When your ride buddy becomes a real drag

A question that comes up from time to time when chatting about aerodynamics stuff is how much impact does another rider in close proximity have on your aerodynamics, or more correctly stated, does having another rider in close proximity change the power required for you to maintain your speed?

We are all familiar with the reduction in power required when riding behind another rider. This "drafting" benefit is substantial and anyone with a power meter can see the big reduction in power when they move from riding directly into the wind to riding behind another rider. Even if you don't have a power meter the difference is certainly large enough to notice the reduction in effort required.

But what about when your buddy is drafting behind you or rides beside you? Does this impact the power needed to maintain the same speed?

The short answer is: yes, both of them do.
But in what way and by how much?

The question as to whether a rider in front gains benefit from having a rider behind them has been researched before, and the consensus is that yes, they gain a small benefit. There is good reason for this slightly counter intuitive result and it's to do with the "bow wave" of air from the rider behind causing a change in the turbulent air flow behind the lead rider and reducing, by a small amount, the depth of the low pressure zone that exists behind the front rider.

This slight reduction in the fore to aft air pressure differential of the lead rider provides a small but measurable gain. This can be expressed as a reduction in apparent CdA, but since a rider's CdA doesn't really change if their position and equipment hasn't, then in reality it's a change in the forces acting on the rider, and as a result, the power demand at the same speed is slightly reduced when compared with having no rider in close proximity (or alternatively, a rider can travel slightly faster for the same power when they have a rider immediately behind them).

In 2010  Andy Coggan examined data from a 2007 track session ridden by his wife, in which she did efforts on the track both with and without having a rider drafting behind her. In Andy's assessment of the data he remarked "having a rider drafting closely behind them apparently lowered their CdA by 3.2%, i.e., from 0.198 to 0.192 m^2.".

The reduction in energy demand will be of a very similar amount to the reduction in apparent CdA. Assuming ~350W, a reduction from a CdA of 0.198 to 0.192 is equivalent to a reduction in power demand at the same speed of ~10W, or 2.8%. In this case the other rider was riding in pursuit set up, and were themselves very "aero" (an elite track pursuit rider).

So that's one example.

This phenomenon has also been reported in the published scientific literature, examples include:

Racing cyclist power requirements in the 4000-m individual and team pursuits, Medicine and Science in Sports and Exercise, v31, no.11, pp 1677-1685, 1999. J.P. Broker, C.R. Kyle and E.R. Burke.
http://www.ncbi.nlm.nih.gov/pubmed/10589873

where amongst their data they report that the lead rider requires 2-3% less power while riding on the front of a 4-man team than if riding solo at the same speed.

Another more recent study examined this using both computational fluid dynamics (CFD) simulations along with wind tunnel validation as described in this paper:
CFD simulations of the aerodynamic drag of two drafting cyclists, Computers & Fluids Volume 71, 30 January 2013, Pages 435–445,. Bert Blocken, Thijs Defraeyeb, Erwin Koninckxc, Jan Carmelietd, Peter Hespelf
http://www.sciencedirect.com/science/article/pii/S0045793012004446

In this paper they report the lead rider of two riders riding in single file receives a reduction in energy demand of 2.6% while riding in the time trial position.

Above are three examples of data from a similar situation, with reported reductions in energy (power) demand to ride at the same speed ranging between 2% to 3% for the lead rider compared with riding solo.

There's another paper that reports a 5% advantage for the lead rider of team time trial, although I'm not able to see more than the abstract:

Aerodynamics of a cycling team in a time trial: does the cyclist at the front benefit?; European Journal of Physics, Volume 30 Number 6, 2009; A Íñiguez-de-la Torre and J Íñiguez
http://m.iopscience.iop.org/0143-0807/30/6/014

Edit: I've now read the paper and it used two dimensional CFD analysis on ellipses as a simple model simulation of multiple riders in a line and is indicative of the principles involved.

I've had the resources to test this for some time but I've hadn't got around to doing the experiment, mainly because exclusive use of track time costs money and I'm focussed on working with clients on answering more important aerodynamics questions for them than doing experiments just for the fun of it.

But today I had the opportunity to do just such an experiment.

I was doing aerodynamics testing as part of a story being written about a woman masters rider preparing for the UCI World Masters track cycling championships being held in Manchester later this year. Cycling NSW kindly offered and arranged for the track time to make this possible, and a client of mine, Rod Wagner, loaned a special power meter to enable the testing on the rider's track bike, while I offered my time for the aero work.

We'd reached the end of our allotted track time, but as luck would have it no one else was ready to ride on the track, so we had some spare time for the experiment, and willing participants.

I won't comment on the primary aero testing session as that's for another to write about for later publication in magazine and online, but I'll expand on the impromptu experiment.

The method of measurement

With the Alphamantis Track Aero System, I record and monitor in real time a rider's aerodynamics as they circulate around the indoor velodrome. Testing is done indoors as this removes the wind variable and provides for a well controlled environment. The system enables us to monitor speed and velocity and along with other key inputs such as air density, track geometry data, centre of mass height, rider mass and rolling resistance variables, the Coefficient of drag x Frontal area (CdA) is also plotted in real time and lap by lap a picture of a rider's aerodynamics is revealed.

I've briefly explained this system before in this post, which also has a video demo. You can also read more on the Alphamantis site linked above.

The experiment

Normally this testing is done with a rider riding solo on the track but for this experiment I asked her coach, another world level master's rider, to join in. His task was to ride in various positions relative to the test rider (who would continuously circulate around the track at approximately 40km/h) while her coach would change his relative position on the track every 4-6 laps as follows and on my instruction, he would:

- ride in front of the test rider to test the level of drafting assistance, then
- ride next to, and on the outside of the test rider, then
- ride immediately behind the test rider, then
- drop off entirely and stop riding, so that we could obtain data from the test rider circulating solo.

This test process was repeated a second time during the long test run to validate the results from the first run.

For reference, the test rider is a slim 60kg female approximately 172cm tall, and the coach weighs approximately 80kg and is ~185cm tall. The test rider was using a track bike with pursuit bars, while the other rider was using a track bike in regular mass start set up.

The system is really reporting the impact on apparent CdA. It's a quick way to measure how beneficial or detrimental having the other rider near you is, and the measurements are not overly sensitive to the changes in speed during the run (this is the nice thing about the process).

