My sprocket size and ratio research.
 

So one day I thought about how the smaller the front chain-wheel is in relation to the pedal-crank arm, the greater leverage the pedal crank arm has on pulling the chain against the resistance of the rear sprocket. Of course I knew that if you kept the same ratio, then the rear sprocket would get smaller at the same time the front chain-wheel did and counter the added leverage the smaller front chain-wheel enjoyed. But I thought, was the gain or loss with a smaller or larger front chain-wheel directly proportional to the gain or loss with the larger or smaller rear sprocket needed to maintain the same ratio ??? Or was there some small but useful gain of leverage with using the combination with the smaller front chain-wheel?

To answer this I dragged two combinations of parts from my junk-pile from which to take measurements from. One a 52/19 , the other a 43/16 combination of wheels and sprockets. I used a 170mm length for the crank-arm and a 13.5" radius for the complete rear wheel and tire assembly. Then I measured the distance from the center of each sprocket and chain-wheel to half the height of their sprocket-teeth where I thought the roller of the chain would bear.

In the end after using all of the measurements to calculate the gain/loss of the leverages of the front crank/chain-wheel combinations and the change in leverage between the different rear sprockets and the radius of the rear wheel, I found that yes it was absolutely proportional and there was no advantage in leverage or power to be had by using either combination. A friend of mine pointed out that there were fewer chain rollers in contact with the smaller sprockets so there may be less friction, but I countered that the fewer rollers on the smaller sprockets would be under more leverage, strain and pressure than the chain rollers on the larger sprockets and that would make the overall change in total friction also zero.

As long as you are having fun nothing is a waste of time......


https://cimg9.ibsrv.net/gimg/bikefor...af644e18cd.jpg

Maybe an easier way.

At Genentech, BITD, we used to say you can save yourself hours in the library by spending a couple months in the lab.

Sheldon Brown is wrong. He states that the crank length changes the gear ratio, which is nonsense. No matter what the crank length of a bike is, it will still go the same distance with one revolution of the crank. All that changes with a longer or shorter crank arm is the torque that the rider's weight produces in foot/pounds. With his thinking if you put shorter cranks on a bike with sprocket ratios that have greater mechanical advantage it is equal to having longer cranks with a gear ratio that has less mechanical advantage, which is absolutely wrong because the rider will have to pedal the bike with the "shorter" gears, the ones with more mechanical advantage, at a higher rpm than they would the bike with the longer cranks to achieve the same top speed, but because any rider has a maximum cadence, with the one setup they will not be able to go as fast, so his theory is garbage and misleading.

There is a lot of useful information in the Sheldon Brown pages and he did a lot of thinking, but I have found instances where his information is bad and this is one of those. Just because something is popular or old does not make it right, and this is why it is better to be able to think for yourself, critically and independently than to rely completely on a "smart" phone or on others for information.

Thanks for bringing up Sheldon B.'s observations on gearing versus crank length. He was wrong, but only because he stuck with the needlessly vague word "gear" in his discussion. He apparently meant "effective gear" instead.

But a better word would have been "leverage." The crank, after all, is a lever, and the chain ring and sprocket serve to change the effective length of that lever with respect to the work done. The choice of a lower gear results in a longer effective lever, for instance.

And thanks again for your post, because otherwise I might never have discovered that, spurred by his irritation at the fact that conventional methods of calculating gear ratios completely ignore crank length, he had invented a radically different calculation method.

Note that he jokingly gives the page where he explains his new method the byline "Sheldon (Quixote) Brown," tacitly (and correctly) acknowledging that his new and improved method would be universally ignored.

Here's the page, and here's an excerpt (where he uses the word "gear" in the last line of the first paragraph below, I suggest reading "leverage," for reasons I explained earlier):

What About Crank Length?

All of these systems share a common inadequacy: none of them takes crank length into account! The fact is that a mountain bike with a 46/16 has the same gear as a road bike with a 53/19 only if they have the same length cranks. If the mountain bike has 175's and the road bike 170's, the gear on the mountain bike is really about 3% lower!

A New Standard Proposed

I would like to propose a new system, which does take crank length into account. This system is independent of units, being expressed as a pure ratio.

This ratio would be calculated as follows: divide the wheel radius by the crank length; this will yield a single radius ratio applicable to all of the gears of a given bike. The individual gear ratios are calculated as with gear inches, using this radius ratio instead of the wheel size.

