Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque |
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Authors: | Marcello Romano |
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Institution: | (1) Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, 700 Dyer Road, Monterey, CA, USA;(2) Space Systems Academic Group, Naval Postgraduate School, Monterey, CA, USA |
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Abstract: | The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of
inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary
initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular,
the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points
on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type,
which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent
hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic
map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to
the small set of special cases for which an exact solution of the rotational motion of a rigid body exists. |
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Keywords: | Rigid body dynamics Kinematics Rotation Integrable cases of motion Spherical ellipsoid of inertia |
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