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One of the requirements when designing a bicycle frame is clearance between the chainstay and the teeth of the chain ring.

This question was asked on bicycles.se, and the answers have me interested in the math behind it. https://bicycles.stackexchange.com/questions/5264/how-do-i-calculate-the-diameter-of-a-chainring-from-the-number-of-teeth

(In the case of bicycle chain rings, there is a fixed size for the tooth, and a fixed distance between teeth related to the pitch of the chain. I suspect that other gear system designs would have a fixed tooth size/spacing as well.)

Does someone here have either a verification of the answers given here, with a more detailed explanation as to why they work or don't work, or a better solution to the problem?

Since the frame design is using a theoretical chainring, simple measurement is not an option, except in test cases to determine a practical solution to the question.

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    The answer is no, you cannot calculate the diameter of a bicycle chainring from the number of teeth because if you take a chainring of double the size it will have the same number of teeth but a different diameter. One solution is to measure the distance between the teeth but that is in practical terms more inaccurate than just measuring the diameter...2011-08-05
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    In the case of bicycle chain rings, this is not the case. There is a fixed size for the tooth, and a fixed distance between teeth related to the pitch of the chain. In the case of other gear systems you may have a point, but I suspect that they too would have a fixed tooth size/spacing in their own system as well.2011-08-05
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    Ok, maybe you should include this to your question because most users will not know this and it is an essential point :-)2011-08-05
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    I will. I suspect it is the same for any gearing system, however.2011-08-05
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    There is not a fixed size for the tooth. The chains we run are a 40 series chain http://en.wikipedia.org/wiki/Roller_chain#Chain_standards and have a 1/2" or 12.7mm pitch between pin centers. The teeth, however, will start out somewhat above the pin centers (varying by manufacturer) and wear down somewhere below the pin centers from use. You could measure your individual ring, but there **is no standard** on tooth spacing.2012-06-03
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    @Ehryk, you're wrong. The size *is* fixed for the teeth, and there is a fixed height, excepting those teeth which are shortened for ramping.2012-06-03
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    What do you mean by _fixed_ then? They should be constant between the other teeth on a (new) chain ring, sure - but the tooth height (and therefore spacing) can easily vary between manufacturer and wear. I could take one and grind 2mm off the tooth tops, or just ride it until they wear down over time. Tooth height (which is proportional to spacing) is **variable** and there is no set standard for it, like there **is** for pitch. I could also make a chain ring with a bit longer teeth - certainly not optimal, but the 'tooth spacing' would be greater. Pitch is the true standard here.2012-06-03
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    What I meant by fixed, is that if you know the pitch the ring was designed for and the number of teeth, you can then calculate the pin center circle/n-gon, but you cannot then assume tooth spacing/height. That must be measured on each chain ring, or an approximation must be assumed. Would it be your claim, then, that all tooth spacings are 12.75mm on new and worn rings? New only? New 'non ramp shortened'? Because if it's not fixed, it's variable by definition.2012-06-03
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    Pitch may be standard, but since the purpose of the measurement is to ensure that the **tips** of the teeth don't grind chunks out of your newly designed frame, the pitch circle diameter, which is smaller than the tooth circle diameter, **doesn't matter to answer the question asked!** As for whether tooth height is fixed, as a practical matter, it is. Any tooth taller than the height of the roller of the chain (which is a function of the desired standard pitch) will not cleanly disengage from the chain while shifting. The shorter ramping pins are few, and for our purpose may be safely ignored.2012-06-03
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    All you've done now is set a maximum value. I'll add a minimum value: if it's below or equal to valley height (which is also a function of... not exactly pitch, but chain series standards) then it's not a tooth. Now there's a range of values that are valid tooth height/spacing which can vary between manufacturer, 'ramp shortened'?, and wear. Sounds variable (antithesis of fixed) to me. Further, I understand what the question is asking, but you can't just make assertions to answer it. Deterministically you can describe the pin center circle, from there you have to make assumptions or add info.2012-06-03
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    I'm not claiming anything, and new or old is not the discussion here. It's about designing a frame and chain ring clearance issues.2012-06-03
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    I'm not necessarily challenging your assertion, and you've already noted there are exceptions. I'm just pointing out there is an assertion there: "All chain rings for a 40 series chain have a tooth height that can be approximated for practical purposes to 12.75mm". 40 series chains only specify that the pitch (from pin center to pin center) is 12.7mm, nothing about tooth height/spacing.2012-06-03
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    And I've said nothing about 40 series chains, or 12.75mm being a standard. I said I measured a large number of chain rings in my shop, and the math I was attempting to use worked to a very close approximation, on all of them. I used Shimano, Campagnolo, FSA, and SRAM rings. For obvious reasons, it didn't work on Rotor rings. In addition, I have already said my math was a practical approximation which would work for the purpose described by the OP, and that Lantius' answer was the correct one mathematically. What exactly do you think rehashing this is contributing to anyone?2012-06-03

