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.

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.

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.

A Geometric Derivation of the Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos
Keyword(s):  
Numerical Methods ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Main Idea ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Motion Constraints ◽  
Stiffness Properties

A new set of equations of motion is presented for a class of mechanical systems subjected to equality motion constraints. Specifically, the systems examined satisfy a set of holonomic and/or nonholonomic scleronomic constraints. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion. The constraints are incorporated one by one, in a process analogous to that used for setting up the equations of motion. This proves to be equivalent to assigning appropriate inertia, damping and stiffness properties to each constraint equation and leads to a system of second order ordinary differential equations for both the coordinates and the Lagrange multipliers associated to the motion constraints automatically. This brings considerable advantages, avoiding problems related to systems of differential-algebraic equations or penalty formulations. Apart from its theoretical value, this set of equations is well-suited for developing new robust and accurate numerical methods.

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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