Singlespeed & Fixed Gear - 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.

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.

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.

For instance, if you add one tooth, are you just increasing the circumference of the sprocket by one inch?

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.

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).

Ohh, right, each tooth adds a half an inch to the circumference.

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...

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.

You can find this information on the Park Tools (http://www.parktool.com/repair/readhowto.asp?id=26) 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.

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.

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...

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]

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!

Oh don't worry, if I really geek out I'll do it all in user forms or some other such sillyness.

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.

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

Not so much.

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?

sheldon brwon has a gear inch calculator

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.

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.

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.

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.

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.

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.

i made an error. arctan((r2-r1)/d) is only half the angle; alpha is 2*arctan((r2-r1)/d). the chain is tangent to the cog and chainring; lines that are orthogonal to the chain at each point of tangency are parallel to each other, go through the centers of the cog and the chainring, and make the same angle (alpha) with the vertical that the chain makes with the horizontal ("similar triangles").

i also need to correct the equation for the bottom, angled length of chain. here is the correct set of equations:

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

you can derive these results by drawing the picture in cartesian coordinates with top dead center of the chainring as the origin. then:

upper point of tangency for the cog is [-d,0];
lower point of tangency for cog is [-d - r1*sin(alpha),-r1*(1+cos(alpha))];
lower point of tangency for chainring is [-r2*sin(alpha),-r2*(1+cos(alpha))].

i don't have access to a good, to-scale picture-drawing tool here, or else i would be happy to draw a picture. you are probably more likely to believe it, however, if you draw it yourself!

I'm still not convinced that the "d" term you have in the denominator of the alpha equation doesn't need to be adjusted for the gear ratio.

I'll think about this this evening. I'm almost done with this spreadsheet and want to finish it before I leave work.

i.e., I think implicit in your derivation you assume that you can alter the frame of reference so that the top of the cog and chainring are parrallel and that this line is parrallel to the line denoted by "d". But I'm not sure if that's what you've done.

wait a minute. by similarity, the "top" and "bottom" lengths of chain should be the same, correct? you are right that "d", as i am using it, is not the actual distance between the centers of the bb and the hub; it is slightly less. sometime later i will look at this figure carefully and determine how to calculate the straight lengths of chain from "d", "r1" and "r2". that is all you need.

okay okay okay - forget everything i wrote before. here's the deal:

let "d" be the actual distance between the bb and hub centers, "r1" be the cog diameter and "r2" be the chainring diameter. than:

alpha = arcsin((r2-r1)/d)
straight lengths = [d^2 - (r2 - r1)^2]^(1/2)
cog arc = r1*(pi - 2*alpha)
chainring arc = r2*(pi + 2*alpha)

i finally realized the significance of the symmetry. sorry for all the confusion.

Start with long chain. Subtract links until it's the proper length. Leave the math alone, that stuff'll kill you.

Start with long chain. Subtract links until it's the proper length.

for my bike, that's what i do. but i also like a math problem, in the abstract.

This is an improvement and more than I could come up with.

However, I'm not entirely sure, but I still have the same objections to your "d" term in your new derivation. I think putting that in and not accounting for the gear ratio is tantamount to accepting the 180 degrees assumption.

Anyway, the spreadsheet will be done on monday. It will show the chainlength and gear inches of a gear choice, determine the number of gearings that would work with vertical dropouts of a given chainstay length, and provide all feasible 'flop' combinations with a given horizontal drop range and primary gearing choice. what fun!

if all goes well i can scan my sketch and post it on monday as well. the 180 ° assumption is out the window. "d" is the hypotenuse of a right triangle, whose long side is parallel with the straight part of the chain, and equal in length. that is where the adjustments to the arc lengths come from: chainring arc greater than 180, cog arc less than 180. also, why the straight lengths of chain are less than "d", by an amount that accounts for gear ratio. cheers!