The Dual Euler Basis: Constraints, Potentials, and Lagrange’s Equations in Rigid-Body Dynamics

[+] Author and Article Information
Oliver M. O’Reilly

Department of Mechanical Engineering,  University of California, Berkeley, CA 94720-1740oreilly@Berkeley.edu

We can also consider more general constraints where the translational and rotational motion of the rigid body are coupled, but this would not significantly add to the present discussion.

This set of Euler angles is used in Refs. 2,6-7. The term “Euler basis" first appeared in Casey.

To calculate the last three of these equations, we have used Eq. 12.

J. Appl. Mech 74(2), 256-258 (Feb 03, 2006) (3 pages) doi:10.1115/1.2190231 History: Received September 23, 2005; Revised February 03, 2006

Given a specific set of Euler angles, it is common to ask what representations conservative moments and constraint moments possess. In this paper, we discuss the role that a non-orthogonal basis, which we call the dual Euler basis, plays in the representations. The use of the basis is illustrated with applications to potential energies, constraints, and Lagrange’s equations of motion.

Copyright © 2007 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Schematic of the 3-2-1 set of Euler angles: ψ, θ and ϕ. The vectors e1′=cos(ψ)E1+sin(ψ)E2, and e3″=cos(θ)E3+sin(θ)e1′.

Grahic Jump Location
Figure 2

A circular rod moving on a horizontal plane. The rotation of this body is an example of a situation where one of the Euler angles is constant. The basis vectors ei are fixed to the rod.




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