
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.
BACKGROUND
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 (selfexplanatory) PD = pitch diameter (see below) P = diametral pitch (see below) OD = outside diameter of gear (selfexplanatory)
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 DEDENDUM.
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 motion.
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=pi*PD1*RPM1/12 SFM2=pi*PD2*RPM2/12
SFM1 = SFM2 leads to:
speed ratio = RPM2/RPM1 = PD1/PD2
but:
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
Thus:
PD1/N1 = PD2/N2
or:
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 is:
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 = ODPD = (N+2)/P  N/P = 2/P
thus:
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.
MAKING SIMPLE GEARS AND WORMS
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, noncritical 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 flatsided 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 threadcutting 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
http://www.myvirtualnetwork.com/mklotz
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 intertooth 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 backlash.
EXPANDING ON THIS PROCESS
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
