[SEL] OT gear making in brief

[Richard Allen]

Making gears, brief summary

Gear tooth geometry for the most part is a nightmare from the
perspective of a mathematician, but a piece of cake for a machinist. I
have my feet planted in both fields and have looked at gear tooth
design from both perspectives. From a real-world practical standpoint I
could care less about the actual mathematics of a gear tooth shape
because knowing the math of its shape is not necessary for generating
that shape, just as I don’t have to have any knowledge of parabolic
curve equations to be able to throw a rock! 

The way that the commonly used involute gear tooth shape is generated
is done quite simply by the interaction of two mating parts rotating in
the way in which they are to be used. The result is that the correct
and exact curved surfaces on the gear teeth are formed naturally. It is
nowhere near as simple for a mathematician to figure this out and then
draw a picture, and in their words, the machinist’s method is “elegant”
compared to theirs. The name of the shape is “involute”, but unlike the
parabolic curve or the circle where there is only one such shape for
each of those (and consequently makes them unique), the involute curve
is actually a whole series of curves that gradually transition in
one-tooth increments from the truly straight-sided triangular teeth
(with the tips cut off) of the flat rack gear to the theoretical
minimum two-toothed pinion gear whose pair of opposed teeth have highly
curved contact surfaces that engage the teeth of a flat gear rack. The
flat gear rack can be thought of as a gear with lots of teeth and a
huge diameter, and in fact on gears with a very great number of teeth,
the teeth are very nearly the same shape as the triangular rack gear
teeth. By the way, one-tooth gears exist and are quite common, they are
called single-helix worms and they run on a worm gear, but that is
cheating because the profile of the worm tooth is the same as a flat
gear rack tooth! A single helix worm is considered a one-tooth gear
because its mating gear advances one tooth per revolution of the worm.
Worms can have more than one helix, though, just like a threaded rod
can have more than one helix cut on it (multiple helix threaded parts
are most commonly used for the screw lids on jars where the engagement
has to be great but the number of turns to unscrew it needs to be few
to save motion).

Gear teeth can be any form you want as long as it is physically
possible for them to mesh. A common everyday non-involute example we
all know is where the teeth are roughly pyramidal on one of the gear
wheels and circular humps on its mating part, which in practice is the
ordinary bicycle sprocket gear and roller chain. The roller chain is
just a long flexible gear and when made into a perfect circle it still
works just fine with its sprocket gear, it fits equally well both
inside and outside! Another non-involute form is the so-called silent
chain used for the timing gear train of engines and also for the
transfer-case gearing in a lot of four-wheel-drive vehicles. The silent
chain tooth shape and method of construction are extremely simple, as
well as rather amazing and efficient in operation, and I think highly
underutilized. 

Involute gears are made by several methods, but the only one that does
not involve specially shaped cutters is by shave-cutting the teeth
using a cutting tool that is a flat gear rack in profile (gear rack, as
in the rack-and-pinion gearing used for a lot of steering mechanisms).
This is highly desirable because the involute curve of a tooth is
different for different numbers of teeth. A cutter with the profile for
a ten-tooth gear is useless for cutting a gear with eleven teeth
because the teeth are different shapes. The only time a specific tooth
profile cutter becomes practical for gears with different number of
teeth is when the number of teeth on a gear becomes very large, which
is when tooth shape begins to approach the flat-sided pyramid shape of
the flat gear rack. For gears with large numbers of teeth a cutter can
be used for a specific range of teeth, one size cutter say for gears
with 24 and 25 teeth,  another for 26, 27, and 28 teeth, another for
29, 30, 31, and 32 teeth, etc., the range increasing as the number of
teeth gets larger. For shave cutting with a gear rack-shaped cutter,
the round gear blank is first “gashed”, which is where the blank is
indexed for the number of teeth it will have by removing most of the
metal between the teeth by sawing or milling. Next, with the gear blank
laying on its side while the gashed slots are rotated in perfect time
with the rack-shaped cutter teeth, the gashed blank is fed under the
rack-shaped cutter, which is also laying on its side. When the
rack-shaped cutter moves down, its teeth mesh at full depth into the
slots gashed into the blank, and the excess metal between the blank
teeth is shaved off by the sharpened side of the rack. The rack is
moved the same way you would drag a comb down the side of a round
biscuit of soft clay so its teeth scrape grooves into it. The gashed
blank is fed in another tiny increment while also being rotated in
time, and then the rack-shaped cutter moves down again, taking off
another small shaving. After the gashed blank has been fed one complete
revolution in time with the rack-shaped cutter, while also having
dozens or even hundreds of downward strokes performed to shave off
unwanted material, it now has its teeth properly formed with the
involute curve. There were no special levers, arms, linkages, cams, or
any other mind-boggling Rube Goldberg contraptions used. The gear
rack-shaped cutter will cut every and all number of gear teeth on any
size gear using that size and shape of tooth profile. All that is
needed for generating the involute curve are a rack-shaped cutter,
which is extraordinarily simple to make, and a method for timing the
rotation between the gashed biscuit blank and the cutter. Another gear
cutting method based on the rack is gear hobbing using a cutting tool
called a hob. The hob is a round cutter with gear rack-shaped teeth
that cuts by being rotated instead of being moved linearly and is the
same basic design shape as a thread-cutting tap. A hob that is
positioned correctly with an accurately-gashed gear blank is
self-timing in very thin gears and automatically rotates the gear blank
as it is fed in from the side, the angled teeth grabbing and rotating
the work piece as it turns. The flat gear rack teeth can have any
amount of truncation and angle of triangle you want, within reason set
by the physical limitations of the parts that mesh with it. Half of the
angle between the legs of the triangular rack teeth is one definition
of the gear tooth pressure angle, and the angle most commonly used for
gears with large numbers of teeth is 14.5 degrees. Gears with few teeth
use 20° or 25° to avoid thin tooth roots from gear tooth tip motion
that requires clearance that “undercuts” into the root of the gear
tooth. The 14.5° angle was very specifically chosen because the sine of
14.5 degrees is extremely close to 0.25 (0.25038 to be exact), making
the measurement calculations for machining the gear rack very easy (in
fact, the 14.5° gear tooth pressure angle is in reality 14.478° because
the actual sine value used is 0.25000, not 0.25038, but in the real
world that value in its application is so close to 14.5° that there is
no meaningful difference between the two angles). A pressure angle of
20° is also very common in gears with a large number of teeth,
especially in German machinery, or for gears made from very soft
materials. As the pressure angle is increased the force pushing the
gears apart increases, so if the squat blunt teeth of a shallow
pressure angle are used, then a great deal of force is generated
pushing the gears apart and it unnecessarily tears up the bearings
supporting the gears. A pressure angle of 30° is occasionally found,
and it is the result of the gears being hobbed using an ordinary
threading tap! The thread tap-hobbed 30° pressure-angle teeth are very
blunt and come to sharp points, which are only good for light torque
loads. 

