GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Gibbs-Appell’s equations of motion are extended to the most generalvariable mass nonholonomic mechanical systems.Then the Gibbs-Appell’s equations ofmotion in terms of generalized coordinates or quasi-coordinates and an integral variationalprinciple of variable mass nonlinear nonholonomic mechanical systems are obtained.Finally,an example is given.     

ON THE NEW FORMS OF THE DIFFERENTIAL EQUATIONS OF THE SYSTEMS WITH HIGHER-ORDER NONHOLONOMIC CONSTRAINTS

In this paper,the new forms of the differential equations of motion of the systems withhigher-order nonholonomic constraints are obtained at first,and then the equivalencebetween these equations and the known equations is demonstrated.Finally an example isgiven to illustrate the application of our new equations.     

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.     

KANE’S EQUATIONS FOR PERCUSSION MOTION OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Kane’s equations for the Routh’s form of variable massnonholonomic systems are established.and the Kane’s equations for percussion motionof variable mass holonomic and nonholonomic systems are deduced from them. Secondly,the equivalence to Lagrange’s equations for percussion motion and Kane’sequations is obtained,and the application of the new equation is illustrated by anexample.     

EXTENSION OF THE MAC-MILLAN′S EQUATIONS TO NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this article, the Mac-Millan's equations are extended to the most general nonholonomic mechanical systems and the generalized Mac-Millan's equations for nonlinear nonholonomic systems are obtained. And then the equivalence between the generalized Mac-Millan's equations and the generalized Chaplygin's equations is demonstrated. Finally an example is given,     

THE EQUATIONS OF MOTION OF A MECHANICAL SYSTEM IN MATRIX FORM

This work recommends methods of construction of equations of motion of mechanical systems in matrix form.The use of a matrix form allows one to write an equation of dynamics in compact form,convenient for the in vestigation of multidimensional mechanical systems with the help of computers.Use is made of different methods of constructing equations of motion,based on the basic laws of dynamics as well as on the principles of D‘Alambert-L(?)range,Hamilton-Ostrogradski and Gauss.     

ON THE RELATIONSHIP BETWEEN THE VACCO MODEL AND CHETAEV MODEL

In this paper,the geometric property and the mechanical property ofnon-holonomic constraints are studied.Vacco equation is obtained by using theclassical Hamilton principle.On the basis of Vacco equation,Chetaev condition isobtained by using mechanics principle of determining the ideal constraint.It is pointedout that the Vacco model is compatible with Chetaev model,so these two kinds ofmodels are unified.     

Gauss white noise perturbations of nonholonomic mechanical systems

The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper.It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case,and that there areε~2-terms but noε-terms in the differential equations of the second-order moments.Two propositions are obtained.Finally,an example is given to illustrate the application of the results.     

GENERALIZED NOETHER’S THEOREM OF NONHOLONOMIC NONPOTENTIAL SYSTEM IN NONINERTIAL REFERENCE FRAMES

The new Lagrangian of the relative motion of mechanical system is constructed,thevariational principles of Jourdain’s form of nonlinear nonholonomic nonpotential system innoninertial reference frame are established,the generalized Noether’s theorem of thesystem above is presented and proved,and the conserved quantities of system are studied.     

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.      

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

The work analyzes energy relations for nonholonomic systems, whose motion is restricted by nonlinear nonholonomic constraints. For the mechanical systems with linear constraints, the analysis of energy relations was carried out in [1], [2], [3], [4], [5], [6] …. On the basis of corresponding Lagrange’s equations, a general law of the change in energy dε/dt is formulated for mentioned systems by the help of which it is shown that there are two types of the laws of conservation of energy, depending on the structure of elementary work of the forces of constraint reactions. Also, the condition for existing the second type of the law of conservation of energy is formulated in the form of the system of partial differential equations. The obtained results are illustrated by a model of nonholonomic mechanical system.     

Forward and inverse dynamics of nonholonomic mechanical systems

This paper deals with the forward and the inverse dynamic problems of mechanical systems subjected to nonholonomic constraints. The intrinsically dual nature of these two problems is identified and utilised to develop a systematic approach to formulate and solve them according to an unified framework. The proposed methodology is based on the fundamental equations of constrained motion which derive from Gauss’s principle of least constraint. The main advantage arising from using the fundamental equations of constrained motion is that they represent an effective method capable to derive the generalised acceleration of a mechanical system, constrained in general by a set of nonholonomic constraints, together with the generalized constraint forces (forward dynamics). When the constraint equations are used to represent the desired behaviour of the mechanical system under study, the generalised constraint forces deriving from the fundamental equations of constrained motion provide the control actions which reproduce the specified motion for the system (inverse dynamics). This approach is systematically extended to underactuated mechanical systems introducing a new method named underactuation equivalence principle. The underactuation equivalence principle is founded on the key idea that the underactuation property of a mechanical system can be mathematically represented using a particular set of nonholonomic constraint equations. Two simple case-studies are reported to exemplify the proposed methodology. In the first case-study the computation of the generalised constraint forces relative to the revolute joint constraints of a physical pendulum is illustrated. In the second case-study the calculation of the control action which solves the swing-up problem for an inverted pendulum is described.     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

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.     

Mei’s symmetry theorem for time scales nonshifted mechanical systems

We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities. Firstly, the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle. Secondly, the definitions of Mei symmetry on time scales are given and its criterions are deduced. Finally, Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems, time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved, and new conserved quantities of above systems are obtained. Results are illustrated with two examples.     

A geometric solution to the general single contact frictionless problem by combining concepts of analytical dynamics and b-calculus

This study presents a systematic approach, leading to a new set of equations of motion for a class of mechanical systems subject to a single frictionless contact constraint. To achieve this goal, some fundamental concepts of b-geometry are utilized and adapted to the general framework of Analytical Dynamics. These concepts refer to the theory of manifolds with boundary and provide a suitable and strong theoretical foundation. First, the boundary is defined within the original configuration manifold of the system by the equality in the unilateral constraint. Then, an appropriate vector bundle is considered, involving only smooth vector fields, even at the boundary. After determining the essential geometric properties (i.e., the metric and the connection) near the boundary, Newton’s law of motion is applied. In this way, the equations of motion during the contact phase are derived as a system of ordinary differential equations. These equations possess a special form inside a thin boundary layer. In particular, the essential dynamics of the systems examined is found to be governed by a single second order ordinary differential equation, which is investigated fully. Moreover, a critical comparison of the present formulation with the classical formulations applied to systems with a frictionless contact is performed. Finally, the effect of the dominant parameters on the dynamics during the contact phase and the steps for the application process to mechanical systems are illustrated by two selected examples, referring to contact of a particle and a rigid body with a plane.     

On the dynamics of a rolling ball actuated by internal point masses

The motion of a rolling ball actuated by internal point masses that move inside the ball’s frame of reference is considered. The equations of motion are derived by applying Euler–Poincaré’s symmetry reduction method in concert with Lagrange–d’Alembert’s principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball’s frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton’s second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system’s dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.     

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”.