Rolf,
Unlike the discussion above I will fix the ring gear and take power
out of
the planetary cluster. The cluster would be attached on the prop shaft
and
be supported front and back on bearings. How would I calculate the
load the
planetary gears will be seeing? Total torque divided by the total
tooth
contact area? My engine will be putting out over 200 ft/lbs of
torque at
max rpm. With a supported prop shaft I believe the loads on the
bearings
will be from torque transmission and the centrifugal force of the
planets
themselves, right?
Also if I understand it correctly the gear line velocity is
circumference
times rpm. Calculating it from the Bell sun gear in feet that is
377mm /
304.8mm = 1.24 ft. 1.24' X 8200 rpm = 10,168 ft per min?
Mistral's unit at a 8000 designed redline with a slightly larger sun
gear
would not be far off this figure. You mentioned maximum line speeds of
5,000 ft/ min. Is that figure for light duty gearboxes?
Thanks
Doug,
You have a fixed ring gear and rotating planets. Give me the dimensions
of all the gears, DP and pitch diameter and we can calculate it
together. As already mentioned, it is pitch diameter times pi times rpm
/12. Pitch diameter in inches, pitch line velocity in fpm.
Since the pitch line velocity of all gears is the same, one need to
calculate just one. Best is the ring gear and the output speed or
propeller speed. The gear ratio is thereby irrelevant. The torque load
is also the same on all gears. The load per tooth is divided by the
number of planets.
The bearing loads are essentially torque divided by radius. The
centrifugal force of the planets only need be considered on the planet
bearings, for the output shaft they cancel each other out through equal
spacing around the circumference.
Rolf
The centrifugal load of the rotating planet assemble is
contained by the ring gear and the separating tooth forces.
Paul Lamar ...No rotor no motor.
Hi Rolf,
Here is the info again on the Bell 47 gear sets.
*Diameter is measured at gear tooth contact point. All measurements in
mm.
Name teeth number diameter gear width
Sun 46 120 27
Planet 23 57 22
Ring mm 92 234 25.4
Proposed input max rpm = 8200
Proposed max hp 343Hp
Proposed max input torque 220 ft/lbs
The planet gears tooth dimensions from the valley to the top of the 2mm
wide flat top is 5.9mm. The teeth are 4.6mm at their widest root.
In application Bell used the inner part of the hardened gears as the
bearing
race. A hardened collar (sleeve?) was placed over the gear shaft to
become
the inner race. This was 1950's technology and designed for low rolling
speeds but great torque. Keeping them center in the race where thick
washers on the outside and a fiber cage on the inside. TBO on the whole
craft was 1200hrs. The fiber cages lasted about 300 hours. Perhaps on
later
models Bell improved the bearings cages.
The inner hardened roller race that fits on the gear shaft is 23 mm in
diameter. The outer part of the bearings (actually the inner part of the
gear itself) is 39mm. The recess axially dimension to the central
inner rib
is 9 mm. Therefore the central rib must be 4mm wide. 22 - (2X9).
Calculations of bearing loadings:
Please check I am on the right track here.
The sun gear has to transfer a maximum of 220 ft/lbs torque at full
chat.
Most likely less. The torque output to the prop shaft with the 3:1 ratio
would increase to about 660 ft/lbs. Each of the six planetary gears
would
see 110 ft lbs. on what is basically a 2.5mm x 22mm (or 55 square mm) one
tooth contact area. The bearing is seeing 110 ft/lbs divided by the gear
28.5mm radius which works out to 3.85 lbs distributed over a line 18mm
long
contact line.
In conclusion. I believe I will have to have new planetary gears made and
ground with different internal bore specs to accept a standard high speed
sleeved roller bearing. Perhaps to increase bearing contact surface
area 22%
the internal gear rib will have to be eliminated. If gear deflection
strength becomes an issue the bore can be reduced in diameter slightly.
Depending on the design of the gear case the gear also can be widened
a bit
more thus giving even more bearing surface area. Presently the hand
inserted
roller bearings are held in place by two 3.2mm (one on each side) thick
spacer washers.
The housing has several ports to direct a stream of oil at the
planetary set
and at the front bearing. Since this design unit was meant to fly on a
helicopter I would like to find a way to introduce pressurized oil
through
the sun gear.
