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Technical FAQ: A deeper dive in the math of measuring chains and chainrings

VeloNews' technical expert take a deeper dive into the math involved in calculating chain tension.

Have a question for Lennard? Please email him at veloqna@comcast.net to be included in Technical FAQ.

Dear Lennard,
Your chainring diameter numbers in your article dated January 19, 2021 looked a bit off to me.

A chain has half-inch-long links. Thus, a 26T chainring should have a circumference of approximately 13 inches. I say approximately because the chain is not a perfect circle when wrapped around a chainring, it is a series of straight lines drawn between the chain pins. But for the discussion here, I will neglect that for now and assume a circle circumference is good enough, this is corrected later.

Therefore, the diameter of a 26T chainring should be (26/2) * 25.4 / 3.14159 = 105.11mm. Note this circle would be drawn through the chain pins, not the tips of the chainring teeth.

The Wolf Tooth measurement of 111.8mm had a difference from my calculation of 6.7mm, which was different enough from my calculation that it could not be a rounding error. The difference was almost 6 percent.

Thus, I looked at the chart you provided the link for. That chart states the purpose of the measurement is for “clearance between the chainring and chainstay”, which I assume is based on the height of the teeth on the chainring.

Since the chain tension would be the tension on the links between chain pins, not the tips of the teeth of the chainring, I think your calculations are slightly off and used an incorrect diameter or radius.

When I used the 52T chainring diameter of 210.21mm using my circle assumption, that is again 6.7mm smaller than their calculated diameter. The same difference for both the 26T and 52T. I did not look at the other rows on their table, but I would not be surprised if all diameters listed are 6.7mm larger than an assumed circle. I think this explains why the Wolf Tooth table difference made no sense to you when you doubled the 26T diameter and it was not exactly equal to the 52T diameter.

Regarding my assumption that assuming a circle was close enough, using the example 26T chainring, I calculated a distance from center of bottom bracket spindle to center of chain pin of 52.68mm (double that for diameter of 105.36mm) using some more complicated trigonometry to get a more precise distance. Thus, the assumed circle diameter that was 105.11mm was probably close enough for most purposes with an error of only 0.25mm, or less than a quarter of a percent.

I enjoy reading your articles, thank you very much. About a year ago you wrote an article on chain wear checkers, and that encouraged me to buy a new Pedro’s chain checker.

Side note: When I have estimated chain tension, I estimated that one pound of force on a pedal multiplied by the ratio of the crank arm length to chainring radius provides the chain tension. For that, I assumed a circle to calculate my chainring radius.
— George

Photo: Wolf Tooth

Dear Lennard,

Dear George,
Thanks for pointing that out. You are one of many who did. Thanks to all of you; I appreciate it.

I wish now that I had read the fine print and noticed that the Wolf Tooth chainring diameters were designed for chainstay clearance. You elegantly explained why Wolf Tooth’s listed diameter for a 52T chainring is not exactly double the diameter of the 26T on the Wolf Tooth list.

Most of the letters I got on this pointed out that the Wolf Tooth measurements were to the tips of the teeth, but few of them calculated chainring diameter. Those that did (including your first calculation) looked at it as a circle. Here’s another reader’s calculation; this one calculates it while looking at the chainrings as polygons, rather than as circles.
― Lennard

Sketch of chain rollers and chainring teeth. Illustration: Andreas A.

Dear Lennard,
In your latest Technical FAQ on calculating chain tension, you write that Wolf Tooth’s reported diameter of a 52 tooth chainring makes no sense to you, as it’s not twice the reported value of a 26 tooth chainring.

As the chain wraps around the chainring, it forms part of a regular polygon rather than part of a circle. The length of any side of that polygon is the length of a (half) link, that is s = 12.7mm (1/2″). Then the diameter of a regular polygon with n sides can be computed as d = s / sin(180/n). So, for a 26 tooth chainring, we get a diameter of 105.4mm, and for a 52 tooth chainring, we get 210.3mm. These measurements correspond to the center of the pins of the chain, which I believe would be the correct measurement to use for calculating the chain tension.

Wolf Tooth seems to provide the table with chainring sizes to determine clearance between the chainring and the chainstay. It would therefore make sense that they measure the diameter of the chainring at the tip of the teeth rather than at the center of the pins of the chain. For the 26 tooth chainring, they report 111.8mm, which gives a difference of 111.8 – 105.4 = 6.4mm. For the 52 tooth chainring, they report 216.9mm, which gives a difference of 216.9 – 210.3 = 6.6mm. So, in both cases, the difference between the two measurements is around 6.5mm (the slightly different values are likely due to rounding). This means that the teeth of the chainring protrude about 3.25mm from the centerline between the pins of the chain (see the attached drawing – please excuse my poor drawing skills).

To me, Wolf Tooth’s measurements make perfect sense.

— Andreas
P.S.: I’m in no way affiliated with Wolf Tooth Components, although I do use one of their chainrings.

Your latest column on chain tension got me thinking practically again: for riding at a given speed, if gearing allows, is it better to use the small chainring or the large chainring to be most efficient? This assumes that there are roughly similar gears which allow for roughly similar cadence. Generally and broadly, is there an advantage for one chainring over another, say on my indoor trainer?
— Jorin

Dear Jorin,
This chain friction test column answers your question. You can see clearly from the graph that, if you achieve the same gear ratio with two different configurations of chainring and cog, the configuration with the larger chainring will have the lower chain friction. So, use the big chainring if you want to minimize frictional losses.
― Lennard

Dear Lennard,
In your last column regarding chain tension, you write:
One pound of force (lbf) is the amount of force (called “weight”) exerted by a one-pound mass due to gravity at the equator…I’d point out that the pound is in fact a unit of force. The corresponding unit of mass is the slug, which on earth “weighs” 32 pounds. Calculations of energy/power/etc. have to use the pound (force). Dynamic calculations would use the slug.
— Jerry

Dear Jerry,
Thanks for reminding me about the slug!
― Lennard

Dear Lennard,
I appreciated the January 19th article on calculating chain tension. I read through it and thought I was getting it all but was stymied by the change in the reference RPM. Why did it change from 95 to 126?
— David

Dear David,
The cadence never changed in the calculation. It consistently incorporated the 126RPM that Brad (who wrote the handwritten original calculation) used in his calculations. The 95RPM you mention was the cadence we used in the test that I referred to.
― Lennard


Lennard Zinn, our longtime technical writer, joined VeloNews in 1987. He is also a custom frame builder (www.zinncycles.com) and purveyor of non-custom huge bikes (bikeclydesdale.com), a former U.S. national team rider, co-author of The Haywire Heart,” and author of many bicycle books including Zinn and the Art of Road Bike Maintenance,”DVD, as well as Zinn and the Art of Triathlon Bikesand Zinn’s Cycling Primer: Maintenance Tips and Skill Building for Cyclists.” He holds a bachelor’s in physics from Colorado College.

Follow @lennardzinn on Twitter.