Friday, December 19, 2014

The Physics recap

In an earlier blog post I examined the influence of altitude on the physics of cycling’s world hour record, and showed how the reduction in air density as altitude increases means one can travel faster for the same power output, or put another way, the power demand reduces at any given speed as altitude increases.
That resulted in this chart, which shows the relationship in power to drag ratio (W/m^2) for speeds ranging from 47km/h up to Chris Boardman's 56.375km/h record. I've slightly amended the chart to extend up to altitude of 3,000 metres. Click on the pic to see a larger version.

Each slightly curved coloured line represents a speed as marked, and from that you can see how the W/m^2 required reduces with increasing attitude. The chart clearly suggests there is an advantage to performing such record attempts at higher altitudes, but it's never that simple of course.

And the Physiological impact...

As we climb to higher altitudes and air density drops, the "thinner" air also means a reduction in the partial pressure of oxygen, which negatively impacts the power output we can sustain via aerobic metabolism. That loss of power can be as much as 20% or more depending on how high we go, and our individual response to altitude.

So the gain in speed from the physics side of the equation is somewhat negated by the reduction in physiological capacity. But by how much, and what might be the optimal or "sweet spot" altitude for a cyclist seeking to set a new record?

The physics side of the equation is easier to predict than the physiological, since the physics applies equally to all, however individual physiological response to altitude is quite variable, both from person to person, but also depending on how well a rider has acclimated to altitude. There are even differences in how altitude affects elite versus non-elite riders.

There have been a few published papers examining the impact of altitude on aerobic athletic performance and from these formulas to estimate the loss of power as a function of altitude have been developed. There was one from the 1989 paper by Peronnet et al, two from the 1999 paper by Bassett et al, one each for acclimated and non-acclimated athletes. Adding to those, I have generated a fourth formula, based on the 2007 study by Clark et al. The relevant papers are:

Péronnet F, Bouissou P, Perrault H, Ricci J.:
A comparison of cyclists' time records according to altitude and materials used.

Bassett DR Jr, Kyle CR, Passfield L, Broker JP, Burke ER.:
Comparing cycling world hour records, 1967-1996: modelling with empirical data.
Clark SA, Bourdon PC, Schmidt W, Singh B, Cable G, Onus KJ, Woolford SM, Stanef T, Gore CJ, Aughey RJ.:
The effect of acute simulated moderate altitude on power, performance and pacing strategies in well-trained cyclists.

Peronnet et al used empirical data from actual world cycling hour records to estimate the impact of altitude on an elite cyclist's power output. The assumptions used in estimating altitude induced power loss may have some error; in particular due to methods used to estimate the power for each rider as neither the power nor coefficient of aerodynamic drag was actually measured.

According to the old Wattage forum FAQ item by Dr David Bassett, Jr, the two Bassett et al formula were derived from earlier papers examining altitude impact on aerobic performance of four groups of highly trained or elite runners. So while these formulas were not derived from cyclists we can still generalise from those to the loss of aerobic capacity for cyclists.

Finally, the study by Clark et al measured the impact on peak oxygen utilisation (VO2), gross efficiency and cycling power output on ten well trained but non-altitude acclimated cyclists and triathletes by testing riders at simulated altitudes of 200, 1200, 2200 and 3200 metres. They examined a number of factors, including maximal 5-minute power output, VO2 and gross efficiency relative to performance at 200 metres, as well as sub-maximal VO2 and gross efficiency.

I used these data to generate a formula similar to those from Peronnet et al and Bassett et al. Of course there is an assumption of an equivalent reduction in 1-hour power as for 5-minute power. Clark et al noted slightly greater reductions in VO2 peak than for 5-minute maximal power, and no change in gross efficiency at 5-min max power with altitude. So there is some anaerobic metabolic contribution presumably making up the difference. There was some loss of sub-maximal efficiency noted at a simulated 3200 metres.

I chose in this instance to use the reduction in 5-minute power rather than fall in VO2 peak as the base data for the formula, and applied an adjustment to offset the formula for sea-level equivalency to bring it into line with the formula by Peronnet et al and Bassett et al. Of course when you look at the reported data there are of course sizeable variations within the test group at each simulated altitude, so the formula is based on group averages for each simulated altitude.

