I am trying to figure out how to calculate either the distance or angle of 2 Idler Gears in a Timing Gear Configuration like so:

enter image description here

I have been able to eyeball things, so they are approximate. I would like to come up with an equation to get an exact number for either dimension A or B. I came up with an angle for A of approx 44.7 deg where all the teeth mesh.


I was able to use CAD to figure out how this needs to be done. Now I just need to figure out how to calculate all of this:

enter image description here

Basically what I ended up doing is all the angles are multiples of the tooth angle for each gear. Next I took a line at the tangent points of each gear perpendicular to a connecting line between each angle. See the blue lines in the above picture.

In this problem the tooth angle can not be used to calculate the spacing directly because of the gear mesh as seen on the bottom of the 22 tooth gear as seen in the top picture. The meshing actually takes place off of the tooth angle.

Just to clarify I am using 64 pitch gears, the diametrical pitch is calculated using standard gear equation:

Dp = N/P


N = Number of teeth

P = Pitch

  • $\begingroup$ Sorry the small idler Gear is 22 Teeth not 20 $\endgroup$ Apr 27, 2015 at 19:34
  • $\begingroup$ Can you also mention how you calculated the angle A to be equal to 44.7° $\endgroup$
    – jNerd
    Apr 28, 2015 at 5:25
  • $\begingroup$ I came up with 44.7 by using a CAD program and eye balling the rotation and incrementing the angle until everything lined up. I know that is not a very good way of doing this but could not figure out how to keep the clocking of the teeth. $\endgroup$ Apr 28, 2015 at 12:24
  • $\begingroup$ Are these gears all rigidly mounted? If so, I'm pretty sure this system won't work, as you'll have two different rotational speeds on the middle two gears, but the connections they share require them to turn at the same speed. $\endgroup$ Apr 29, 2015 at 17:20
  • $\begingroup$ @TrevorArchibald Yes they are all rigidly mounted. I was curious about the speeds and ratio's of the gears also but after running a simulation it works. $\endgroup$ Apr 29, 2015 at 20:43

1 Answer 1


You can solve this by modeling the gears as abutting circles with their cicumferences proportional to the number of teeth. It will be easier to then work with the radii instead of circumferences directly, but the rest is just basic geometry.

You will have to decide one more constraint than pictured. There are several ways to express this degree of freedom, but one is how far apart the two middle gears are. Or, you could specify how far apart the top and bottom gears are.

  • $\begingroup$ That certainly covers how to get the pitch circles in contact, but doesn't address the issue of meshing the teeth. I would have thought that only certain values of that remaining constraint would make it possible for all four gears to interface correctly simultaneously. $\endgroup$
    – Dan
    Apr 29, 2015 at 15:51
  • $\begingroup$ Modeling the Gears was the only way that I was able to come up with something close but not exact, I am concerned about wear so if the distance A is too close it might put unneeded stress on the gears causing premature wear. $\endgroup$ Apr 29, 2015 at 20:41

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