Constrained Gears - how to calculate workable placements.

dbb-the-bruce

Dave
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So when you have a bunch of gears arranged in a way that they form a connected loop - they become constrained. Constrained in the sense that tooth counts and placements have to work out such that the gear train can actually work.

I've been puzzling over this for a while. I have a project that has an arrangement that breaks down to 3 gears that engage with a rack. I knew intuitively that the gears could only be placed at specific contact angles in order to mesh correctly. However I couldn't work out how to determine what angles worked and what didn't. It was bad enough that I waited until I had cut the racks and gears so I could play with them and try to figure it out.

After a lot of head scratching, I finally did some more google-ing and found a great resource:


This page shows a number of different common constrained systems with formulas on how to check if they work. The last example on the page was exactly what I was looking for. Although the explanation of the formula was sparse.

However, the cool thing is that because I'd been thinking about it and had gears to play with it all became clear. The gist is that for any constrained system there is a connected path or loop of teeth and it just ain't gonna work if the length of that path is not an even number of teeth. Cool. That's all you need to know.

In the rack example they don't show the rack in the diagram (looks like it just got clipped) however, it's clear that the rack path length is the same as the spacing of gears 'a'. In their formula, the racks contribution to tooth count is - a / pi m

I couldn't get it to make sense with m as the module - so I just used CP instead - which is the tooth spacing on a rack.

Here is a screen shot of a spreadsheet I worked up. I'm interested in the contact angle, so for now I just stuff an angle in and tweak it until the tooth count is as close to an integral number as possible. I'm going to write a little script that will just find all workable angles and spit them out.

gears with rack.png

This board won't let me attach the spreadsheet - it's no big deal, but if anyone wants a copy just ask.
I just figured that other people here would appreciate the info.

-Dave
 
Fixed a few things and put into a sharable google-sheets doc:


My issue with the module calc was that the book I referenced included metric/english conversion in the module / DP formulas. One is just the inverse of the other, so if you don't change units: DP = 1/m and m = 1/DP.

I also changed the spreadsheet to handle different sizes on the two gears.

Check out the shared link if you want. Setup values are highlighted in green, play with contact angle (red highlight) until you get an integral number of total teeth.

Would be easy to do something like this for other arrangements.

The KHK gears site has a LOAD of good info for working with gears. Strongly recommend it if you are trying to build something with a gear train.

-Dave
 
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