Page 291 - Analysis and Design of Machine Elements

Shafts
deflection y and slope . The equivalent diameter of a stepped shaft can be obtained 269
from [4]
√
√ L
√
v
d = √ z (10.13)
4 √ ∑ l i
d 4
i=1 i
where
l , d –length and diameter of the ith segment in a stepped shaft;
i i
L –calculated length of stepped shaft. When a load is applied between bearings,
L = l; for an end loaded cantilever beam, L = l + K,where l is the span and K is
the cantilever length;
z –number of segments in the stepped shaft;
The calculation of slope and deflection are based on successive integration of differ-
ential equations for the elastic beam deflection curve expressed as [5, 6]
2
d y M
= (10.14)
dx 2 EI
dy M
= = dx (10.15)
dx ∫ EI
d M
y = = dxdx (10.16)
dx ∫∫ EI
The integration may be performed either analytically, graphically or numerically. For

a shaft of given length and loading, the bending deflection is inversely proportional to
the product EI, as indicated in Eq. (10.16). Therefore, the effective way to increase the
rigidity of a shaft is to increase the diameter of shaft.
The bending rigidity criteria are then
y ≤ [y] (10.17)
≤ [ ] (10.18)
The allowable misalignment of a shaft is determined by the requirements of mounted
specific gears or bearings by checking gear or bearing catalogues. As a rough guideline,
the allowable bending deflections for a transmission shaft is (0.0003–0.0005)l,where l is
span between bearings. When a shaft supports gears, the allowable bending deflections
is selected as (0.01–0.03)m ,where m is the normal module of mounted gears. The
n
n
allowable slopes should be checked at locations where gears and bearings are mounted.
At the cross section where a gear is mounted, the allowable slopes should be within
0.001–0.002 rad. The allowable slopes for the shaft where a deep groove ball bearing, a
cylindrical roller bearing and a tapered roller bearing, is mounted should be less than
0.005, 0.0025 and 0.0016 rad, respectively [7].

10.3.2.2 Torsional Deflections
Torsional rigidity is less important unless in some special applications. If a shaft has
uniform diameter over its whole length, the unit length angular deflection may be readily
calculated from formula in Mechanics of Materials [5], repeated here as
180 T 3 4 T
= × × 10 = 5.73 × 10 (10.19)
GJ GJ