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用具有负定矩阵的梯度系统构造稳定的变质量力学系统

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李彦敏1,*,, 章婷婷2, 梅凤翔3
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

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

LiYan-Min1,*,, ZhangTing-Ting2, MeiFeng-Xiang3
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
中图分类号:O316
文献标识码:A

通讯作者:*通讯作者:李彦敏, 教授, 主要研究方向: 分析力学. E-mail: hnynmnl@163.com
收稿日期:2017-08-17
接受日期:2017-11-10
网络出版日期:2018-02-20
版权声明:2018《力学学报》编辑部《力学学报》编辑部 所有
基金资助:国家自然科学基金(批准号: 11372169,11272050, 11572034)

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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.

关键词: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.

Keywords:variable mass system;gradient system;stability

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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[J]. 力学学报, 2018, 50(1): 109-113 https://doi.org/10.6052/0459-1879-17-283
Li Yan-Min, Zhang Ting-Ting, Mei Feng-Xiang. Stable variable mass mechanical systems constructed by using a gradient system with negative-definite matrix[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(1): 109-113 https://doi.org/10.6052/0459-1879-17-283

1 引言

随着科学技术的进步,变质量系统动力学的研究显得越来越重要, 喷气飞机,火箭,卫星,航天器等一般都是变质量系统, 有关变质量系统动力学的研究已取得重要进展,如专著[1-3]. 对变质量系统总希望它的解是稳定的或渐近稳定的. 梯度系统是一类重要的动力系统, 动力学方程的梯度性质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, 在流体力学、固体力学、热力学、光学、经典与量子场论以及工程科学等诸多领域得到应用[4,5]. 梯度系统适合用Lyapunov函数来研究[4,5,6,7,8]. 有关约束力学系统与梯度系统的关联研究已取得一些进展,如文献[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. 专著[31]主要涉及通常梯度系统,斜梯度系统,具有对称负定和半负定矩阵的梯度系统等. 本文提出了一类梯度系统,其中的矩阵是负定非对称的. 适当选取梯度系统中的矩阵和函数使这类梯度系统能够较好地研究解的稳定性. 把这类梯度系统和变质量力学系统有机结合,由这类梯度系统来构造解为稳定的变质量力学系统.

2 变质量系统的运动微分方程

变质量完整力学系统的微分方程有形式
$\frac{\text{d}}{\text{d}t}\frac{\partial T}{\partial {{{\dot{q}}}_{s}}}-\frac{\partial T}{\partial {{q}_{s}}}={{Q}_{s}}+{{P}_{s}}\quad \quad \left( s=1,2,\cdots ,n \right)$ (1)
其中
$T=\frac{1}{2}{{m}_{i}}{{\mathbf{\dot{r}}}_{i}}\cdot {{\mathbf{\dot{r}}}_{i}}$
为系统的动能,${{Q}_{s}}={{Q}_{s}}\left( t,\mathbf{q},\mathbf{\dot{q}} \right)$为广义力,${{P}_{s}}$为广义反推力,有
${{P}_{s}}={{\dot{m}}_{i}}\left( {{\mathbf{u}}_{i}}+{{{\mathbf{\dot{r}}}}_{i}} \right)\cdot \frac{\partial {{\mathbf{r}}_{\mathbf{i}}}}{\partial {{q}_{s}}}-\frac{1}{2}v_{i}^{2}\frac{\partial {{m}_{i}}}{\partial {{q}_{s}}}+\frac{\text{d}}{\text{d}t}\left( \frac{1}{2}v_{i}^{2}\frac{\partial {{m}_{i}}}{\partial {{{\dot{q}}}_{s}}} \right)$ (2)
其中ui为微粒相对第i个.质点的速度,${{\mathbf{v}}_{\mathbf{i}}}$为第i..个质点的速度. 研究${{m}_{i}}={{m}_{i}}\left( t \right)$,即质量.仅依赖时间的情形,此时有
${{P}_{s}}={{\dot{m}}_{i}}\left( {{\mathbf{u}}_{\mathbf{i}}}+{{{\mathbf{\dot{r}}}}_{\mathbf{i}}} \right)\cdot \frac{\partial {{\mathbf{r}}_{\mathbf{i}}}}{\partial {{q}_{s}}}\quad \quad \left( s=1,2,\cdots ,n \right)$ (3)
方程(1)比常质量系统的Lagrange方程要复杂的多,不仅Ps与质量变化有关,而且动能T.也与质量变化有关. 假设系统(1)非奇异,即设
$\det \left( \frac{{{\partial }^{2}}T}{\partial {{{\dot{q}}}_{s}}\partial {{{\dot{q}}}_{k}}} \right)\ne 0$ (4)
此时可由方程(1)求得所有广义加速度,记作
.${{\ddot{q}}_{s}}={{\alpha }_{s}}\left( t,{{q}_{k,}}{{{\dot{q}}}_{k}} \right)\quad \quad \left( s,k=1,2,\cdots ,n \right)$ (5)

