing: Beijing Institute of Technology Press, 2007 (in Chinese)
[20]	梅凤翔, 李彦敏, 吴惠彬. 关于Gauss原理. 动力学与控制学报, 2016, 14(4): 301-306 (Mei Fengxiang, Li Yanmin, Wu Huibin. On the Gauss principle. Journal of Dynamics and Control, 2016, 14(4): 301-306 (in Chinese) doi: 10.6052/1672-6553-2016-08
[21]	杨兆光. 变质量质点系的高斯最小约束原理与赫兹最小曲率原理. 上海力学, 1986, 7(4): 13-21 (Yang Zhaoguang. Gauss’ principle of least constraint and Hertz’s principle of least curvature of the systems with variable mass. Shanghai Journal of Mechanics, 1986, 7(4): 13-21 (in Chinese)
[22]	黎邦隆, 宋福磐. 关于冲力情况下的高斯最小约束原理. 湖南大学学报, 1995, 22(4): 23-28 (Li Banglong, Song Fupan. On Gauss’ principle of least constraint for impulsive motion. Journal of Hunan University, 1995, 22(4): 23-28 (in Chinese)
[23]	Ivanov AP. On the variational formulation of the dynamics of systems with friction. Regular and Chaotic Dynamics, 2014, 19(1): 100-115 doi: 10.1134/S1560354714010079
[24]	Wang LS, Pao YH. Jourdain’s variational equation and Appell’s equation of motion for nonholonomic dynamical systems. American Journal of Physics, 2003, 71(1): 72-82 doi: 10.1119/1.1514239
[25]	Yan CC. Hamilton’s principle and Schrodinger’s equation derived from Gauss’ principle of least squares. Foundations of Physics Letters, 2000, 13(1): 79-87 doi: 10.1023/A:1007773720600
[26]	Zegzhda SA, Soltakhanov SK. Application of the generalized Gaussian principle to the problem of damping vibrations of mechanical systems. Journal of Computer and Systems Sciences International, 2010, 49(2): 186-191 doi: 10.1134/S1064230710020036
[27]	Lewis AD. The geometry of the Gibbs-Appell equations and Gauss’ principle of least constraint. Reports on Mathematical Physics, 1996, 38(1): 11-28 doi: 10.1016/0034-4877(96)87675-0
[28]	Yunt K. Gauss’ principle and principle of least constraints for dissipative mechanical systems//The 7th Vienna International Conference on Mathematical Modelling, Vienne, Austria. 2012: 842-847
[29]	Cveticanin L. A review on dynamics of mass variable systems. Journal of the Serbian Society for Computational Mechanics, 2012, 6(1): 56-73
[30]	Hurtado JE. Analytical dynamics of variable-mass systems. Journal of Guidance, Control, and Dynamics, 2018, 41(3): 701-709 doi: 10.2514/1.G002917
[31]	钱志源, 赵言正, 付庄. 基于变质量系统力学理论的爬壁机器人动力学分析. 机器人, 2007, 29(2): 106-110 (Qian Zhiyuan, Zhao Yanzheng, Fu Zhuang. Dynamics analysis of wall-climbing robots based on variable mass system mechanics theory. Robot, 2007, 29(2): 106-110 (in Chinese) doi: 10.3321/j.issn:1002-0446.2007.02.002
[32]	Musicki D, Cveticanin L. Generalized Noether’s theorem in classical field theory with variable mass. Acta Mechanica, 2020, 231: 1655-1668 doi: 10.1007/s00707-019-02526-4
[33]	Guttner WC, Pesce CP. On Hamilton’s principle for discrete systems of variable mass and the corresponding Lagrange’s equations. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2017, 39(6): 1969-1976 doi: 10.1007/s40430-016-0625-4
[34]	Jiang WA, Han XJ, Chen LQ, et al. Probabilistic solutions of a variable-mass system under random excitations. Acta Mechanica, 2020, 231(7): 2815-2826 doi: 10.1007/s00707-020-02674-y
[35]	Jiang WA, Liu K, Xia ZW, et al. Algebraic structure and Poisson brackets of single degree of freedom non-material volumes. Acta Mechanica, 2018, 22(9): 2299-2306
[36]	Casetta L. Theorem on a new conservation law for the dynamics of a position-dependent mass particle. Acta Mechanica, 2017, 228(1): 351-355 doi: 10.1007/s00707-016-1697-z
[37]	Casetta L. Poisson brackets formulation for the dynamics of a position-dependent mass particle. Acta Mechanica, 2017, 228: 4491-4496 doi: 10.1007/s00707-017-1924-2
[38]	Casetta L. Geometric theory on the dynamics of a position-dependent mass particle. Acta Mechanica, 2016, 227: 1519-1532
[39]	杨来伍, 梅凤翔. 变质量系统力学. 北京: 北京理工大学出版社, 1989 Yang Laiwu, Mei Fengxiang. Mechanics of Variable Mass Systems. Beijing: Beijing Institute of Technology Press, 1989 (in Chinese)
[40]	梅凤翔. 非完整动力学研究. 北京: 北京工业学院出版社, 1987 Mei Fengxiang. Studies on Nonholonomic Dynamics. Beijing: Beijing Institute of Technology Press, 1987 (in Chinese)

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THE GENERALIZED GAUSS PRINCIPLE FOR MECHANICAL SYSTEM WITH VARIABLE MASS AND ITS GENERALIZATION TO HIGHER ORDER NONHOLONOMIC SYSTEMS

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