Open Access

ARTICLE

The Computations of Large Rotation Through an Index Two Nilpotent Equation

by ,

1 Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2006, 16(3), 157-176. https://doi.org/10.3970/cmes.2006.016.157

Abstract

To characterize largely deformed spin-free reference configuration of materials, we have to construct an orthogonal transformation tensor Q relative to the fixed frame, such that the tensorial equation Q^˙ = WQ holds for a given spin history W. This paper addresses some interesting issues about this equation. The Euler's angles representation, and the (modified) Rodrigues parameters representation of the rotation group SO(3) unavoidably suffer certain singularity, and at the same time the governing equations are nonlinear three-dimensional ODEs. A decomposition Q = FQ₁ is first derived here, which is amenable to a simpler treatment of Q₁ than Q, and the numerical calculation of Q₁ is obtained by transforming the governing equations in a space of RP³, whose dimensions are two, and the singularity-free interval is largely extended. Then, we develop a novel method to express Q₁ in terms of a noncanonical orthogonal matrix, the governing equation of which is a linear ODEs system with its state matrix being nilpotent with index two. We examine six methods on the computation of Q from the theoretical and computational aspects, and conclude that the new methods can be applied to the calculations of large rotations.

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