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The Computations of Large Rotation Through an Index Two Nilpotent Equation

Chein-Shan Liu1

Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail:

Computer Modeling in Engineering & Sciences 2006, 16(3), 157-176.


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.


Cite This Article

Liu, C. (2006). The Computations of Large Rotation Through an Index Two Nilpotent Equation. CMES-Computer Modeling in Engineering & Sciences, 16(3), 157–176.

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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