On the problem of reduction in the dynamics of a gyrostat

We consider the motion relative to the fixed pole of the gyrostat, which occurs in the automatic control mode according to R. Grammel. The gyrostat is acted upon by a system of specified non-stationary forces caused by the influence of the external environment. A description is given to the procedure for reducing the dynamic system of the gyrostat, for which there must be a first integral linear in the components of the angular sorption of its carrier. The reduction is realized as a result of constructing a nonlinear integrodifferential equation that determines the dependence of one of the components of the absolute angular velocity vector of the gyrostat carrier. Some special cases of reduction associated with the structural and dynamic features of the gyrostat and the types of a given analytical dependence of the components of the gyrostat moment vector are considered.