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TECHNICAL PAPERS

# A Discrete Quasi-Coordinate Formulation for the Dynamics of Elastic Bodies

[+] Author and Article Information
G. M. T. D’Eleuterio

Institute for Aerospace Studies, University of Toronto, Toronto, ON, M3H 5T6, Canadagabriele.deleuterio@utoronto.ca

T. D. Barfoot

Institute for Aerospace Studies, University of Toronto, Toronto, ON, M3H 5T6, Canadatim.barfoot@utoronto.ca

There is potentially a confusion in using the word order for it may refer to differential order as well as algebraic order. To avoid this, we shall restrict the use of the term order to the former and cmploy degree, which is customarily used in reference to polynomials, for the latter.

J. Appl. Mech 74(2), 231-239 (Jan 30, 2006) (9 pages) doi:10.1115/1.2189873 History: Received July 12, 2004; Revised January 30, 2006

## Abstract

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

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## Figures

Figure 2

Transverse deflection (full model, Q=3)

Figure 3

Axial deflection (full model, Q=3)

Figure 4

Effect of higher-degree terms on transverse deflection (Ne=1)

Figure 5

Effect of higher-degree terms on axial deflection (Ne=1)

Figure 1

An elastic body

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