D’Alembert’s Principle and the Equations of Motion for Nonholonomic Systems

2006 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Variational Methods ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Rigid Bodies ◽  
Virtual Power ◽  
Lagrangian Equations ◽  
Conservative Systems ◽  
Principle Of Virtual Power ◽  
D'alembert's Principle

D'Alembert's principle is manipulated in the presence of nonholonomic constraints to derive the principle of virtual power in nonholonomic form, and Lagrange's equations for nonholonomic systems. The Lagrangian equations had been expressed previously for conservative systems, derived by variational methods. The D'Alembert derivation confirms the roles of constrained and unconstrained Lagrangians directly by the presence of constrained and unconstrained velocities in D'Alembert's principle. The constrained form of nonconservative generalized forces is also determined for both particles and rigid bodies. An example is a rolling disk.

Unified Approach for Holonomic and Nonholonomic Systems Based on the Modified Hamilton’s Principle

2009 ◽  
Author(s):  
Keisuke Kamiya ◽  
Junya Morita ◽  
Yutaka Mizoguchi ◽  
Tatsuya Matsunaga
Keyword(s):  
Equations Of Motion ◽  
Time Derivative ◽  
Nonholonomic Systems ◽  
Hamilton’S Principle ◽  
Virtual Power ◽  
Unified Approach ◽  
Hamilton's Principle ◽  
Basic Principles ◽  
Principle Of Virtual Power ◽  
Holonomic And Nonholonomic Systems

As basic principles for deriving the equations of motion for dynamical systems, there are d’Alembert’s principle and the principle of virtual power. From the former Hamilton’s principle and Langage’s equations are derived, which are powerful tool for deriving the equation of motion of mechanical systems since they can give the equations of motion from the scalar energy quantities. When Hamilton’s principle is applied to nonholonomic systems, however, care has to be taken. In this paper, a unified approach for holonomic and nonholonomic systems is discussed based on the modified Hamilton’s principle. In the present approach, constraints for both of the holonomic and nonholonomic systems are expressed in terms of time derivative of the position, and their variations are treated similarly to the principle of virtual power, i.e. time and position are fixed in operation with respect to the variations. The approach is applied to a holonomic and a simple nonholonomic systems.

Explicit Poincaré equations of motion for general constrained systems. Part I. Analytical results

2007 ◽  
Vol 463 (2082) ◽  
pp. 1421-1434 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Analytical Results ◽  
Multi Body ◽  
D'alembert's Principle

This paper gives the general constrained Poincaré equations of motion for mechanical systems subjected to holonomic and/or nonholonomic constraints that may or may not satisfy d'Alembert's principle at each instant of time. It also extends Gauss's principle of least constraint to include quasi-accelerations when the constraints are ideal, thereby expanding the compass of this principle considerably. The new equations provide deeper insights into the dynamics of multi-body systems and point to new ways for controlling them.

Dynamics of Multirigid-Body Systems

1978 ◽  
Vol 45 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. L. Huston ◽  
C. E. Passerello ◽  
M. W. Harlow
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Computer Algorithms ◽  
Euler Parameters ◽  
Closed Loops ◽  
Explicit Formulation ◽  
Relative Coordinates ◽  
D'alembert's Principle

New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multibody mechanical systems are presented and discussed. These ideas include the use of Euler parameters, Lagrange’s form of d’Alembert’s principle, quasi-coordinates, relative coordinates, and body connection arrays. The mechanical systems considered are linked rigid bodies with adjoining bodies sharing at least one point, and with no “closed loops” permitted. An explicit formulation of the equations of motion is presented.

A Variationally Consistent Approach to Constrained Motion

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
John T. Foster
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonholonomic Systems ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Constrained Motion ◽  
Consistent Approach ◽  
Holonomic And Nonholonomic Systems ◽  
D'alembert's Principle

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.

What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems?

2002 ◽  
Vol 69 (3) ◽  
pp. 335-339 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Analytical Mechanics ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.

scholarly journals Ehresmann connection in the geometry of nonholonomic systems

2012 ◽  
Vol 91 (105) ◽  
pp. 19-24
Author(s):  
Aleksandar Baksa
Keyword(s):  
Dynamic System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Geometrical Properties ◽  
Ehresmann Connection ◽  
Affine Constraints

This article deals with a dynamic system whose motion is constrained by nonholonomic, reonomic, affine constraints. The article analyses the geometrical properties of the ?reactions" of nonholonomic constraints in Voronets?s equations of motion. The analysis shows their link with the torsion of the Ehresmann connection, which is defined by the nonholonomic constraints.

Closed and Expanded Form of Manipulator Dynamics Using Lagrangian Approach

1985 ◽  
Vol 107 (2) ◽  
pp. 223-225 ◽  
Author(s):  
T. Wang ◽  
D. Kohli
Keyword(s):  
Equations Of Motion ◽  
Equation Of Motion ◽  
Rigid Bodies ◽  
Lagrangian Approach ◽  
Alternative Derivation ◽  
Manipulator Dynamics ◽  
Lagrangian Equations ◽  
A Chain ◽  
Expanded Form ◽  
Simplified Form

An alternative derivation of the equations of motion of a chain of rigid bodies using Lagrangian equations of motion is presented. In an effort to reduce the complexity of the coefficients appearing in the equations of motion, a modified form of Lagrangian equations due to Silver [3] are utilized. This approach leads to a simplified form of coefficients of the equation of motion.

Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations

1994 ◽  
Vol 116 (4) ◽  
pp. 1058-1064 ◽  
Author(s):  
S. Yoon ◽  
R. M. Howe ◽  
D. T. Greenwood
Keyword(s):  
Numerical Simulation ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Geometric Constraints ◽  
State Variables ◽  
Step Size ◽  
Constraint Equations ◽  
Lagrangian Equations ◽  
Small Step Size ◽  
Gradient Based

Conventional holonomic or nonholonomic constraints are defined as geometric constraints. The total enregy of a dynamic system can be treated as a constrained quantity for the purpose of accurate numerical simulation. In the simulation of Lagrangian equations of motion with constraint equations, the Geometric Elimination Method turns out to be more effective in controlling constraint violations than any conventional methods, including Baumgarte’s Constraint Violation Stabilization Method (CVSM). At each step, this method first goes through the numerical integration process without correction to obtain updated values of the state variables. These values are then used in a gradient-based procedure to eliminate the geometric and energy errors simultaneously before processing to the next step. For small step size, this procedure is stable and very accurate.

Dynamic Analysis of Mechanisms Using Constrained Lagrangian Bond Graphs

1992 ◽  
Author(s):  
Y. A. Khulief
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Bond Graph ◽  
Rigid Bodies ◽  
Interconnected System ◽  
Lagrangian Equations ◽  
System Of Rigid Bodies ◽  
Reaction Forces ◽  
Bond Graph Modeling ◽  
Generalized Constraint

Abstract A method for dynamic analysis of mechanisms using the Lagrangian equations of motion for an interconnected system of rigid bodies is presented. The method stems from a recent extension to the bond graph modeling technique. Intrinsically, this approach allows the formulation of the final form of equations for holonomic systems without recourse to the Lagrangian function. Consequently, the burdens of deriving the expressions for kinetic and potential energies, and performing the necessary differentiations have been eliminated. This method calls only for constructing the Jacobian matrix of constraints, and then employing a bond graph that accounts for the generalized constraint reaction forces.

Virtual Work & d’Alembert’s Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Euler Equations ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Gibbs Function ◽  
Constraint Force ◽  
Appell Equations ◽  
Jourdain's Principle ◽  
D'alembert's Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

Sign in / Sign up

Export Citation Format

Share Document