Efficient Solution of Initial-Value Problem of Torque-Free Rotation

Details Efficient Solution of Initial-Value Problem of Torque-Free Rotation Efficient Solution of Initial-Value Problem of Torque-Free Rotation

May 2009

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009dda....40.1402f&link_type=abstract

American Astronomical Society, DDA meeting #40, #14.02; Bulletin of the American Astronomical Society, Vol. 41, p.908

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Scientific paper

We present an efficient solution of the initial-value problem of torque-free rotation. It covers all the parameter cases and three rotation modes of a general triaxial rigid body. For the given initial values of the rotational angular momentum vector in an inertial reference frame, L0, and the direction cosine matrix (DCS) specifying the orientation of the rotating body in the same inertial reference frame, E0, the solution provides their values after the time duration, Delta t, as, L = L0, and E = E0 F. Here F is the transition matrix being a 3x3 special orthogonal matrix. In general, F is of the form of products of three matrices as F = W0T R W. Here R is a rotation matrix around L and W and W0 are the wobble matrices relating E and E0 with an intermediate DCS, one axis of which is along L, respectively. The computation of the wobble matrices is simplified by using the fact that one column or row vector of the wobble matrix is in parallel with a vector consisting of the body-fixed components of the rotational angular momentum vector, LA, LB, and LC. Since these body-fixed components are analytically expressed in terms of Jacobian elliptic functions, the evaluation of W from W0 reduces to a problem to obtain the values of elliptic functions after Delta t when their initial values are known. Similarly the calculation of R needs the value of the pi amplitude function. The evaluation of these elliptic functions is dramatically accelerated by their addition theorems. Further, the computational time of the elliptic functions of the incremental argument is significantly shortened by their Maclaurin series expansion. The resulting formulation is a counterpart, in rotational dynamics, of the method of f- and g-functions to solve the initial-value problem of a Keplerian orbit.

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