Is it possible for a bicycle chainwheel to have a fractional number of teeth?

Question

Background

In the world of bicycle motocross, also known as BMX racing, gearing is a hotly-debated topic.

Since the bikes are all single-speed, gear ratio is a fixed number defined as chainwheel / cog (front gear divided by rear gear). Altering your gear ratio is understood as an immediately-noticeable tradeoff between acceleration and top-end speed.

Here is a series of common gear ratios:

╔════════════╦═════╦════════╗
║ Chainwheel ║ Cog ║ Ratio  ║
╠════════════╬═════╬════════╣
║         43 ║  16 ║ 2.6875 ║
║         41 ║  15 ║ 2.7333 ║
║         44 ║  16 ║ 2.75   ║
╚════════════╩═════╩════════╝

In 2012, a company called Rennen Design Group created a supposed breakthrough innovation called "decimal gearing". The claim is that through manipulations of tooth profile and ring diameter, in-between gear ratios can be created - for example:

╔════════════╦═════╦════════╗
║ Chainwheel ║ Cog ║ Ratio  ║
╠════════════╬═════╬════════╣
║ 43         ║  16 ║ 2.6875 ║
║ 45.7       ║  17 ║ 2.6882 ║
║ 37.7       ║  14 ║ 2.6929 ║
║ 43.1       ║  16 ║ 2.6938 ║
║ 41         ║  15 ║ 2.7333 ║
║ 41.1       ║  15 ║ 2.74   ║
║ 52.2       ║  19 ║ 2.7473 ║
║ 44         ║  16 ║ 2.75   ║
║ 44.2       ║  16 ║ 2.7625 ║
╚════════════╩═════╩════════╝

Note: Table is not exhaustive.

For example - a 44.2 tooth gear actually only has 44 teeth, but the tooth spacing, tooth profile, and chainwheel diameter is supposed to have been manipulated to create a larger gear.

In the world of BMX racing, the existence of in-between gear ratios like this is a Really Big Deal. Since the man behind Rennen has a Master's from MIT - and since most BMXers would rather hit jumps than do math or measure things - nobody has really ever checked up on whether or not this is valid. Some questions were asked a long time ago in the dusty corners of a BMX forum, but the testing methods didn't properly control for all variables and the thread descended into a bunch of name-calling and ad-hominem attacks.

The Actual Question

Is this physically possible?

I understand "gear ratio" to be defined as:

For a given gear ratio x / y, one rotation of the gear with x teeth will result in x / y rotations of the gear with y teeth.

For a gear ratio of 44/16, one full rotation of the 44 tooth gear (chainwheel) should result in 2.75 rotations of the 16 tooth gear (cog).

So for a "decimal ratio" of 44.2/16, one full rotation of the 44.2 tooth gear (which again - only has 44 teeth) is supposed to result in 2.7625 rotations of the 16 tooth gear.

My biggest reservation is the fact that a chain-driven drivetrain is a TIMED DRIVETRAIN. No matter how big or small you make the teeth on the chainwheel, if they fit the chain, they're only going to push as many links through per rotation as the chainwheel has teeth.

For a true 44.2 tooth chainwheel, one would expect that 442 links get pushed through over 10 full rotations of the chainwheel - but that's not the case. Only 440 links will ever get pushed through to the cog because only 44 links get pushed through per full rotation of the chainwheel. I actually spent my whole afternoon yesterday taking video and counting links and measuring.

But I'm not a scientist. My high school didn't even offer a physics course. I'm just a racer that trains really hard and knows how to do basic math.

If this were a belt-driven system, I would completely understand how a manipulation of the chainwheel diameter would change the effective ratio - but it's not. It's a timed drivetrain, limited by the physical dimensions of the chain.

I have several hundred dollars and months of training and metrics invested in these stupid chainwheels. If someone could confirm or deny my theories, I would really appreciate it. I just want some closure.

Here's a photo of the teeth from a 41 tooth chainwheel on top of the teeth from a 41.2 tooth chainwheel - both are Rennen gears:

Here's a 41t on top of a 41.2t:

Here's the 41.2t on top of the 41t, from behind:

Answer 1

Gear ratio

This is a Roller Chain drive system, and as such it's a timed system. The ratio between the two connected pieces depends entirely on the number of teeth on either end.

Even if you altered the diameter of the root of each tooth so the chain can sit higher or lower, at the end of 1,000 revolutions the chain will have moved the same amount on gears with the same number of teeth as the altered gears, irrespective of the root diameter, or tooth profile. As long as the chain is engaged and doesn't skip links, then the gear ratio is the same as any other gear with the same number of teeth.

If you take your "normal" 41t gear and rotate it 10 times, it will move the same amount of chain (to within tolerances for the gears and chain) as the modified 41t gear.

