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.     

Noether's theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints in event space

Ｔｈｅｓｔｕｄｙｏｆｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｄｙｎａｍｉｃａｌｓｙｓｔｅｍｉｓａｓｉｇｎｉｆｉｃａｎｔａｓｐｅｃｔｏｆｍｏｄｅｒｎａｎａｌｙｔｉｃａｌｍｅｃｈａｎｉｃｓ.Ｉｎ1918,ｕｓｉｎｇｔｈｅｉｎｖａｒｉａｎｃｅｏｆＨａｍｉｌｔｏｎａｃｔｉｏｎｕｎｄｅｒｔｈｅｉｎｆｉｎｉｔｅｓｉｍａｌｔｒａｎｓｆｏｒｍａｔｉｏｎｓ,Ｎｏｅｔｈｅｒ[1]ｓｔｕｄｉｅｄｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍａｎｄｐｒｅｓｅｎｔｅｄｔｈｅｆａｍｏｕｓＮｏｅｔｈｅｒ’ｓｔｈｅｏｒ…     

One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

Equations of motion for nonholonomic mechanical systems with unilateral constraints

IntroductionThemotionofamechanicalsystemwithunilateralconstraintsismoregeneralthanwithbilateralconstraints1andyetitsinvestigationismoredifficult[1,2].Thesystemswithunilateralconstraintsinvolvetheholonomicmechanicalsystemswithunilateralholonomicconstraints[3～6]andthenonholonondcmeChanicalsystemswithunilateralholonondcconstraintS[7]andthenonholonondcmechanicalsystemswithunilateralnonholonondcconstraintS[2]andsoon.ThispaperstUdiesatypeofmoregeneralsystemswithunilateralconstfaints:nonholonondcs…     

Integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frames

This paper establishes the integral theory for the dynamics of nonlinearnonholonomic system in noninertial reference frame.Firstly,based on the Routhequation of the relative motion of nonlinear nonholonomic system gives the firstintegral of the system.Secondly,by using cyclic integral or energy integral reduces theorder of the equation and obtains generalized Routh equation and Whittaker equationrespectively.Thirdly,derives canonical equation and variation equation and by usingthe first integral constructs integral invariant.And then,establishes the basic integralvariants and the integral invariant of Poincare-Cartan type.Finally,we give a series ofdeductions.     

NOETHER’S THEORY FOR NONHOLONOMIC DYNAMICAL SYSTEMS RELATIVE TO NON-INERTIAL REFERENCE FRAME

The new variational principle of Gauss’s form of nonlinear nonholonomicnonpotential system relative to non-inertial reference frame is established byconstructing generalized inertial potentials.Noether’s theorem and Noether’s inversetheorem of the system above is presented and proved.Finally,one example is given toillustrate the application.     

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.     

A METHOD FOR SOLVING THE DYNAMICS OF MULTIBODY SYSTEMS WITH RHEONOMIC AND NONHOLONOMIC CONSTRAINTS

ＡＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＤＹＮＡＭＩＣＳＯＦＭＵＬＴＩＢＯＤＹＳＹＳＴＥＭＳＷＩＴＨＲＨＥＯＮＯＭＩＣＡＮＤＮＯＮＨＯＬＯＮＯＭＩＣＣＯＮＳＴＲＡＩＮＴＳ￥ＳｈｕｉＸｉａｏｐｉｎｇ（水小平）ＺｈａｎｇＹｏｎｇｆａ（张永发）（Ｄｅｐａｒ...     

Robust adaptive control of nonholonomic systems with uncertainties

Robust adaptive control of nonholonomic systems in chained form with linearly parameterized and strongly nonlinear disturbance and drift terms is dicussed. The novelty of the proposed method is a combined use of the state-scaling and the back-stepping procedure.     

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

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

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.     

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…     

A field method for solving the equations of motion of nonholonomic systems

In this paper, the field method^[1] for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass. Two examples are given to illustrate its application. The project supported by the National Natural Science Foundation of China.     

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.     

Noether's theory of mechanical systems with unilateral constraints

Ｎｏｅｔｈｅｒ’ｓｔｈｅｏｒｅｍｒｅｖｅａｌｓｔｈｅｉｎｎｅｒｃｏｎｎｅｃｔｉｏｎｂｅｔｗｅｅｎｔｈｅｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓａｎｄｔｈｅｄｙｎａｍｉｃａｌｓｙｍｍｅｔｒｙｏｆｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ．Ｉｎｔｈｅｒｅｃｅｎｔｔｗｅｎｔ...     

About the basic integral variants of holonomic nonconservative dynamical systems

In this paper, we prove that for holonomic nonconservative dynamical system the Poincaré and Poincaré-Cartan integral invariants do not exist. Instead of them, we introduce the integral variants of Poincaré-Cartan's type and of Poincare's type for holonomic nonconservative dynamical systems, and use these variants to solve the problem of nonlinear vibration. We also prove that the integral invariants introduced in references [1] and [2] are merely the basic integral variants given by this paper. Project supported by National Natural Science Foundation of China     

Parametric equations of nonholonomic nonconservative systems in the event space and the method of their integration

In this paper, the parametric equations with multipliers of nonholonomic nonconservative systems in the event space are established, their properties are studied, and their explicit formulation is obtained. And then the field method for integrating these equations is given. Finally, an example illustrating the application of the integration method is given. The Project is supported by the National Natural Science Foundation of China.     

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 with higher-order nonholonomic constraints are obtained at first, and then the equivalence between these equations and the known equations is demonstrated. Finally an example is given to illustrate the application of our new equations.     

The theory of relativistic analytical mechanics of the rotational systems

I'IntroductionThespinningmotionisintrinsicattributeofmicroscopicparticle.In1979,R.BengtssonandS.Frauendorfaccuratlymeasuredthemaximumvolumesofthespinningrotativevelocityof14kindsofnucleons,andtheresultsshowedthatthemaximumvolumesofthespinningrotativevelocityofeachnucleonaredifferent[ll.Withthede\'elopingofmodernscienceandtechnology,moreandmoreexperimentalfactsrelatedtothehighspeedrotationquestions,theEinstein'srelativitytheoryandtherotationaltheoryofclassicalmechanicsarenotsuitabletothesepro…