## Thursday, January 28, 2010

### 1152x12 vs 52x11 Gearing : A Look At Chordal Action

If you're the type who likes to gun it down the line in high gears, you may have wondered more than once - what is really better in terms of crankset chainring-rear sprocket size, would it be a 52T x 11T or a 52T x 12T?

At the same RPM, you get a slightly higher top end speed with the 11T at the sacrifice of some torque. But if you're asking this from an efficiency standpoint (ratio of input power over output power), its a slightly tougher nut to crack if you don't have actual measuring equipment.

I won't talk directly about efficiency but I'll talk about something else that you may want to start connecting, perhaps more with equipment durability than efficiency.

A bicycle chain has links, connected by a distance called pitch which is usually 1/2 inch in bicycles. When you're riding your bike at a cadence of 100 RPM, the chain has an average velocity called pitchline velocity. At the sprocket though, some interesting things happen with chain velocity.

For one link in your chain to engage a teeth in your sprocket, the link has to swing about an angle before the roller is seated between tooth. This is called Angle of Articulation, calculated by using the relation : 180/T, where T is the tooth count of the sprocket.

For a 11 tooth sprocket, the angle of articulation is 16.36 degrees, while for a 12 tooth sprocket, it is 15 degrees, a reduction of 8.3%.

Because the chain is turning at these sharp angles at the same time impacting the teeth, the velocity of the chain is not constant, but infact fluctuates between the maximum and a minimum each cycle. The maximum occurs before engagement, the minimum occurs after the link has swung in engagement. This change in velocity is called Chordal Action.

The point is that chordal action results in fluctuations in chain transmission and may be minimized by reducing the angle of articulation, which decreases with increasing sprocket size. For 11T and 12T sprockets, which are about the smallest standard sizes you can find in the bicycles, the angles are tight which results in uneven exit velocities.

But is it something you should worry about? The math is complicated to present here so I just did all the calculations myself elsewhere to look at chordal action. Perhaps you can decide whether it matters or not after looking at the following theoretical numbers.

Say you're riding hard at a cadence of 100 RPM with a 52 front chainring. The RPM at the rear sprocket is multiplied by the gear ratio, which results in 433 RPM with 12 T and 482 RPM 473 RPM with 11 T. With a 12T sprocket, one cycle of tooth engagement sees a 3.4% variation of chain linear speed, whereas an 11T sprocket sees 4.1% change in linear speed for the same because of the tighter angle the link has to swing.

Here are two graphs I generated :

Note : Chain velocity is a function of pitch circle diameter, RPM and articulation angle

What this does for transmission efficiency in bicycling is moot. What I wanted to present before you is the fact that chordal action is real in small sprocket sizes, and it has an effect on your tooth wear and pulsating motions at high RPMs. But it may not matter in low cadence (60 and below) and if you're using wear resistant, hardened sprockets. Feel free to discuss.

Dan Connelly : Drivetrain Losses

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1. There is a larger change in velocities, but you will be going faster. If you look at the ratio of gear inches between the (52x12)/(52x11) you get ~.91. Then if you look at the ratio of the delta V's you get ~.83. So what would the ratio be if you adjusted the RPM to yield the same speed. I imagine in delta V the difference would be much less between the gears.

2. A few points.... one of which Hightower just made.

One is the assumption of constant cadence. I think this makes the most sense if you assume the same gear ratio (for example, 44/11 versus 48/12) With a fixed chainring, I'd scale the cadence in inverse proportion to the cog (ie constant wheel rotation rate).
With this assumption, the chain is engaging teeth more slowly at the rear cog. On the other hand, there's fewer teeth, so each tooth is engaged at the same rate. So if you're calculating cog wear, your analysis actually applies more to the case of constant wheel speed than constant cadence.
That said, how does the speed fluctuation relate to wear? Do you have a functional relationship?

In my simple-mindedness I'd have been tempted to adopt a model where wear is proportional to chain tension times the amount the chain bends upon contacting the the tooth times the rate at which the tooth engages chain links.

