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Singlespeed & Fixed Gear "I still feel that variable gears are only for people over forty-five. Isn't it better to triumph by the strength of your muscles than by the artifice of a derailer? We are getting soft...As for me, give me a fixed gear!"-- Henri Desgrange (31 January 1865 - 16 August 1940)

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Old 09-06-05, 08:50 AM   #1
herst
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How does Chain Length Relate to Gearing?

Hi Guys, this is a simple question about the relation between chain length and gearing choice. Mainly, if I have a 52 x 16 gearing, would this use the same length of chain as a 40 x 28 gearing? The logic is that if you take away one tooth from the chain ring and add it to the rear cog you maintain chain length, but I may be way off.
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Old 09-06-05, 09:38 AM   #2
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I tried to derive a formula for calculating chain length increase when adding teeth a while back. Unfortunately, I got way bogged down and didn't feel like trying to simplify. It was messy.

Anyhow, while it's true that they do wrap the same amount of chain, it may also be important to recognize that the path of the chain from ring to cog is the hypotenuse of a right triangle, approximately.
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Old 09-06-05, 10:51 AM   #3
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Hmm, I get it. I wonder how much that varies, given the low angles we're dealing with. Is there any good source on the web that has the diameter of sprockets as a function of their number of teeth? I assume, but do not know, that this is constant across sprocket types.

I would be down for trying to hook up a spreadsheet if I got some initial input.

Here's why I'm asking. I'm about to do a ss conversion on an old road bike that has a cassette freehub in back and a 40/52 double up front. I'm building this for one specific ride, a 40 miler in western mass that has a great 10 mile climb and 8 mile descent, so I was going to gear this 40 x 30 (its a big hill). But then I got to thinking how annoying this gearing would be for the rest of the ride (long rolling hills), and so I wondered whether I could put a second sprocket on the freehub that lined up with the 52 tooth chainring. I figured that perhaps there's some chain length conservation law or something, so I might be able to fit an 18 tooth sprocket and not even have to adjust the rear wheel. 52 x 18, btw, would be a perfect gear for the rest of the ride I'm thinking of.

Unfortunately, I haven't yet bought the tools to do the conversion so I can't just test this out.

Anyway, like I said, I'd be happy to pursue this further and try to make some calculations if someone gave me some direction.
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Old 09-06-05, 10:54 AM   #4
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For instance, if you add one tooth, are you just increasing the circumference of the sprocket by one inch?
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Old 09-06-05, 11:20 AM   #5
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OK, to go a little further, let's denote the distance between the center of my rear hub to the center of my bb by d. Then call the radius of the rear sprocket r1 and the radius of the chainring r2. Why wouldn't the chain length be defined as:

L = Pi*r1 + Pi*r2 + 2*( (|r1-r2|^2 + d^2)^1/2)

I can explain my reasoning if its unclear. Note this is assuming perfect chainline.

C'mon, there must be an engineer or a physicist on this board that deals with this sort of thing all day.
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Old 09-06-05, 11:21 AM   #6
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I'm thinking the circumference of your chainring is (#teeth/2) inches, so the change in circumference would be 1/2 inch. As for diameter as a function of teeth, circumference equals 2*pi*R, so the radius R = #teeth/(4*pi), and thus diameter D = 2*R = #teeth/(2*pi). If you add one tooth, the change in diameter would be 1/(2*pi).
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Old 09-06-05, 11:23 AM   #7
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Ohh, right, each tooth adds a half an inch to the circumference.
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Old 09-06-05, 01:18 PM   #8
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Herst-

I've also looked into the problem before. The main issue is that unless the cog and chainring are the same size, neither is wrapped 180 degrees by the chain. It ends up breaking down to an extremely complex system if you want to get it right. I'm not even convinced that there's an analytical solution. It may have to be broken down into a linear system, but I'm not entirely sure yet. I haven't counted all my "unknowns". you'd probably have to solve a generalized linear system (I think I'd suggest gauss-jordan method for this) before getting your equations to put into your spreadsheet.

I haven't gotten that far yet... and now you have me working on it again too!

I'll see what I come up with...
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Old 09-06-05, 01:36 PM   #9
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Yeah, that was my fear.

Is there any way of bounding it though? I mean, you're right that you won't have 180 degree wrap, but you also don't need the chain to be exactly the same length for the two gears to work, especially because this is a singlespeed, not a fg.

Also, I'd be interested in gaining some insight into how you determine the extent of wrap given a gear ratio and distance.

If an analytic solution proves messy, is there any way of just rephrasing this as a constrained minimization problem and solving numerically? Ultimately, you're just looking for the smallest amount of chain subject to the constraints that the chain is always a certain distance away from two poles, those distances determined by the radii of the chainring and cog and the position of the poles represented by the variable d above.

Unfortunately, I dropped physics in college and am therefore no good with this sort of thing.

Last edited by herst; 09-06-05 at 01:47 PM.
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Old 09-06-05, 02:00 PM   #10
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You can find this information on the Park Tools web site.

They provide two formulas. You have to round the answer to the nearest whole number, obviously. The simple one, for a fixed-gear/single-speed:

L = 2C * [(F+R)/4]

where

L = chain length in inches
C = chainstay length in inches, from center of bottom bracket to center of rear hub axle, to nearest 1/8"
F = number of front chainring teeth
R = number of rear cog teeth

A month or two ago I put the equations into an Excel spreadsheet, which is pretty handy.
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Old 09-06-05, 02:06 PM   #11
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I'm working on it... one of the more interesting problems I've had to deal with recently. I think I may have stumbled onto something analytical...

