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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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他类型的动力学系统.
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.
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.
引文
1Novoselov VS.Analytical Mechanics of Systems with Variable Mass.Leningrad:LU Press,1969(in Russian)
2 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))
3Ge ZM.Advaneed Dynamics for Variable Mass Systems.Chinese Taipei:Gaulih Book Company,1998
4Hirsch MW,Smale S.Differentical Equations,Dynamical Systems and Linear Algebra.New York:Academic Press,1974
5 Mc Lachlan RI,Quispel GRW,Robidoux N.Geometric integration using discrete gradients.Philosophical Transactions of the Royal Society A,1999,357(1754):1021-1045
6陈向炜,赵永红,刘畅.变质量完整动力学系统的共形不变性与守恒量.物理学报,2009,58(8):5150-5154(Chen Xiangwei,Zhao Yonghong,Liu Chang.Conformal invariance and conserved quantity for holonomic mechanical systems with variable mass.Acta Physica Sinica,2009,58(8):5150-5154)
7 李彦敏.变质量非完整力学系统的共形不变性.云南大学学报(自然科学版),2010,32(1):52-57(Li Yanmin.Conformal invariance for nonholonomic mechanical systems with variable mass.Journal of Yunnan University(Natural Sciences),2010,32(1):52-57(in Chinese))
8郑世旺,王建波,陈向炜.变质量非完整系统Tzénoff方程的Lie对称性与其导出的守恒量.物理学报,2012,61(11):111101(Zheng Shiwang,Wang Jianbo,Chen Xiangwei.Lie symmetry and their conserved quantities of Tzénoff equations for the vairable mass nonholonomic systems.Acta Physica Sinica,2012,61(11):111101(in Chinese))
9 楼智美,梅凤翔.力学系统的二阶梯度表示.物理学报,2012,61 (2):024502(Lou Zhimei,Mei Fengxiang.A second order gradient representation of mechanics system.Acta Physica Sinica,2012,61 (2):024502(in Chinese))
10梅凤翔,吴惠彬.一阶Lagrange系统的梯度表示.物理学报,2013,62 (21):214501(Mei Fengxiang,Wu Huibin.A gradient representation of first-order Lagrange system.Acta Physica Sinica,2013,62(21):214501(in Chinese))
11Chen XW,Zhao GL,Mei FX.A fractional gradient representation of the Poincaréequations.Nonlinear Dynamics,2013,73(1):579-582
12 Mei FX,Wu HB.Skew-gradient representation of generalized Birkhoffian system.Chinese Physics B,2015,24(10):104502
13梅凤翔,吴惠彬.广义Birkhoff系统与一类组合梯度系统.物理学报,2015,64(18):184501(Mei Fengxiang,Wu Huibin.Generalized Birkhoff system and a kind of combined gradient system.Acta Physica Sinica,2015,64(18):184501(in Chinese))
14 吴惠彬,梅凤翔.事件空间中完整力学系统的梯度表示.物理学报,2015,64(23):234501(Wu Huibin,Mei Fengxiang.A gradient representation of holonomic system in the event space.Acta Physica Sinica,2015,64(23):234501(in Chinese))
15Tomá?B,Ralph C,Eva F.Every ordinary differential equation with a strict Lyapunov function is a gradient system.Monatsh Math.,2012,166:57-72
16 Marin AM,Ortiz RD,Rodriguez JA.A generalization of a gradient system.International Mathematical Forum,2013,8:803-806
17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))
18 Lin L,Luo SK.Fractional generalized Hamiltonian mechanics.Acta Mechanica,2013,224(8):1757-1771
19Luo SK,He JM,Xu YL.Fractional Birkhoffian method for equilibrium stability of dynamical systems.International Journal of NonLinear Mechanics,2016,78(1):105-111
20 陈向炜,曹秋鹏,梅凤翔.切塔耶夫型非完整系统的广义梯度表示.力学学报,2016,48(3):684-691(Chen Xiangwei,Cao Qiupeng,Mei Fengxiang.Generalized gradient representation of nonholonomic system of Chetaev’s type.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):684-691(in Chinese))
21Mei FX,Wu HB.Two kinds of generalized gradient representations for holonomic mechanical systems.Chinese Physics B,2016,25(1):014502
22 Chen XW,Zhang Y,Mei FX.An application of a combined gradient system to stabilize a mechanical system.Chinese Physics B,2016,25(10):100201
23Hirsch MW,Smale S,Devaney RL.Differential Equations,Dynamical Systems,and an Introduction to Chaos.Singapore:Elsevier,2008
24 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))
25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))
26 Chen XW,Mei FX.Constrained mechanical systems and gradient systems with strong Lyapunov functions.Mechanics Research Communications,2016,76:91-95
27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))
28 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))
29陈向炜,曹秋鹏,梅凤翔.广义Birkhoff系统稳定性对双参数的依赖关系.力学季刊,2017,38(1):108-112(Chen Xiangwei,Cao Qiupeng,Mei Fengxiang.Dependance of stability of generalized Birkhoff system on two parameters.Chinese Quarterly of Mechanics,2017,38(1):108-112(in Chinese))
30 张晔,陈向炜.弱非线性耦合二维各向异性谐振子的动力学行为.动力学与控制学报,2017,15(5):410-414(Zhang Ye,Chen Xiangwei.Dynamics behavior of weak nonlinear coupled two-dimensional anisotropic harmonic oscillator.J of Dynam.and Control,2017,15 (5):410-414(in Chinese))