Chain tension = P / (C × Nf × L), where P is power, C is the cadence, Nf is number of chainring teeth, and L is the chain pitch = 1/2 inches.
bending angle = twice articulation angle = 2 π / Nr, where Nr is the number of cog teeth.
Rate at which teeth get engaged = C × Nf / Nr²

Multiplying these yields:

2 π P / (L × Nr³)

which is curious in that it depends neither on the chainring teeth nor the cadence. Shifting to a smaller chainring and increasing cadence increases chain speed and decreases tension, and the effects cancel, assuming wear is proportional to the product.

But I think you have better training in this area than I do. My model is surely too simple.

3. Whoops! Correction: shifting to a smaller chainring and increasing cadence keeps the same chain speed and same tension. There's nothing to cancel. The chain speed-tension tradeoff comes when the road grade changes at the same gear, changing cadence at the same power. Sorry for the clutter.

4. Gentlemen : I made an idiotic error, in 52/12 gear ratio, its actually 473 RPM as opposed to 482 (I may have done 53/12) so I just edited that part and consequently the graph for 11-T changes but yeah delta V doesn't change at all.

Yes it is a constant cadence analysis, pretty simple actually. What I wanted to highlight though is that with smaller cogs like the 12T and 11T, the delta in chain velocity each cycle doesn't change and it has been well established that with sprocket sizes this small, chordal action is 3.5% to 4% and it progressively becomes smaller as the number of teeth on the cogs increase and is negligible for sprockets with 20-25 teeth and more. So if you drew out a graph with number of teeth on x axis and % change in chain velocity on your y axis, you get an exponentially decreasing graph.

@Hightower : I see where you're getting at.

Let's assume the chain is 95% efficient (not a bad assumption). Now let's say I want to sprint in a 52-12 or a 52-11 to achieve a top speed of 35 mph. I will have to pedal at 110 RPM in 52-12 as opposed to 100-101 RPM in 52-11. Multiplying those RPM adjustments to yield the chain linear speeds with articulation angle variation changes my Vmax and Vmin in both cases, however the delta V doesn't change. Its a law of physics for gearing :) What is better between a 12 and an 11? You get higher speed for lesser RPM in 11T, which may probably mean that you you won't raise your heart rate as much to achieve the same speed but I can't describe all that physiological stuff very well.

5. @ DJ : No I don't have a functional relationship between chordal action and tooth wear, although it'd be interesting to have one and put it up on the blog.

It doesn't seem too wrong to assume that repeated chordal rise and fall of the chain in such small sprockets over many months of riding can accelerate tooth wear if those gears are used frequently. Afterall, its the roller impacting on the tooth.

I personally know a chap who rides in such high gears (52-15, 52-12 etc) more than 70% of the time, even to go 15 mph. It puzzles me, did he forget that there were 9 other gears before that? :)

6. Anonymous3:34 PM

chordal action causes chain whip and most of your sprocket wear happens then.

7. Anonymous3:54 PM

Math is hard.

52t chain rings are for pansies.

Go for a bike ride.

Xo,

Cousin Brucie

8. Anonymous10:22 AM

Hi - I've been having a long time argument with friends on gear inches and efficiency. Is 60 gear inches on the inside chainring any different than a 60 inch with large chainring combination, assuming no strange chain alignments?

9. To understand how power losses occur in chains, you have to look at what the chain goes through on its way to transfer power from the chainring to the sprocket. The article here explores one of the mechanisms of wear and perhaps efficiency loss. Others are there, although minimal, like friction in the links of the chain and if we assume friction does work, then there is a power loss factor from that if you calculate the work done per chain rotations etc. All in all, its a pain in the ass to crunch these numbers out! :)

10. http://roostershamblin.wordpress.com/ would you please spend a few minutes of your time and check out my new blog. I am a farmer who has been raising more than 50 breeds of chickens for forty years.

11. Rooster : Not sure what a discussion on chains have to do with chickens, but oh well, I did check it out. :) Not a great way to solicit interest though :)

Thank you. I read every single comment.