3 具有负定矩阵的梯度系统

系统的微分方程有形式
${{\dot{q}}_{s}}={{c}_{sk}}\frac{\partial V}{\partial {{q}_{k}}}\quad \quad \left( s,k=1,2,\cdots ,n \right)$ (6)
其中相同指标表示求和,矩阵$\left( {{c}_{sk}} \right)$是负定非对称的. 按方程(6)求$\dot{V}$,得
$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{q}_{s}}}{{c}_{sk}}\frac{\partial V}{\partial {{q}_{k}}}$ (7)
如果函数$V=V\left( t,\mathbf{q} \right)$在解的邻域内正定,为研究解的稳定性,总希望$\dot{V}$负.定或半负定. 因此,首先希望二次型
${{F}_{m}}=\left( {{q}_{1}}\ {{q}_{2}}\cdots {{q}_{m}} \right)\left( \begin{matrix}{{c}_{11}} & {{c}_{12}} & \cdots & {{c}_{1m}} \\{{c}_{21}} & {{c}_{22}} & \cdots & {{c}_{2m}} \\\cdots & \cdots & \cdots & \cdots \\{{c}_{m1}} & {{c}_{m2}} & \cdots & {{c}_{mm}} \\\end{matrix} \right)\left( \begin{matrix}{{q}_{1}} \\{{q}_{2}} \\\vdots \\{{q}_{m}} \\\end{matrix} \right)$ (8)
是负定或半负定的. 因为矩阵$\left( {{c}_{sk}} \right)$负定,还不能保证二次型(8)负定或半负定. 因此,在选取矩阵$\left( {{c}_{sk}} \right)$时,应避免二次型(8)为变号的情形. 函数V应选.为正定的. 这样,才有利于研究解的稳定性问题.

4 构造稳定的变质量系统

为求得与梯度系统(6)相应的变质量系统(5),首先,需将一阶方程(6)化成二阶方程,然后再对照方程(5). 将方程(6)等式两边对时间t求导,可得
${{\ddot{q}}_{s}}={{\dot{c}}_{sk}}\frac{\partial V}{\partial {{q}_{k}}}\text{+}{{c}_{sk}}\left( \frac{{{\partial }^{2}}V}{\partial {{q}_{k}}\partial t}+\frac{{{\partial }^{2}}V}{\partial {{q}_{k}}\partial {{q}_{s}}}{{{\dot{q}}}_{s}} \right)\quad \left( s,k=1,2,\cdots ,n \right)$ (9)
方程(6)和方程(9)结合消去其他变量,就可得到其中一个变量的二阶方程. 有的梯度系统(6)可化成二阶系统,有的则不能. 因此,在选梯度系统(6)时需避免后一情形. 同时,方程(5)中包含动能$T$,广义力${{Q}_{s}}$和广义反推力${{P}_{s}}$,它们都与质.量变化相关,构造这些函数是相当复杂的. 为简单起见,研究下述问题:对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.