If tolerances allow the chain to sit higher or lower in the teeth and "modify" the diameter, the reality is that the same amount of chain will be moved, but you'll experience more noise as the chain is rattling (loose) or creaking (tight) around the modified gear.

So, the gear ratio between the pedals and the wheel does not change. If so, what does change?

Torque ratio

If the tooth profile is modified so it's a little loose, then the chain will start to disengage under tension, and pull itself up out of the tooth a little as it approaches the top of the gear on the way forward to the pedals.

This means that the torque transfer to the gear is actually a little further away from the center of the gear, which provides more torque given the same chain tension as a comparable gear which is fully engaged.

This does decrease how well the chain engages in the gear, and increases both chain and gear wear. This might be an acceptable tradeoff if the torque is increased.

Keep in mind that these gears are used at relatively low speeds, and torque matters more than the gear ratio. In fact, for mountain biking, you might say the torque ratio is more important than the gear ratio.

So if the gear is designed to push the chain away from the center of the gear as it approaches the top, and thus increases the torque ratio, then this claim is possible, and the effect should be measurable.

Testing

Put a 200 lb scale on the pedal and tie a very heavy weight to string wound around the rim of the wheel. You should find that one ring produces a higher or lower torque transfer. Note that if you do this with lighter weights, the tension on the chain might not be sufficient to pull the chain up the slopes designed into the gear that pull the chain away from the center of the gear.

Answer 2

I suspect that the answer to this is that, ultimately the gear ratio comes from the ratio of diameters of the gears rather than the number of teeth, although in most circumstances practicality dictates that they are proportional.

Say you have a 10 tooth cog and a 40 tooth chainwheel. It's fairly simple to imagine that you could remove every other tooth from the 40 tooth wheel while keeping the diameter the same and maintain exactly the same gear ratio. Similarly you could have a completely gearless wheel (putting aside issues of slippage) driving a chain which drove a geared cog.

With this is mind, my suspicion would be that what they have done is increased the gear wheel diameter slightly but kept the same number of teeth by spacing them out slightly more and that there is enough tolerance in the whole system to get away with it. It may even be that the tooth spacings are not identical.

This implies that every so often a chain link is skipped and also that only a small proportion of teeth are directly engaged with the chain at any one time

As long as there is a possible geometry which avoids a collision between teeth and chain pins I think it should work.

More thoughts

It has also occurred to me that it is possible that the gear wheel itself is slightly elliptical; i.e., it has the same circumference as a conventional gear but is slightly squashed in one axis. I initially dismissed this as it wouldn't give a constant ration, but it also occurred to me that the torque input to a bike crank isn't completely constant anyway so it may not matter and apparently eliptical or otherwise non-circular chainrings are a thing.

I've mentioned a few time in comments that the relationship between number of teeth and gear ratio is not absolute if the tooth pitch is not consistent. While it is usually desirable for meshing gears/chains to keep a constant pitch and profile there is no fundamental principle which ties N teeth to a particular diameter and it is the ratio of diameters that ultimately determines gear ratios, as this is defined by the relationship between moments, torque and angular velocity. A trivial proof of this is that you can remove an arbitrary number of teeth from a given gear and it will still provide the same torque ratio (as long as it does not slip).

With this in mind it seems reasonable that you could achieve a small fraction ratio change by increasing the effective diameter and retaining the same number of teeth as long as the tolerances in the system allowed it.

To illustrate this consider a gear of large diameter D1 with one tooth meshing with a cog of diameter D2 via a chain. In this case it is self evident that this system behaves as a zero slip pulley and you can change the size of D1 to get any ratio (D1/D2) you want as long as the single tooth is engaged in the chain (say between 12 o'clock and 6 o'clock).

Answer 3

A "44.2 tooth" gear still can only push 44 links of chain per revolution, which pushes a 16 tooth cog around 2.75 times. There's no getting around that.

I think that is the question that has everyone's head smoking. So I made a visualization.

Here we have small gear with 40px diameter and 4 cogs and a large gear with 80px diameter and 8 cogs, respectively. The chain is indicated by red dots and fits perfectly onto the gears. As one would expect, chain links are picked up at the top of gear and released at the bottom.

In the second picture, I increased the diameter of the larger gear to 90px, but it still has only 8 cogs (I call it the 8.9999/4 ratio). As you can see, I had to move the gears closer together to adjust for the larger circumference (with a less significant increase, as from 41 to 41.2, the chain might only sit slightly tighter).

So, does increased radius mean that the smaller gear moves faster? I'd say: No.
As you can see in the second picture, the chain links only grip at the very bottom of the gear. The cogs move faster than the chain links, which means the gear subtly skids under the chain.

Some other answers suggested that this setup might lead to a higher torque. However, I don't think so.

Firstly, increasing the size of the larger gear should decrease the torque. Secondly, according to Wikipedia work is $W = \tau\phi$. Thus, as long as the angular speed of the chain ($\phi$) does not change, you will get the same torque($\tau$) for your work.