The results

Here's a table summarising the results of all the data runs. Click on images to see larger versions.

In the case of the support rider riding behind the test rider, the test rider gained a benefit of a reduction in apparent CdA of around 0.008m^2, or about 3.8%. Note (i) the error range and (ii) the support rider was riding in a more upright mass start position (and consequently has a larger "bow wave") and is somewhat larger than the test rider.

Also shown are the results of the traditional drafting, being a reduction in apparent CdA to nearly half of the solo value, and interestingly, the apparent CdA increase of ~ 0.013m^2, or nearly 6% when the other rider was riding alongside the test rider.

Since apparent CdA differences are a little harder to understand, I've flipped the data around to show, at a normalised velocity of 40km/h, what the power demand for the solo rider would be for each scenario:

The table below summarises the chart data, and also shows the difference in power compared with riding solo:

Compared with riding solo, the test rider gains a ~7W (3%) benefit from having her ride buddy directly behind her; a 76W (39%) benefit from drafting behind her ride buddy; and a 10W (5%)  penalty when her ride buddy is riding alongside.

So in this experiment, I found a 3% energy demand benefit from having a trailing rider, and that's right in line with (but at the top of) the range found by the other reported data referenced earlier.

This result of a 10W penalty when riding alongside another rider is more novel, although it doesn't surprise me it may be news to some.

It is something to ponder when riding in team formation events, especially when the lead rider pulls aside to make their way to the back of the line of riders. They and their team are better off (at least in low yaw conditions) if the rider pulls over and moves well away from their companions until they are near the back and can return to be in the draft of the other riders. 10W is nothing to sneeze at.

Conclusion

So it would seem that if you wish to ride alongside your ride buddy, it might cost you ~10W give or take. If speed is of the essence, then ride in single file, you'll both go quicker that way.

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Aero for slower riders. Part II

A couple of years ago in this blog item I explained how there really aren't riders too slow to gain speed benefit from an aerodynamic improvement. I demonstrated how the same aero benefit actually resulted in greater time savings for slower riders over a fixed distance course.

That might seem counter intuitive to begin with, but it's simply because the relative speed gains are almost the same for everyone, and that the slower riders are on course for longer, thereby shaving more time from their ride.

Of course as I mentioned in my previous item the development priorities for every rider will be different, and whether or not spending time, effort, money or other resources on improving aerodynamics is a priority depends very much on your objectives and what your other development priorities are. Keep in mind it is possible to work on various aspects of performance simultaneously, it's not an either/or proposition.

That said, this is really just to cover the physics, which shows us that it really doesn't matter what level of rider you are, there is a speed benefit to improving aerodynamics, and the benefit is pretty much the same for everyone.

So here's the chart*:

It shows three sets of data. The lines plot the speed an rider would sustain on flat road at various power outputs from 100 watts to 400 watts. Put out more power, you go faster. That's pretty obvious.

I plot two of those lines, one each for a given coefficient of drag area (CdA) of 0.32m^2 and one for a CdA of 0.30m^2. Note that these CdA values are approximately midway between values typical for a rider of the size modelled on a road bike and position and a time trial bike and position.

A 0.02m^2 (6.25%) reduction in CdA is entirely possible with clothing, helmet and wheel choices. Of course it's also possible to attain such a drop from positional changes.

How much any individual can reduce their CdA depends on many factors, mostly how (un)aerodynamic they are to begin with. Some people have a greater opportunity for improvement than others.

In any case, the line with the same lower CdA shows a higher speed for each of the power outputs which is to be expected.

Below those lines I show with the red columns the proportional increase in speed attained from that 6.25% reduction in CdA. It ranges from 1.96% increase in speed at 100W to 2.09% increase in speed at 400W.

So while a faster/more powerful rider gains more speed from the same drop in CdA, the relative speed gains are pretty much the same at around 2% across a wide spectrum of power outputs.

OK, as I said last time, putting on some flash aero wheels and a skinsuit won't turn a local club amateur into a pro bike rider, but suggesting that a rider is too slow to gain speed from an aerodynamic improvement is nonsense.

And what's interesting is that all riders, be they fast or slow, benefit almost equally from the same aerodynamic improvement.


* And once again the data is derived using the same model as described in this paper:

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W/m^2, Altitude and the Hour Record. Part III

In my previous posts on this topic I explored the impact of altitude on the hour record. You can recap by clicking on the links here:

W/m^2, Altitude and the Hour Record. Part I
W/m^2, Altitude and the Hour Record. Part II

In summary, the primary impacts on the speed attainable (or distance attainable for an hour) are:

1. Physiological - the reduction in sustainable aerobic power as altitude increases due to the reduced partial pressure of Oxygen, and

2. Physical - the reduction in aerodynamic drag as altitude increases due to the reduction air density.

Of course there are other factors - variable track surfaces and geometry, logistical, financial, physiological and so on, but for the purpose of this exercise I have confined analysis to the primary physiological and physical impacts.

These primary competing factors - reduced power and reduced drag combine to mean that in general an increase in altitude means a greater speed is attainable. In other words, the benefit of the lower air resistance at higher altitude typically outweighs the reduction in power. But not always.

The level of impact to speed is individual and is a function of each individual's physiological response to altitude - while the physics side of the equation is the same for everybody. I covered this in more detail in Part II of this series, and used data from several studies which provide four formula for the average impact of altitude on power output.

I plotted the different formula depending on whether athletes had acclimatised to altitude or not.

This chart should be fairly intuitive - further up in altitude you go, the more power you lose compared with sea level performance. The vertical scale of the chart amplifies the differences between them, which are not large, but also not insignificant either. A key element was the difference between athletes that had acclimated to altitude and those who had not.

Then I layered on that the physics impact of reducing air resistance, but the resulting chart was not quite as intuitive to follow and so I decided to revisit this another way.

Hence exhibit A below (click on the image to view larger version):

This should be reasonably straightforward to interpret, but even so I'll  provide some explanation.

The horizontal axis is altitude and the dark vertical lines represent the altitude of various tracks around the world.

The vertical axis is the proportion of sea level speed attainable.

The curved coloured lines represent the combined impact of both a reduction in power using each of the formula discussed in Part II of this series, combined with the reduction in air resistance.

