Chain length calculator

I’m wondering what teams use to calculate the correct center spacing for sprockets when an idler isn’t used. Obviously this is trivial if the sprockets are the same size. It gets more difficult if the sprockets are different sizes, and when you realize that a sprocket is not a circle, but a polygon. For several of our mechanisms last year we used short runs of 25 chain, and tried to design in the correct spacing with Dr Joes chain path spreadsheet. Unfortunately, the chain was loose in every case.
I recently compared Dr Joes spreadsheet result with several online calculators, and they all came up with a center spacing that was larger. On one of the pages, it explained that a simple calculation assumes the chain follows a circular path around the sprocket, but this introduces some error. A better method assumes the chain rollers will be at the sprocket pitch diameter, but they are connected with straight lines, not arcs. For simple chain runs, I’ll try this method in the future, but the great thing about Dr Joes tool is you can use more than two sprockets, and it shows a graphic representation of the chain path. Does anyone know of other tools out there that can do this? Should I ask Dr Joe for an upgrade?

Martin Gears & Sprockets gives a lookup table for chain lengths as a function of center spacing and sprocket sizes (approximation) and center spacing as a function of chain length and sprocket sizes (more accurate). (page 17)

I know our team has used this is the past.

Let me quote myself from here…

You can also use the excel spreadsheet from that post to assist you.

Thanks everyone for the input. I made a quick spreadsheet using the formula provided in the Martin Sprocket and Gear document, and it seems to match various online calculators I have seen. For our application, it gives a center distance about 0.010" greater than Dr. Joes spreadsheet. Would this have been enough to take out the slack in our chain last year? I don’t know, but I’m sure it would help.
My understanding is that the formula most people use is still an approximation.
This page here alludes to a more accurate method:
It gives me a center spacing that is 0.015" greater. I know we are splitting hairs at this point, but I figure the more accurate the spacing, the less need we’ll have for tensioners.

.010 extra in length at about 12 inch centers gives .212 sag in the chain at midspan if I did the math right (using 12 for the linear distance and 12.010 for the arc length).

If you use Inventor, the Design Accelerator will also allow you to verify dimensions for chain and sprockets. Having never used SolidWorks, I can’t speak for it, but it’s the kind of thing I’d be surprised it doesn’t have.