变加速动力学系统的广义高斯最小拘束原理

张毅; 宋传静; 翟相华

doi:10.6052/0459-1879-23-030

变加速动力学系统的广义高斯最小拘束原理

doi: 10.6052/0459-1879-23-030

张毅^*, ,,
宋传静^†,
翟相华^*, ,

*.
苏州科技大学土木工程学院, 江苏苏州 215011
†.
苏州科技大学数学科学学院, 江苏苏州215009

基金项目: 国家自然科学基金资助项目(12272248, 11972241, 12172241, 12002228)

详细信息

通讯作者:
张毅, 教授, 主要研究方向为分析力学. E-mail: zhy@mail.usts.edu.cn

翟相华, 讲师, 主要研究方向为分析力学. E-mail: zxh@mail.usts.edu.cn

中图分类号: O316
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出版历程
- 收稿日期: 2023-02-01
- 录用日期: 2023-03-07
- 网络出版日期: 2023-03-08

GENERALIZED GAUSS PRINCIPLE OF LEAST COMPULSION FOR VARIABLE-ACCELERATION DYNAMICAL SYSTEMS

Zhang Yi^{*
, ,},
Song Chuanjing^†,
Zhai Xianghua^{*
, ,}

*.
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China
†.
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China

摘要

摘要: 变加速运动在日常生活和工程问题中普遍存在. 变加速动力学又称牛顿猝变动力学, 因其在混沌理论和非线性动力学中的应用而获得广泛关注. 高斯原理是一个具有极值性质的微分变分原理. 因此, 研究变加速动力学系统的广义高斯原理在理论和应用两方面都有重要意义. 文章提出并研究变加速动力学系统的广义高斯原理. 首先, 引入急动度空间的广义高斯变分概念, 将质点的达朗贝尔原理对时间求导数后与广义高斯变分点乘, 并利用高斯意义下的理想约束条件, 建立了变加速动力学系统的广义高斯原理. 在此基础上, 通过构造广义拘束函数建立并证明变加速动力学系统的广义高斯最小拘束原理, 并给出原理的阿佩尔形式、拉格朗日形式和尼尔森形式. 其次, 研究原理对变质量力学的推广. 从密歇尔斯基方程出发, 将它对时间求导并与广义高斯变分点乘, 建立了具有理想约束的变质量变加速动力学系统的广义高斯原理. 通过构造变质量系统的广义拘束函数, 建立并证明变质量力学系统变加速运动的广义高斯最小拘束原理. 文中以开普勒−牛顿空间问题为例, 利用所得的广义高斯最小拘束原理方法进行计算, 验证了方法的有效性.
- 变加速运动 /
- 牛顿猝变动力学 /
- 高斯最小拘束原理 /
- 变质量力学
Abstract: The motion with variable acceleration is common in daily life and engineering problems. Variable acceleration dynamics, also known as Newtonian jerky dynamics, has gained wide attention due to its application in chaos theory and nonlinear dynamics. Gauss principle is a differential variational principle with extreme value characteristics. Therefore, it is of great significance to study the generalized Gauss principle of dynamical systems with variable acceleration in both theory and application. In this paper, the generalized Gauss principle for dynamical systems with variable accelerated motion is presented and studied. Firstly, we introduce the concept of the generalized Gaussian variation in the jerky space. We take the derivative of d’Alembert principle of a particle with respect to time, and then calculate its dot product with the generalized Gaussian variation. By using the condition of ideal constraints in the sense of Gauss, we establish the generalized Gauss principle for dynamical systems with variable acceleration. On this basis, the generalized Gauss principle of least compulsion is established and proved by constructing the generalized compulsion function. The Appell form, Lagrange form and Nielsen form of the principle are given. Secondly, the extension of the principle to variable mass mechanics is explored. Starting from Meshchersky equation and taking its derivative with respect to time, and then calculating its dot product with the generalized Gaussian variation, we establish the generalized Gauss principle for variable-mass variable-acceleration dynamical systems with ideal constraints. The generalized compulsion function in the case of variable mass is constructed and the generalized Gauss principle of least compulsion for variable-mass variable-acceleration mechanical systems is established and proved. We take the Kepler-Newton problem as an example, and use the approach of the generalized Gauss least compulsion principle we presented to calculate, and verify the effectiveness of the method.
- variable acceleration motion /
- Newtonian jerky dynamics /
- Gauss principle of least compulsion /
- variable mass mechanics

