Thursday, July 29, 2010

VeloNews aero wheel ranking formula

In the September 2010 VeloNews Magazine, which I just got in the mail today, there's a review by Zack Vestal of six aero wheel pairs Check out the magazine for all the numbers. The topic of this blog post is the rating system. That system is to rank each of the six wheels from 1 to 6 in the following categories, then to average those rankings to get a final score:
  1. Aerodynamic drag
  2. weight
  3. front wheel stiffness
  4. rear wheel stiffness
  5. rear wheel rotational inertia
  6. front wheel rotational inertia
Rotational inertia describes how hard it is to change the rotation rate of an otherwise stationary wheel. For a given unit mass, its contribution to weight is obviously proportional to the mass, while the contribution to rotational inertia is proportional to the mass multiplied by the square of the distance of that mass from the axis of rotation. So mass at the rim contributes more to rotational inertia than mass at the hub.

First, note that three of these rankings depend on mass, while only one depends on aerodynamics. The implication is keeping wheels light is three times more important than making them cut smoothly through the wind. Does this make sense?

The wheels are ranked at a 10 degree yaw angle for the wind (the wind coming typically at an angle 10 degrees off head-on). At this yaw angle, the aero wheels (front wheel only) saved between 18 and 33 watts relative to a "standard" 32-spoke wheel, a spread of 15 watts, at 30 mph. At 25 mph, a more typical time trial speed, the spread would be 9 watts. So I'll use that as the relevant number. Note this neglects the rear wheel, which will add even more to the difference depending on how effectively the wheel drafts the frame.

Now to the effect of mass on power due to accelerations. Suppose a rider has to slow from 25 to 20 mph every minute, then accelerate back. Sure, there's also "micro-accelerations" from nonuniform pedal stroke or small undulations in terrain, but inertia reduces the magnitude of these fluctuations (a heavier bike or wheel moves at a more constant speed), so we should only count the accelerations from forced decelerations (typically due to safety cornering or changing speed to match another rider). That's 0.374 W/kg from these accelerations.

Rotational inertia from the wheels varies by approximately 134 kg-cm² for rear wheels and 90 kg-cm² for front wheels, which for a rolling radius of 33.4 cm, yields an effective inertia difference of 120 grams for the rear and 80 grams for the front. So in this example, the power cost from these accelerations every 60 seconds from 20 to 25 mph varies among the wheels, based on their rotational inertia differences, by approximately 0.045 W for rear wheels and 0.030 W for front wheels.

Total mass varies among the wheelsets by 357 grams. This also affects the energy required to accelerate: the effect of total mass of the acceleration on the wheels (357 grams of inertia difference) is bigger than the effect of rotational inertia differences (200 grams of inertia difference). So it's already wrong the rotational inertia was counted as more important. Add in that total mass affects climbing power and rolling resistance power and total mass is clearly more important in this case than rotational inertia.

So we start with the power for accelerating out of our hypothetical corners: 0.13 W. Then assume a average speed of around 25 mph with a CRR = 0.33% (Kraig Willett recommends using a number around 50% higher than Al Morrison measures on rollers, so I assume a typical-excellent rolling resistance measured at 0.25%). With the mass difference of 357 grams, this yields a power difference from rolling resistance of 0.13 W as well. So the variation among the wheels in power due to total mass becomes 0.26 W, half from acceleration, half from rolling resistance.

Then there's an effect from net and gross climbing. Obviously this is course-dependent. I'll assume a course with only gradual climbing and descending, so speed lost on uphills from heavier mass is regained on downhills (to first order). Therefore I assume the primary effect of heavier mass is on rolling resistance and on accelerations. Rolling terrain is trickier to model.

Then there's stiffness. Stiffness affects cornering, perhaps. This, it seems, is more of a speed thing than a power thing. I don't have any sort of reasonable model for how stiffness affects speed, nor am I sure it actually has any significant effect. In the article it's stated "our heaviest tester thought the (least stiff wheels tested) were plenty stiff for his powerful sprint". In light of this, it's fairly surprising stiffness was counted in the results twice as heavily as aerodynamic drag. In any case, I'll stick for the purposes of this blog to aero drag, mass, and rotational inertia.

So we get the following for the power cost from the variation among the tested wheels:
  1. front wheel rotational inertia: 0.030 W
  2. rear wheel rotational inertia: 0.045 W
  3. total mass: 0.26 W
  4. aerodynamic drag @ 25 mph: 9 W
VeloNews counts each of these components equally. But it's clear that even cursory analysis shows they're not even close: aerodynamic drag is more than an order of magnitude more important.

Obviously for a hill climb time trial things would be different: mass takes on larger importance. But it's physically impossible from these numbers that the relationship between mass and rotational inertia can flip. Unless, as I previously suggested, the total bike is being limited to a UCI-mandated 6.8 kg total mass, something which rarely applies to riders in the United States.

Also, for a criterium, for example, accelerations may happen more than once per minute. Still, you can see from the huge difference between the aero power difference and the inertial power differences that rotational inertia will match aerodynamics in importance only in a pathological case. Indeed, even at the World Championship level, riders in sprint events on the track use disk wheels or other deep dish wheels due to the relatively larger importance of aerodynamics. And on the road, sprints typically start from a relatively high speed, so acceleration takes on lesser importance.

So how would I improve things? Instead of using ranking, VeloNews should have come up with a set of scale factors for each test result, then summed the normalized ratings. For example, I might think that 200 grams of mass is as important as 60 grams of rotational inertia which is as important as ‒0.02 mm/N of compliance which is as important as 1 watt of aerodynamic drag. Come up with whatever numbers you want based on sound, physics-based criteria. Then sum 'em up and only rank the wheels at the end. This will more closely correlate ranking with speed.

4 comments:

aahunter said...

Yet, you go to any bike shop and they will try to sell you on the importance of rotational "weight" and ceramic bearings. I once did a similar calculation as yours and came to the same conclusion. Thanks for publishing.

djconnel said...

Rotating weight of bearings: that's funny. I also read about how pedals are rotating, chains are rotating, hubs are rotating, chainrings... r² is a big deal: all that is actually "rotating" with any inertial consequence is rims, tire, tube, and rim strip, and even there, total mass is still more important (2 grams on the frame is still significantly worse than 1 gram on the rim).

ragweed said...

Great blog, Dan. Can you add a RSS feed gadet to it? (If there's already one I couldn't find it and the "Subscribe to: Post Comments (Atom)" feed doesn't do it in My Yahoo page.)

THX

djconnel said...

The RSS address for the blog is:

http://djconnel.blogspot.com/feeds/posts/default?alt=rss

Is this what you need?

Dan