Here are the formulas:

x = kilometres above sea level:
Peronnet et al:
Proportion of sea level power = -0.003x3 + 0.0081x2 - 0.0381x + 1
Bassett et al Altitude-acclimatised athletes (several weeks at altitude):
Proportion of sea level power = -0.0112 x2 – 0.0190x + 1
R2 = 0.973
Bassett et al Non altitude-acclimatised athletes (1-7 days at altitude):
Proportion of sea level power  = 0.00178x3 – 0.0143x2 – 0.0407x + 1
R2 = 0.974
Simmons’ formula based on Clark et al:
Proportion of sea level power  = -0.0092x2 – 0.0323x + 1
R2 = 0.993

So how do each of these estimates of power reduction at altitude compare? Well here's a plot of these formula:

There is some variance between each formula's estimates, although the gap between the Non-acclimatised athlete estimates by Bassett et al and by Simmons based on Clarke et al is not all that large, ranging up to a ~2% variance.

Had I chosen to use the reduction in peak VO2 for 5-min max power, then I'd expect those two lines to be closer. In any case, these data by Clark et al reasonably match earlier reported findings of the impact of altitude on sustainable aerobic power. And once again - the individual response varies - these are simply averages based on the limited data available and for the cohorts tested. As always, YMMV.

The formula by Peronnet et al is the least aggressive at reducing the estimate of a cyclist’s power at higher altitudes, and that may be due to various not insignificant assumptions used in calculating each rider’s power outputs.

OK, so now we have estimates of both the physics upside and the physiological downside of altitude, What happens when we merge the two?

Well if I recreate the chart showing the physics, and overlay on that the curve showing power output as a function of altitude, this is what we get if we examine a rider capable of sustaining 51km/h at sea level:

Let me explain how to interpret the chart.

First of all, the vertical axis scale has been changed for clarity – the slightly curved coloured lines still represent the power to drag ratio required to attain a given speed at various altitudes.

So let's examine the case for a rider capable of sustaining 51km/h at sea level.

The thick orange line represents the power to drag required to sustain 51km/h. At sea level that's ~1,800 W/m^2 (Red circle 1). The exact value depends on a few other assumptions of course, so let's just use that as our "baseline" W/m^2 value.

Now if we apply the Bassett et al formula for power reduction for an altitude-acclimatised athlete, then their baseline sea level power (and with it their power to aero drag ratio) falls with increasing altitude. This drop in sustainable power with increasing altitude is indicated by the black dotted line.
We can see the power to drag ratio resulting from the physiological impact of altitude (the dotted black line) doesn't fall as quickly as the power to drag ratio required to sustain 51km/h (the thick orange line).

If you trace the black dotted line from left to right, we can see that at Red Circle 2, the power to drag ratio crosses the line marked 52km/h at an altitude of ~700 metres. Then as you trace the dotted line further to the right, we can see it cross the 53km/h line at ~1,500 metres. Tracing the line to the right hand edge of the chart out to 3,000 metres altitude, we can see it doesn't quite reach the 54km/h line, falling a little short at 53.9km/h. So for this altitude-acclimatised athlete, they can gain an extra 2km on their hour record simply by choosing to ride at an altitude of 1,500 metres.

OK, so what happens if the athlete is not acclimatised to altitude?

This time the non altitude-acclimatised power line is indicated by the lower black dashed line. It starts at 1,800 W/m^2 at sea level indicated at Blue circle 1, but as we trace that line to the right, it falls away more quickly than for the altitude-acclimatised athlete, crossing the 52km/h line at ~1,000 metres altitude (Blue circle 2) and not reaching the 53km/h line by the time the athlete is at 3,000 metres, where in this case the athlete would be estimated to achieve a speed of ~ 52.9km/h (Blue circle 3).

So while the acclimated athlete can improve their speed by 1km/h by going from sea level to 700 metres, and increase speed by 2km/h by going up to 1,500 metres, to achieve the same speed gains the non-acclimated athlete would need to ride at an altitude of 1,000 metres and would not be able to attain a 2km/h speed gain even at 3,000 metres.

We can see that as the altitude increases, the extra speed gains begin to diminish, and there are risks in going too high, especially if you are not acclimated, or experience an above average decline in power with altitude.
Conversely, if you are well acclimated and/or have a below average decline in power with altitude, then there are benefits in going higher if maximising speed is your primary objective.

Any rider considering an hour record would do well to consider the opportunity presented by tracks located at altitude. Of course costs, logistics, regulations all factor into the choice of venue, and how much time a rider may need to acclimate to altitude, and their individual response to altitude.