So for example, if we look at the green line (Basset et al acclimated), this shows that as an cyclist increases altitude, they are capable of attaining a higher speed up until around 2,900 metres, and any further increase in altitude shows a decline in the speed attainable, as the power losses begin to outweigh the reduction in air density.

The track in Aigle Switerland represents around a 1% speed gain over London, while riding at Aguascalientes would provide for between a 2.5% to 4% gain in speed. Head to Mexico City and you might gain a little more, but as the chart shows, the curves begin to flatten out, and so the risk v reward balance tips more towards the riskier end of the spectrum.

Altitude therefore represents a case of good gains but diminishing returns as the air gets rarer. Once you head above 2,000 metres, the speed gains begin to taper off, and eventually they start to reduce, meaning there is a "sweet spot" altitude.

Caveats, and there are a few but the most important are:
-  any individual's sweet spot altitude will depend on their individual response to altitude - the plotted lines represent averages for the athletic groups studied;
- the formula used have a limited domain of validity, while the plotted lines extend beyond that, a point I also covered in Part II of this series;
- these are not the only performance factors to consider, but are two of the most important.

I suspect that the drop off in performance with altitude might occur a little more sharply for many than is suggested here. Nevertheless, the same principles apply even if your personal response to altitude is on the lower end of the range, and it is hard to imagine why anyone would suggest that heading to at least a moderate altitude track is a bad idea from a performance perspective.

Alex Dowsett rode 52.937km at Manchester earlier this year. At Aguascalientes he could reasonably expect to gain ~3.5% +/-0.5%  more speed, or just about precisely what Bradley Wiggins attained in London.

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Density matters

I saw a question today from someone who read recent comments about how high air pressure resulted in Brad Wiggins' hour distance being less than it might otherwise have been with more favourable conditions.

He was wondering if you can control air pressure in velodromes, or choose a time of year when it is lower. So can we do that?

Climate control

While there are velodromes where the inside air temperature is controllable (mostly northern hemisphere tracks located in cold climates), the control of air pressure is not something possible at any currently existing track that I'm aware of.

It would require quite a deal of engineering, in particular to provide an air lock / sealed environment that enables lots of people (and service vehicles) to enter /exit the building without affecting inside pressures and which meets emergency evacuation requirements for a large crowd, as well as fresh air to breathe. I don't see that happening any time soon.

Air locks do exist, e.g. at Aguascalientes velodrome in Mexico they use an air pressure differential to support the roof, but that means the air pressure inside the velodrome needs to be higher than outside. Not by much, but it will always need to be higher relative to local weather conditions, and inside the velodrome air pressure will still vary relative to outdoors.

So what about picking a better time of year?

Well let's look at the daily barometric pressure readings near London for the past three and a half years. Source for these charts is the National Physical Laboratory in the UK.

Barometric pressure London Jan-Dec 2012

Barometric pressure London Jan-Dec 2013

Barometric pressure London Jan-Dec 2014

Barometric pressure London Jan-June to date 2015

Looking at the above, it's pretty clear there is no obvious pattern to suggest a time of year when barometric pressure will be, on the balance of probabilities, lower.

Air density is what matters.

Air pressure of course is not the only variable. What really matters is attaining as low an air density as is physiological sensible. Air density along with a rider's aerodynamics, ie. their CdA, determines the energy demand for riding at a given speed, and lower air density is desirable for greater speed, provided of course the means to achieve that lower density doesn't reduce a rider's power to the extent performance ends up being worse. e.g. by riding at such high altitudes or temperatures that the rider's power output is compromised to a greater extent than the air density benefit provides.

Air density is a function of:
- air temperature
- barometric pressure
- altitude
- relative humidity

You can pretty much discount the latter as the changes in air density is very small with changes in humidity, although for the record humid air is slightly less dense than dry air (at same temperature, pressure and altitude).

Air density reduces with increasing temperature and altitude, and with reducing barometric pressure.

Since attempting to reduce air pressure either via climate control or by picking suitable times of year is not really an option, that leaves us with adjusting the other two variables - temperature and altitude.

I've discussed altitude before in this item. I'm going to revisit it in a future post in an attempt to simplify the impact of the variables involved.

Heating the air inside a velodrome is common, and this was attempted with some powerful portable heating devices during Jack Bobridge's unsuccessful attempt earlier this year, and in the case of attempts at most northern hemisphere tracks, the temperature has been dialled up to the rider's desired level.

Wiggins did specific heat acclimation work and reports are the temperature inside the velodrome was around 28-30C. That's pretty warm - going too hot can be detrimental as power losses can occur with inadequate cooling. As I said earlier, it's a balance between a physical benefit and a potential physiological cost.

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Wiggo's Hour

Just a short one today to update the chart from the one I posted here and on other social media forums. Click to see bigger version.

54.526km

Different reports of barometric pressure of 1031-1036hPa and air temp of 30.3C inside the track mean that Wiggins must have been exceptionally aerodynamic and recent work on his bike and position at the track suggest some good aero gains were made.

I estimate a power to CdA ratio of 2500-2550W/m^2 was required.

There are of course a range of assumptions:
Total mass: 82kg
Crr: 0.0023
Drivetrain efficiency: 98%
Altitude: 50m
Relative Humidity: 60%

If drivetrain efficiency is better, say 99% and Crr at 0.0020, then it drops the power to CdA ratio down to 2200-2220W/m^2.

and perfect pacing.

Just on that, my colleague Xavier Disley has once again produced a lap pacing chart - here it is:

That's a very slight fade over the course of an hour, which in my humble opinion is pretty much perfect. Opening few laps a bit hard, but that's understandable as a rider seeks to control the adrenaline rush with thousands in the crowd watching on and cheering.

The high air pressure did cost distance, and on another day perhaps 55km was within reach

As for going to high altitude, well there are many variables, but another 1-2km is feasible. See this item for more on that.

Well done to Brad Wiggins. That's sure a fine ride.

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Pressure on the Hour

My colleague Xavier Disley did up a neat chart showing the impact the daily variability of barometric pressure can have on the distance attainable for an hour record, and how it's looking given the weather forecast when Xav last did the chart:

Nice - it shows how much breaking a record can still come down to a bit of luck with weather.

I think in Wiggins' case, assuming no major execution (i.e. totally crummy pacing) or mechanical issues, he'll break Dowsett's current mark no matter the weather as his power to drag ratio is sufficiently higher than Dowsett to overcome a slow air day.

