The Maggi or Canonical Form of Lagrange’s Equations of Motion of Holonomic Mechanical Systems

1990 ◽  
Vol 57 (4) ◽  
pp. 1004-1010 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Linear Algebra ◽  
Nonlinear Equations ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Holonomic System ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
New Variables

This paper formulates the simplest possible, or canonical, form of the Lagrangean-type of equations of motion of holonomically constrained mechanical systems. This is achieved by introducing a new special set of n holonomic (system) coordinates in terms of which the m ( < n) holonomic constraints are expressed in their simplest, or uncoupled, form: the first m of these new coordinates vanish; the remaining (n-m) (nonvanishing) new coordinates of the (n-m) degree-of-freedom system are then independent. From the resulting equations of motion: (a) The last (n-m) are reactionless canonical equations (the holonomic counterpart of the linear or nonlinear equations, either of Maggi (in the old variables), or of Boltzmann/Hamel (in the new variables)) whose solution yields the motion, while (b) the first m supply the system reactions, in the old or new coordinates, once the motion is known. Special forms of these equations and a simple example are also given. The geometrical interpretation of the above, in modern vector/linear algebra language is summarized in the Appendix.

Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics

2006 ◽  
Vol 462 (2071) ◽  
pp. 2097-2117 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Explicit Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constrained Motion ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
Mass Matrices ◽  
Body Dynamics ◽  
Multi Body ◽  
Explicit Equations Of Motion

We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

On a Consistent Application of Newton’s Law to Constrained Mechanical Systems

2013 ◽  
Author(s):  
Elias Paraskevopoulos ◽  
Sotirios Natsiavas
Keyword(s):  
Riemannian Manifolds ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constrained Mechanical Systems ◽  
Motion Constraints ◽  
Newton's Law ◽  
Newton’S Law ◽  
Correct Application ◽  
Practical Implications ◽  
Consistent Application

An investigation is carried out for deriving conditions on the correct application of Newton’s law of motion to mechanical systems subjected to constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and anholonomic constraints. The focus is on establishment of conditions, so that the form of Newton’s law remains invariant when imposing an additional set of motion constraints on a system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold. The latter is weaker than a similar condition employed frequently in the literature, but holding on Riemannian manifolds only. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space (and not on the dual space) of a manifold. Finally, the Euler-Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated.

Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints

2000 ◽  
Vol 68 (3) ◽  
pp. 462-467 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Lagrangian Mechanics ◽  
Constrained Motion ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Lagrangian Formulations ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

The Hamiltonian dynamics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 564-583 ◽  
Keyword(s):  
Nonholonomic System ◽  
Integrable Function ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Holonomic System ◽  
Principal Function ◽  
Holonomic Constraints ◽  
Second Derivatives ◽  
General Dynamical System ◽  
Hamilton Jacobi Equation

A formalism is developed which makes it possible to express the equations of motion of a nonholonomic system in Poisson bracket form. The main difficulty which has to be overcome arises from the fact that the Lagrangian co-ordinates and their corresponding momenta do not form a canonical set. However, at each instant of time, these variables can be expressed in a unique way as functions of a canonical set called the locally free co-ordinates and momenta. Poisson brackets can be formed with respect to the locally free variables, and it is shown that these lead to the correct equations of motion for a general dynamical system subject to a given set of non-holonomic constraints. Hamilton’s principle applies to a non-holonomic system , so a principal function can be formed, and its properties are studied in the second part of this paper. In addition to the usual Hamilton-Jacobi equation, the principal function satisfies a set of equations corresponding to the set of constraints. It is shown that these equations imply an indefinite or non-integrable principal function. A non-integrable function is one for which the order of double differentiation is not reversible. A precise method is given for defining the principal function for a non-holonomic system, and it is shown how this leads to indefiniteness in its second derivatives.

scholarly journals Selecting Control Parameters of Mechanical Systems with Servoconstraints

2021 ◽  
Vol 264 ◽  
pp. 04085
Author(s):  
Kahramanjon Khusanov
Keyword(s):  
Mathematical Model ◽  
Mechanical System ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Control Parameters ◽  
Holonomic Constraints ◽  
Research Results

The research results on the derivation of equations of motion (a mathematical model) of a mechanical system constrained by holonomic constraints of the first and second kind, which contains only tangential components of the constraint of the second kind, are presented in the article. These tangential components are taken as control parameters. Besides, the controllability of the plate, considered in the Appel problem, is investigated.

Application of the Vector-Network Method to Constrained Mechanical Systems

1986 ◽  
Vol 108 (4) ◽  
pp. 471-480 ◽  
Author(s):  
Tai-Wai Li ◽  
Gordon C. Andrews
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Computer Applications ◽  
Kinematic Chains ◽  
Constrained Mechanical Systems ◽  
3 Dimensional ◽  
Methodical Approach ◽  
Dynamic Formulation ◽  
Reaction Forces ◽  
New Formulation

The vector-network technique is a methodical approach to formulating equations of motion for unconstrained dynamic systems, utilizing concepts from graph theory and vectorial mechanics; it is ideally suited to computer applications. In this paper, the vector-network theory is significantly improved and extended to include constrained mechanical systems with both open and closed kinematic chains. A new formulation procedure is developed in which new kinematic constraint elements are incorporated. The formulation is based on a modified tree/cotree classification, which deviates significantly from previous work, and reduces the number of equations of motions to be solved. The dynamic equations of motion are derived, with generalized accelerations and a subset of the reaction forces as solution variables, and a general kinematic analysis procedure is also developed, similar to that of the dynamic formulation. Although this paper restricts most discussions to two-dimensional (planar) systems, the new method is equally applicable to 3-dimensional systems.

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