If a sea level based rider were considering a fly-in / fly-out attempt without much acclimation time, then I'd suggest choosing a good track that is not too high, as the risks of a larger than expected power decline increase significantly, and the potential speed gains diminish as well increasing complexity of execution as nailing pacing gets trickier. Of course the more experience a rider has with altitude and its impact on their performance, the more confident they can be with predicting an ideal location.

So what tracks are there at altitude?

Indoor laminated wooden 250m tracks at altitude include:
• Aguascalientes, Mexico: 1,887 metres above sea level
• Guadalajara, Mexico: 1,550 metres above sea level
• Aigle, Switzerland: 415 metres above sea level
• Astana, Kazakstan: 349 metres above sea level
• Grenchen, Switzerland: 340 metres above sea level
There are track at much higher altitudes, but they are 333 metre outdoor tracks with concrete surfaces:
• La Paz, Bolivia: 3,340m
• Cochabamba, Bolivia: 2,571m
• Arequipa, Peri: 2,295m
• Mexico City, Mexico: 2,260m
Of the above listed tracks, Aguascalientes is a venue well worth considering. Eddy Merckx's October 1972 hour record of course was set in Mexico City, as were Francesco Moser's two hour records in January 1984. Most hour records since then have been set at or near sea level, with the recent rejigged rule records set by Jen's Voigt and Matthias Brändle at the Aigle and Grenchen tracks in Switzerland respectively.

So what's actually possible by the bigger guns of the sport. e.g. Wiggins, Martin, Bobridge and company?

I'll save that analysis for a future post, as well as a look at generating a formula to estimate the range of potential speed gains as a function of altitude, given an estimated sea level performance.

Thursday, December 11, 2014

The sum of the parts II

Warning. Bike geek talk follows involving maths and physics, although I'll spare you the calculations and just provide the charts summarising the outcomes.

In this February 2013 post "The sum of the parts", I discussed the relative importance of wheel mass and wheel aerodynamics during accelerations using a technique known as forward integration, and based on the mathematical model of road cycling power by Martin et al.

As I said back then, aero and weight are not the only wheel performance factors to consider, but these are the two I am going to examine here, since  they are the two factors most often conflated with respect to their relative importance. Read my earlier post for a long list of other factors to consider.

In that item I made some assumptions about the scenarios being compared and said I would return one day with an updated version using better assumptions applied to the forward integration model.

I’ve now had the chance to do the revised modelling, and I was prompted by some recent forum postings once again promulgating the wheel rotating mass is really important myth, since it’s an often misunderstood concept.

So I’m going to run some numbers through the model for three scenarios:

1. A standing start acceleration lasting 10-seconds on flat windless road,
2. A rolling start acceleration lasting 10-seconds from an initial speed of 30km/h
3. A rolling start acceleration lasting 10-seconds from an initial speed of 15km/h and heading up a 4% gradient

For the first I will examine the impact of adding mass to the wheel rims, and then for each when adding that extra rim mass comes with an aerodynamic benefit.

This time I have added extra features to the model to improve its realism. These are:

i. Including calculation of the rotational kinetic energy of the wheel in the model. Previously I had simply overstated the wheel mass difference as a (not unreasonable) means to compensate for not including a calculation of changes in rotational kinetic energy.

ii. Using a power curve in each scenario that is more realistic. In the original model I used a flat 1000W power curve, but of course no one generates power in such a manner. So this time I used power curves for a standing start and a rolling acceleration based on sample accelerations from my own (post amputation) data.

Rotational Kinetic Energy demand of accelerating wheels is small

The amount of additional energy or power required to accelerate a heavier rim at the same rate is very small, but nonetheless, I added it to the model for the sake of completeness. In this revised model I only considered the case of adding extra mass at the rim, since this is the “worst case scenario” for adding wheel mass, and happens to also be easier to calculate the moment of inertia.

If you want to really examine the difference between two wheelsets, you'd need to know the moment of inertia of each wheel, but it will always be less than if all the extra mass is added at the rim.

So just how much is the difference in energy demand from adding extra mass at the rim?

Well if you consider a 5-second long acceleration of an 80kg road bike + rider on a flat windless road from 30km/h to 40km/h, it requires a 5-second average power of over 700W. Adding 250 grams to the wheel's rims will require a whopping extra 2.7 watts to attain the same rate of acceleration, of which half is the additional power required for the translational (linear) acceleration component, and half for the wheel's rotational acceleration component.

Like I said, the extra energy demand to accelerate additional rim mass is not much. But every little bit counts. Sum of the parts.

