Environmental Engineering Reference
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and
2
TR
2
2
2
c
fd
= fd R
= QK R
(13 )
a
s
(
)
where d p is the planet wheel pitch diameter, d s the sun wheel pitch diameter, d a the
annulus pitch diameter, f the sun wheel face width, N s the sun wheel tooth number,
C the planet tooth bending criterion, K the sun wheel surface criterion, T s the sun
wheel torque, T c the planet carrier torque and Q is the number of planets.
From Fig. 5 it can be seen that in effect, an annulus has a negative diameter
exemplifi ed by the concave fl anks on internal teeth. This means that given the
same pressure angles, the product of the annulus and planet base tangent lengths is
R
times that of the planet and sun whereas the sum of the respective base tangent
lengths are equal and opposite so that algebraically, the relative radius of curvature
at the planet/annulus mesh point is precisely R times that of the planet/sun. Given
the same face widths its K value is reduced accordingly by the reciprocal of R . The
internal tooth root thickness is also somewhat thicker due to its concavity so that
lower grade material and/or a smaller face width may be used.
The signifi cance of the above is illustrated by comparing the annulus volumes
of two planetary gears having the same carrier torque and sun wheel surface stress
but with R equal to either 2 or 3, i.e. planetary ratios of 3 and 4 having either 8 or
5 planets respectively viz.
2
a
(14a )
fd
= T
(4 / 8)
=
0.5
T
c
c
or
2
a
fd
= T
(9 /10)
=
0.9
T
(14b )
c
c
The larger ratio annulus is therefore, 1.8 times the volume!
The comparable volumes of a simple wheel subject to the same torques, surface
stress and ratios are viz.
2
wc
(15a )
fd
= T
(3
+
1)
=
4
T
c
or
2
wc
(15b )
fd
= T
(4
+
1)
=
5
T
c
Without considering the pinion offset, the fi rst is 8 times and the second 5.56 times
the volumes of the annuli of the alternative planetary gears.
Even with only three planet wheels, the volume of the annuli is always 30% of
an equivalent parallel shaft wheel for any ratio from 3 to 12.
4 B earings
Rolling element bearings are the type most commonly used in wind turbines for both
parallel shaft and epicyclic gears. Generally, the design criteria for such bearings
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