Friday, July 09, 2010

Ascension Rates and Power to Body Mass Ratios

As many of you know, the cycling speed attained when ascending steep climbs is primarily a function of a rider's sustainable power to weight ratio. More power and/or less weight means a rider can climb faster. Pretty simple really.

But it's not the only factor to consider. For instance, wind can still play a sizable role in speed attained. And of course when climbing in a race, race tactics will play a role, with attacks, surges and pacing by team mates (or motorbikes!) all serving to alter ascension rates for a given power output.

Recently there has been some (typically annual) discussion on a few cycling forums about ascension rates of pro riders, estimated power to weight ratios and whether or not such performances are plausible sans-doping, even suggesting that some ascension rates cross over some level of "sans-doping plausibility" and should be a red flag to anti-doping authorities.

Well I'm not going to delve into all aspects of this issue other than to say that, in essence, there are so many variables that such an approach is really a pretty futile exercise.

In the end, all the pro riders that demonstrate such tremendous physical acts will already be under the eyes of anti-doping authorities, so I really don't see how such an idea really adds any value to the issue of doping in cycling.

Many issues of a physiological nature have been batted about, and there are a couple of excellent summaries of some of the science demonstrating the massive variations in such estimations in these two items by Dr Andy Coggan:
Superhuman Performance? Part I
Superhuman Performance? Part II

Nevertheless, I thought I would look at the challenge of estimating power to body mass ratios from ascent times up one of the most famous climbs in Tour de France history - Alpe d'Huez. Below is a chart summarising (click on the picture to see a larger version):

In modelling of cycling power and speed, I used the mathematical model as per the 1998 Martin et al paper:
Validation of a Mathematical Model for Road Cycling Power.

The equation in question is shown below:

For the purposes of this exercise, I have simplified the equation a little. The main assumption being that of reasonably steady state cycling and no change in kinetic energy from start to finish (which is reasonable assumption given that the difference in speed from start to finish would be negligible and over ~40-minutes is a tiny proportion of overall energy demand). If there are a lot of surges or changes of pace along the way then a little more of the overall energy demand may go into changes in kinetic energy.

Then there is the climb itself. I have used a course elevation profile which, as far as I can tell, corresponds to the timing points which have been used to time the ascension up Alpe d'Huez since 1999. Before then different measuring points were used. My data indicates a climb of 13.93km with 1085m of vertical ascent (there are a few metres of marginally negative gradient right at the top).

I divided the climb into 56 segments of 250 metres (final segment a balance), with each segment having a gradient and wind vector assigned. The modeling then applied the maths to the segmented climb.

The following additional assumptions were used for the modeling:
- Rider mass: 70kg
- Bike + gear mass: 8kg
- A coefficient of rolling resistance (Crr): 0.0045
- A coefficient of drag x effective frontal area (CdA): 0.300m^2
- Air density: 1.046kg/m^2

Importantly, I have also assumed an even application of power for the duration of the climb. Of course no rider applies power perfectly evenly up a climb, although climbs with relatively consistent gradients generally produce consistent power outputs (if you inspect power meter files, you can usually pick the climbs as the power line is smoother and speed is low).

Then what I did was to calculate the ascension times up the Alpe d'Huez course profile for various power to body mass ratios, with a 2.5 m/s tailwind (9km/h), with no wind and with a 2.5 m/s headwind.

One can then see the quite sizable role that wind can play in estimating W/kg from ascent times.

To read the chart, for instance, take a time of 40-minutes flat (40:00) on the vertical axis and see where that time intersects the diagonal lines marking the tail-, no- and head-winds. The horizontal axis then marks the corresponding W/kg required for that time.

So, for 40:00, depending on wind conditions and assuming even pacing (and other assumptions as listed in the chart), then the power to body mass ratio required would range from 5.6W/kg for a tailwind to 6.35W/kg for a headwind.

Alternatively, if you are a 5.9W/kg rider then you could attain a time anywhere from 38:10 with a 2.5 m/s tailwind through to 40:00 with no wind and 43:00 with 2.5m/s headwind (off the chart).

I then added lines to mark the ascent times for various riders I selected from this Alpe d'Huez Wikipedia reference. Note that the times from 2004 were an individual time trial, the rest are final ascents during a TdF stage race. As we can see, the estimated power to body mass ratio for Armstrong’s super quick ascent time in the 2004 Individual Time Trial falls in the 6.00-6.85W/kg range, depending on overall wind direction.

Times for other riders in earlier tours such as Pantani were not taken using the same timing points, hence I have excluded them.

Of course the course winds its way up the ascent in various directions due to the famous switchbacks, and any wind vector would naturally vary accordingly, so by putting an indicator of reasonably modest but noticeable winds, at least one can see that any given ascent time will still end up with quite a wide range of possible power to body mass ratios.

All I can say is, given that some believe there is a performance level that is beyond plausibility sans-doping (some have suggested 6.2W/kg, some less, some more) then all the climb times listed in the chart straddle such "plausibility levels" with such a large range of uncertainty that it is simply not possible to draw any firm conclusions on power to mass ratios from ascent times alone.

Keep in mind that the highest ever 1-hour power to body mass ratio known and recorded is 6.4W/kg by, as far as is understood, a non-doped rider.

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