But to set an outstanding mark such as Rominger's record, he'll need luck on his side. High pressure days are not good for speed.

Below is another version of this relationship between barometric pressure and distance attainable for four combinations of power and aerodynamic drag (CdA) values.

The chart is pretty self explanatory. For each combination of power and CdA chosen, the distance attainable reduces as barometric pressure increases.

That's because higher air pressure means a higher density of air molecules, and more air molecules to push out of the way requires more power.

A 60hPa difference in barometric pressure is equivalent to about 1km difference in distance attainable for the hour for the same power and CdA. That's a wide range of barometric pressure though, and variations are not normally quite that wide in most locations.

But a variation of half that is certainly possible over just a few days of varying weather as can be seen in Xavier's chart above.

I chose two power outputs: 430W and 450W, and two CdA values: 0.20m^2 and 0.19m^2. I don't know what Wiggins' power nor CdA value actually is or will be on the day, but for the sort of speeds he's likely to attain, these are in the ballpark.

It's the ratio of power (W) to aero drag coefficient CdA (m^2) that primarily determines the speed or distance attainable. hence why we refer to power/CdA ratio as measured by W/m^2. This chart covers a power to drag ratio range of 2150-2370 W/m^2.

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Where will Wiggo wind up?

Chart showing the progress of the UCI hour record since 1893 (click on it to view a bigger version):

The chart shows all the successful hour records recorded by the UCI. It doesn't show failed attempts.

The blue dots show the incremental increase in what is the absolute furthest distance attained.

The red dots show successful records for various categories of hour record but that did not surpass the furthest record for all categories up to that date.

For example, up until the early 1990s, the UCI had separate hour record categories for:
- amateur and professional riders
- above and below 600 metres altitude
- indoor and open air tracks

As a result, there were six categories of hour record for the period from about 1940 to the early 1990s.

And of course there have been bike/equipment regulation changes at times, most notably after Obree's and Boardman's records in the mid 1990s,

So where will Bradley Wiggins end up?

I'm pretty sure it'll be another red dot and not get close to Boardman's 1996 record and I doubt he'll beat Rominger's 1994 mark either. But he will likely beat Alex Dowsett's record (52.937km - the currently recognised record) by 1km or so.

I think anything above 54km will be very tough going. 54.5km perhaps if things go well. Closer towards 55km if everything is perfect.

Power 440-460W

CdA - who knows?

Say 0.200m^2.

Such a power range would net him around 53.5 - 54.4km at typical air density. 

On a low air density day that range would stretch to 54.5 - 55.4km. 

Weather forecast suggests low air density is unlikely although there is plenty of chat that they will raise the air temperature a lot, even up to 32C (yikes!).

So if velodrome air is heated to say 30C and air pressure is say 1020hPa, then at that power range and guessed CdA, the distance for a well paced effort will be in the 53.9 - 54.7km range.

Of course his CdA is the big unknown. Looks like he's been doing some work on it.

Drop that to 0.190m^2 and we can add about another lap (260 metres) to those estimated ranges.

Best of luck to Wiggo!

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Some kilometres are longer than others

With the spate of attempts at the UCI world hour record over late-2014 and into 2015 due to the revised UCI rules making the record within reach of more riders, it has naturally sparked interest in discussing what matters for best performance in the event.

Jens Voigt started the latest round of hour record attempts at the UCI's Aigle track

I recently saw some chat on a triathlon forum speculating about who could do what distance and so on. All in good fun, but none of them actually go to a track to find out. If they did, they'd realise it's not quite as simple (or as hard) as they might make out.

It pretty much comes down to optimising four main elements:

• maximising sustainable power output for an hour
• minimising the physical resistance factors of riding on the track
• technical execution / skill
• logistics & resources

Some might add psychological factors to that list, but ultimately I consider these to be expressed within the outcomes of each of the above.

Regarding logistics, there are of course UCI requirements to be permitted an official attempt an hour record, e.g.: minimum time in anti-doping bio-passport program is mandatory at elite level or dope testing at age group level, application submitted in advance for approval to relevant levels of cycling administrations, all the technical requirements including international level commissaires to supervise, a UCI approved track, use of timing equipment, start gates, specified date and time of attempt, etc. You can't just rock up and ride whenever you like. Well you could but it would never be a sanctioned attempt.

Then of course you need to factor in enough solo rider time on track for preparation, and that costs money and time as well. Quality indoor tracks are not always local, and even if they are, getting solo time on the track is not always so easy, let alone cheap. For an elite professional rider whose job is to race on the road, it may be difficult to devote sufficient time to the task of preparing properly for a track event.

Of course assuming the paperwork is all in order and you can do your training, then sustainable aerobic power and aerodynamics are king and the rider's ratio of power to aerodynamic drag area is the single most important factor for how far they will go in the hour. But W/m^2 is not the only factor.

There are other physical resistance force factors, like the influence of air density which is a function of altitude, temperature and barometric pressure (and to a much lesser extent, humidity) and the rolling resistance of the track and tyres chosen. I discuss some of these in the following items:
Altitude and the Hour record Part I
Altitude and the Hour record Part II

Which leaves us with technical execution and skill factors, of which there are a couple of key items, namely:

• pacing, which I covered in this case study of the masters hour record by Jayson Austin, and 
• riding a good lines around the track.

Riding good lines

Riding a good line involves a couple of components, one is pretty obvious and involves not riding further than you need to around the bends. Ride wide and you ride further. Pretty simple given the track is all but two semi-circles joined together with two straight sections. OK, the actual shape of tracks are more subtlety curved but that's close enough to describe why riding wide adds distance to your travels around a lap.

Design of the Glasgow Velodrome

If you ride 10cm wider in the turns, you add 10cm x 2 x PI = 62.8cm per lap.

If the extra width is measured on the track's surface, well the actual addition to the distance the wheel travels is reduced by the cosine of the banking angle. e.g. say the track's turns are, on average, banked at 40 degrees, and you ride 10cm above the black line. Then the actual additional track radius ridden is cosine (40 degrees) x 10cm = 7.7cm, and the additional distance per lap = 7.7cm x 2 x PI = 48.1cm. Nearly half a metre.

