切塔耶夫型非完整系统的广义梯度表示

非定常非完整力学系统的稳定性研究是重要而又困难的问题,直接从微分方程出发来构造李雅普诺夫函数往往很难实现.本文给出了一种间接方法.提出了10类广义梯度系统的定义,并分别给出了10类广义梯度系统的微分方程.进一步研究一般切塔耶夫型非完整系统的广义梯度表示,给出该系统分别成为这10类广义梯度系统的条件,从而将切塔耶夫型非完整系统化成各类广义梯度系统.最后利用广义梯度系统的性质来研究切塔耶夫型非完整系统零解的稳定性.这种方法在直接构造李雅普诺夫函数发生困难时,显得更为有效.举例说明结果的应用.     

Nonholonomic systems with non-linear constraint equations

The development of a form of Lagrange's equations applicable with nonholonomic systems with non-linear constraint equations is presented and discussed. The analysis is based upon, and is an extension of. a method developed by the authors for nonholonomic systems with linear constraint equations in the generalized coordinate derivatives. The method is illustrated with the problem of the “balancing pole”.     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…     

Complete identification of chaos of nonlinear nonholonomic systems

The chaos of nonholonomic systems with two external nonlinear nonholonomic constraints where the magnitude of velocity is a constant and the magnitude of the velocity is a constant with a periodic disturbance, respectively, is completely identified for the first time. The scope of the chaos study is extended to nonlinear nonholonomic systems. By applying the nonlinear nonholonomic form of Lagrange’s equations, the dynamic equation is expressed. The existence of chaos in these two nonlinear nonholonomic systems is first wholly proved by all numerical criteria of chaos, i.e., the most reliable Lyapunov exponents, phase portraits, Poincaré maps, and bifurcation diagrams. Furthermore, it is found that the Feigenbaum number still holds for nonlinear nonholonomic systems.     

New methods to find solutions and analyze stability of equilibrium of nonholonomic mechanical systems

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives \({\ddot{q}}_0 ,{{\dddot{q}{} }}_{\!0} ,\ldots \) is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples.     

Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints

Based on the dynamical theory of multi-body systems with nonholonomic constraints and an algorithm for complementarity problems, a numerical method for the multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints is presented. In particular, a wheeled multi-body system is considered. Here,the state transition of stick-slip between wheel and ground is transformed into a nonlinear complementarity problem(NCP). An iterative algorithm for solving the NCP is then presented using an event-driven method. Dynamical equations of the multi-body system with holonomic and nonholonomic constraints are given using Routh equations and a constraint stabilization method. Finally, an example is used to test the proposed numerical method. The results show some dynamical behaviors of the wheeled multi-body system and its constraint stabilization effects.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

ON THE STABILITY OF NONHOLONOMIC MECHANICAL SYSTEMS WITH RESPECT TO PARTIAL VARIABLES

ＯＮＴＨＥＳＴＡＢＩＬＩＴＹＯＦＮＯＮＨＯＬＯＮＯＭＩＣＭＥＣＨＡＮＩＣＡＬＳＹＳＴＥＭＳＷＩＴＨＲＥＳＰＥＣＴＴＯＰＡＲＴＩＡＬＶＡＲＩＡＢＬＥＳＺｈｕＨａｉ－ｐｉｎｇ（朱海平）ＭｅｉＦｅｎｇ－ｘｉａｎｇ（梅凤翔）（ＢｅｉｊｉｎｇＵｎｉｖｅｒｓｉｔ...     

