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A -F -E -F A -D -E -D c with D, E, F as small quantities, and the vertical forces N, K between the rotor and the upper or lower bearing, respectively. tor in the direction of the 3-axis. The theorem of moment of momentum is now establish ed for the rotor and the theorem of impulse is applied to the motion of the upper and lower bearing and the center of mass of the rotor in a similar way as it was done in Chapt. Z for the rotor on overhung shaft. Hence, in inertially fixed coordinates the equations of motion show the well-known form {Chapt.

3. 2). e. that the system is statical- ly stable. + cur) c0 ch] + (4. 4) +. c 0 c h cu. ,. ( c +d) 2 > 0 holds. As the damping matrix is only positive semidefinite asymptotical stability on the basis only of the damping matrix is not secured but it can be supposed. k - 8 k + ~ real parts oo k for a numerical example. ale. It turns out that the are all negative and thus the solutions are asymp- totically stable. The natural frequencies Wk are shown in Fig. 7 (see next page} depending on the rotor frequency Q.

1 H 2. a1 + 2. g1 For H ~ oo both the damping H 2. 8 P "-~ 1/ H2 and the pre- cession frequency U>p"'l/ H tend to zero. In Chapter 4 a gyroscopic system is presented the technical performance of which is decisively governed by its damping behavior for high rotor velocities. 3. 3. Critical speed and resonance. Disturbing forces that act on the system are caused first of all by small rotor unbalances and lead in general to a disturbance vector f ( t) on the right side of the equation of motion (3.

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General relativity and gravitation Vol. 25


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