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参考文献(42)

[1]	Linz SJ. Newtonian jerky dynamics: Some general properties. American Journal of Physics, 1998, 66(12): 1109-1114 doi: 10.1119/1.19052
[2]	黄沛天, 马善钧, 徐学翔等. 变加速动力学纵横. 力学与实践, 2004, 26(6): 85-87 (Huang Peitian, Ma Shanjun, Xu Xuexiang, et al. A review of the dynamics of varying accelerated motion. Mechanics in Engineering, 2004, 26(6): 85-87 (in Chinese) doi: 10.3969/j.issn.1000-0879.2004.06.032
[3]	de Winkel KN, Irmak T, Happee R, et al. Standards for passenger comfort in automated vehicles: Acceleration and jerk. Applied Ergonomics, 2023, 106: 103881 doi: 10.1016/j.apergo.2022.103881
[4]	Steiner W. On the impact of the jerk on structural dynamics. Proceedings in Applied Mathematics and Mechanics, 2021, 21(1): e202100202
[5]	Kenmogne F, Noubissie S, Ndombou GB, et al. Dynamics of two models of driven extended jerk oscillators: Chaotic pulse generations and application in engineering. Chaos, Solitons and Fractals, 2021, 152: 111291 doi: 10.1016/j.chaos.2021.111291
[6]	Schot SH. Jerk: The time rate of change of acceleration. American Journal of Physics, 1978, 46(11): 1090-1094 doi: 10.1119/1.11504
[7]	谈开孚, 赵永凯, 郭小弟. 谈加加速度. 力学与实践, 1988, 10(5): 46-51 (Tan Kaifu, Zhao Yongkai, Guo Xiaodi. On the jerk. Mechanics in Engineering, 1988, 10(5): 46-51 (in Chinese)
[8]	Linz SJ. Nonlinear dynamical models and jerky motion. American Journal of Physics, 1997, 65(6): 523-526 doi: 10.1119/1.18594
[9]	Ismail GM, Abu-Zinadah H. Analytic approximations to non-linear third order jerk equations via modified global error minimization method. Journal of King Saud University – Science, 2021, 33(1): 101219 doi: 10.1016/j.jksus.2020.10.016
[10]	Rech PC. Self-excited and hidden attractors in a multistable jerk system. Chaos, Solitons and Fractals, 2022, 164: 112614 doi: 10.1016/j.chaos.2022.112614
[11]	Braun F, Mereu AC. Zero-Hopf bifurcation in a 3D jerk system. Nonlinear Analysis: Real World Application, 2021, 59: 103245 doi: 10.1016/j.nonrwa.2020.103245
[12]	Singh PP, Roy BK. Pliers shaped coexisting bifurcation behaviors in a simple jerk chaotic system in comparison with 21 reported systems. IFAC PaperOnLine, 2022, 55(1): 920-926 doi: 10.1016/j.ifacol.2022.04.151
[13]	Karawanich K, Prommee P. High-complex chaotic system based on new nonlinear function and OTA-based circuit realization. Chaos, Solitons and Fractals, 2022, 162: 112536 doi: 10.1016/j.chaos.2022.112536
[14]	Tang DD, Zhang SR, Ren JL. Dynamics of a general jerky equation. Journal of Vibration and Control, 2019, 25(4): 922-932 doi: 10.1177/1077546318805583
[15]	梅凤翔, 刘端, 罗勇. 高等分析力学. 北京: 北京理工大学出版社, 1991 Mei Fengxiang, Liu Duan, Luo Yong. Advanced Analytical Mechanics. Beijing: Beijing Institute of Technology Press, 1991 (in Chinese)
[16]	马善钧, 徐学翔, 黄沛天等. 完整系统三阶Lagrange方程的一种推导与讨论. 物理学报, 2004, 53(11): 3648-3651 (Ma Shanjun, Xu Xuexiang, Huang Peitian, et al. The discussion on Lagrange equation containing third order derivatives. Acta Physica Sinica, 2004, 53(11): 3648-3651 (in Chinese) doi: 10.3321/j.issn:1000-3290.2004.11.005
[17]	Ma SJ, Liu MP, Huang PT. The forms of three-order Lagrangian equation in relative motion. Chinese Physics, 2005, 14(2): 244-246 doi: 10.1088/1009-1963/14/2/004
[18]	Ma SJ, Ge WG, Huang PT. The three-order Lagrange equation for mechanical systems of variable mass. Chinese Physics, 2005, 14(5): 879-881 doi: 10.1088/1009-1963/14/5/003