Kinda weird to decide that everything comes down to input RPM instead of pedaling circumference per minute. Use the latter and it recognizes the influence of the crank length. Which we should, since people aren't motors with drive shafts.

Am I right in saying that you were trying to prove that a gear ratio of 52/19 is more or less the same as 43/16?

So you aren't happy with the gearing on your bike?

What is it not giving you that something else might do way better?

Over many decades (some) people have argued that "leverage" is somehow a magical number to be superimposed on gear ratio. It's not. Crank length studies abound and none have shown anything other than it is easier to spin higher rpm with shorter cranks. From that, if you want to spin, use lower gears and shorter cranks, and if you want to mash use taller gears and longer cranks. And if you do the latter expect to have more knee problems and have more difficulty holding speed for long distances. Note that every hour record set in the modern era has been at 100-110 rpm.

I've thot about this a lot and it seems that on my bikes a smaller cog and a bigger ring goes faster than the other way around.

Sheldon Brown is wrong period. Nobody including him mentions different size riders. For a short rider 165mm cranks would feel more friendly than the 180mm cranks that a tall rider like me likes, both size cranks feel the same to appropriately sized riders. All riders seem to like a cadence between 80rpm-90rpm, backed up by a poll on this very forum not long ago.

All I was trying to do was research if there was that if a 1:2.7 drivetrain ratio was good for me to use, if there was any advantage to using smaller sprockets or larger sprockets because their different diameters might offer more torque, and my measurements said there was no advantage. And that thought and research has nothing to do at all with anything Sheldon Brown talks about at all, zero. Finding flaws in his thinking was just a bonus because the forum member linked to his site when they mistakenly thought the page had something to do with my research, which it did not.

Actually, I think he proved that the number of teeth on a chain ring is proportional to its diameter. Shocking discovery.

I'm certainly not qualified to comment on the merits of the op's 'research' or of his 'critical, independent' thought, but far as I can see the main thing here is that the op rather enjoys saying 'Sheldon Brown is wrong'.

Seems a harmless enough little pastime??

Repeating what I explained in my earlier post, in other and simpler words:

Sheldon B. was wrong. Trivially wrong. On a technicality.

The technicality: in some instances where he wrote "gear," he should have written "leverage."

Here's that quote again, from the page I linked to in my earlier post. He's discussing systems that use equations to calculate gear inches and systems that use equations to calculate gear development. All equations in both systems use wheel diameter, chainring size, and sprocket size. All ignore crank length. I've bolded the relevant instance of "gear," where "leverage" is meant.

----All of these systems share a common inadequacy: none of them takes crank length into account! The fact is that a mountain bike with a 46/16 has the same gear as a road bike with a 53/19 only if they have the same length cranks. If the mountain bike has 175's and the road bike 170's, the gear on the mountain bike is really about 3% lower!

Sheldon then went on to provide a method of calculation that incorporates crank length and is irrefutable (to everyone but the unfortunately innumerate). See the link in my earlier post for the formula.

Now that the question of whether Sheldon was "wrong" in some sense that merits dismissal of his ideas on the subject has been shown to be a non-issue:

The 3% change in leverage in his example comparing 170-mm and 175-mm cranks would likely be all but undetectable to most cyclists in terms of expended effort---probably on the order of the effect of balanced versus unbalanced wheels on how a bike rides. That being the case, it's no surprise that the cycling world long ago settled on gear calculation systems that ignore crank arm length.

beng1 noted in a recent post that he had satisfied himself that there would be no effective difference between a combination of a smaller chainring with a smaller sprocket representing a particular ratio and a combination of a bigger ring and bigger sprocket representing the same ratio. That's certainly true mathematically. The only other pertinent consideration would be the fact that mechanical: frictional losses within the chain increase as the sprocket size decreases. Unless frictional losses also accrue from the use of a combination with a large offset in chain line between the chainring and sprocket, and those losses are greater.

That really has little to do with the effective gear you are using.

Why do we need anything other than the ratio between the the chainring and the cassette sprocket here? And more importantly, can I score some of ya'lls stash...
I will say that there is a difference in friction (and wear), all be it small, and bigger is better in both cases.

As a mechanical engineer, I can confirm that the ratio of chainring to cassette teeth is all that matters in relation to torque/leverage. The OP went to the trouble of proving this by measuring the gear diameters, but it wasn't really necessary. At least he got the right answer!

