Internal vs. External Gear teeth - same profile?

I was just looking at some pictures of internal gears, and now I’m wondering, is it the exact same teeth profile as a spur gear? Like a “negative” in a sense. Like if I took a spur gear and (analogously) put in in a bucket of plaster, and then the plaster hardened and I pulled the spur gear out, would I have an internal gear of correct dimension/shape?

As I said in your other post, the negative of an external gear does not result in the proper profile of an internal gear.

The profile of an internal gear may also need to be undercut:

When the number of teeth in a gear is small, the tip of the mating
gear tooth may interfere with the lower portion of the tooth profile.
To prevent this, the generating process removes material at
this point. This results in loss of a portion of the involute adjacent
to the tooth base, reducing tooth contact and tooth strength.
On 14-1/2°PA gears undercutting occurs where a number of
teeth is less than 32 and for 20°PA less than 18. Since this
condition becomes more severe as tooth numbers decrease, it
is recommended that the minimum number of teeth be 16 for
14-1/2°PA and 13 for 20°PA.

In a similar manner INTERNAL Spur Gear teeth may interfere
when the pinion gear is too near the size of its mating internal
gear. The following may be used as a guide to assure proper
operation of the gear set. For 14-1/2°PA, the difference in
tooth numbers between the gear and pinion should not be less
than 15. For 20°PA the difference in tooth numbers should not
be less than 12.

I have to disagree with Jack on this one. The proper tooth form for an internal involute gear is indeed the inverse of the external involute. The involute is formed on the common tangent between the base circles of the exterior and interior gear. This is identical to the involute formation for two external gears, except that the tangent does not cross the line of centers between the base circles. This configuation causes the relationship to invert, but in all other respects the formation action parallels the formation of the standard external involute.

So your idea of “if I took a spur gear and (analogously) put in in a bucket of plaster, and then the plaster hardened and I pulled the spur gear out, would I have an internal gear of correct dimension/shape?” is theoretically correct. In actual fabrication, the addendum of the internal tooth is slightly shortened to prevent interference and reduce the need for undercutting on relatively large pinions (thus, the note above regarding having internal teeth - external teeth > 15).


Thanks dave. After six weeks of hard thinking, my brain is too tired to understand the first paragraph of your post, but I am almost understanding this part. I have a couple more questions about the theoretical vs. actual profile. Specifically, how critical is the shortening of the addendum of the internal tooth?

Say I have a large sheet of steel, say 1/4" thick. I put this sheet in a theoretical machine (like a wire EDM, or a watrerjet, or a laser) that can cut with absolutely no radius whatsoever. I cut the profile of a standard (external) involute gear. Remember, this fancy theoretical machine has no dimension whatsoever to the cutting tool (laser, water, etc.)

If I took small piece that falls out of the sheet when done, I would most definitely have a nice external gear.

Now here’s my question, if I discarded this small piece that falls out, would the sheet have a nice internal gear ? If so, would this internal gear be suitable for use in something like a small planetary, not unlike the AndyMark?

Would the gears eventually “wear in” or would they not mesh at all to begin with?

If this procedure would work on this fancy theoretical machine, would it work on an actual machine?

Thanks for your help.

The criticality of the reduction in the height of the addendum is dependent on the relationship between the internal gear and the pinion. If the pinion is small in relation to the internal gear (e.g. 10 tooth pinion, 60 tooth internal gear), then then reduction to the addendum can be small (but there still needs to be some). But if the gears are close in the number of teeth, then the potential for interference is significant unless the addendum is reduced. There is a specific formula for this that is based on the separation between the two base circles and the height of the addendum, but I don’t remember it right now. If I can find it, I will let you know.

With regard to cutting the internal gear with your theoretical cutting machine, yes this would work. However, as this theoretical cutting machine has the ability to cut the part with a kerf of zero dimension, the resulting gear would be a perfect involute with no backlash. This may not be a desirable situation, as zero backlash setups can lead to rapid wear of the tooth surfaces.


