Another form of equations of motion for constrained multibody systems

This paper presents a new and simplified set of explicit equations of motion for constrained mechanical systems. The equations are applicable with both holonomic and nonholonomic systems and the constraints may, or may not, be ideal. It is shown that this set of equations is equivalent to governing equations developed earlier by others. The connection of these equations with Kane's equations is discussed. It is shown that the developed equations are directly applicable with controlled systems where the controlling forces and moments may be subject to constraints. Finally, a procedure is presented for determining which control force systems are equivalent. Examples are presented to demonstrate the advantages, features, and range of application of the 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.     

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 Whittaker equations to non-holonomic mechanical systems

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation^[1].In this article, Whittaker equations are extended to non-holonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Kiel-sen’s form.Finally an example is given.     

Finite and impulsive motion of constrained mechanical systems via Jourdain's principle: discrete and hybrid parameter models

The main purpose of this paper is to present a unified analytical dynamics framework for the analysis of finite and impulsive motion of mechanical systems using Jourdain's principle. Emphasis is given to the general case when a mechanical system is described by a hybrid (discrete-distributed) parameter model. A large group of finite and impulsive, generally non-holonomic, constraints are analysed in detail and a so-called extended Appellian classification is presented for these constrained motion problems. The fundamental dynamic equation of constrained systems is developed in terms of velocity variations (Jourdain's principle). Based on this equation and the constraints, the methods of quasivelocities and Lagrangian multipliers are adopted and interpreted for the finite motion of hybrid parameter models of mechanical systems; and the methods of independent quasivelocity variations and Lagrangian multipliers are introduced for the analysis of impulsive motion of such models. To illustrate the proposed material, an example of a one-link flexible arm intercepting and capturing a moving target is considered.     

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.     

On null controllability of linear systems with recurrent coefficients and constrained controls

Given a family of time-dependent linear control processes, we study conditions under which local null controllability implies global null controllability. This is done by employing methods of dynamical systems and the Sacker-Sell spectral theory. We show that the above implication holds almost surely for recurrent families provided the spectrum of the associated linear system is contained in (–,0].     

Vertical integration of three-phase flow equations for analysis of light hydrocarbon plume movement

A mathematical model is derived for areal flow of water and light hydrocarbon in the presence of gas at atmospheric pressure. Vertical integration of the governing three-dimensional, three-phase flow equations is performed under the assumption of local vertical equilibrium to reduce the dimensionality of the problem to two orthogonal horizontal directions. Independent variables in the coupled water and hydrocarbon areal flow equations are specified as the elevation of zero gauge hydrocarbon pressure (air-oil table) and the elevation of zero gauge water pressure (air-water table). Constitutive relations required in the areal flow model are vertically integrated fluid saturations and vertically integrated fluid conductivities as functions of air-oil and air-water table elevations. Closed-form expressions for the vertically integrated constitutive relations are derived based on a three-phase extension of the Brooks-Corey saturation-capillary pressure function. Closed-form Brooks-Corey relations are compared with numerically computed analogs based on the Van Genuchten retention function. Close agreement between the two constitutive models is observed except at low oil volumes when the Brooks-Corey model predicts lower oil volumes and transmissivities owing to the assumption of a distinct fluid entry pressure. Nonlinearity in the vertically integrated constitutive relations is much less severe than in the unintegrated relations. Reduction in dimensionality combined with diminished nonlinearity, makes the vertically integrated water and hydrocarbon model an efficient formulation for analyzing field-scale problems involving hydrocarbon spreading or recovery under conditions for which the vertical equilibrium assumption is expected to be a satisfactory approximation.     

A simple physical approach for deriving the characteristic equations of fluid dynamics

A simple physical approach for deriving the characteristic equations of fluid dynamics is presented. The approach is based on the physical concept that information propagates through a flowfield along pathlines due to particle motion and along wavelines due to acoustic wave motion. The characteristic equations and compatibility equations are derived in vector forms which are valid in any co-ordinate system.     

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.     

Equations of motion for nonholonomic mechanical systems with unilateral constraints

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

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper,Routh’s equations for the mechanical systems of the variable masswith nonlinear nonholonomic constraints of arbitrary orders in a noninertial referencesystem have been deduced not from any variational principles,but from the dynamicalequations of Newtonian mechanics.And then again the other forms of equations fornonholonomic systems of variable mass are obtained from Routh’s equations.     

UNCONDITIONAL STABLE SOLUTIONS OF THE EULER EQUATIONS FOR TWO－AND THREE－D WINGS IN ARBITRARY MOTION

ＵＮＣＯＮＤＩＴＩＯＮＡＬＳＴＡＢＬＥＳＯＬＵＴＩＯＮＳＯＦＴＨＥＥＵＬＥＲＥＱＵＡＴＩＯＮＳＦＯＲＴＷＯ－ＡＮＤＴＨＲＥＥ－ＤＷＩＮＧＳＩＮＡＲＢＩＴＲＡＲＹＭＯＴＩＯＮＧａｏＺｈｅｎｇｈｏｎｇ（高正红）（ＲｅｃｅｉｖｅｄＪａｎ．１２,１９９５,Ｃ...     

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.     

Differential equations of motion for a variable-mass holonomic system with a uniformly rotating constraint

In this paper, the differential equations of motion will be established for variable-mass holonomic mechanical systems constrained to the uniform rotation, including the equations in Lagrange form. Nielsen form and Appell form. The application of these new equations is illustrated with an example.     

Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems

The automatic derivation of motion equations is an important problem of multibody system dynamics. Firstly, an overview of the matrix calculus related to Kronecker product of two matrices is presented. A new matrix form of Lagrangian equations with multipliers for constrained multibody systems is then developed to demonstrate the usefulness of Kronecker product of two matrices in the study of dynamics of multibody systems. Finally, the equations of motion of mechanisms are derived using the proposed matrix form of Lagrangian equations as application examples.     

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.     

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.