its known that if a torque is applied to a spinning gyro, the gyro will pitch. There's a lot physics on the pitch speed, but what is the torque to induce this pitching?
ps. I am building a gyro stabiliser who's pitch is adjusted by a stepper motor, like that of a boat. I am trying to work out the torque out of adjusting the pitch as well as the torque required to generate the required pitch speed.
I will first provide a link, and then I will go to your question specifically.
For discussion of the underlying mechanics I refer to my 2012 answer: gyroscopic precession. (That discussion, illustrated with diagrams, capitalizes on symmetry to explain the phenomenon.)
(Most attempts at explanation invoke the abstract concept of angular momentum vector. That rather defeats the purpose, because the angular momentum vector is itself a complicated concept.)
As discussed in the linked answer: the pitching motion occurs in response to rotation rate along the axis perpendicular to the pitching axis. That is: the applied torque causes a motion, and the gyroscopic effect occurs in response to that motion. As discussed in the linked answer: the pitching motion itself causes the gyro wheel to exert a torque. The magnitude of that responsive torque is proportional to the rate of gyroscopic precession.
As to what that means for engineering a device:
I expect that marine gyro stabilizers just have a large surplus of actuating capability, and only the rate of change of angle is measured. I expect that the software that controls the actuators is set up to act in rapid response, increasing the power output of the actuator to whatever value it takes to enforce the intended rate of motion.
Generally: the rate of gyroscopic precession elicits a corresponding torque. The elicited torque opposes the torque that led to the gyroscopic precession response.
For a gyro stabiliser: to keep producing the same effect the actuator must overpower the elicited torque. Assuming the spin rate of the gyro wheel will be high: at high rate of spin of the gyro wheel overpowering the elicited torque will be, by far, the dominant factor.
Gravity provides a torque that would pull the gyro over if it is not spinning. When the gyro is spinning, in the non-inertial frame precessing with the gyro, fictitious forces are present. Consider the top and bottom of a rapidly rotating wheel; the centrifugal forces are equal and opposite and cancel out, but the Coriolis forces produce a torque that balance the torque from gravity that prevents the gyro from falling down. You want the gyro to spin rapidly, that is the spin angular velocity much greater than the precession angular velocity.
The motion is developed in many intermediate/advanced physics mechanics textbooks. Also, a text available free online, Dynamics by Kochmann at ETH Zurich, has a very clear mathematical and physical description of tops and gyros; see https://ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/Dynamics_LectureNotes.pdf. You can search this site for tops and gyros for more explanations, especially the 2012 discussion referenced by @Cleonis in his answer to this question.
Hope this helps.