Do that over 200 laps or so for an elite hour record and you'll ride ~100 metres more than you need to. And that's for riding only a hand's width above the black line.

London Velodrome used for the 2012 Olympics

Another more subtle ride line factor involves the shape and design of the banking and in particular the transitions from the straights to the turns and back again, and whether it's advantageous to ride a slightly wider line in the straights to aid the transitions. On the straights you don't suffer the same severe distance penalty of riding a wider "radius" as you do when riding wide in the turns, so you can explore marginal gains in this manner.

However there is no simple or single answer to this, it depends on the rider and the track geometry - all of which have subtle differences. This is a somewhat more complex optimisation problem and I'm not going to delve into it here.

So putting aside these subtleties, the shortest distance around the turns is to ride the track's black measurement line* - ride any further out from the black line and you ride more distance each lap than is necessary. For the hour record you only get credit for the official lap distance each lap, which is typically 250 metres per lap on most modern standard indoor velodromes although some tracks are shorter and some are longer.

* it is possible to ride inside the black line, however in such timed track events like the hour there are foam blocks placed around the inside line of the track to ensure the riders don't. Very skilled riders can however ride fractionally under the black line on some tracks but it is risky as hitting the foam blocks can disrupt your effort and wash off some speed. The shape of the track in that small space between the black line and the wide blue section varies from track to track and it can be good or not so good to ride in.

Foam blocks discourage riders from riding inside the black measurement line.

Now why are some kilometres longer than others?

Office distance for the hour record =
(Official lap distance)  x  (Number of full laps completed within the hour)
+ a pro-rata distance calculated for the final incomplete lap

I won't go into the formula used by the UCI to calculate the pro-rata distance of the final lap (that's actually deserving of a blog post on its own as the regulations are remarkably confusing).

It matters not how far you actually ride, you'll only be credited with the official minimum lap distance per lap. This is why track riders and coaches are focussed on lap times and not with bike speed, since lap times are the integral of all performance elements. Power meter and other data loggers are of course valuable in parsing out the individual elements of performance that go into attaining lap times, and helping to prioritise development opportunities.

How good are riders at riding the minimum distance necessary?

It varies. Quite a lot. Skilled track riders are typically much better, which is what you'd expect. But what sort of penalty would an unskilled rider face if they started out on a track effort?

Of course we can do lots of maths to figure out how much extra distance on average a rider might cover if they ride wide by so much, but in reality riders move up and down the track, sometimes riding a good line, other times not so good. Some riders are just better at it than others and some adapt to the track more quickly than others.

It'd be so much better if we could simply measure what people actually do rather than speculate.

Which had me thinking. I have some data like that already...

Not so long ago I was doing some performance testing involving half a dozen pro-continental road racers at an indoor 250m velodrome. One of the features of the data logging system used for the tests is an ability to calculate the distance ridden per lap using the wheel's speed sensor data combined with track timing tapes to know precisely when they pass a specific points on the track. With some clever maths this is enough to nail the actual distance ridden each lap to high precision.

In amongst the test data were some solo efforts of at least 10% of the distance of an elite hour record attempt (i.e. 20+ laps of consistent effort) and several such runs by each of the six riders. I figured the runs needed to be long enough to reasonably approximate what a rider might be expected to do over a longer distance/duration.

Of course absolute accuracy of the distance the wheels travel depends on having an accurate wheel circumference value and that value not changing a lot while riding. So I'm not going to assume that the absolute accuracy was perfect, even though the absolute error might typically be somewhat less than 1%. More than that would require an error in tyre circumference assumption of 20mm, which is a lot for those used to measuring such things. However in our favour is that even if such an error existed, it would be a consistent bias error.

So rather than concern myself with absolute accuracy, I thought I'd compare the measured average lap distance for each run with the shortest recorded legitimate lap. In this way if there is any bias error, it's impact on this analysis is minimised (i.e. both measurements would be out by the same proportional amount). By legitimate lap, I mean a full lap not ridden below the black line.

Here's a table summarising data collected from the six riders (in no particular order). Each rider has multiple runs although I haven't identified the riders in the table. What the first column shows is the average lap distance per run less the minimum legitimate lap distance for that same run. The distances are of course distance travelled by the wheel.

Now the riders possibly could ride a tighter line than they actually did for their shortest legitimate lap, meaning that these distances likely underestimate the extra distance ridden when compared with riding very tight to the black line.

For the moment though let's assume the shortest lap they rode during each run was the best they are capable of doing. Since they actually did it, I think that's a reasonable assumption.

The average extra distance ridden per lap varies from one rider to another. One rider consistently rode only 0.3-0.4 metres more per lap than their shortest distance lap, while another was consistently riding more than 2 metres extra per lap on average compared with their shortest legitimate lap. The rider with smallest extra distance per lap had a track racing background.

The second column shows what that average extra lap distance would mean if extrapolated to riding 200 laps of a 250m track (an official distance of 50.000km). For one rider they would be riding nearly half a kilometre further than their track skilled team mate. Yet if both completed exactly 200 laps in the hour, each would be credited with riding precisely 50km, even though one rider's wheels had travelled nearly 500m further than the other's.

In this case, 50.5km = 50.0km. Some kilometres are longer than others.

So what's that extra 500m cost in power terms?

Well for a rider with a CdA of ~0.23m^2, that extra 500 metres travelled requires they output ~11-12 watts more than if they were able to ride a a better line.

Or they'd need to find a 3% reduction in CdA to make up for their skill deficiency.

Remember these were well skilled, well trained and experienced pro-continential road racers and finding an extra 10W or losing another 3% of aero drag coefficient isn't such an easy thing to do.

So no matter your current skill level and experience, if you're expecting to ride such an event yet you have never trained to become proficient riding on the track, well you might want to chop half a kilometre or so from your estimated distance covered based on your power and aero data alone.

Better still, just get to a track a find out what you can actually do.

Likewise, when estimating power, or W/m^2 from the official hour record distances, you might need to add some watts for the technical proficiency of the rider. The less proficient, the more power is required to attain the same official distance.

Read More......

The Sin of Crank Velocity

Crank and bike speed variations during a pedal stroke

This topic occasionally comes up in discussion in cycling forums – just how much does crank speed vary during a pedal stroke? And how much does this affect the accuracy of power meters?