[19]	Ma SJ, Huang PT, Yan R, et al. Three-order Pseudo-Hamilton canonical equations. Chinese Physics, 2006, 15(10): 2193-2196 doi: 10.1088/1009-1963/15/10/001
[20]	Ma SJ, Yang XH, Yan R. Noether symmetry of three-order Lagrangian equations. Communications in Theoretical Physics, 2006, 46(2): 309-312 doi: 10.1088/0253-6102/46/2/026
[21]	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)
[22]	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)
[23]	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)
[24]	Udwadia FE, Kalaba RE. Analytical Dynamics - A New Approach. New York: Cambridge University Press, 2008
[25]	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)
[26]	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)
[27]	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)
[28]	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
[29]	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
[30]	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
[31]	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
[32]	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
[33]	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
[34]	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
[35]	张毅, 陈欣雨. 变质量力学系统的广义高斯原理及其对高阶非完整系统的推广. 力学学报, 2022, 54(10): 2883-2891 (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 (in Chinese) doi: 10.6052/0459-1879-22-202
[36]	Zhang BY, Gavin HP. Gauss’s principle with inequality constraints for multiagent navigation and control. IEEE Transactions on Automatic Control, 2022, 67(2): 810-823 doi: 10.1109/TAC.2021.3059677
[37]	Yang LS, Xue SF, Yao WL. Application of Gauss principle of least constraint in multibody systems with redundant constraints. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2021, 235(1): 150-163 doi: 10.1177/1464419320975301
[38]	Boyer F, Lebastard V, Candelier F, et al. Statics and dynamics of continuum robots based on Cosserat rods and optimal control theories. IEEE Transactions on Robotics, 2023, doi: 10.1109/TRO.2022.3226112
[39]	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) doi: 10.13209/j.0479-8023.2021.035
[40]	Konz M, Rudolph J. Gauss’s principle and tracking control of underactuated mechanical systems. IFAC PapersOnLine, 2021, 54(19): 365-370 doi: 10.1016/j.ifacol.2021.11.104
[41]	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)
[42]	张相武. 变质量完整力学系统的高阶运动微分方程. 物理学报, 2006, 55(4): 1543-1547 (Zhang Xiangwu. Higher order differential equations of motion for holonomic mechanical system of variable mass. Acta Physica Sinica, 2006, 55(4): 1543-1547 (in Chinese) doi: 10.7498/aps.55.1543

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变加速动力学系统的广义高斯最小拘束原理

doi: 10.6052/0459-1879-23-030

通讯作者:
张毅, 教授, 主要研究方向为分析力学. E-mail: zhy@mail.usts.edu.cn

翟相华, 讲师, 主要研究方向为分析力学. E-mail: zxh@mail.usts.edu.cn

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GENERALIZED GAUSS PRINCIPLE OF LEAST COMPULSION FOR VARIABLE-ACCELERATION DYNAMICAL SYSTEMS

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变加速动力学系统的广义高斯最小拘束原理

doi: 10.6052/0459-1879-23-030

通讯作者: 张毅, 教授, 主要研究方向为分析力学. E-mail: zhy@mail.usts.edu.cn 翟相华, 讲师, 主要研究方向为分析力学. E-mail: zxh@mail.usts.edu.cn

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GENERALIZED GAUSS PRINCIPLE OF LEAST COMPULSION FOR VARIABLE-ACCELERATION DYNAMICAL SYSTEMS

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通讯作者:
张毅, 教授, 主要研究方向为分析力学. E-mail: zhy@mail.usts.edu.cn

翟相华, 讲师, 主要研究方向为分析力学. E-mail: zxh@mail.usts.edu.cn