Standing Starts

So let’s begin with the standing start scenario (i.e. an acceleration from 0 km/h).

How much does adding 250g to the rim affect acceleration?

Well here are the model assumptions comparing each set up:

Below is shown the respective speed curves if we apply a standing start power curve typical for me, i.e. power rises quickly to a little over 1000W after 2.5-seconds, holds near that level for about 5-seconds, and begins dropping away after ~8-seconds as neuromuscular fatigue sets in. For reference, the 10-second average power as shown is 924W. You can click on the image to see a larger version.

The difference in speed curves for each scenario is almost impossible to discern - there are actually two speed curves but they overlap very closely, hence appear as only one.

So in order to assess the differences, I plot the difference in cumulative distance travelled at each moment in time. In other words, this chart plots how far ahead or behind the second set up (heavier rim) is after X seconds.

We can see the addition of extra mass at the rim reduces the acceleration slightly and after 10-seconds there’s a loss of 22cm compared with using the lighter rim. Cool, let's all rush out to get lighter wheels. Well, just hang on a minute...

Heavier but with better aerodynamics

As mentioned in my previous post, I’ve already shown a drop in CdA of 0.023m^2 in low yaw conditions between using a set of low profile spoked wheels, and a deep section aerodynamic wheelset. But you don’t need to believe me, there is plenty of wheel test data in the public domain, for example the one done many years back by Roues Artisanales showing the power absorbed for various wheels. For good aero wheels, the advantage increases significantly in cross winds. There are many other wheel aero tests available, and Jack Mott of Aeroweenie has put together a neat list of such datasets.

So what happens if that extra 250 grams of rim mass comes with an aerodynamic bonus reducing CdA by 0.023m^2?

Since the plot of speeds still shows very little discernible difference, and I’m applying the same power curve, I’ll go straight to the cumulative distance difference chart:

Initially the lighter wheel takes an advantage and gradually pulls ahead of its heavier but more aerodynamic rival, gaining a maximum advantage of 5cm after 5-seconds. However, as the acceleration progresses, the bike/rider with the heavier but more aero wheel begins to catch up, draws level and passes the light wheel rider after 8-seconds and finishes the 10-second effort with an advantage of 11cm.

So, not a lot in it, but remember that this is a standing start scenario, which is the quickest acceleration scenario there is on a bike, and where the impact of wheel rim mass has the greatest (albeit minimal) impact on performance.

In my previous post using simpler model assumptions I said that the lighter but less aero wheel set was good for a standing sprint of up to six seconds. Well with a power curve that’s a little more realistic (for me), that advantage extends to all of eight seconds. Beyond then and it won’t matter, the bike/rider with the heavier but more aero wheel will pull away. And the greater your standing power curve, the earlier the advantage tips to the aero wheel.

Rolling accelerations

OK, so what about accelerations from a rolling start? Well you should be able to guess the outcome of this one before reading on.

Here are the power and speed curves for a rolling start sprint effort.

In this case you’ll note that the peak power is higher than for the standing start, closer to 1250W occuring again after about 2.5 seconds (and incidentally sees me add about 1000W above the baseline effort of ~250W), but thereafter drops away consistently. This sort of power curve is normal for me from a rolling start as I can get pedal speed higher and more rapidly but I also experience a quicker decline from that peak than in the standing start. 10-second average power in this scenario is 989W (about 65W higher than the standing start).

Here’s is the cumulative distance difference:

The bike/rider with the heavier but more aero wheel pulls ahead as soon as they start their sprint and never looks back, ending up with a 58cm advantage after 10 seconds, or nearly a full wheel ahead.

Now of course the assumption with these comparisons is that all other properties of the wheels are the same, even so my original conclusion stands, even in races with hard accelerations.

The model can be run with anyone’s individual power curve, mass, CdA and Crr assumptions, as well as considering other factors such as gradient and wind.

Hotdog crit anyone? Sprint up a hill?

Let’s say we have the same bike, rider and wheel sets as above but this time the acceleration begins from only 15km/h and goes up a 4% gradient. Nasty.

Here the lighter wheel gains a maximum advantage of just under 2cm after 3.8 seconds, thereafter the heavier aero rim catches back up after 5.7 seconds and ends the 10 seconds sprint 20cm ahead.

If the finish line were 90-100m or more from the turn, I sure know which wheel I’d prefer to be using. If it was only 45 metres though, well it’d sure be a tight race and you'd need a high speed finish line camera to pick the winner!