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The math in lantius' answer is correct, and your comment underneath that is spot on: The difference between lantius' answer and yours arises from the fact that your solution assumes a circle, which contains an arc between two teeth instead of the straight line of the $n$-gon. Since you presumably measured the distance of $12.75$mm between the tips of the teeth along a straight line, the $n$-gon gives the correct solution. As lantius already pointed out, the difference this makes in the circumference is quite small in this case, only $.005$ inches, compared to the $.05$ inches discrepancy with the value calculated from the diameter. Since the error in your measurement of the distance between adjacent teeth is multiplied by $53$, a discrepancy of only $.05$ inches is very good; it means the error in your distance measurement for the teeth was only about $.001$ inches, or about $.025$ mm, which seems almost a bit too good to be true.

The difference between the $1/2$ inch chain standard and your $12.75$mm measurement is due to the fact that the centres of the chain pins sit slightly below the tips of the teeth, and the $1/2$ inch standard refers to the distance between the centres of adjacent chain pins (see here and here).

There's actually a further potential source of error that hasn't been mentioned yet in the other thread. You didn't specify how you measured the diameter. Since it's a 53-tooth chain ring, there are no opposite teeth to measure it at. Assuming that you measured between the tip of one tooth and the tip of either or both of the teeth almost but not quite opposite to it, what you would then actually measure is not the diameter, but slightly less. By the law of cosines, this distance is

$$\sqrt{r^2+r^2-2rr\cos\frac{52}{53}\pi}=r\sqrt{2-2\cos\frac{52}{53}\pi}=2r\sin\frac{26\pi}{53}\approx0.99956d\;,$$

where $r$ is the radius and $d$ is the diameter. So that makes a difference of around one twentieth of a percent in the diameter, or about $.1$mm, which is about one tenth of the discrepancy of $.05$ inches. However, this goes in the "wrong" direction; to correct for it, you'd have to add that difference to the measured diameter, which would move it very slightly away from the one calculated from the tooth tip distance.

Here's something fun you could do: For each number $k$ between $1$ and $26$, measure the distance between the tips of a tooth and another tooth $k$ teeth along the ring. By the law of cosines, the values you get should be

$$a_k=\sqrt{r^2+r^2-2rr\cos\frac{2k\pi}{53}}=r\sqrt{2-2\cos\frac{2k\pi}{53}}=d\sin\frac{k\pi}{53}\;,$$

so

$$d=\frac{a_k}{\sin\frac{k\pi}{53}}$$

in each case. Note that this establishes a connection between lantius' calculation of the diameter from a measurement of the distance between adjacent tooth tips ($k=1$) and my calculation of the diameter from a measurement of the distance between almost opposite tooth tips ($k=26$); they're both special cases of the same thing. You can make all $26$ measurements and see how the values scatter. You should be able to see that the scatter decreases with increasing $k$, since the measurement error is being magnified less. If you want to go the whole cog, you can do a least-squares fit on these measurements to get a more accurate estimate of the diameter. In doing that, you shouldn't just take the average, but weight the measurements according to their precision (see here for an explanation), so your best estimate for the diameter would be

$$d\approx\frac{\sum_{k=1}^{26}d_k\sin^2\frac{k\pi}{53}}{\sum_{k=1}^{26}\sin^2\frac{k\pi}{53}}=\frac4{53}\sum_{k=1}^{26}\frac{a_k}{\sin\frac{k\pi}{53}}\sin^2\frac{k\pi}{53}=\frac4{53}\sum_{k=1}^{26}a_k\sin\frac{k\pi}{53} \;. $$

[Edit in response to the comment:]

I'll break down some of the things I'm guessing you might be having trouble with; let me know specifically if you want anything else explained.

