Application
A pendulum the length of the Earth's radius is impractical, so Schuler tuning doesn't use physical pendulums. Instead, the electronic control system of the inertial navigation system is modified to make the platform behave as if it were attached to a pendulum. The inertial platform is mounted on gimbals, and an electronic control system keeps it pointed in a constant direction with respect to the three axes. As the vehicle moves, the gyroscopes detect changes in orientation, and a feedback loop applies signals to torquers to rotate the platform on its gimbals to keep it pointed along the axes.
To implement Schuler tuning, the feedback loop is modified to tilt the platform as the vehicle moves in the north-south and east-west directions, to keep the platform facing "down". To do this, the torquers that rotate the platform are fed a signal proportional to the vehicle's north-south and east-west velocity. The turning rate of the torquers is equal to the velocity divided by the radius of the Earth R:
So:
The acceleration a is a combination of the actual vehicle acceleration and the acceleration due to gravity acting on the tilting inertial platform. So this equation can be seen as a version of the equation for a simple gravity pendulum with a length equal to the radius of the Earth. The inertial platform acts as if it were attached to such a pendulum.
Schuler's time constant has other applications. Suppose a tunnel is dug from one end of the Earth to the other end straight through its center, a stone dropped in such a tunnel oscillates with Schuler's time constant. It can also be proved that the time is the same constant for a tunnel that is not through the center of Earth also.
Read more about this topic: Schuler Tuning
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