5 算例

例1 已知梯度系统为
$\begin{align}& \left( \begin{matrix}{{{\dot{q}}}_{1}} \\{{{\dot{q}}}_{2}} \\\end{matrix} \right)=\left( \begin{matrix}-1 & 1 \\-1 & 0 \\\end{matrix} \right)\left( \begin{matrix}\frac{\partial V}{\partial {{q}_{1}}} \\\frac{\partial V}{\partial {{q}_{2}}} \\\end{matrix} \right) \\& V=\frac{1}{2}{{\left( q{}_{1} \right)}^{2}}+{{\left( {{q}_{2}} \right)}^{2}}
\end{align}$ (10)
试求相应的变质量系统.
解 方程(6)给出
$\begin{align}& {{{\dot{q}}}_{1}}=-{{q}_{1}}+{{q}_{2}} \\& {{{\dot{q}}}_{2}}=-{{q}_{1}} \\\end{align}$ (11)
按方程求$\dot{V}$, 得
$\dot{V}=-{{\left( {{q}_{1}} \right)}^{2}}$
它在解${{q}_{1}}={{q}_{2}}=0$的邻域内是半负定的,而$V$是正定的,因此,解${{q}_{1}}={{q}_{2}}=0$是稳定的.
对变质量系统的单自由度运动,方程(5)给出
${{\left( m\dot{x} \right)}^{\bullet }}=Q+\dot{m}\left( u+\dot{x} \right)$ . (12)
其中u为微粒分离的相对速度.
将方程(11)对t求导数.,得
${{\ddot{q}}_{1}}=-{{\dot{q}}_{1}}-{{q}_{1}}$
.令${{q}_{1}}=x$,则有
$\ddot{x}=-\dot{x}-x$ (13)
对照方程(11)、(12),得
.$\frac{Q}{m}+\frac{{\dot{m}}}{m}u=-\dot{x}-x$ .. (14)
这就是广义力$Q$和微粒分离的相对速度$u$应满足的关系. 取$u=0$,则有
$Q=-m\left( \dot{x}+x \right)$
取$u=-\dot{x}$,则有
$Q=-m\left( \dot{x}+x \right)+\dot{m}\dot{x}$
.等等.
例2 已知梯度系统为
$\left(\begin{matrix}{{{\dot{q}}}_{1}} \\{{{\dot{q}}}_{2}} \\\end{matrix} \right)=\left( \begin{matrix}-1 & 0 \\ 1 & -2 \\ \end{matrix} \right)\left( \begin{matrix} \frac{\partial V}{\partial {{q}_{1}}} \\ \frac{\partial V}{\partial {{q}_{2}}} \\ \end{matrix} \right)$ (15)
$V={{\left( {{q}_{1}} \right)}^{2}}+{{\left( {{q}_{2}} \right)}^{2}}-{{q}_{1}}{{q}_{2}}$
试求相应的变质量系统.
解方程(6)给出
$\begin{align} {{{\dot{q}}}_{1}}=-2{{q}_{1}}+{{q}_{2}} \\ {{{\dot{q}}}_{2}}=4{{q}_{1}}-5{{q}_{2}} \\ \end{align}$ (16)
按方程求V,得
$\dot{V}=-8{{\left( {{q}_{1}} \right)}^{2}}-11{{\left( {{q}_{2}} \right)}^{2}}+17{{q}_{1}}{{q}_{2}}$
它在${{q}_{1}}={{q}_{2}}=0$的邻域内负定,而$V$正定,因此,解${{q}_{1}}={{q}_{2}}=0$是.渐近稳定的.
方程(16)可化成一个二阶方程
${{\ddot{q}}_{1}}=-7{{\dot{q}}_{1}}-6{{q}_{1}}$
令${{q}_{1}}=x$,则
$\ddot{x}=-7\dot{x}-6x$
将x代入式(12),得
$\frac{Q}{m}+\frac{{\dot{m}}}{m}u=-7\dot{x}-6x$ . (17)
这就是广义力$Q$和微粒分离相对速度$u$应满足的关系. 在这样的$Q$,$u$下.,变质量系统的解$x=\dot{x}=0$是渐近稳定的.
例3 已知梯度系统为
$\left( \begin{matrix}{{{\dot{q}}}_{1}} \\{{{\dot{q}}}_{2}} \\\end{matrix} \right)=\left( \begin{matrix}-\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right] & -1 \\ 1 & -2 \\ \end{matrix} \right)\left( \begin{matrix}\frac{\partial V}{\partial {{q}_{1}}} \\\frac{\partial V}{\partial {{q}_{2}}} \\\end{matrix} \right)$ (18)$V=\frac{1}{2}{{\left( {{q}_{1}} \right)}^{2}}\left( 1+\frac{1}{1+t} \right)+\frac{1}{2}{{\left( {{q}_{2}} \right)}^{2}}$
