Ben-Amots: Stability of rotation of high-speed spin-axisymmetric rotor

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 θ_s and θ_F is small. For external moments the equations yield the angular velocities θdot_s and θdot_F. The slow precession diverges for positive θdot_s and converges for negative θdot_s; the fast precession - for positive and negative θdot_F, respectively. Moreover, the values of θdot_s and θdot_F 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 axes.

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 D. Bousso (1933-1971), who acted as a supervisor at the first stage of the study

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