## Thursday, January 28, 2010

### 52x12 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.