I'll get what I can and make some drawings later. I know that most of the equations used to calculate the angles (usually for celestial observations) generalize about the radius as though the difference is negligible, but if you modify them to include an "effective radius" in order to increase accuracy, the equations are recursive functions of the angle and some other parameters.

I'll have no problem applying numerical methods, provided I can get to the point where I can begin to apply them! I hope to be finished by the end of the work day, and I may typeset them in PDF format (with diagrams describing my logic) and put them on my webpage. I'll keep you posted.
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Old 09-06-05, 02:10 PM   #12
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Good link. Their "rigorous equation" (bottom of the page) may be more appropriate in this situation. It'll be a good check to see if I'm close...
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Old 09-06-05, 02:19 PM   #13
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Hey thanks for the link.

So it appears Park has no problem making the generalizations I have above; the simple equation even ignores Trevor's point about the slope of the chainline, while I think that the "rigorous" equation corresponds to what I put up.

But what's up with the +1 that they throw in there? A fudge factor to make sure you don't cut too short a chain? Or maybe I'm missing something.

[edit: Oops, didn't read the fine print. They in fact suggest for s/s bikes that you eliminate the +1]
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Old 09-06-05, 02:22 PM   #14
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you cats are the gonest. two things to keep in mind as you proceed:

1) chain length is not continuous, but rather an integer multiple of link length;

2) the distance between the center of the bottom bracket and the rear hub (what herst refers to as "d") is not constrained to be constant, unless maybe if you have vertical dropouts.

give us a nice little vba script when you're finished!
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Old 09-06-05, 02:29 PM   #15
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Oh don't worry, if I really geek out I'll do it all in user forms or some other such sillyness.
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Old 09-06-05, 02:30 PM   #16
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to correct for angle:

call the angle the lower part of the chain makes with the horizontal "alpha". then:

alpha = arctan((r2-r1)/d)
arc around chainring = r2*(pi + alpha)
arc around cog = r1*(pi - alpha)
length on top = d
length on bottom = [(r2-r1)^2 + d^2]^1/2

divide by length of link to get number of links.

round up.
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Old 09-06-05, 07:47 PM   #17
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my rule of thumb for calculating gear inches, and ratios is that 1 tooth on the cog in the back equals 4 on the chainring. i believe this woeld carry over to your question. an example for my theory would be that the ratio for a 52-15 is pretty much equal to the ratio for a 48-14
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Old 09-06-05, 07:58 PM   #18
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Not so much.
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Old 09-06-05, 08:11 PM   #19
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Quote:
Originally Posted by cicadashell
to correct for angle:

call the angle the lower part of the chain makes with the horizontal "alpha". then:

alpha = arctan((r2-r1)/d)
Could you explain this more?
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Old 09-06-05, 08:58 PM   #20
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sheldon brwon has a gear inch calculator
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Old 09-07-05, 07:54 AM   #21
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Guys, just to clarify, our objective here is to find a formula for the length of chain a given gearing will require on a specific bike. While related, this isn't the same thing as the gear inches of the gearing.

DiegoFrogs, any luck? I'm, like, eagerly awaiting your solution. I was thinking that beyond just helping me with my issue (can I make a two-gear s/s bike that doesn't require rear wheel adjustment upon changing gears), this could also help show the range of gearing one could use on a flip flop hub. All you'd have to do is measure the range of distances your dropouts allow and enter that in along with your preferrred gearing.
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Old 09-07-05, 08:29 AM   #22
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Sorry, real work interfered. Something about working full time and taking 20 credits to finally finish school... I left all my work at home, too. I got pretty close to an expression, but it includes an expression for the angle of wrap.

Cicadashell: Your expression for alpha doesn't seem to account for the fact that the radius that touches the "end" of the chain as it leaves the top/bottom of the chainring/cog isn't strictly at the pi/2 and 3*pi/2 position. Specifically, I expect the denominator in that expression to be strictly less than d (or equal to d if the rings are the same size...)

I suggest you draw a diagram with drastically different sized cogs very close together, that way you can see where exactly (in general...) the tangent lines touch the two circles. That was how the problem began to unravel for me, until I had to get back to real engineering work.
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Old 09-07-05, 09:05 AM   #23
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I figured this out once before, and the difference between jsut assuming 180 degrees of chain wrap around a gear, and actually figuring out the actual amount of wrap is insignificant. The largest source of error in this problem is in the measurement of chainstay length.
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Old 09-07-05, 09:12 AM   #24
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very true, it is hard for most people to measure between "imaginary" centers, especially the bottom bracket. I just figured it would be helpful for crafty folks to convert bicycles with vertical dropouts. I was thinking about getting an old hardtail mountainbike for this very purpose...

Mostly, I'm just curious. The answer may also have some application in my everyday work, to an extent, especially if I can keep my solution in the most general form possible.
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Old 09-07-05, 09:44 AM   #25
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Bummer that you're busy DiegoFrogs, I'm very much not so presently.

I've put together a simple spreadsheet that shows chainlength using the formula shown above (which is equivalent to Park's "rigorous" equation). It calculates gear inches as well.

I'd like to add a few more tabs for specific items, like finding whether you can chose a gearing to get near a certain gear ratio on vertical drops and what kind of gearing choices you can make for a flip-flop hub. However, if anyone wants to take a look at this and give me some feedback, PM me.
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