Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque |
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| Authors: | Marcello Romano |
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| Institution: | (1) Mechanical and Astronautical Engineering Department, 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: | New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of
inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for
the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular
to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial
angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed
with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case
of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix
in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of
a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of
an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body
with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic
solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion
time length and rotation amplitude. The present paper adds further elements 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 Revolution ellipsoid of inertia Analytical reduction |
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