Twin-gyroscopic systems are designed for ocean-wave energy converters and ship roll-stabilizers to double desirable gyroscopic effects and eliminate undesirable reaction torques. In deriving analytical equations of motion, the configuration spaces of gyroscopic systems are defined by using body-attached moving frames. The moving frame of each constituent body is defined by its inertial coordinates of the center of mass and a rotation matrix which expresses the attitude of its coordinate axes from the inertial coordinate axes. Therefore, to utilize powerful Lagrange’s method, it is extended to accommodate rotation matrices in configuration spaces and allow angular velocities as generalized velocities.
First, in the paper, to identify undesirable reaction torques of gyroscopic systems and find a scheme to eliminate them, we present the basics of a reaction wheel. Second, to identify the desirable gyroscopic effect, we consider a control moment gyroscope and derive the equations of motion using the extended Lagrange’s method. In addition, the equations of motion are also derived by using the Newton-Euler method, where action and reaction torques are explicitly expressed. The comparison of the resulting equations derived by the two methods reveals the simplicity of Lagrange’s method in treating actuating motor torques and how the effects of reaction torques are implicitly included in the variationally derived equations. Finally, the equations of motion for a twin-gyroscopic system are obtained by incorporating the scheme to eliminate the undesirable reaction torques.