This article studies the mathematical model of the oscillation behavior of a micromechanical gyroscope under angular vibrations of the base. The motion equations are presented in the form of the Mathew-Hill differential equations. By applying the Krylov-Bogolyubov averaging method to analyze the slow-variable dynamics of the gyroscope, it is found that angular base vibrations occurring at resonant frequencies significantly affect the accuracy of the gyroscopic measurement.