And I have to disagree with Dave. If made that way, the gears would mesh perfectly, but only in one position - locked in that position unable to rotate. When you were done filing not only the addendum but the whole depth and possibably pitch angle (involute) to get them to mesh, then you’d end up with the proper profile for the internal, which is the mirror image - not the negative - of the external gear.

From my experience, the main issue with the idea of making an internal gear by boolean subtraction of an external gear from an internal gear is the clearance issues.

 There are 3 clearance issues to think about.  
 The tip of the pinion clearing the valley of the internal gear
 The tip of the internal gear clearing the valley of the pinion
 There being enough room along the pitch diameter to fit both teeth and enough backlash to  allow things not to bind.

#1 and #2 often can cause problems because of naming conventions. What is the “addedendum” for an interal gear? What is the “Dedendum”? There are definitions but folks don’t always say what they mean yet alone look up the proper definitions in an AGMA book or a DIN standard.

 #3 basically means thinning one or both of the teeth from the line to line dim.   
 All three of these problems can cause problems if you use a boolean of an external gear subtracted from block.  
 This is what you must do to the "tool" (external gear) that you subtract from the block to make the internal gear:
 THICKEN the gear teeth by the amount of backlash you want (0.002-0.005" say)
 Add to the addendum (1.25 / DP  rather than 1/DP)
 Subtract from the dedendum (1/DP rather than 1.25/DP)
 From my experience, if you do this, you will be just fine.  
 Joe J.

It took me a while to find the reference I was trying to remember earlier, but here goes. The following is extracted from my old, dogeared, and (until 20 minutes ago) previously misplaced copy of The Handbook of Gears (Dr. George Michalec, Hoboken NJ, 1995). I am using this reference mostly because it is the one sitting in front of me on my desk right now. Anyway,

The gears considered so far can be imagined as equivalent pitch circle friction discs which roll on each other with external contact. If instead, one of the pitch circles rolls on the inside of the ether, it forms the basis of internal gearing. In addition, the larger gear must have the material forming the teeth on the convex side of the involute profile, such that the internal gear is an inverse of the common external gear, see Figure 1.33a.

The base circles, line of action and development of the involute profiles and action are shown in Figure 1.33b. As with spur gears there is a taut generating string that winds and unwinds between the base circles. However, in this case the string does not cross the line of centers, and actual contact and involute development occurs on an extension of the common tangent. Otherwise, action parallels that for external spur gears.

Because the internal gear is reversed relative to the external gear, the tooth parts are also reversed relative to the ordinary (external) gear. This is shown in Figure 1.34. Tooth proportions and standards are the same as for external gears except that the addendum of the gear is reduced to avoid trimming of the teeth in the fabrication process.

Tooth thickness of the internal gear can be calculated with equations 9 (T = pc/2 = pi/2Pd) and 20 (B = Tstd - Tact = Delta T)), but one must remember that the tooth and space thicknesses are reversed, (see Figure 1.35). Also, in using equation 10 (T2 = T1 * (R2/R1) - 2R2 (inv theta2 - inv theta1)) to calculate tooth thickness at various radii, (see Figure 1.36), it is the tooth space that is calculated and the internal gear tooth thickness is obtained by a subtraction from the circular pitch at that radius, Thus, applying equation 10 to Figure 1.36,

So I think that Joe and I are saying the same thing. The tooth form for an internal gear is indeed the inverse of a “normal” external involute tooth, and not a rotation of the external tooth. Practical considerations associated with the insertion of desired backlash and preventing the addendums of either gear from “bottoming out” cause slight modifications to the addendum height and dedundum depth for the internal gear in manufacturing (particularly when the pinion is large relative to the internal gear - e.g. the difference in tooth count approaches 15). But the shape of the inverted involute curve is not modified. The shape of the involute curve itself is designed to provide clearance for the external tooth “n” to pass by internal tooth “n-1” while establishing rolling contact with internal tooth “n+1”.