In this paper the stability of motion of an unbalanced gyroscope with a flexible shaft in gimbal suspension is considered.First,we consider linear Hamiltonian systems of differential equations.A solution stability criterion is obtained in one case.Next,we explore the gyroscope motion.It consists of an internal and external framework.In the normal position,the axes of the frames and the rotor are mutually perpendicular and intersect at a point that is a fixed point of suspension.Friction in the bearings of the rotor and the frame is missing.The rotor motion system equations and gyroscope frames are considered in the first approximation.The case is considered when the characteristic equation corresponding to the motion equations in the first approximation has eigenvalues with a zero-real part.The stability criterion is obtained.
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