Design and Stress Analysis of Extraterrestrial ... - The Black Vault

Critical angular velocities- of a singledisk
rotor with allowance for gyroscopic
niornen t
Let the disk in a shaft-disk system be attached to the shaft
asymmetrically (Fig. 3.68); therefore, during precession gyroscopic
moment will act on- the shaft in addition to centrifugal force. Let
us ,make- the following assumpti6ns..
1)' The shaft is weightless -but it
has elas-tic properties (the
mass of the shaft is negligible as compared with the mass of the disfk).
To reduce calculation error the weight of the shaft can be reduced
to the weight of the disk.
2) The shaft supports are absolutely rigid.
3) The forces of resistance to the motion of the disk and the
shaft are negligible.
During the precession of the disk its instantaneous position
in space can be determined by coordinates x, y, and z and projections
0 and 4 on the xOz and yOz planes of the angle between the tangent to
the elastic line at the disk attachment point and the z-axis.
To obtain equations of motion for the studied system we shall
ise a Lagrange equation of the 2nd kind;
- .(3.105)
dUti Oq Oq
where q are the generalized coordinates (in our case, x, y, wand o);
q = dq/•t-i-tie- gen-~lized velocity;
T is the kinetic energy of the precessing disk;
n is the potential energy of the system.
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