Minute differences in friction due to tighter chain-wrap etc on smaller sprockets are not worth considering.

I think the differences in friction between practical sprocket sizes sometimes gets blown out of proportion. The entire drivetrain typically has a power transmission efficiency of around 96%, give or take 1%. So at 200W you could expect to lose around 6-10W max depending on how clean your drivetrain is. While that is still significant, there isn't a lot you can do about it. For example a 2x drivetrain might save you a couple of Watts over an equivalent 1x due to chain line differences, gear sizes etc. Basically not very much. Then when you start looking at differences between things like 10t vs 11t sprocket or a 52t vs 48t chainring on the same drivetrain, it's going to be fractions of Watts that really don't matter at all. Basically you stand to gain more from cleaning your drivetrain!

No, the woman's record was in the 90s--see:
https://www.cyclingnews.com/news/ell...speed-concept/

"Van Dijk is reported to be running a 58-14 ratio to allow a cadence of between 93-97 RPM over the duration of her attempt."

Seems that everyone is forgetting the different frictional coefficients of steel, titanium, and aluminum alloy cogs. And chain ring hardness.

Let's go all in--do different lubes work better at different temperatures? Does sunspot activity affect the efficiency of electronic shifting?

That's just an assumption you are making. I'm 71" (180 cm) tall with 34.5" (87.6 cm) legs. I ride 165 mm cranks. Tried longer and never liked them.

I never believed the myth of proportional crank sizing. About all it might do is suggest the longest crank length you might can use. Not the crank length you must use.

I never said any one "must" use any crank length. You are under six-feet tall, not that tall. I was 6'3" a lot of my adult life, might have shrunk a bit as am in my 60s now.

A larger, heavier leg will take more energy than a shorter lighter leg to move at the same rate and same distance, that is physics and engineering 101. So if the large and small rider are in condition to generate the same watts of power, the only way physics will let the larger rider go as fast and as long as the smaller rider, is if they can have a longer pedal-crank arm which they can turn at a lower rate, a lower rpm, but still get the same power to the rear wheel. Power equals rpm x torque, so you can have less rpm if you are a large rider and get the same power for the same amount of time only if you are given a larger lever or crank arm, which you can pedal more slowly.

In the last year I have ridden road bikes I own with pedal cranks of 165mm, 170mm, 172.5mm, 175mm and 180mm. I have tested all of these bikes over the same 12.1 mile, mostly level circuit a number of times, some of them dozens of times, in the last year.. The bike with 170mm cranks, which I rode most often, had 165mm cranks the first half of the riding season, and changing them for the 170mm units gave me the fastest time around the circuit to that date. Toward the end of last riding season, when I got a bike with 180mm cranks rideable again, I chopped two entire minutes off the time it took me to go around the circuit with the 170mm crank-equipped bike, dropping the time from 37.5 minutes to 35.5 minutes, and raising the average mph from 19.5 to 20.4. I could run a faster gear ratio around the entire circuit with the long cranks, something I could not do with the same bike with shorter cranks, nor any of the other bikes with their shorter cranks.

26 years ago when I was younger and ran a lot of TT events, I was always fastest with 180mm cranks too.

I'm alway much better near the end of my riding season that at the first of my riding season. So if you didn't do comparable rides of the cranks in the same part of your season then the points are questionable.

As for leg length I feel I have the leg length of most persons 6' 2". I run across many that height have quite shorter legs. So I think my experiences should be comparable as my leg should be of the same weight range as a 6' 2" person with 34.5" inseam for leg length.

While some of your engineering and physics suggestions seem reasonable, they also seem a little askew. You only talk about length of crank to correct for more power needed. However rpm and gear ratios also account for power too. Some people have a wide range of rpm's they can pedal, and some people have a narrow range of rpm's they can or will attempt to pedal.

Also a larger heavier rider will need more power to match the acceleration and speed of a lighter rider. So you baffle me when you say

I'll just add my $0.02 concerning crank arm length here.
In a way, Sheldon was right, shorter cranks do affect gearing - from a certain point of view. Instead of thinking of gear inches, consider how far down the road you go per inch of foot travel. Shorter cranks result in a smaller pedal circle, so a crank revolution represents the feet moving a shorter distance with a shorter crank. Less foot travel per unit of distance moved means you must expend more energy per foot travel. Things equal out since you move your feet less - and isn't that what gearing does?