Below we explain the basic physics of gyroscopes and how we apply gyro-dynamics to eliminating boat roll. We will use SI units in all discussions.
Units
Moment of inertia:
Angular velocity:
Angular momentum:
Torque: |
kg-m2 (kilogram meters squared)
rad/s (radians per second)
N-m-s (Newton meter second)
N-m (Newton meter) |
The primary component of the gyro is a flywheel that has a Moment of Inertia, J, and an angular velocity wF. The moment of inertia of the flywheel is determined by its mass and the distribution of its mass relative to its shaft. Increasing mass at the flywheel’s outside diameter and/or increasing the flywheel’s diameter increases its moment of inertia. The Angular Momentum, L, of the flywheel, the product of its moment of inertia and its angular velocity, is a measure of the extent to which the flywheel will continue to rotate about that point unless acted upon by an external torque. The higher the angular momentum the more ability for the flywheel to react to external torques i.e. in our case more ability to cancel boat roll.
A gyroscope has three axes; a spin axis, and input axis, and an output axis. The spin axis is the axis in which the flywheel is spinning and for our gyro the spin axis is vertical. The input axis is the axis on which inputs are applied. In our case, the principal input axis is the longitudinal axis of the boat since that is the axis around which the boat rolls. The principal output axis is the transverse (athwartship) axis about which the gyro rotates or precesses in reaction to an input.
Let’s look at numbers on a Model 7000 Gyro system.
Moment of inertia J = 6.7 kg-m2
Angular velocity: wF = 1047 rad/s (10,000 rpm)
Angular momentum: L = J × wF = 6.7 kg-m2 × 1047 rad/s = 7015 N-m-s
When the boat rolls, it acts as an input to the gyro. This input causes the gyro to generate rotation around its output axis such that the spin axis rotates to align itself with the input axis. This output rotation is called precession and in our case the gyro is rotating fore and aft around the output or gimbal axis.
Two hydraulic cylinders are coupled to the gyro’s gimbal axis to act as a brake which controls the gyro precession rate. Rate sensors and a digital processor control the precession rate using the hydraulic cylinders.
The maximum output force applied to counteract the boat roll (the input to the gyro) is governed by the following equation:
To = wP × L
where To = output torque about the gimbal axis (N-m)
wP = maximum output rotation rate or precession rate (rad/s)
L = angular momentum (N-m-s)
Plugging in the numbers for the Model 7000 Gyro system:
wP = 2.09 rad/s, limited by the digital processor using the hydraulic brake
L = 7015 N-m-s
To = wP × L = 2.09 rad/s × 7105 N-m-s = 14,661 N-m
This output axis torque is coupled back into the boat’s hull to counteract the roll of the boat.
