The stability of the generalized stationary motions of a gimbal-mounted gyroscope

A.S.Andreev

Ulianovsk State University

Leo Tolstoi str., 42, Ulianovsk, 432970, Russia

C.Risito

Universita'degli studi, Dipartimento di Matematica

Parco Area delle Scienze 53/A, 43100 Parma, Italy

 

In work the problem on stability and stabilization of the generalized stationary motions ofљ symmetric gimbal-mouted gyroscope is investigated. External framework of a gyroscope rotates ith variable angular speed around of a vertical axis.

This problem concerns to problems on stability of motions of mechanical systems with non-stationary connections, where the part of generalized coordinates is cyclic. Such systems can have the generalized stationary motions, in which the positional coordinates are constant, the cyclic velocities are variable, and the cyclic coordinates accordingly change in time under the nonlinear law. On an example it is shown, that for non-autonomous mechanical system direct carry of the Rause theorem on stability of stationary motions (L.Salvadori, V.V.Rumyantsev) is impossible, and the additional research is required. It is in detail enough carried out in early works (A.S.Andreev, C.Risito).

In a considered problem on gyroscope there is only one class of motions, in which the positional coordinates are constant. It is the motions, in which plane of external and internal frameworks coincide, and the axis of a rotor thus is directed vertically. The sufficient conditions of stability of these motions are defined on the basis of one of the theorems (A.S.Andreev, C.Risito). The analysis of these conditions will be carried out in the form of dependence cyclic constant (on a corner of rotation of a rotor) from limits of change of angular velocity of an external framework. Is shown, that at presence of viscous friction in axes of an internal framework it is easy to define the control moment enclosed to a rotor, so that the considered motion of gyroscope would be asymptotically stable on all variables.

The problem on definition of the control moment enclosed to axes of an internal framework of a gyroscope is in addition solved at which the unstable stationary motion of a gyroscope becomes asymptotically stable.