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Substituting in Sin^-1 relationship

Hi, I'm trying to work out the relationship of geometry for centre distance between 2 pulley's using a fixed length of belt. Please have a look at the <Centres Calc> page of attached spreadsheet where I have the formula for lengths around the belt system.

I'm trying to solve for C when I know total L and dia's D & d. All this needs is a 'simple' substitution of rows 2 & 3 into row 4. I can do that bit, but can't remember how to extract C from the Sin^-1 terms.

It's for a very high powered motorcycle that I'm helping to design & build. The goal is a major speed record machine, where we are designing a transmission that can be set up with wide range of final drive ratios. We have many options to refine as (for various reasons) the machine will have 2 stages of toothed belt with pulley's in the order of 44-64 teeth. I thought this spreadsheet would give us a neat tool to explore these options and minimise the number of pulleys we have to make.

With help from this forum, I hope we can give the 'nut behind the wheel' a decent transmission to race with.

(Edit - Found some errors in the formula's - and have changed the attachment)

Thanks, Kev.