**Theory of operation:**

The primary component of the gyroscope is a rotor which has a moment of inertia J and an angular velocity ω. The inertia moment of the rotor is determined by the distribution of its own mass relative to its axis of rotation. By increasing the mass or the diameter of the rotor the moment of inertia will therefore increase. The angular momentum L of the rotor is given by the product of its moment of inertia for angular velocity and indicates the trend of the rotation axis to remain parallel to itself and to oppose any attempt to change its orientation.

Greater is the angular momentum and greater will be the ability of the rotor to react to external torques and, in our case, the greater will be the ability of the stabilizer to cancel the roll.

A gyroscope has three axes:

- a “spin axis”, an “input axis” and an “output axis”.
- The spin axis is the axis around which the rotor rotates, in this case is the vertical axis.
- The input axis is the axis on which inputs are applied, which will be coincident to the axis around which the boat rolls (longitudinal axis).
- The output axis, the transverse axis, the one around which the gyroscope rotates or precesses in response to the input.

When the boat rolls, it has the function of input to the gyroscope. This input causes the gyroscope to generate a rotation around its output axis such that the axis of rotation changes orientation to align with the input axis. The rotation about the transverse axis is known as precession.

Two hydraulic cylinders are coupled to the output axis of the gyroscope with the purpose of curbing, and then handle, the angle of precession.

The maximum force applied to counteract the rolling of the boat, input of the gyroscope, is formed by the following equation: