Friday, November 16, 2012

For the Analy Retentive Cycist: Part III, Chordal Action

So far, we've looked at the expensive and mostly irrelevant ways of improving mechanical efficiency on your bike:  installing ceramic bearings and lubing them (often) with oil, selecting larger gears, and keeping those components clean and precision-fit.

We've looked at hubs, bottom brackets, chain pulleys, cranks, cassettes, and pedals--but we've avoided the chain, except to say that if Boardman had run a 106/22, he'd have been even faster than he was on his 53/11, because of the reduced energy lost to chordal action on the chain.

Boardman went 4,944.1 meters in an hour.  Throw on the 106/22 and, in hypothetical mode, we tack on 100 meters more (5,044.1 meters).   That's a 0.98% improvement.  Factoring in the increased weight of the twice-as-large drive train, that probably puts the gain at well over 1%.

But there's a problem here.  Don't bigger gears require longer chains, and, yes, while the increased size of the gears means the links will pivot at a slightly less acute angle, more links will be pivoting!  This means friction should be nearly the same, big gears (106/22) or small (53/11). 

So, if it's not from reduced friction, where does Boardman on the 106/22 get his energy savings?  

Apparently, something called chordal action.  

Chordal Action
Anecdotally, I became aware of chordal action before I knew what it was.  I borrowed a single speed for DCCX and felt a strange humming coming from the drive train.    At the time, I thought maybe I could solve the issue with better lube.  In fact, I should've consulted a physicist or mathematician.

I don't totally understand the physics of it, and even if I did, a full explanation would take a while, but the basic idea is really cool and related to music, so here's a little digression on how guitar strings are like bike chains, and why that's cool for guitar lovers, but bad for cyclists.  

Go to Guitar Center and pull down an an electric guitar.  Don't plug it in.  Pluck a string.  

Now go to the wood-walled acoustic guitar section.  Take one down.  Pluck a string.

You're putting the same energy into each string, but the acoustic guitar is much louder, right?

The reason it's louder is because the body of the acoustic guitar is shaped in a way that takes advantage of the natural shape of sound waves.  It's sides are shaped like this:  ~ .  It's also hollow and thin, which allows its sounding board to vibrate easily and to transmit energy from the string into the air as sound waves.  Ever wonder why violins, violas, cellos, and double basses are all shaped like ~ ?  Well, because that shape is the same shape as the strings, and that's why it resonates--that is, it takes on the same vibrations that the strings are producing.

When strings vibrate, they don't just sway back and forth, like this:
1st Harmonic

They also do this:

2nd Harmonic

 And this:
3rd Harmonic
And so on.  

This all, of course, requires energy.  

What's crazy about this is that, in nature, you don't get the 1st harmonic or any single harmonic alone. Ever.  Instead, you get all of them at once!  And all of these are mathematically related to each other--as you can see above--the second harmonic produces two waves at the same distance as the first harmonic; the third harmonic produces exactly three waves at the same distance as the third harmonic.  You can mess around with this stuff here.

The body of acoustic instruments is designed to resonate--that is, it transfers the energy from its strings to the air.  The optimal size of acoustic instruments is mathematically set by the length of the strings.  

One more thing to note (heh heh)--strings transmit energy in oscillating waves (like those shown in the figure above) because their ends are fixed.  And that energy moves the air because one of the ends is fixed to a sounding board.

Chordal Action
Take a bike chain that's also fixed--in a way--to a crank rotating at a constant speed (as you pedal).  When a string oscillates, it pulls, repeatedly at its fixed ends.  

But bike chains aren't fixed like guitar strings.  They are, in fact, continually sliding over two pivoted surfaces.

Chains are most efficient when the links of the chain fit precisely into the teeth of the crank and cassette cogs.  They work poorly when chains stretch (as the Rohloff study showed) or they are not lubricated and require more energy to slide into place.

Now, take a chain that is continually being pulled forward and backward by oscillations.  The links are continually being slid, ever so slightly, forward and backwards, so they do not fit precisely into cassette and chain ring teeth.  

Maximizing mechanical efficiency requires reducing the oscillations of your bike chain.

How do you do this?  Well, one way is to do the math--that's what Cozy Beehive did when he recommended Boardman double the size of his gears.  

Notice in harmonics 2 and 3 (above) how there are certain points in the wave that do not move--they're called "nodes," and are also shown in this diagram:  
Nodes:  Stationary Red Dots

If you could place the pivot points of the chain somewhere close to these nodes, rather than at the peaks/valleys of the wave, you'd be at your most efficient.  Just make sure your shaft center distance is not a multiple of chain pitch + 0.5, which will put you right in the middle of the peaks valleys of oscillations.  

You can do the math if you want. 

Takeaway #1: Reduce chordal action by placing the contact points of your chain on nodes; that is, making your shaft's center distance not a multiple of chain pitch + 0.5.

I totally don't understand what that means, but it's true.  In short, forget about it.  There's little you can do about chordal action except what was suggested yesterday--use bigger gears (at the same ratio).  If you singlespeed race, instead of using a 39x17 (as I did), run a 53x23.  They're the same ratio (see Sheldon Brown's calculator), but you'll reduce chordal action and improve efficiency.

You knew that yesterday, but now you know why.

You also know why, now, loose chains sap energy, because it's the same reason chordal action saps energy--chains don't slide on teeth smoothly, requiring you to push harder on the pedals to make them fit.

When your mechanic uses his chain gauge and finds stretch in your chain, that stretch is not a result of the links stretching; rather, it a result of friction wearing down pins and the contact points between pins and the plates.  That worn away metal--that space--allows your chain to bounce and oscillate, and at each point of contact with a gear, it fits less precisely than it should.  That means you have to pedal slightly harder to make it go around.

Sadly, bike mechanics are sometimes right.

Another thing you can do is get a well-made chain in the first place; Friction Facts find that Shimano's Dura-Ace CN-7901 is the best--remember, the best chain for speed.  That's all we care about now.  Not longevity or looks or durability.  If there was a chain made of nanobots or semiconductors or the bones of unicorns that was more efficient, we'd recommend that, but such chains don't yet exist, as far as I am aware.

Takeaway #2:  Reduce energy loss from "chain slide" by using a precisely machined, new chain.

So far, here's what we've got:
  • Spend loads of money on ceramics
  • Spend loads of time oiling those ceramics, because grease isn't quite slippery enough
  • Spend loads of money on huge, nonstandard gears (my guess is, having a machinist smelt you an aluminum 106-tooth chainring will cost a few grand, at a minimum)
  • Spend loads of money on new, Dura-Ace chains
Next, we'll look at how you can get fractionally faster by spending loads of time and probably money lubing your chain:  what kinds of lubes make you fastest, and why those are--surely--not the easiest nor most protective of your expensive equipment.  Wet, dry, wax, oily, greasy, Omega 3, Omega 6, partially hydrogenated, and so on.  We'll take a slippery slide down the slope of lube, and see which is slickest.  

1 comment:

Anonymous said...

You need to move the decimal place on Boardman's distance record, about 49441 m or 49.441 km

And you haven't figured in the aerodynamic aspect of a 106/22 - more frontal area.