For pressurized oil into the prop shaft the Bell unit offers several
possible locations. The easiest manner would be to feed it into a
slightly
thicker outer collar that contained the proper shaft seals. This would
be a
'two birds killed with one stone' possibility. The present collar is
about
25mm thick.
All in the all the unit is rugged, compact and the case designed to
take the
large lateral transient loads placed on it which I believe my plane will
require.
Doug Fir
And too big to run at 8200 RPM.
Doug,
>From the table above, the DP seems to be 10, assuming 'American
National Standard Coarse' gearing with a straight cut, full length tooth.
If you have a helix angle the DP might be 12. It could also be Metric
Module gearing, m = 2.5 or so, but likely not, being in the Bell
helicopter.
The gear ratio is 1 + 92/46 = 3 : 1
Torque is given at 220 ft lbs (not ft / lbs)
Input speed is 8200 rpm, output speed = 8200 / 3 = 2733 rpm.
Using the 234 mm given by you as pitch diameter of the ring gear, we have a
pitch line velocity of 234 * pi * 2733 / 60000 = 33.5 m/s,
In imperial units = 234 / 25.4 * 3.14 * 2733 /12 = 6590 fpm
At 6500 rpm the speed would be 6.5 / 8.2 * 6590 = 5220 fpm or 26.5 m/s.
Output torque = 220 * 3 = 660 ft lbs.
F = T / r = 660 ft lbs * 12 / (9.2" PD / 2) = 1722 lbs,
with 6 planets, each planet gear carries 1722 / 6 = 287 lbs. or 1275 N.
The pitch line velocity is the same on all gears. After all, one tooth
forward on one gear is one tooth forward on the mating gear. The same is
true for the force between the gears. The force on one gear is the same
as the force on its pairing tooth. Since the gears all intermesh,
velocity and force is equal on all teeth of all gears. No need to do
more calculations.
The highest tooth bending load is on the smallest gear, here the planet
gears of 23 teeth.
Tooth shape:
Here I have to make assumptions, which are: full depth tooth ANSI,
coarse, DP 10, straight cut tooth. This may by no means correct.
If so, tooth working height = 2 / 10 = .2 inch or 5.08 mm, full depth =
2.25 / 10 = .225 in or 5.72 mm, tooth width at pitch line = 1.57 /10 =
.157 in = 3.99 mm.
The calculated working root width at 23 teeth is a bit more than that
measured by you of 4.6 mm. (perhaps there is a helix angle?)
The bending load is full tooth height * force = .2 " * 287 lbs = 57.4 in
lbs.
The section modulus is w* h^2 / 6 = 22/25.4 * (4.6/25.4)^2 / 6 = .004735
in^3
Stress = bending load / section modulus = 57.4 in lbs / .004735 in^3 =
12,120 psi,
Same in metric:
M bending = 5.08 * 1275 = 6477 Nmm
Z tooth = 22 * 4.6^2 / 6 = 77.6 mm^3
Stress = 6477 / 77.6 = 83.5 MPa, (* 145 = 12,100 psi.)
(Having a wider root width would reduce the stress.)
Keep in mind that the stress of 12,000 psi is when tooth contact is at
the top of the tooth. The stress reduces to zero when contact is at the
bottom. The mean stress is only half, but you need to design for the
full load.
Now you have to find a gear steel that has an allowable stress of 12,000
psi for say 3000 hrs.
The speed should be acceptable when using the maximum rpm during
take-off and generally short duration. I suppose cruising rpm would be
lower, say the 6500 rpm mentioned above where the pitch line speed
becomes more acceptable. Remember also that case hardness and core
strength determine gear life.
At a somewhat higher speed life gets a bit shorter. That can be
compensated by a higher quality steel for the gears that prolong life.
And the always important good lubrication will also serve to extend the
gear life.
Regarding gear pitch, preferred and practical shop sizes are DP 64, 48,
32, 24, 20, 16, 14, 12, 10, 9 etc. The smallest in the fine series is
120; coarse from 20 going down to 1, below that circular pitch is used
rather than diametrical pitch. In between sizes do exist as well.
All in all the gearing should be acceptable as likely you will not run
flat out at 8200 rpm continuously. Even so, it would only affect its
life cycle.
The bearing load on the output shaft near the ring gear is equal to the
force as calculated torque / radius, F = T / r = 1722 lbs.
Greetings
Rolf
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