TITLE: Approximate analytical solution for high-speed
spin-axisymmetric rotor, using coordinate system linked
to precession and nutation
AUTHOR: Ben-Amots, N.
SOURCE: Acta Mechanica, v. 25,
, pp. 111-119 (
Smithsonian/NASA Astrophysics Data System (ADS)
Physics Abstracts Service
ABSTRACT: A solution for symmetric external moments is considered, taking
into account a sequence of approximations and their convergence for a
high-speed motor. Precessions and perturbations in the Lagrange-Poisson
case are compared. It is found that the perturbation to the fast
precession and slow precession involve the same phenomenon. A
description of rotor motion is provided and attention is given to the
limitations inherent in the analysis, the stability of the precessions,
and a numerical-simulation study.
SECOND ABSTRACT SOURCE: Scopus
SECOND ABSTRACT: An approximate solution associated with the fast and slow
precessions of the rotor is found for the nonlinear equations. This
approximation is valid for a high-speed rotor in the ranges where at least
one of the nutation angles θs and θF is small.
For external moments the equations yield the angular velocities
θdots and θdotF. The slow precession diverges for
positive θdots and converges for negative θdots; the
fast precession - for positive and negative θdotF, respectively.
Moreover, the values of θdots and θdotF themselves
permit quantitative comparison of the effects of external moments (applied
simultaneously or separately) on the rotor stability.
FIRST REVIEWER: Greenwood, D.T.
REVIEW SOURCE: Applied Mechanics Reviews, v. 30, p. 736, No. 4630 (1977)
FIRST REVIEW: Beginning with the solution of the classical problem of the
heavy axisymmetric top, extensions are made to cases involving more general
applied moments. An iteration procedure is developed to obtain the fast
and slow precession rates in terms of successively higher powers of 1/n,
where n is the total spin rate about the axis of symmetry. The fast
precession is considered as a perturbation on the slow precessional motion,
or vice versa.
The conditions for the stability of the system are analyzed by noting the
changes in amplitude of the fast and slow components of the motion. The
stability of the slow precession is governed primarily by moment about a
transverse axis which is also perpendicular to the nutation axis; on the
other hand, the stability of the fast precession depends primarily upon
moments about an axis normal to the plane of the precession and nutation
SECOND REVIEWER: Anonymous referee (not in Acta Mechanica).
SECOND REVIEW YEAR: 1975
SECOND REVIEW: The motion of an axisymmetric rotor about a fixed point in
the presence of various kinds of moments is one of the central problems of
classical rigid-body dynamics. It has been studied extensively and solved
exactly for virtually every case where the equations of motion can be
linearized. However, only a few solutions of nonlinear systems are known,
and these of a highly restricted kind.
The author of the present paper has undertaken the analysis of the nonlinear
system of dynamical equations that results where the applied moments are
restricted only insofar as they are required to remain invariant in
precession angle and co-spin angular acceleration. This represents a useful
excursion into the unknown, and one with many potential applications.
The work is carried out in a logical manner and seems accurate. It is
especially impressive that limitations are carefully noted, and that
numerical verifications were carried out. Nor have interesting insights
(e.g., those regarding perturbations of slow and fast precessions in the
Lagrange-Poisson case) been neglected, over and above the conclusions
regarding stability to be expected from such a study.
Publication is recommended.
● COMMENT: The paper, dedicated to the memory of Professor
who acted as supervisor at the first stage of the study
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