Page 460 - Wind Energy Handbook

434 COMPONENT DESIGN

ó
ó alt
¼ 1 (7:52)
ó lim ó ult
where ó lim is the permitted alternating bending stress with zero mean, ó is the
mean bending stress and ó ult is the ultimate tensile strength. Setting ó ¼ ó alt =2
results in:

ó lim ó ult
ó alt ¼ (7:53)
(ó ult þ ó lim )

If the ó lim =ó ult ratio is 0.2, then ó alt ¼ 0:833ó lim and the permitted peak bending
stress at the endurance limit is 1:667ó lim . In epicyclic gearboxes, by contrast, the
gear teeth on the planet wheels are loaded in both directions, so the permitted peak
bending stress at the endurance limit is only 0:5ó lim . As the number of teeth on the
smallest gear cannot be reduced indefinitely, this means that tooth bending is more
likely to govern in the case of epicyclic gearing.
The minimum total gear volume for an infinite series of epicyclic gear stages with
fixed annuli is obtained for a gear stage ratio of two, which implies that the radius
of the sun gear is the same as that of the annulus gear and that there are an infinite
number of planets! This is not realistic, and the annulus radius is in practice
typically double the sun radius, giving a gear ratio of three. It is instructive to
compare the volume of gears for an epicyclic and parallel gear stages with this ratio,
assuming that tooth bending stress governs in each case.
For the parallel stage, it can be shown using Equation (7.47a) that the volume of
the pinion is

ð 2 F t d 2 z 1 2T LSS z 1
bd ¼ k B F t d 1 z 1 =ó alt ¼ k B ¼ k B (7:54)
1
4 ó alt u 0:833ó lim u
where k B is a constant. This gives a volume for gear wheel and pinion of
2
2:4k B T LSS z 1 (1 þ 1=u )u=ó lim ¼ 8k B T LSS z 1 =ó lim for u ¼ 3. For the epicyclic stage, the
volume of the planet, which is assumed to have the same number of teeth as the
pinion of the parallel stage, i.e., the minimum permissible, is

ð 2
bd Pl ¼ k B F t d PL z 1 =(0:5ó lim ) (7:55)
4

If the low-speed shaft drives the planet carrier and the N planets are spaced at 1.15
diameters, then the low speed shaft torque is

T LSS ¼ F t N(r A þ r S )

where

ð(r A þ r S )
N ¼
1:15(r A r S )