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National Advisory Committee for Aeronautics, Technical Notes - Analysis of Stresses and Deflections in a Disk Subjected to Gyroscopic Forces

The differential equation governing the deflection of a disk of var-
iable thickness subjected to gyroscopic loading is derived. For the case
of a disk of constant thickness, solutions are obtained by a finite-
difference method for a range of centrifugal loading parameter M from
O to 50, and the ratio of shaft to disk radii varying from 0.1 to 0.5.
Results are presented in dimensionless form suitable for design purposes.
The method for solving the prdblem of a disk of variable thickness with
a temperature gradient is also presented.

When the axis of rotation of a disk is itself rotated, forces are
set up normal to the disk. These forces, or gyroscopic loads, occur on
Jet—engine comressor and turbine disks whenever the airplane changes
direction either in the air or on the ground. These gyroscopic forces
will deflect a disk out of its plane of rotation, induce vibratory bend—
ing stresses, and produce a bending moment that will increase the shaft
bearing loads. This problem may be even more serious for the high-speed
disks used in missiles undergoing high accelerations . Although the
stresses thus produced may not themselves be very large, the combined
effect of these stresses with the already existing stress distribution
may be sufficient in some cases to cause a failure. It is therefore nec-
essary to investigate both the stresses and the deflections of a disk
under gyroscopic loading.

By considering the gyroscopic forces and centrifugal forces acting
on an element of a disk of variable thickness, the general differential
equation describing the deflection is derived. A finite—difference solu-
tion of this differential equation is obtained for a constant—thickness
disk rotating about its center at a constant angular velocity. The com-
puted deflections and the corresponding stresses are plotted in dimen-
sionless form for a wide range of a centrifugal—loading parameter, and
for a number of different ratios of shaft to disk radii.