Equivalence of free and driven rotor problems of a two-degree-of-freedom gyroscope

It is shown that the behavior of a two-degree-of-freedom gyroscope with a driven rotor is mathematically equivalent to the behavior of a gyroscope with free rotor under all conditions, that is, the gyroscope is assumed to be mounted in a vehicle moving in an arbitrary manner. It is noted that when evaluating the kinetic energy of the free-rotor system, the moment of inertia of the inner gimbal about the spin axis of the rotor should be replaced by the difference in the moments of inertia of the inner gimbal and the rotor. The angular velocity of the rotor relative to the inner gimbal should be replaced by the sum of the value of this angular velocity and the component of the angular velocity of the inner gimbal along the spin axis of the rotor at some specified instant of time.