Sprocket gears used with roller chains are another example where the
shape of one of the parts is used to generate the other. Sprockets are
cut by advancing a round cutter into a gear blank that is rotated in
time with it. The round cutter travels the same path as a roller in the
roller chain and therefore cuts out the shape of its contact path. 

The silent chain used for timing in a lot of engines is an example
where there are no special curves or oddball calculations needed! The
geometry of the links eliminates all of that. Another feature that is
peculiar to silent chain is that the lengthening of pitch from wear is
self-compensating because as the chain is re-tensioned it climbs
outward on the gear teeth, where the pitch is greater. Because of this
they are good where rotational velocity needs to be very uniform,
making it perfect for running camshafts that have the ignition timing
attached. They are also good for tank tracks where high speed
chattering from a mis-matched pitch between the track and drive
sprocket caused by excessive wear would be a problem. 

Hypoid and other gears with compound curved teeth and odd-angle contact
seem to be a real puzzler about how they are cut, but again the motions
needed to form the contact paths are generated while being cut by
timing the cutters with the rotation of the gear blanks. 

Standardized gear tooth sizes are determined by making the generator
flat gear rack with tooth spacings specified by using a formula based
on pi-inches. One pi-inch is simply 3.14159 inches long. A gear with 20
teeth per pi-inch has the teeth 0.1571 inches apart, or 1/20 of an inch
multiplied by pi.  A gear with three teeth per pi-inch has them 1.0472
inches apart. Instead of calling them teeth per pi-inch it is
abbreviated as a 20-pitch, 9-pitch, or 3-pitch, etc., gear. All gears
with the same size pitch number will all fit each other. So a gear with
127 teeth will fit a 50 tooth gear if they both have the same pitch
number. 

The only special calculations that need to be made for the machining
operations generated by the natural functions of gear tooth contact
paths are for making adequate clearance and minimizing backlash. The
final operation in some gearing is by selecting the gears that fit
together the best because things like minor differences from one
cutting tool to the next can amplify differences that make selected fit
necessary in spite of careful measurement and good machine shop
practice. That is why straight cut spur gears are a favorite where
noise is of no concern since their simplicity in both design and
manufacturing methods produce very precise parts where little or no
matching needs to be done to ensure a good fit. Helical and hypoid
gears introduce angles where the set-up measurements are based on their
tangents, and tiny errors of 1/10,000 inch will introduce
angle-amplified cutting errors of up to 0.020-inch that are totally
unavoidable, even by the best equipment and operators. This is where
custom fitting of parts becomes necessary, and is why automobile
differentials with hypoid ring and pinion gears must be in custom
matched sets.

There is still a lot more to gears and their designs, but this covers
most of the real basics. Simple spur gears can be cut in a home
workshop on an ordinary lathe with some special attachments.

Rich Allen