试求与之相应的变质量系统.
解方程(6)给出
$\begin{align} {{{\dot{q}}}_{1}}=-{{q}_{1}}\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right]\ \left( 1+\frac{1}{1+t} \right)-{{q}_{2}} \\ {{{\dot{q}}}_{2}}={{q}_{1}}\left( 1+\frac{1}{1+t} \right)-2{{q}_{2}} \\ \end{align}$ (19)
按方程求$\dot{V}$,得
$\dot{V}=-\frac{1}{2}{{\left( {{q}_{1}} \right)}^{2}}\frac{1}{{{\left( 1+t \right)}^{2}}}-{{\left( {{q}_{1}} \right)}^{2}}{{\left( 1+\frac{1}{1+t} \right)}^{2}}\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right]-2{{\left( {{q}_{2}} \right)}^{2}}$
它在${{q}_{1}}={{q}_{2}}=0$的邻域内负定,而$V$正定且渐减,因此,解${{q}_{1}}={{q}_{2}}=0$是一致渐近稳定的. 将式(19)对$t$求导数,得
$\begin{align}& {{{\ddot{q}}}_{1}}=-{{{\dot{q}}}_{1}}\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right]\,\left( 1+\frac{1}{1+t} \right)-2{{{\dot{q}}}_{1}}{{\left( {{q}_{1}} \right)}^{2}}\left( 1+\frac{1}{1+t} \right)+{{q}_{1}}\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right]\frac{1}{{{\left( 1+t \right)}^{2}}} \\& \quad \,\quad -{{q}_{1}}\left( 1+\frac{1}{1+t} \right)-2{{{\dot{q}}}_{1}}-2{{q}_{1}}\left[ 1+{{\left( {{q}_{1}} \right)}^{2}} \right]\,\left( 1+\frac{1}{1+t} \right) \\\end{align}$.令${{q}_{1}}=x$,则有. $\ddot{x}=-\dot{x}\left\{ 2+\left( 1+3{{x}^{2}} \right)\left( 1+\frac{1}{1+t} \right) \right\}-x\left\{ 1+\frac{1}{1+t}+2\left( 1+{{x}^{2}} \right)\left( 1+\frac{1}{1+t} \right)-\left( 1+{{x}^{2}} \right)\frac{1}{{{\left( 1+t \right)}^{2}}} \right\}$.将其代入(12),得.. $\begin{align}& \frac{Q}{m}+\frac{{\dot{m}}}{m}u=-\dot{x}\left[ 2+\left( 1+3{{x}^{2}} \right)\left( 1+\frac{1}{1+t} \right) \right] \\& \quad \quad \quad \quad -x\left[ 1+\frac{1}{1+t}+\left( 1+{{x}^{2}} \right)\left( 2+\frac{2}{1+t}-\frac{1}{{{\left( 1+t \right)}^{2}}} \right) \right] \\\end{align}$ (20)这就是变质量系统的广义力$Q$和微粒分离相对速度$u$应满足的关系. 此时,系统的解$x=\dot{x}=0$是一致渐近稳定的.

6 结论

梯度系统具有较强的物理意义, 二阶梯度系统是物理上标准的振动模型. 如果一个力学系统可以化成梯度系统,那么就可借助梯度系统的性质来研究该力学系统的积分、解的稳定性和渐进行为. 约束力学系统稳定性研究是一项重要而又困难的课题, 从微分方程出发直接构成Lyapunov函数往往很难实现. 本文利用矩阵为负定的梯度系统(6)构造出解为稳定的或渐近稳定的一些变质量系统. 为变质量力学系统解的稳定性研究提供了一种新的方法. 所举例子是简单低阶的. 对复杂高阶的系统,构造起来困难较大,但方法是一样的. 本文的构造方法也适合其它类型的动力学系统.
The authors have declared that no competing interests exist.

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