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,
Quote:
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,
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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".
-dave