# Internal teeth involute gear pitch circles

I'm having some trouble figuring out how to design the corresponding internal gear for any particular spur gear. With spur gears, meshing is accomplished by having two gears with the same modulus (MOD) or, in the US, the same diametral pitch. The spacing between the two gears is such that their pitch circles touch.

Looking at an internal gear the tooth profile for an internal involute gear is the inverse of the external involute. However if I try to create such a system (in Fusion 360), I get some interference issues:

What I've done is create two spur gears, both MOD = 1, one with 24 teeth which I've just simply inverted to create an internal ring gear and the other with 12 teeth. My questions:

• The pitch circles are marked in green. Is the pitch circle for the internal gear the same as the spur gear that I used in inverse to create it? Or is it moved somewhat. The proper relative gear positions should be where the pitch circles touch so that's a main issue.

• Although it doesn't look too bad, the left and right teeth in the image are interfering. I think if the addendum of the internal gear's teeth was reduced this would help matters. It seems that the larger the ratios (eg: if the spur gear was 8 teeth instead) the more the gears would crash and perhaps the more the addendum should be reduced.

• In addition I could add backslash, would this help the interference issue?

Are there specific rules to design an internal involute gear for a particular MOD?

• Just found engineering.stackexchange.com/questions/4105/… which is related. – carveone Feb 6 '17 at 15:26
• I think I'm rediscovering the, er, wheel. There are books covering this exact topic which I should go read! (eg: The Machinery Handbook, The Handbook of Gears). – carveone Feb 6 '17 at 15:33
• Look at the difference between tooth forms between small pinions, large gears, and rack. As in the set of 8 gear cutters covering 12 teeth and up, for example, metalwebnews.com/howto/gear/gear1.html I believe you have to extrapolate further, i.e. "beyond rack" but I don't know anything specific to internal gears. Maybe the Ivan Law book referenced here might : mikesworkshop.weebly.com/making-gear-cutters.html – user_1818839 Feb 6 '17 at 15:41

First of all, yes the pitch circle is the same for an internal gear as it is for the equivalent external gear. Second, note that indeed the addendum refers to how much the teeth extend inward of the pitch circle for an internal gear (instead of outward, as is the case for external gears). Vice versa for the dedendum. For standard gears, both internal and external, the addendum $a$ and dedendum $d$ are as follows:
$$a=m=\frac{1}{D_p}$$
$$d=1.25m=\frac{1.25}{D_p}$$
$m$ is the module (in millimetres) and $D_p$ is the diametral pitch (in inches$^{-1}$). The units of the addendum and dedendum are consistent with that of the module or diametral pitch, depending what you chose.
Therefore, it looks like you will need to increase the dedendum for the internal gear to $1.25m$, and make sure the addendum is only $m$. This will then prevent these inference issues, and will generally be fine as long as any external gears are not significantly larger than half the diameter of the internal gear they are inside of.