Page 55 - Mechanical Engineer's Data Handbook

44 MECHANICAL ENGINEER’S DATA HANDBOOK

Annulus bonded to sleeve: 1.6.6 Critical speed of whirling of shatits
~RLG
When a shaft rotates there is a certain speed at which, if
there is an initial deflection due to imperfections, the
centripetal force is equal to the elastic restoring force.
At this point the deflection increases to a large value
and the shaft is said to ‘whirl’. Above this speed, which
depends on the shaft dimensions, the material and the
loads carried by the shaft, the shaft whirling decreases.
Shafts must be run well below or well above this speed.
It can be shown that numerically the critical speed is
the same as the frequency of transverse vibrations.
Formulae are given for several common cases.
Critical speed for all cases:
Annulus bonded to disks:
1
ZG Nc=j+ii
(r:
s == -rf)
where: g=acceleration due to gravity, y= ‘static’
deflection at mass.

Cantilevered shaft with disc at end

Mass of shaft neglected.
N, = 1/2x,/m

Hyperbolic contour:

Central disc, ‘short’ bearings

N,= 1 / 2 R J w ~

Non-central disc, short bearings

N, = 1 / 2 R , / m