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Bicycle Crank Length: Derivation of the Formula
Being a cyclist, a mechanical engineer, and 6'-4", I suppose it is logical that I'd be the one to develop such a formula on my own. Somebody had to.
Basically, the formula is the mathematical result of three assumptions and one bit of empirical data. The first assumption occurred to me when I realized that my bike, with a 25-inch frame, came fitted with exactly the same crankset as shorter bikes. The first assumption:
Taller people should generally be using longer cranksets.
This seems to be a fairly revolutionary concept. If you ask cyclists or bike shops, you may get some discussion but the recommendation will always be the same -- you need a 170mm crank. Taller people using longer cranks is sacrilege.
When you get right down to it, there are really only three assumptions to choose from at this point:
A taller person should have a shorter crank -- a somewhat silly proposition;
All people should be using the same crank size, regardless of their size. I consider this to also be a rather silly proposition, but it is in fact essentially the presently accepted wisdom. More cyclists are concerned about changing crank length for the application (track, road, MTB, etc.) than adjusting it for the size of the cyclist.
A taller person should have a longer crank. This is the conclusion that I would come to, and I fail to see how any thinking person could come to any other. Obviously, very small children need a shorter crank than adults; why would anyone assume that all adults need the same size, regardless of their leg length?
The second assumption:
Crank length should be proportional to the size of the cyclist.
-- or to some part of the cyclist's anatomy, anyway. This may sound similar to the first assumption, but it is in fact a further and separate assumption. It means that if one cyclist is 10% larger than another, he should be using a crankset 10% longer. This is, in fact, a rather major leap; it may make sense to some that one cyclist 10% larger than another should have a longer crank, but not necessarily 10% longer. Alternative suggestions might include that the crank length should be proportional to the square root of the rider's size, or some other power. Or, the crank should be some fixed length plus some amount for each inch of inseam.
There is, in fact, some compelling reasoning to feel this second assumption might be incorrect. A taller human being is not necessarily simply a scaled-up version of an average size human being; his stride, flexibility, musculature, etc., may not simply be proportionally larger. However, for the derivation of this formula it was assumed that a person with longer or shorter legs would be comfortable with a crankset proportionally longer or shorter, and in each case where the formula has been applied the cyclist has been elated with the results. At this point, I feel confident in asserting that even if the crank length should not be exactly proportional to the size of the rider, it should be very close.
The third assumption, of course, deals with exactly which part of the cyclist's anatomy should be measured to determine the correct crank length.
The third assumption:
Crank length should be related to inseam measurement.
I chose the inseam for purposes of ease of measuring and repeatability. When thinking about pedalling motion, it becomes fairly evident that the crank length should be a function of the length of the upper part of the leg; the lower part is simply a "connecting rod", and really doesn't figure into the stroke length. Those paying attention will note that the length of the foot is also a factor, since proper pedalling motion also involves moving the ankle.
However, measuring the length of the upper leg -- from the pivot point of the hip to the pivot point of the knee -- is a fairly serious problem. Even if I were to present a detailed method for doing it, chances are good that different people will come up with considerably different results, probably by fairly large margins. All of this would tend to make any formula based on such a measurement a **** shoot, we might as well simply choose a crank length at random. The same can pretty much be said for measuring the foot, since the important measurement is between the ankle pivot and the ball, the length of the toes doesn't figure in, and therefore shoe size may not be the ideal indicator.
For most cyclists, the length of the upper leg and the length of the foot will be roughly proportional to the inseam; in other words, if one cyclist has an upper leg that is 10% longer than another cyclist's and a foot that is also 10% longer, usually his inseam measurement will also be 10% longer. If this assumption is correct, then deriving a formula based on inseam measurement is perfectly valid; from a mathematical point of view, if you could safely assume the important lengths were proportional to the thickness of the cyclist's earlobe, then basing the formula on earlobe thickness would be just as valid. It does not matter if you are measuring exactly the specific length you need, as long as what you are measuring is proportional to that length.
The use of the inseam measurement has worked very well in application, but it must be noted that it does introduce a margin of error into the calculations. If a particular cyclist has unusual proportions in his lower extremities (i.e., his lower leg appears unusually long while his upper leg is not, or he's a small guy with big feet or a big guy with small feet, etc.), it may be better to "fudge" the indicated value for crank length to compensate for this error.
For those who feel that this amount of potential error is unacceptable in such a formula, let me remind you of the conventional wisdom in crank length: "You should be using a 170mm."
Mathematically, plugging all these assumptions into one equation means that you should be able to measure one's inseam and multiply that measurement by some constant to find one's ideal crank length. Just what is that constant? Well, that's where the empirical data is necessary. We must find an example of a crank length that is correct for a particular inseam length; considering all the hooey out there about proper crank length, that might be a challenge. Once a correct length is determined for at least one inseam size, we can back-calculate from there to determine what the proper value for this constant is.
After much qualitative observation and some informal testing, I have concluded that:
The standard crank length of 170mm is optimum for a cyclist with a 31-inch inseam.
Hence, by dividing 170 by 31 we get a value for this constant: 5.48. So, the formula for crank length is: multiply the inseam measurement by 5.48.
Use of this constant in the formula has been very successful, even for very tall and very short people. However, I will add this note: While every single rider I am aware of that has applied this formula has been very happy with the results, I have gotten a few comments from some (tall) riders that they felt they could actually use a longer crank than called for. This may, of course, merely be a reaction to the improvement gained from the first change; if a crank this much longer works so much better, an even longer one has gotta be good! Since they are usually already using a much longer crank than convention would suggest anyway, there has been no serious attempt to find out if they really could use an even longer crank long term. On the other hand, I haven't gotten any comments from users that felt they should be using a shorter crank than the formula calls for. In conclusion, it may be possible that the 5.48 number should be increased slightly, perhaps up to 5.6 or so; it is fairly certain that it should not be any smaller.
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