By Gerhard Schweitzer
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Extra resources for Critical Speeds of Gyroscopes: Course Held at the Department for General Mechanics September 1970
2). The behavior is, in the last analysis, already given by the solutions of the homogeneous system (Chapt. 3. 2), but nevertheless it shall be dis cussed briefly. Disturbances by rotor unbalances are caused by the rotation . ~ 1c = Q 1c of a slightly unsymmetrical rotor. P 1c remains as assumed a cyclical variable, when the equations are linearized. The unbalance forces can form, quite similar 32 Gyrospopic system to the deviation of Fig. 4, a complex quantity. Fig. 5 For the sake of slmplicity it is assumed that the angular velocities of the rotors are all alike of disturbances of ( 3.
The investigation has to show whether and how much the system suits to this aim. At first the equations of Fig. legrees of freedom will be established. There are Elastically supported rotor 38 six translational degrees of freedom, two for the upper and the lower bearing and the center of mass of the rotor, respectively, the rotor has two rotational degrees of freedom, the angular velocity about its axis of symmetry be a constant. For a symmetrical gyro the rotation about the vertical is a particular solution.
For a symmetrical gyro the rotation about the vertical is a particular solution. It has to be asymptotically stable. For the nearly symmetrical gyro the deviations of its axis of symmetry from the vertical shall be small. Therefore the highly nonlinear equations of motion can be linearized provided the variables are chosen suitably. According to Fig. 6, two coordinate systems are used. The transformation of the inertially fixed 1 2 3 - system into the body-fixed i 23 - 2 by the cardanic angles a,j), y system is described as in Chapt.
Critical Speeds of Gyroscopes: Course Held at the Department for General Mechanics September 1970 by Gerhard Schweitzer