Equations of motion for mechanical systems: A unified approach

This paper presents a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems. The unified approach extends the applicability of the first two approaches to systems where the constraints are non-linear functions of the generalized velocities and are not necessarily independent. Furthermore, the approach leads to the Explicit Gibbs-Appell Equations.     

Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems

Some new theoretical results are presented on modeling the dynamic response of a class of discrete mechanical systems subject to equality motion constraints. Both the development and presentation are facilitated by employing some fundamental concepts of differential geometry. At the beginning, the equations of motion of the corresponding unconstrained system are presented on a configuration manifold with general properties, first in strong and then in a primal weak form, using Newton׳s law of motion as a foundation. Next, the final weak form is obtained by performing a crucial integration by parts step, involving a covariant derivative. This step required the clarification and enhancement of some concepts related to the variations employed in generating the weak form. The second part of this work is devoted to systems involving holonomic and non-holonomic scleronomic constraints. The equations of motion derived in a recent study of the authors are utilized as a basis. The novel characteristic of these equations is that they form a set of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the constraint action. Based on these equations, the corresponding weak form is first obtained, leading eventually to a consistent first order ODE form of the equations of motion. These equations are found to appear in a form resembling the form obtained after application of the classical Hamilton׳s canonical equations. Finally, the new theoretical findings are illustrated by three representative examples.     

STABILITY OF MOTION FOR A CONSTRAINED BIRKHOFF'S SYSTEM IN TERMS OF INDEPENDENT VARIABLES

I.IntroductionInl927,AmericanmathematicianG.D.Birkh0ffgaveakindofequationsofdynamicswhichweremoregeneralthantheHamilton'sequationsinhiswork"DynamicalSystems-l'].TheequationsarecalledBirkhofCsequationssuggested.byAmericanphysicianR.M.SantiIIiinl978[2j.Inl9…     

Equations of motion for nonholonomic mechanical systems with unilateral constraints

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

Connections and geodesic characteristic of equations of motion for constrained mechanical systems

I.IntroductionSinceEinsteinestablishedgeneralrelativityatthebiginningofthiscentury,differentialgeometry,especiallythemodernditTerentialgeometry,hasbeenextellsivelyappliedtomanyfieldsofphysics.Thestudyofregularholonomicmechanicalsystemsinthemodernsettingofdifferentialgeometryhasahistoryofmorethanthirtyyears.Andtheresearchtendstoperfectgraduallyt'~'l.Sinceearlyin1980'sthegeometrizationaboutconstrainedmechanicalsystemsandsingularmechanicalsystemshasbeenattachedimportanceextensivelyandsomeresult…     

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

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Mixed variational principle in elasticity theory and canonical systems of equations

The paper proposes a modification of the mixed variational principle from which stationarity conditions are derived in the form of a mixed system of equations resolved for the first derivatives of the displacement and stress components acting in a plane perpendicular to one of the coordinate axes. The variational principle allows decreasing the dimension of the problem of elasticity thus reducing the system of equations to a canonical form. The modified mixed principle helps immediately obtain a canonical system of equations for various applied theories. This possibility is demonstrated with the example of the Timoshenko theory of plates __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 55–62, May 2007.     

Existence and stability of bushes of vibrational modes for octahedral mechanical systems with Lennard-Jones potential

The special non-linear dynamical regimes, “bushes of normal modes”, can exist in the N-particle Hamiltonian systems with discrete symmetry (Physica D 117 (1998) 43). The dimension of the bush can be essentially less than that of the whole mechanical system. One-dimensional bushes represent the similar non-linear normal modes introduced by Rosenberg. A given bush can be excited by imposing the appropriate initial conditions, and the energy of the initial excitation turns out to be trapped in this bush.In the present paper, we consider all possible vibrational bushes in the simple octahedral mechanical system and discuss their stability under assumption that the interactions between particles are described by the Lennard-Jones potential.     

Lagrangian description and numerical analysis of a discrete model of flexible systems

An approach is proposed to derive the equations of motion for one-dimensional discrete-continuous flexible systems with one-sided deformation characteristics. To implement this approach, the stationarity principle is generalized to dynamic problems. Solution algorithms are based on cubic spline functions. The capabilities of the approach are demonstrated by the example of a beacon buoy connected by a flexible tether to a submersible that moves along a prescribed trajectory.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 107–116, December 2004.     

The motion equations of rotors,gyrostats and gyrodesics: application to powertrain-vehicle mechanical systems

The kinematics of wheels and rotors is described using a new auxiliary frame called gyrodesic frame or simply gyrodesic. By this, the absolute motion of the wheel becomes a serial composite of two motions: (1) the gyrodesic motion and (2) the wheels eigenmotion (or spin), i.e., the motion relative to the gyrodesic. The eigenmotion is described by an equation called rotor-equation. Gyrodesic coordinates turn out to be a particular useful tool in powertrain- and vehicle-dynamics as well as for general multibody systems. They allow a proper separation of the rotor- from the vehicle-equations and provide a rigorous method of coupling the powertrain-model into full spatial multibody-systems vehicle model. Some common misconceptions regarding this subject are identified and dispelled. The method is generalized to be applicable to the study of motion of general systems of rigid bodies with gyrostats or rotors as subsystems. The usefulness of the formalism is demonstrated by means of an illustrative example of non-trivial nature: the gyrostatic chain. Gyrodesic coordinates lead to a better grasp and deeper understanding of the structure of the dynamic equations of spatial vehicles in particular and of the motion of multibody-systems with rotors in general. The investigation reveals an interesting analogy to concept of parallel transport of vector fields in the sense of Levi-Civita. Dedicated to Prof. J. Wittenburg at the occasion of his 70th birthday.