13 research outputs found
Exact Analytic Solution for the Rotation of a Rigid Body having Spherical Ellipsoid of Inertia and Subjected to a Constant Torque
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.Comment: "Errata Corridge Postprint" In particular: typos present in Eq. 28 of
the Journal version are HERE correcte
Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque
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.Comment: "Errata Corridge Postprint" version of the journal paper. The
following typos present in the Journal version are HERE corrected: 1)
Definition of \beta, before Eq. 18; 2) sign in the statement of Theorem 3; 3)
Sign in Eq. 53; 4)Item r_0 in Eq. 58; 5) Item R_{SN}(0) in Eq. 6
Binary Collisions and the Slingshot Effect
We derive the equations for the gravity assist manoeuvre in the general 2D
case without the constraints of circular planetary orbits or widely different
masses as assumed by Broucke, and obtain the slingshot conditions and maximum
energy gain for arbitrary mass ratios of two colliding rigid bodies. Using the
geometric view developed in an earlier paper by the authors the possible
trajectories are computed for both attractive or repulsive interactions
yielding a further insight on the slingshot mechanics and its parametrization.
The general slingshot manoeuvre for arbitrary masses is explained as a
particular case of the possible outcomes of attractive or repulsive binary
collisions, and the correlation between asymptotic information and orbital
parameters is obtained in general.Comment: 12 pages, 7 figures, accepted for publication Dec'07, Celestial
Mechanics and Dynamical Astronom
Searches for solar-influenced radioactive decay anomalies using Spacecraft RTGs
Experiments showing a seasonal variation of the nuclear decay rates of a
number of different nuclei, and decay anomalies apparently related to solar
flares and solar rotation, have suggested that the Sun may somehow be
influencing nuclear decay processes. Recently, Cooper searched for such an
effect in Pu nuclei contained in the radioisotope thermoelectric
generators (RTGs) on board the Cassini spacecraft. In this paper we modify and
extend Cooper's analysis to obtain constraints on anomalous decays of
Pu over a wider range of models, but these limits cannot be applied to
other nuclei if the anomaly is composition-dependent. We also show that it may
require very high sensitivity for terrestrial experiments to discriminate among
some models if such a decay anomaly exists, motivating the consideration of
future spacecraft experiments which would require less precision.Comment: 8 pages, 4 figures (to appear in Astroparticle Physics
Explicit guidance of drag-modulated aeroassisted transfer between elliptical orbits
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77072/1/AIAA-20103-582.pd