Contribution to the projection method of obtaining equations of motion
type: Item
instance of: scholarly article
title: Contribution to the projection method of obtaining equations of motion
zbMATH Open document ID: 0724.70010
DOI: 10.1016/0093-6413(91)90012-L
author: W. Blajer
published in: Mechanics Research Communications
publication date: 1991-01-01T00:00:00Z
review text: In Am. J. Phys. 56, 451-456 (1988), \textit{D. Scott} proposes an alternative technique, called by him the projection method, to obtain equations of motion for mechanical systems. He compares the technique against Lagrange's equations, and emphasizes the advantages of his method; its ease of derivation, its intuitive nature, the physical insight it gives and the fact that it is somewhat more general than Lagrange's equations. The Scott's method has then been discussed and a bit generalized by \textit{J. Storch} and \textit{S. Gates} [J. Guid. Control Dyn. 12, No.4, 593-595 (1989; Zbl 0712.70026)], and the equivalence of the approach and Kane's method was shown. The crux of Scott's method lies in the projection of Newton's laws into the tangent and orthogonal subspaces relative to the constraints imposed, and intuitively appeals as a generalization of methods used in simple dynamics problems (a particle moving on a smooth surface, for instance). The tangent projection gives the equations of motion, whereas the orthogonal projection determines the constraint reactions. The present paper generalizes Scott's method, and gives a unified approach to the dynamic analysis of constrained systems. The formulation reported enables one to carry out analyses in generalized coordinates and/or quasi-velocities, and systems subject to holonomic and nonholonomic constraints are given equal treatment (usually all the cases are considered separately, which often brings a lot of inconvenience). Simplifications due to the choice of independent generalized coordinates/quasi-velocities are also treated. Moreover, it is shown that the present formulation of the projection method comprises many other well-known methods for derivation of constraint reaction-free equations of motion; the orthogonal complement method, Kane's method, and Maggi's method. Additionally, the formulation for the determination of constraint reactions is provided, which is usually omitted by the other methods.
Mathematics Subject Classification ID: 70F20
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4195244
zbMATH Keywords: projection method
zbMATH Keywords: Lagrange's equations
zbMATH Keywords: Kane's method
zbMATH Keywords: orthogonal subspaces
zbMATH Keywords: tangent projection
zbMATH Keywords: dynamic analysis of constrained systems
zbMATH Keywords: nonholonomic constraints
zbMATH Keywords: reaction-free equations
zbMATH Keywords: Maggi's method
MaRDI profile type: Publication
full work available at URL: 0093-6413(91)90012-l
OpenAlex ID: W2073097675
description: scientific article; zbMATH DE number 4195244