Worm Gears and Worms

Worm Gears and Worms

At the February meeting I discussed my 'hobbyist method' for making miniature
worm gears and worms.  There seemed to be some genuine interest in what I said
so I decided to write a bit about it for those who may have missed some of what
I presented rather quickly.


First we need to discuss a bit of (spur) gear terminology and mathematics.  In
the system used in America (metric gears are very similar but the nomenclature
is somewhat different), gears are characterized by:

N = number of teeth on gear (self-explanatory)
PD = pitch diameter (see below)
P = diametral pitch (see below)
OD = outside diameter of gear (self-explanatory)

Picture two cylinders which are rolling against each other without slipping.
This is the model for all gears.  The diameters of these cylinders correspond
to the pitch diameter of the gears.  To form gears, we imagine adding teeth to
the cylinders.  These teeth extend above the pitch diameter by an amount
called the ADDENDUM and below the pitch diameter by an amount called the

The meshing of the teeth during the rolling of the cylinders does two things.
First it guarantees that the rolling will occur without slipping - this is why
gears are used in applications where the angular relationship of the axes must
be maintained, as in the leadscrew drive on a lathe for threading.  Second, the
meshing of the gears allows the transmission of far more torque than would be
possible in two rolling cylinders that depend on only friction to transmit

To reduce wear on the teeth when they mesh and unmesh, we would like the
contact line between two meshing teeth to remain fixed during contact - no
relative motion => no wear.  Although I don't intend to prove it here, this can
be accomplished by forming the tooth face in the form of an epicycle.  Imagine
a small circle rolling around the large circle formed from the pitch diameter.
The curve traced out by a point on this small circle will be an epicycle.

The diametral pitch of a gear is analogous to the pitch of a screw thread.
In mathematical terms,

P = N/PD

It is the ratio of the number of teeth to the pitch diameter of the gear.

Theoretically, P can take on any value (just as any screw thread pitch is
possible) but, in production, gears are made to a few standard pitches (just
as screws come in standardized values of threads/inch).  *Some* example
standards are:

20, 24, 32, 48, 64, 72, 80, 96, 120, 200

If making your own gears, you can, of course, use any pitch you wish, though,
in the interest of preserving the ability to incorporate commercially
available gears, you may want to consider using standard values.

For two gears to mesh, the pitches must be the same.  This fact is what allows
us to calculate gear ratios in terms of the number of teeth on each gear.
The speed ratio of two gears is the ratio of the two pitch diameters that
describe the theoretical cylinders rolling against each other, e.g.:

For two cylinders, diameters PD1 and PD2 inches, to roll without slipping, the
surface speed of the two cylinders must be the same.


SFM1 = SFM2 leads to:

speed ratio = RPM2/RPM1 = PD1/PD2


PD1 = N1/P1 and PD2=N2/P2

As mentioned, the pitches of the two gears must be the same, so:

P1 = PD1/N1 must equal P2 = PD2/N2


PD1/N1 = PD2/N2


PD1/PD2 = N1/N2

That is, the ratio of the pitch diameters is the same as the ratios of the
number(s) of teeth for two gears of the same pitch.

If one intends to make gears, one must know the size of the blank to cut -
what is referred to above as the OD of the gear.  For spur gears, the relation

OD = (N+2)/P

The addendum is the amount that a given tooth projects above the pitch circle
so it must be true that the addendum, A, satisfies:

2*A = OD-PD = (N+2)/P - N/P = 2/P


A = 1/P

The distance between adjacent tooth flanks along the pitch circle circumference
is known as the circular pitch, CP.  In mathematical form:

CP = pi*PD/N = pi/P

Occasionally you will see CP used to characterize a gear instead of the more
commonly used diametral pitch, P.  With the above relation, obtaining the
diametral pitch from a know circular pitch is trivial.

Gear teeth are also characterized by a pressure angle, PA.  The pressure angle
is the angle between a tooth flank and the gear radius, measured at the pitch
circle.  For older gears, this angle was typically 14.5 deg.  This rather odd
value was chosen because the fact that sin(14.5 deg) ~= 0.25 made gear
calculations easier in the days before electronic calculators and computers.
More modern equipment uses a PA of 20 deg although other values (e.g., 25 deg)
are occasionally encountered.


I mentioned that a properly formed gear tooth has a flank in the shape of an
epicycle, a curvilinear profile.  However, as the pitch diameter of the gear
increases, this curvilinear form becomes straighter and straighter.  In the
extreme case of a rack (a gear with an infinite pitch diameter), the flanks of
the teeth are planar surfaces.