If you are pedalling along at a steady rate and maintaining a consistent power output (in other words you are not attempting to accelerate or slow down) and are using circular chainrings, then the short answer is: not a lot.

But crank rotational velocity during a pedal stroke is not totally constant, there is some variation.

Of course anyone who has ridden a bike behind a derny or motor pace bike on flat terrain or at a velodrome knows they are able to ride consistently close to the rear guard or roller of the motorbike and not experience significant fore-aft movements during each pedal stroke.

Riders following a motor pacer manage to remain very close
to the pacer and don't experience large fore-aft relative motion.

Riders in a highly skilled team pursuit formation riding at high power outputs are also able to ride within centimetres of the wheel in front without experiencing major changes in velocity between riders during each pedal stroke, which if it did happen would of course be somewhat disastrous. Note the riders below are all at different phases in the pedal stroke:

Track Cycling World Championships 2014: GB women's team pursuit team
Image from: www.telegraph.co.uk

So even without any examination of the research, or doing any fancy modelling of the physics involved, we already know empirically there really isn't going to be large variations in bike and crank velocity.

So how much variation is there and what influences bike and crank speed during a pedal stroke?

When pedalling in a steady state manner, the main factors influencing the variability of crank velocity are:

• The average power being applied
• The manner and level of power variation during a pedal stroke
• The inertial load of the system (i.e. the speed and mass of the bike + rider, plus a little rotational inertia from rotating components)
• The resistance forces in play (i.e. air and rolling resistance, gravity, and changes in kinetic energy) and whether for example air resistance is dominant (e.g. when riding on flat terrain), or overcoming gravity is dominant (e.g. climbing a steep grade)
• The shape of the chainrings, or as in the case of some weirdo bike crank and pedal sets ups, the variable effective radius of the crank arms

Why do we care about crank velocity variation?

Apart from satisfying our curiosity on this somewhat esoteric matter, the answer does have a few practical applications with respect to cycling performance, one of which is to do with power meter accuracy. Others pertain to efforts seeking to eek out minor performance gains though examining mechanical adaptations to the bicycle drive train, such as design of non-circular chainrings. Whether these designs result in a performance improvement is debatable (the research is equivocal on the matter for sustainable aerobic power) and not the topic of this post, so I'll leave it there for now.

Power meter accuracy and crank rotational velocity

Many power meters, e.g. SRM, Quarq, Power2Max, Garmin Vector and Stages, rely on the assumption that crank speed during a pedal stroke does not vary (Powertap on the other hand assumes wheel speed does not vary during their fixed duration torque sampling period of 1 second).

This is an important assumption, since torque is sampled at a fixed frequency (e.g. at 200Hz for an SRM or somewhat lower, e.g. at 50-60Hz for other brands) and then those torque samples are averaged over a complete crank revolution. This averaging of torque over a full revolution to calculate power will only be accurate if the crank velocity does not vary much during the pedal stroke.

If however a rider pedals significantly more slowly during one part of a pedal stroke compared with another, then that slower part of the pedal stroke will be over-weighted in the average of total torque samples. Hence interest in examining the assumption about constant or near constant crank velocity during a pedal stroke.

So let’s look at some of the crank velocity variability factors I mentioned earlier.

Power application during a pedal stroke

It’s well known that when we pedal we apply torque to the cranks in a pulse-like manner, with each leg’s down-stroke moving from a phase of minimal propulsive power when the crank is vertical with the pedal at top dead centre, increasing propulsive force and power as the crank moves down towards the horizontal, and diminishing as the crank moves towards the vertical again with the pedal at bottom dead centre. It looks a little like this:

Image courtesy of:
http://www.rohloff.de/en/company/index.html

In steady state cycling we apply little if any propulsive force on the upstroke and some may in fact apply a little negative force (this is not necessarily a bad thing and is something often misunderstood about pedalling dynamics - but I digress).

This pulse-like power cycle is repeated by the opposite leg and crank arm, so that for each full revolution of the crank, we apply two pulses of power, mostly from the down-stroke push of each leg.

This pulsating nature of power application has been measured and is well reported in the scientific literature, and typically shows a wave-like sinusoidal pattern (i.e. it looks like a sine wave), which should come as no surprise given our legs act like two pistons pushing down on rotating crank arms.

Studies examining this go back many, many decades, e.g. this 1968 paper by Hoet at al as an example reported the following finding:

Those summary findings by Hoes et al have been consistently replicated in many subsequent studies.

Such analysis goes back even further, to the late 1800s. Pedal pressure was measured and shown in this book: Sharp, A. (1896). Bicycles and tricycles: an elementary treatise on their design and construction.

This fabulous old book (which covers a huge range of cycling physics and performance matters over 32 chapters and which Lee Childers at Alabama State University kindly referred me to) is available in scanned form online here:

https://archive.org/details/bicyclestricycl02shargoog

Here are image scans from four of the book's 561 pages showing pedal pressure measurements from various riding conditions on a fixed gear bicycle (flat track riding, ascending and even back pedalling when descending). Read from the bottom of page 268 - Section titled: 214 - Actual Pressure on Pedals.

The charts plot the measured pedal pressure, which is not the same as the tangential (propulsive) forces, a point made on page 270 - but even so we can see the basic shape of pedal forces follows this basic pattern. Indeed Sharp refers to such tangential pedal force measurement pedals designed by Mallard and Bardon, their "dynamometric pedals". I don't have a link to show these unfortunately. This was the late 1800s. What's old is new again.

Below is an image from the 1991 Coyle et al paper showing the measured crank torque applied by one leg through a full pedal stroke for each of the 15 riders in the study. We don’t need to inspect the plots too closely; the main point is we can readily see the approximately sinusoidal shape of torque applied to the cranks during the down-stroke phase, and the minimal torque applied during the upstroke phase.

The primary differences between riders are in the amplitude of the down-stroke phase of curve, and the shape of the upstroke phase, which mostly hovers around the zero line. As always, you can click on an image to see a larger version.

If you then imagine the opposite leg repeating the same type of torque application, then we can see we apply torque to the cranks in a pulsating wave-like, or sinusoidal, manner. This sinusoidal curve was also demonstrated in the 2007 Edwards et al paper which included the following plot of typical pedal torque applied by both legs for a full crank revolution:

So while not perfect sine wave-like application of torque, thinking of the force applied to the cranks as being sinusoidal-like is a very good first order approximation.

