Nutational stability and core energy of a quasirigid gyrostat

Abstract

The asymptotic nutational stability of a quasirigid gyrostat is analyzed. The primary purpose of this analysis is to resolve a debate concerning the use of the energy-sink method of analysis for systems containing driven rotors. It is shown that when the work done by the motor torque is not taken into account, the analysis leads to a contradiction even when the total energy is dissipative. A proper application of Landon's original idea yields a relationship between the time rate of change of Hubert's "core energy" and the energy dissipation rate of the damping mechanisms in the spacecraft. The analysis shows that the core energy might increase during a rotor despin condition; hence, the minimality of core energy-a previous criterion-is not always guaranteed. A criterion for the design of the damper to insure dissipation of the core energy is presented; this condition is always satisfied for the case of a constant relative rotor spin speed that facilitates a "closed-form" solution to the nutation angle time history of an axisymmetric gyrostat. The stability condition resulting from this analysis is consistent with the Landon-Iorillo stability criterion. Nomenclaturefl/ = dextral orthonormal triad fixed on the platform but aligned along //, / = 1, 2, 3 b 3 = unit vector along the rotor spin axis and parallel= distance between the damper particle and the spin axis (see Fig. 3) c -linear viscous damper constant E -total energy of the gyrostat (considered as a two-body system) EC = core energy E D = energy dissipation due to damping mechanisms E P = energy dissipation rate in the platform E R = energy dissipation rate in the rotor H = (central) angular momentum of the gyrostat // = principal moments of inertia of a gyrostat, / = 1, 2, 3 I s = moment of inertia of an axisymmetric gyrostat about the spin axis I t = moment of inertia of an axisymmetric gyrostat about the transverse axis J = axial moment of inertia of the rotor k = linear spring constant m G = mass of the gyrostat m p = mass of the particle in the discrete damper case Q = quantity defined in Eq. (30) q = spring deflection of the discrete damper T P = net torque on the platform TP/M = motor torque on the platform T R = net axial torque on the rotor TR/M = motor torque on the rotor W = rate of work done by the motor torque 77 = nutation angle A 0 = inertial nutation frequency X P = platform nutation frequency \ R = rotor nutation frequency Q = relative spin rate of the rotor co/ = components of the inertial angular velocity of the platform along the principal axes of the gyrostat, / =1,2,3Presented as Paper 91-112 at the AAS/AIAA Spaceflight Mechanics Meeting, Houston, TX, Feb. 11-13, 1991; received March 11, 1991; revision received Sept. 23, 1992; accepted for publication Oct. 1, 1992. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States.*Assistant Professor, Department of Aeronautics and Astronautics, Mail Code AA-Ro. Member AIAA. Introduction SINCE the pioneering work of Vernon D. Landon, ...

Escape from Gyrostat Trap States

“…The correspondingLagrangemultiplier¸o is determined from the rst-order conditions [Eq. (20)]. Thus, the ¹ D 0 starting points are as given in Table 1.…”