Wheelbase vs gearing question. Whiz kids and rocket surgeons, help!

[andyb]

Long question, and the correct answer is not going to be "just buy a long chain and cut it to length", okay?

I know the math is doable, but I'm pretty pooped and could use a poke in the proper direction.

Assuming you have a 530 chain, you have 5/8" chain pitch between the rollers.  You can then calculate the diameter (and thus radius) of the sprockets, where t = number of teeth, and d = nominal diameter of the sprocket (as would be measured from c-c on the pins opposing, with the formula:  (((5/8)t)/pi).  Obviously this isn't a dead-on calculation, because each link of the chain actually has a chord length of 5/8" rather than being curved as this figures out.  If anyone wants to do the math for me (or show me the formula), I'd love to see the right way to calculate that.

So we now can make a chart that gives a nominal diameter of each size of sprockets (for 530 chain, anyhow).  Using that and a given length of chain, can we calculate the distance between the sprockets?  If we draw it out, one run can be exactly flat, and because the sprockets are different sizes, we end up with an angle between the two chain runs.  I need to be able to calculate the exact distance between the sprockets for a project that I'm doing, and need the formula so that I can choose the correct chain length and gearing (at both ends) to get the length I want while staying within the swingarm.

Complicated a pinch, I realize!!  A few minutes of fighting google didn't help, as 98% of the answers were "just buy a longer chain".  I'm more interested in the interaction between gear sizes and length, so that I can figure out the closest gearset to meeting my goals, while keeping the axle firmly mounted!  Obviously tensioning the chain is going to adjust things a little, but that should prove to be a fairly direct ratio (and I can figure that out, I think, once I have the rest of the formula in front of me... just adding the distance you want and making the flat chain run a triangle with that deflection as the peak of it at the measurement spot of your choice).

[Arnie]

andyb,

This would be fairly easy if both sprockets had equal # of teeth.  Then your chain would be # of teeth/links to fully wrap ONE sprocket plus 2x the distance in # of links you want the center of each sprocket apart.

In fact, with UNequal tooth sprockets the chain wrap would still be 1/2 the sum total # of teeth on both sprockets plus the 2x the distance apart in # of chain links.

I'm not sure how to easily provide a means to determine the angle variation from flat with different sized sprockets.

Hope some of above is a help.  I'll look at your question again in the morning when I'm sharper?? (after coffee)

Arnie

[andyb]

That may be the missing link (hah!  haha!  see what I did there!) that I'm missing.  Once you can figure out how much of the sprocket is being wrapped, you can figure out how much sprocket isn't being wrapped as well, and then calculate the free distance between the two.

Dunno, gonna get a good nap in, a bucket of coffee, and look at that again.  I get the feeling the answer is going to be one of those "it can't be this easy" sort of things.  Though I started looking at the required math to figure out exact sprocket diameters and it isn't difficult so much as involved.  Gotta sit down and brush up on stuff that I haven't used in years and years (and distinctly remember telling my teachers, parents, and anyone else who'd listen that I'd never use the stuff... well, I was right for a long damned time!).

[Arnie]

andyb

Sprocket diameter is actually easy (I think).
To get sprocket dia to the pin center of the chain (mid tooth)   #t-1  x  5/8" = Circumference
So, 40t-1  x  .625 = 24.375" cir
Cir = pi D  or D = 24.375/3.1415....  =  7.7588"  (197mm)  might be slightly less as the links are chords not arcs.

Go measure and tell me if this is correct.

Also, with only 2 sprockets and a single loop of chain, 1/2 of the circumference of BOTH sprockets is wrapped by the chain.  Problem I have is figuring how much wrap is on each sprocket if they are not equal in size (tooth count).

Arnie

[tmkaos]

Also, with only 2 sprockets and a single loop of chain, 1/2 of the circumference of BOTH sprockets is wrapped by the chain.  Problem I have is figuring how much wrap is on each sprocket if they are not equal in size (tooth count).

Arnie

Would that be a factor of the sprockets size in relation to each other?

IE if the circumference of both sprockets added together represents  100%, one sprocket being 60% of this, the other being 40%, (for example) would the wrap on each be able to be worked out that way?

[Arnie]

No, because the wrap and angle of the chain run will be affected by how close together the sprockets are.  The further apart the less the angle from tangential.

if the axle centers of the sprockets are fixed, there would be a proportional angular deflection you could probably figure out based on relative diameters.

Arnie

[Ned]

Arnie
Pop your head in here http://www.machinehead-software.co.uk/bike/chain_length/chain_length_calculator.html or http://www.gearingcommander.com// or http://www.ducatimonster.org/forums/tech/109492-gearing-guru-tech-article-calculating-chain-length-when-regearing.html


Various ideas - the amount of chain wrap variation due to distance between sprockets is probably negligible for the sort of distance you get on a bike

Ned

[Arnie]

Thanks Ned,
This isn't my goal in life - It's andyb's.  I was just trying to help him out.

I will go take a squizz at those links just for interest, but I'm not planning anything that would be quite that critical.

Cheers,
Arnie

[andyb]

Got it sussed, after I had a good nap and a pot of coffee.

The trick was to set both sprockets so that they were on the ground, then compare their heights, making a right triangle with the vertical portion as tall as the difference between them.  That gives the angle, the chain angle can be figured out pretty easily afterwards (and there's nearly nothing in it unless you're making huge wheelbase changes), and then I just worked out where the axle would be based on various combinations of chain length and sprocket sizes, including the current one, and subtracted the current one from the calculated ones, giving me the total change in wheelbase for each adjustment.  A little surprising how little change there is on some of them!

But, it helped me nail down the proper set for next season's racing.  Just gotta measure quick to ensure I've got the room, then do the math on how the suspension will react to having the new sizes.  Should be fun!