The law of cosines states that if $a$, $b$ and $c$ are the sides of a triangle and $\gamma$ is the angle opposite $c$, then

$$c^2=a^2+b^2-2ab\cos\gamma\;.$$

I applied this to triangles formed by two tips and the centre; two sides $a$ and $b$ are the distance from the tips to the centre, which is the radius of the circle, and the third side $c$, the distance between the two tips, is determined by taking the square root of the above equation. When the teeth are $k$ teeth apart, the angle is $k$ times one $n$-th of a full circle, where $n$ is the number of teeth; since a full circle is $2\pi$, this is $2\pi k/n$.

I then used the trigonometric identity $\sqrt{2-2\cos\gamma}=2\left|\sin\frac\gamma2\right|$; since all our angles are positive, I dropped the absolute value.

At the end, I simplified the denominator in the least-squares result as follows:

$$ \begin{eqnarray} \sum_{k=1}^m\sin^2\frac{2k\pi}{2m+1} &=& \sum_{k=1}^m\left(\frac1{2\mathrm i}\left(\mathrm e^{\mathrm i\frac{2k\pi}{2m+1}}-\mathrm e^{-\mathrm i\frac{2k\pi}{2m+1}}\right)\right)^2 \\ &=& -\frac14\sum_{k=1}^m\left(\mathrm e^{2\mathrm i\frac{2k\pi}{2m+1}}+\mathrm e^{-2\mathrm i\frac{2k\pi}{2m+1}}-2\right) \\ &=& -\frac14\left(\sum_{k=1}^m\left(\mathrm e^{2\mathrm i\frac{2k\pi}{2m+1}}+\mathrm e^{-2\mathrm i\frac{2k\pi}{2m+1}}\right)-\sum_{k=1}^m2\right) \\ &=& -\frac14(-1-2m) \\ &=& \frac{2m+1}4\;, \end{eqnarray} $$

where $n=2m+1$ is the odd number of teeth, $53$, so $m=26$, and the left-hand sum evaluates to $-1$ because the $(2m+1)$-th roots of unity sum to $0$ and the sum contains all of them exactly once, except $1$. I suspect you might not understand that last part; alternatively, you can just ask Wolfram|Alpha.

Regarding your measurement of the diameter: Yes, that does affect the conclusions. The distance between a tooth and the almost opposite teeth is already a bit shorter than the diameter; the distance from a tooth to the line between the almost opposite teeth is a bit shorter yet; the difference is roughly twice as much. The distance you measured is one radius plus the distance from the centre to the line connecting two teeth, which is the cosine of half the angle between two teeth times the radius, so you measured

$$r+\cos\frac\pi{53}r=\left(1+\cos\frac\pi{53}\right)r=\frac{1+\cos\frac\pi{53}}2d\approx0.99912d\;,$$

compared to $0.99956d$ above, so this would add about $.2$mm instead of $.1$mm to the discrepancy.

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    I like the answer, but I don't know enough to follow the math. Can you break it down for me a bit, or link to something that will help me work through it? I get it conceptually, I think, but the math is beyond my education currently. You're correct about there being no way to measure the diameter directly. I traced a line around the chain ring, and connected the teeth, then measured the distance from one tooth to the opposite line between teeth. Does that affect your conclusions in any way? At least as they refer to additional error sources?2011-08-05
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    Thanks. I'll be working through this for a bit. :)2011-08-05
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    Can you repost this at the question above on bicycles.se? Or do you mind if I do, giving you credit of course?2011-08-05
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    Feel free to give it a try, but I don't think they have $\TeX$ over there?2011-08-05
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    Yeah, didn't work out so well. Thanks again.2011-08-05