Edit:

In this setup, the chain only grips at the top of the chainwheel (larger gear); I've visually verified this. The other rollers float between the teeth as you continue on down the gear.

If we look closely at your pictures, we can see that the cogs are very close together so I suspect the chain links fit perfectly in-between. Towards the top, the cogs become slimmer. This might cause the tight chain links to slide upwards and become more loose.

Here is another illustration (again, I greatly exaggerated the effect):

Answer 4

My interpretation of the closeup images such as provided by kivetros is that the teeth themselves are wider or narrower at the top, and that the rate at which the teeth widen down to the "bottom" of each interval controls how far towards the axis the chaing links can drop. So what you get is teeth which maintain the link spacing but which allow the chain to drop to a specific radius on the chainwheel. In this way the link spacing is maintained so as not to destroy the chain, but the actual operational diameter is controlled to the equivalent of a fractional tooth.

This assumes, I guess, that all "normal" chainwheels have a standard tooth-depth.

This article provides more detail.

Answer 5

Altering your gear ratio is understood as an immediately-noticeable tradeoff between acceleration and top-end speed

In general this is true.

However, as I said in a comment, you're conflating two things under the convenient heading of the gear ratio.

top-end speed

is limited by development: when you spin out at your maximum rpm (of the cranks), the tooth ratio - coupled with wheel+tyre circumference - tells you exactly how fast you're going.

However, at least at low speeds,

acceleration

is limited by how much force you can put through a small number of teeth over less than a whole revolution. This is limited by the mechanical advantage of the crank->chainring->sprocket->wheel/tyre system. You have 4 different effective radii affecting the leverage in there, and if the crank length alters it (which it does), I don't see why chainring radius shouldn't as well.

Now I have no idea whether allowing tooth count and radius to diverge like this actually works, works well, or has any other disadvantages. It seems like it would be a pretty small effect if it does work.

To be fair, it's also misleading to give fractional tooth counts, because it encourages this conflation: that 44.2 is as you say exactly like a 44-tooth ring for top speed purposes, but gives (or implies) slightly more leverage for acceleration. Your measurement of development (chain links moved per revolution) doesn't address this.

Answer 6

@Chris Johns' answer pretty much nails it, though I will add my two cents as well. Apparently, in the world of BMX racing the difference between a 44 tooth and a 45 tooth chain wheel is large enough that there's a demand for intermediate ratios.
Short of making a new chain with different link spacing, I think the only way to do this is by taking advantage of the mechanical tolerances in link spacing to create a tooth pattern that doesn't line up perfectly with the links. I would guess that Rennen's teeth are spaced a little farther apart than normal (to increase the sprocket diameter), but the ability of the chain and its links to stretch a little bit allows this. About half the teeth on the sprocket must be engaged at any time, so you spread the mismatch between tooth and link spacing out over 22 teeth or so.

Answer 7

By making the overall gear slightly smaller while keeping the same number of teeth, they give you a little more torque, with the same ratio.

However, this looks a little dangerous, as if the space between the teeth is shorter than the distance between the links, then only the last tooth that connected to a link is really connected. It also mean that that tooth starts pulling a little earlier.

On the other hand, by making the gear larger, only the first tooth that connects really pulls, the torque is slightly lower, and the chain kind of falls on the next tooth when the connected link leaves the gear.

Answer 8

A very interesting question!

I think what has been done here is commonly known as a Gear Profile Shift .

This procedure changes the tooth profile and the working pitch diameter of a gear.

This article provides information about the profile shift, and this article¹ is a technical reference on gears.

¹ _{Forewarning, the gear guide is 13 MB in size.}

Edit:

Something else: It sounds like you are doing BMX racing on a very professional level, so I assume you already did that - still, I have to ask:

When you were testing the new decimal chainwheels, did you replace all other transmission components (rear hub cog and chain) with new parts as well?

Because a used bike chain becomes permanently stretched, and therefore has a larger pitch, so it won't fit a new chainwheel well.

Answer 9

Is it possible for a bicycle chainwheel to have a fractional number of teeth?

TLDR: Yes, but they are not suitable for BMX.

Expanding chainwheels can provide continuously variable transmission. They have been re-invented several times over the years. A recent example is the Wavetrans Bicycle Transmission. The ratio is adjusted by moving segments of gears radially on an carrier. Non-integer effective ratios occur when the disengaged sector has a fractional effective length. A chain tensioner is needed to take up the slack as the ratio is varied.

This system has not been a commercial success over the years as derallieur and hub gears have proved more effective.

A review is: http://bikeretrogrouch.blogspot.co.uk/2016/01/new-is-old-again-expanding-chainring.html

A thought: If gearing is critical for the discipline and cost is unimportant then having a range of rear wheels/tyres with slightly varying diameter will provide intermediate gear progression, as will slightly varying the tyre pressure.