Theorem of the gyroscopic moment

The theorem of the gyroscopic moment recovers in fact two different theorems:

the theorem of the gyroscope maintained , at clean angular velocity ( $\ dowry {\ phi} = R = cste$ ) compared to its casing angular velocity $\ Omega$ .
the theorem of the free gyroscope on its axis of revolution which binds it to the casing.

These two theorems must in the Russian school (Lyapunov, Chetaev, etc), but Kelvin and Touches knew them too.

Theorem of the maintained gyroscope

Stated: One can suppose that the gyroscope is made interdependent of its casing and to add a couple known as gyroscopic which will take account exactly of the effect of rotation: = Cr couples. K \ $\ Omega$ .
Demonstration:

It is thus about the couple of the forces of Coriolis of the spinning top in the turning reference frame of the casing.

Theorem of the free gyroscope on its axis

Stated: One can suppose that the gyroscope is made interdependent of its casing. To take account of clean rotation, it should be considered that C = 0 for the solid gyroscope-interdependent of the casing AND it is necessary to add the couple ( L.K ). K \ $\ Omega$ .
Demonstration:

Applications

Very many and under development full until 1990: at this time, the GPS, definitely more powerful appears like means of positioning. As for stability, one counted already more on the chain sensor-control. As for the angular velocity measurement, the gyro-lasers détrôné the gyro mechanics. Nevertheless, the concept of gyroscopic moment continuous to be useful in grinds circumstances and it is advisable to detect of it the presence in the qualitative analysis of any movement where the three components of the vector rotation are coupled.

See too

Central inertial
gyroscopic Compass
kinetic Gyroscope
Moment embarked
Movement of gyroscopic Lagrange
Pendulum
Wheel of gyroscopic mill
Stabilization
Stability of attitude of the satellites

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