Gear teeth are properly cut with expensive cutters and hobs that faithfully
generate the epicyclic form.  However, if you look at small gears of
moderately large pitch (e.g. 50 and up), you'll see that the tooth flanks are
very close to being flat.

What this means for the hobbyist is that we can grind simple, planar tools and
use them to cut small, non-critical gears without the need to invest in
complex and expensive gear cutting equipment.  Obviously, I'm talking here of
hobbyist applications where gear speeds and torques are low, wear is not a
major issue, and some degree of backlash is acceptable.  Applications
requiring serious power or accuracy should use conventional gear cutting
methods or employ commercial gears.

For a gear with flat-sided teeth (same as a rack of the same pitch), the angle
that the tooth side makes with a radius will be equal to the pressure angle.
Therefore, if we wish to make gears with a 20 deg pressure angle, we grind a
lathe tool that looks much like a thread-cutting tool with the difference that
the included angle of the tip is twice the pressure angle or 40 deg.

The next step is to determine the size of the blank (OD) we need for the gear.
Although one can work with the equations outlined above, it's easier to use
the GEARSPUR program available on my web page


This program allows you to enter any two of (N, OD, P, PD) and it will compute
all the other gear specifications.

Let's say we want to make a 64 pitch gear with 36 teeth.  Running the program
produces an output that looks like:

<C:\mwk>  gearspur
Enter whatever data you know.  Enter zero (0) for unknowns.
You must enter two data items to obtain an answer.
OD of gear [2.35 in] ? 0
Number of teeth [45] ? 36
Diametral Pitch [20 in] ? 64
Diametral Pitch = 64.0000
Module = 2.5197
Number of teeth = 36
Outside Diameter = 0.5938 in = 15.0813 mm
Addendum = 0.0156 in = 0.3969 mm
Dedendum = 0.0188 in = 0.4763 mm
Whole Depth = 0.0344 in = 0.8731 mm
Circular Pitch = 0.0491 in = 1.2468 mm
Tooth Thickness = 0.0236 in = 0.5985 mm
Pitch Diameter = 0.5625 in = 14.2875 mm

We need a blank with an OD = 0.5938 in.  This blank is mounted in an indexer (I
used a Spindex) capable of indexing every 360/36 = 10 deg.  The gear cutting
tool is mounted in a holder (I used a boring head) such that it will rotate in
a horizontal plane in the mill.  Adjust such that the tool tip is on the
centerline of the blank.  Set up such that the depth of cut of the tool will be
0.0344", the Whole Depth of the tooth reported by the computer program.

Cut a groove, increment by 10 deg, repeat until 36 grooves have been cut and
you'll have a serviceable 64 pitch gear.  You can check your results by meshing
the gear you cut with a commercial 64 pitch gear/rack.  They should mesh with
no binding.

I needed a worm to mesh with this gear.  If you look at the program output,
you'll see that the Circular Pitch for this gear is 0.0491 in.  As mentioned
above, this is the distance between adjacent teeth (or inter-tooth spaces)
measured on the pitch circumference of the gear. 

This pitch (0.0491 in) is almost exactly the pitch of a 20 tpi thread.  (It
wasn't by accident that I chose to cut a 64 pitch gear :-) So, the worm is made
by cutting a 20 tpi thread on a blank (3/16 in diameter but that isn't
critical) using exactly the same tool used to make the gear.  Depth of thread
on this worm is the same as the tooth depth on the gear, 0.0344 in.

As I showed with the prototype at the meeting, this process produced a very
servicable worm gear and worm that meshed smoothly with only a small amount of


If you're going to make only gears (and not worms), you'll need to have the
ability to do the angular dividing necessary for the number of teeth you need.
I was fortunate to be able to live with a number of teeth (36) that led to a
simplistic angular increment, 10 deg.  Depending on your choice of N, you may
need a dividing head or rotary table to do the necessary dividing.

If you're going to make worms, you'll need to choose a combination of primary
gear parameters (N, P, PD, OD) such that the resulting Circular Pitch is close
to a thread pitch that you can cut on your lathe.

Aside:  There is an opportunity for experimentation here.  I *believe* there
are some commercial gears made with a 30 deg PA, although you might prove me
wrong.  If so, a conventional threading tool could be used to cut gears and
worms could possibly be cut with a conventional die, which opens up the
possibility of a wider selection of thread pitches than might be doable with
the lathe.  (Think of using metric dies to get pitches 'in between' the
Imperial pitches.)

Marv Klotz