爬壁机器人系统的Noether对称性和守恒量

爬壁机器人的运动是一种模仿壁虎爬行的运动, 爬壁机器人的运动可分解为四肢带动身体的运动, 先前的研究都是基于牛顿力学的方法. 本文采用Lagrange 力学的方法建立爬壁机器人系统的运动方程, 并运用Lie群分析方法建立该系统的Noether对称性理论, 得出爬壁机器人的运动规律. 首先, 给出非完整爬壁机器人系统的动能、势能和Lagrange函数以及所受的非完整约束, 从而建立了非完整爬壁机器人系统的Lagrange方程; 其次, 引入关于时间和广义坐标的无限小变换, 提出了非完整爬壁机器人系统的Hamilton作用量和Hamilton作用量的基本变分公式; 第三, 给出爬壁机器人系统 Noether对称性变换和广义准对称变换的定义, 判据和存在的Noether守恒量, 并提出了非保守完整系统和非保守非完整爬壁机器人系统的Noether定理; 最后, 以圆锥面上爬壁机器人为例, 对给出的守恒量直接进行积分给出圆锥面上爬壁机器人整体运动的精确解和四肢运动的数值解, 发现了该爬壁机器人的运动规律, 很好地验证了非完整爬壁机器人系统的Noether对称性理论. 本文的研究为Lie群分析方法应用于其他复杂的机器人系统以及柔性机器人系统的对称性求解提出了一种新的对称性求解方法.      

包含伺服约束的非完整系统的对称性摄动与绝热不变量

建立包含伺服约束系统的运动微分方程,基于包含伺服约束系统对称性与不变量理论研究包含伺服约束系统的对称性摄动与绝热不变量 问题,提出了包含伺服约束系统的高阶绝热不变量概念,证明了精确不变量与绝热不变量存在的条件及形式,并举例说明结果的应用。     

Numerical simulation of nonholonomic rigid-body systems

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D’Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 106–115, September 2006.     

Szebehely's problem for nonholonomic systems with a given first integral

I.IntroductionTheinverseproblemofdynamicsisoneoftheimportantsubjectsinmechanics.In1977,Szebehelysetforthaninverseproblemforthedeterminationofthet'orcefunctiontoamaterialpointintheplanefromparametricfamilyoftrajectories,andobtainedalinearfirstorderpartialdifferentialequationfortheforcefunction.Later,Erdil'l,MellsandPirast=l,MellsandBorgherol'l,BoilsandMertnsl4]extendedSzebehely'sproblemtoboththreeandndimensionalholonomicsystem.Recently,theauthorandProfessorMetFengxiangl'1studiedtheSzebehe…     

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate the application of the result.     

非完整系统稳定性的若干进展

介绍非完整系统稳定性理论的某些近代进展,包括非完整系统平衡位置关于全部变量的稳定性,平衡状态流形的稳定性,平衡位置关于部分变量稳定性及其与关于全部变量的稳定性的关系,平稳运动的稳定性,以及非完整控制系统的镇定．同时,讨论非完整系统稳定性的几个主要应用,并给出几个未来研究方向的建议      

A field method for integrating the equations of motion of nonholonomic controllable systems

This paper presents a field method for integrating the equations of motion of nonholonomic controllable systems. An example is given to illustrate the application of the method.     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

Quasi-momentum theorem in Riemann-Cartan space

The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.     

非完整系统的Lie对称性与Noether对称性

研究非完整系统的Lie对称性与Noether对称性及其间的关系,具体研究了Chetaev型变量质量非完整系统和非Chetaev型非完整系统的Lie对称性与Noether对称性。给出Lie对称性导致Noether对称性及Noether对称性导致Lie对称性的条件。     

非完整约束系统几何动力学研究进展:Lagrange理论及其它

近10年来, 非完整力学的发展主要集中在两个相互关联的方向上, 一个是非完整运动规划, 另一个则是非完整约束系统的几何动力学, 这两个研究方向都充分地利用了现代几何学, 如纤维丛理论、辛流形和Poisson流形结构等等.本文主要综述非完整约束系统几何动力学的外附型和内禀型Lagrange理论, 包括非定常力学系统所需要的射丛几何学的基本概念、射丛按约束的直和分解、约束流形上的水平分布、D'Alembert-Lagrange方程与Chaplygin方程的整体描述、以及Riemann-Cartan流形上的非完整力学, 文中对Chetaev条件和d-δ交换关系的几何意义作了深入讨论.除此之外, 简要评述非完整力学的Hamilton理论与赝Poisson结构、Noether对称性和Lie对称性、动量映射与对称约化、Vakonomic动力学等几个非常重要专题的研究进展.