Users of indoor bike training systems like Computrainer, Wattbike or SRM's Torque Analysis System may have also seen similar torque output curves. Here’s a random example of an image of a Computrainer SpinScan plot I found with a quick Google search:

Now SpinScan is not as fine a resolution measurement tool as used in scientific study, but at least you can see that the same basic shape of the power output curve during a pedal stroke. Wattbike charts are typically displayed in polar chart format, so to avoid confusing things I won’t post a picture here but the same overall pattern of torque and power application is repeated, as they are with SRM's Torque Analysis system (although this is a very low power example from SRM's website):

One thing we can see with all these examples is how power doesn't generally drop all the way to zero while pedalling, and that it is often not symmetrical for left and right legs.

I have generated a sample sinusoidal power curve to reasonably approximate this pedal power pattern which I'll use for modelling I'll discuss later on. In this case it shows the power curve when pedalling with an average power of 250 watts at 90rpm (click on pic to see larger version):

Now this is by no means a perfect model of actual power application, and as can be seen from the various charts shown earlier, everyone pedals slightly differently, but it’s certainly a very good first order approximation to examine the issue of how much bike speed is affected by application of such a pulsating power curve. It matches the general shape of actual torque delivery delivery and the peak power is nearly double the average power, which also matches the torque profile measured experimentally many times, and at least since Hoet at al reported this in the 1960s.

Naturally the power curve is a function of both crank torque and crank velocity, but as we shall eventually see, the latter does not vary all that much, certainly not enough to make this first order approximation invalid for the purpose of answering this question.

So what effect does this pulsating variable power output have on bike and crank velocity?

Firstly I’ll look at an example of what’s actually been measured and reported in the scientific literature, and then I’ll examine the physics with some modelling.

Actual measurement of crank speed variations

Crank speed variations during a pedal stroke were reported in this paper by Tomoki Kitawaki & Hisao Oka: A measurement system for the bicycle crank angle using a wireless motion sensor attached to the crank arm, J Sci Cycling. Vol. 2(2), 13-19.

Here’s a link to a pdf of the paper I found in a Google search:
http://tinyurl.com/qapfoek

I'm referring to this paper in particular as it provides some helpful images and a full version is available online for anyone interested in examining it in more detail. Other studies have also measured crank velocity during pedalling and found similar results, but the data may not be available in the freely available public domain nor presented in such a convenient and helpful manner as needed for this discsussion.

Figure 6 of this study (shown below) summarises the measured variability in crank velocity over a range of power outputs for riders in three groups categorised as beginner, intermediate and expert level by their relative power to weight ratio being approximately 1.5W/kg, 2.25W/kg and 3W/kg respectively.

Crank velocity for each group is shown at two different cadences (70rpm and 100rpm) and was measured using two different crank velocity measurement systems (the study was primarily to examine the validity of a crank angle and velocity measurement system compared with a standard. As it turns out the two methods matched quite well). They also measured the crank velocity variations at the riders’ freely chosen cadence but did not provide charts for those data.

In this study, riders used their own bike on a CycleOps Powerbeam Pro trainer, which is a fairly typical and relatively low inertia indoor trainer.

It shows six charts. In each the crank rotational velocity relative to the average crank rotational velocity (i.e. the normalized crank rotational velocity) is plotted by crank angle for a complete pedal stroke. There are a series of dots and lines plotted in the charts, and these represent the normalized crank rotational velocity as measured using two different measurement systems.

The maximal variance in normalized crank rotational velocity occurs in the most powerful group of riders, and at the lowest pedal rate (70rpm). The variance in this case was around +/-3% and in all other cases the crank velocity variance was less than +/-2%.

Keep in mind this experiment was performed on a low inertia trainer. Why does this matter? 

It matters because a mechanical system (or an object) with lower inertial load will accelerate (or decelerate) more rapidly when a net force is applied to it compared with a system with greater initial inertial load. Many indoor trainers, like the Powerbeam Pro used in this study, have lower crank inertial loads for the same rear wheel speed compared with what a bike and rider riding out on the road typically possesses.

So what happens when we examine the case of a more realistic road-like inertial load scenario?

Since I’m unaware of published and validated crank velocity measurements readily available from actual on road riding (if anyone can point me to any please let me know), we can instead examine the physics of what happens to a bike’s speed when power is applied in this pulsing sinusoidal-like manner.

Forward integration - Balance sheet accounting for energy

To do that I use the technique of forward integration, a technique that’s really no more complicated than an accounting balance sheet but for energy rather than money.

On one side of the balance sheet is the energy supply coming via our legs, and the other side of the balance sheet is the energy demand, i.e. the power required to overcome various resistance forces such as air resistance, rolling resistance, gravity if changing height, and importantly resistance to changes in speed (kinetic energy). That energy balance sheet must remain in balance. It’s a fundamental law of nature.

At any point in time if we know a rider’s speed, their mass and that of their bike, the moment of inertia of their wheels, their coefficients of air and rolling resistance, the road gradient, and other small frictional loss factors, then we can calculate how much power is required for each of these various resistance forces, and when added together they tell us how much power is required to maintain that speed.

If a rider’s actual power output is different to that required to maintain the same speed, then the balance of power supply must result in a change in kinetic energy, and hence a change in speed.

If a rider is in deficit on the energy balance, then that deficit must come from somewhere and that means their kinetic energy (and hence speed) must reduce by the same amount accordingly. Put another way, they are not supplying sufficient power to maintain their original speed and must therefore slow down.

Conversely, if the rider is providing power in excess of that required to maintain their original speed then they will accelerate, and at a rate so that the increase in kinetic energy matches that energy surplus.

In the model these calculations are made for each brief moment in time, the net energy surplus or deficit for that initial speed is determined and converted to the exact change in speed required to maintain the energy balance.

Let’s examine what happens to a rider’s speed in a sample scenario.
250W, flat road @ 90rpm

Riding along at 90rpm on a dead flat road with no wind and with an average power of 250W at around 36km/h with:

• Bike + rider mass: 80kg (including wheel rim mass of 1kg)
• Coefficient of rolling resistance (Crr) of 0.005 (fairly typical for good road tyres on asphalt/chip seal surface)
• Coefficient of drag area (CdA) of 0.35m^2 (e.g. road bike on the hoods)
• Air density of 1.20kg/m^3 (sea level, 1020hPa, temp 21C, relative humidity 77%)
• Drivetrain efficiency:  100% (just to keep it simple, although ~97% is typical)

Using the forward integration model with the simulated sinusoidal-like power output during a pedal stroke, we can now plot the impact on bike speed during a pedal stroke. Click to view a larger image.

