用具有负定矩阵的梯度系统构造稳定的变质量力学系统

李彦敏; 章婷婷; 梅凤翔

doi:10.6052/0459-1879-17-283

用具有负定矩阵的梯度系统构造稳定的变质量力学系统

doi: 10.6052/0459-1879-17-283

1.
商丘师范学院物理与电气信息学院, 商丘 476000
2苏州科技大学数理学院, 苏州 215009
3北京理工大学宇航学院, 北京 100081

基金项目: 国家自然科学基金(批准号: 11372169,11272050, 11572034)

详细信息

作者简介:
*通讯作者：李彦敏, 教授, 主要研究方向: 分析力学. E-mail: hnynmnl@163.com

通讯作者:
李彦敏

中图分类号: O316;
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出版历程
- 收稿日期: 2017-08-17
- 刊出日期: 2018-01-18

Stable variable mass mechanical systems constructed by using a gradient system with negative-definite matrix

1.
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

摘要

摘要: 随着科学技术的发展,对喷气飞机、火箭等变质量系统动力学的研究显得越来越重要, 并且总是希望变质量系统的解是稳定的或渐近稳定的. 而通用的研究稳定性的Lyapunov直接法有很大难度, 因为直接从微分方程出发构造Lyapunov函数往往很难实现. 本文给出一种研究稳定性的间接方法, 即梯度系统方法. 该方法不但能揭示动力学系统的内在结构, 而且有助于探索系统的稳定性、渐进性和分岔等动力学行为. 梯度系统的函数V通常取为Lyapunov函数, 因此梯度系统比较适合用Lyapunov函数来研究. 列写出变质量完整力学系统的运动方程,在系统非奇异情形下,求得所有广义加速度. 提出一类具有负定矩阵的梯度系统, 并研究该梯度系统解的稳定性. 把这类梯度系统和变质量力学系统有机结合,给出变质量力学系统的解可以是稳定的或渐近稳定的条件, 进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的变质量力学系统. 通过具体例子,研究了变质量系统的单自由度运动,在怎样的质量变化规律、微粒分离速度和加力下,其解是稳定的或渐近稳定的. 本文的构造方法也适合其它类型的动力学系统.
- 变质量系统 /
- 梯度系统 /
- 稳定性
Abstract: With the development of science and technology, it is more and more important to study the dynamics of variable mass system such as jet aircraft and rocket, and it is always hoped that the solutions of the variable mass system are stable or asymptotically stable. It is difficult to study the stability by using Lyapunov direct methods because of the difficulty of constructing Lyapunov functions directly from the differential equations of the mechanical system. This paper presents an indirect method for studying stability, that is, gradient system method. This method can not only reveal the internal structure of dynamic system, but also help to explore the dynamic behavior such as the stability, asymptotic and bifurcation. The function V of the gradient system is usually taken as a Lyapunov function, so the gradient system is more suitable to be studied with the Lyapunov function. The equations of motion for the holonomic mechanical system with variable mass are listed, and all generalized accelerations are obtained in the case of non-singular system. A class of gradient system with negative-definite matrix is proposed, and the stability of the solutions of the gradient system is studied. This kind of gradient system and variable mass mechanical system are combined, then the conditions under which the solutions of the mechanical systems with variable mass can be stable or asymptotically stable are given. Further the mechanical system with variable mass whose solution is stable or asymptotically stable is constructed by using the gradient system with non-symmetrical negative-definite matrix. Through specific examples, it is studied that the solutions of the single degree of freedom motion of a variable mass system are stable or asymptotically stable under some conditions of the laws of mass change, particle separation velocity and force. The method is also suitable for the study of other constrained mechanical systems.
- variable mass system /
- gradient system /
- stability

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参考文献(1)

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