Stability Criterion for Stationary Solutions of Multi-Current Model Equations for a Synchronous Gimbal-Mounted Gyroscope. Part 1

Abstract

We study the dynamics of a gimbal-mounted gyroscope, which has a vertical external gimbal axis and is equipped with a synchronous electric drive that rotates a gyroscope (electric motor rotor). A mathematical model of the electric motor that includes differential equations for electric currents in the rotor windings is used. Both friction and control torques on the gimbal axes are assumed to be absent. Friction torque on the rotor axis is a nonlinear function of the angular velocity of its rotation relative to the stator. The differential equations of motion for such an electromechanical system have an infinite set of stationary (steady-state) solutions that describe its stationary motions, namely, regular precessions and uniform rotations of the rotor. Due to the verticality of the outer axis of the gimbal suspension, the considered system of differential equations has a first integral, which expresses the constancy of the projection of the angular momentum of the device onto the vertical axis. In the present article we have proved that the presence of an isolated minimum of the total reduced potential energy is the necessary and sufficient condition for the stability of stationary (steady-state) motions. Thus, the previously obtained result using a simplified non-current model of a synchronous electric motor is generalized to the case of the multi-current model. Content of the study is presented in two articles. In the first part, we consider the cases when the mechanical reduced potential energy is not identical constant with respect to the internal gimbal angle. In the second part, we analyze the case of constancy for the mechanical reduced potential energy.