In the chart above we have the power output (yellow line) varying during the pedal stroke, which takes a total of 0.667 seconds to complete each revolution. The plot show 1 full second of pedalling, and so the graph actually shows one and half revolutions of the crank.

The initial velocity is set to a little under 36km/h. The speed resulting from that pulse like power input is also plotted by the blue line with the left hand axis being speed. Note the speed scale ranges from 24km/h to 40km/h and so is already zoomed in a little to amplify the variation. At that slightly zoomed scale we can see that the speed line wavers up and down just a little during the pedal stroke.

Since it’s a little hard to see how much variation in speed happens, let’s zoom in much more by adjusting the speed scale to amplify that speed variation curve.

Note the scale of the speed axis on the left side – each horizontal grid-line represents 0.05km/h, and the variation in speed is easier to see.

When power is higher than that required to maintain a constant speed, then speed increases. Eventually though the power drops below the level required to keep accelerating, and speed levels off then begins to decline as power has dropped such that there is now an energy deficit compared with that required to maintain that speed, and so kinetic energy (speed) must fall. As a result, the speed plot follows a similar sinusoidal-like curve, but out of phase with the power plot.

Here’s a table summarising the average, minimum, maximum and amount of variation in power and speed.

The normalized speed variation is less than +/-0.2%.

As I said, not a lot of variation in bike speeds during a pedal stroke, even though the power is ranging from a low of 20W up to a maximum of 480W twice each pedal stroke.

That's bike speed - what about crank velocity?

Now the next logical step is to assume that this “all-but” constant bike speed is also matched by constancy of chain speed and hence crank speed. Since the chain is moving around the rear wheel's circular cog and the upper drive section of the chain has positive tension then that is what you would expect.

So provided the front chainring is circular, this will also result in a nearly uniform crank rotational velocity.

Hence in this steady state cycling scenario we need not be concerned with any inaccuracies in power measurement due to the assumption used by power meters that crank velocity is constant during a pedal stroke.

For crank velocity not to closely match the bike’s velocity it would require the part of the chain under constant tension to stretch and contract significantly during each half pedal stroke, or for the chainring to not be circular. Chains just don’t do that but chainrings may be all sorts of curved shapes.

Sensitivities and assumptions

As I said earlier, this input power model is only a first order approximation; people apply power slightly differently, and not necessarily symmetrically or consistently. Some of the other assumptions may not necessarily hold, e.g. aerodynamic and rolling resistance coefficients, and drivetrain efficiency remaining constant during a pedal stroke when they may in fact vary a little during pedal stroke. The rider’s legs also slightly vary their kinetic and potential energy as they move through a pedal stroke,

These are second and third order effects that would only make minor changes to the shape of the modelled speed curve, and we can see that the speed variation is already so small such that second and third order modifications are not going to change the outcome to any significant degree.

What about when climbing a steep hill?

When climbing, average speed for same power will be significantly less than when on a flat road. On an 8% gradient our 250W rider will be travelling at closer to 13km/h instead of 36km/h. That’s means a much reduced kinetic energy – which is dominated by translational KE of the rider.

KE = 0.5*mass*velocity^2,  plus a little bit from wheel rotational inertia (which is very small).

The translational KE of our 80kg bike and rider at 36km/h is 4,000 joules, and at 13km/h it’s 522 joules, or only 13% of the KE at 36km/h, even though velocity is 36% of flat road speed.

The next big difference is the resistance forces are now dominated by overcoming gravity rather than overcoming air resistance. This also has an impact on the size of bike speed variations during a pedal stroke, the result being we should expect changes in speed to be greater.

Finally, many riders have a tendency to pedal at a lower cadence when climbing steep hills. Not everyone does of course, but sometimes the available gearing means a lower cadence is inevitable. So for the sake of this scenario, let’s assume the rider’s cadence has dropped to 60rpm (that's about what cadence would be with a 39x23 gear at 13km/h).

This is the power and speed plot:

Since the cadence is 60rpm, it take a full second for each pedal revolution. We can see even with the low zoom level on the speed axis that the bike’s speed line does indeed vary more than when on the flat road.

Here’s what the speed variation looks like using the same zoomed-in view setting as before:

So when climbing there is a much greater variation in bike speed during a pedal revolution than when riding on a flat road at higher speed, but it’s still less than +/-2%.

Is a +/-2% crank speed variance during a pedal stroke something to be concerned with for power meter accuracy? 

As a rough rule of thumb, the error this would introduce to the calculation of power would be approximately 40% of the crank speed variation, or less than 1%. Whether that 1% error matters to you I can't really say, but it's a couple of watts and for most people it's not a significant factor for the purposes they might be using power meter data from climbs for.

If you are doing some aerodynamic field testing though, then such an error would be of concern. Fortunately we don't do such testing on steep climbs all that much, but rather mostly on flatter terrain where any error due to crank rotational velocity variation as we have seen is tiny.

That’s a 3.5W/kg rider. What about a more powerful 5.5W/kg rider climbing that 8% grade?

More powerful riders climb faster, and likely pedal at a higher cadence as well, so let’s assume our 400W rider climbs the 8% grade and pedals at 75rpm (that's roughly the cadence if pedalling a 39x19 gear at 19 km/h). This is the resulting power and speed plots:

So even though the rider is more powerful and has much greater variation in power output, the increased cadence and higher speed means the normalized crank rotational velocity variation is only +/-1%, and power meter error is likely to be less than 0.4%.

Summary

While pedalling in a steady state manner out on the road with circular chainrings, crank speed does not vary all that much. It varies more when climbing than when riding along flatter terrain, but the amount of variation is still small such that the basic assumption of non-varying crank velocity used by power meters to calculate power is sufficiently valid and within their generally stated margins of error.

Crank speed variation is larger when riding on low inertia trainers, such that the level of potential error in reported power may begin to approach the limit of the devices' stated error margins.

Read More......

W/m^2, Altitude and the Hour Record. Part II

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.

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