31梅凤翔,吴惠彬.约束力学系统的梯度表示(上下册).北京:科学出版社,2016(Mei Fengxiang,Wu Huibin.Gradient Representations of Constrained Mechanical System,(Vol 1,2).Beijing:Science Press,2016(in Chinese))
2 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))
3Ge ZM.Advaneed Dynamics for Variable Mass Systems.Chinese Taipei:Gaulih Book Company,1998
4Hirsch MW,Smale S.Differentical Equations,Dynamical Systems and Linear Algebra.New York:Academic Press,1974
5 Mc Lachlan RI,Quispel GRW,Robidoux N.Geometric integration using discrete gradients.Philosophical Transactions of the Royal Society A,1999,357(1754):1021-1045
6陈向炜,赵永红,刘畅.变质量完整动力学系统的共形不变性与守恒量.物理学报,2009,58(8):5150-5154(Chen Xiangwei,Zhao Yonghong,Liu Chang.Conformal invariance and conserved quantity for holonomic mechanical systems with variable mass.Acta Physica Sinica,2009,58(8):5150-5154)
7 李彦敏.变质量非完整力学系统的共形不变性.云南大学学报(自然科学版),2010,32(1):52-57(Li Yanmin.Conformal invariance for nonholonomic mechanical systems with variable mass.Journal of Yunnan University(Natural Sciences),2010,32(1):52-57(in Chinese))
8郑世旺,王建波,陈向炜.变质量非完整系统Tzénoff方程的Lie对称性与其导出的守恒量.物理学报,2012,61(11):111101(Zheng Shiwang,Wang Jianbo,Chen Xiangwei.Lie symmetry and their conserved quantities of Tzénoff equations for the vairable mass nonholonomic systems.Acta Physica Sinica,2012,61(11):111101(in Chinese))
9 楼智美,梅凤翔.力学系统的二阶梯度表示.物理学报,2012,61 (2):024502(Lou Zhimei,Mei Fengxiang.A second order gradient representation of mechanics system.Acta Physica Sinica,2012,61 (2):024502(in Chinese))
10梅凤翔,吴惠彬.一阶Lagrange系统的梯度表示.物理学报,2013,62 (21):214501(Mei Fengxiang,Wu Huibin.A gradient representation of first-order Lagrange system.Acta Physica Sinica,2013,62(21):214501(in Chinese))
11Chen XW,Zhao GL,Mei FX.A fractional gradient representation of the Poincaréequations.Nonlinear Dynamics,2013,73(1):579-582
12 Mei FX,Wu HB.Skew-gradient representation of generalized Birkhoffian system.Chinese Physics B,2015,24(10):104502
13梅凤翔,吴惠彬.广义Birkhoff系统与一类组合梯度系统.物理学报,2015,64(18):184501(Mei Fengxiang,Wu Huibin.Generalized Birkhoff system and a kind of combined gradient system.Acta Physica Sinica,2015,64(18):184501(in Chinese))
14 吴惠彬,梅凤翔.事件空间中完整力学系统的梯度表示.物理学报,2015,64(23):234501(Wu Huibin,Mei Fengxiang.A gradient representation of holonomic system in the event space.Acta Physica Sinica,2015,64(23):234501(in Chinese))
15Tomá?B,Ralph C,Eva F.Every ordinary differential equation with a strict Lyapunov function is a gradient system.Monatsh Math.,2012,166:57-72
16 Marin AM,Ortiz RD,Rodriguez JA.A generalization of a gradient system.International Mathematical Forum,2013,8:803-806
17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))
18 Lin L,Luo SK.Fractional generalized Hamiltonian mechanics.Acta Mechanica,2013,224(8):1757-1771
19Luo SK,He JM,Xu YL.Fractional Birkhoffian method for equilibrium stability of dynamical systems.International Journal of NonLinear Mechanics,2016,78(1):105-111
20 陈向炜,曹秋鹏,梅凤翔.切塔耶夫型非完整系统的广义梯度表示.力学学报,2016,48(3):684-691(Chen Xiangwei,Cao Qiupeng,Mei Fengxiang.Generalized gradient representation of nonholonomic system of Chetaev’s type.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):684-691(in Chinese))
21Mei FX,Wu HB.Two kinds of generalized gradient representations for holonomic mechanical systems.Chinese Physics B,2016,25(1):014502
22 Chen XW,Zhang Y,Mei FX.An application of a combined gradient system to stabilize a mechanical system.Chinese Physics B,2016,25(10):100201
23Hirsch MW,Smale S,Devaney RL.Differential Equations,Dynamical Systems,and an Introduction to Chaos.Singapore:Elsevier,2008
24 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))
25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))
26 Chen XW,Mei FX.Constrained mechanical systems and gradient systems with strong Lyapunov functions.Mechanics Research Communications,2016,76:91-95
27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))
28 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))
29陈向炜,曹秋鹏,梅凤翔.广义Birkhoff系统稳定性对双参数的依赖关系.力学季刊,2017,38(1):108-112(Chen Xiangwei,Cao Qiupeng,Mei Fengxiang.Dependance of stability of generalized Birkhoff system on two parameters.Chinese Quarterly of Mechanics,2017,38(1):108-112(in Chinese))
30 张晔,陈向炜.弱非线性耦合二维各向异性谐振子的动力学行为.动力学与控制学报,2017,15(5):410-414(Zhang Ye,Chen Xiangwei.Dynamics behavior of weak nonlinear coupled two-dimensional anisotropic harmonic oscillator.J of Dynam.and Control,2017,15 (5):410-414(in Chinese))
31梅凤翔,吴惠彬.约束力学系统的梯度表示(上下册).北京:科学出版社,2016(Mei Fengxiang,Wu Huibin.Gradient Representations of Constrained Mechanical System,(Vol 1,2).Beijing:Science Press,2016(in Chinese))
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