Tuesday, February 05, 2013

The sum of the parts

A perennial favourite argument on cycling forums is the cost-benefit of choosing a wheelset with superior aerodynamics vs a wheelset that is lighter (or an aero vs a lighter frame).

It is of course a false dichotomy that one must chose only one or the other. But that does not stop people having fun arguing the merits of each, or of holding onto beliefs/myths/folklore handed down through the generations. Of course there are a multitude of things that go into what is a suitable choice of wheels, and I'm not going to delve into those, suffice to say they involve a range of factors aside from aerodynamics and mass, including, inter alia (and not in any particular order):

  • strength
  • durability
  • ability to stay round and true
  • lateral stiffness
  • cost
  • repair-ability and service cost
  • suitability for the purpose/race/riding situation
  • braking demands
  • handling characteristics
  • available tyre choices
  • bearing and freehub quality etc
  • rules of competition
  • suitability for the bike (e.g. will it fit?)
  • sex appeal / bling factor
  • and so on.....
Then one needs to weigh up those factors and apply their own personal judgement as to which factors matter most. That will of course be different for everyone. It's no wonder wheel manufacturers have a field day with all the various possible points of difference available when marketing their wares.
But let's get back to the issue of wheel mass and aerodynamics, and what actually matters if for instance we could assume that all other factors between two wheel sets were identical.

Just before diving into that - to slightly complicate matters, one might assume the rotational inertia of a wheel plays a big part in its performance during accelerations (over and above the simple difference in wheel mass itself). Well of course one should expect some difference between wheels with different moments of inertia, but is it really a factor of significance when it comes to acceleration performance?
Now this question has already been examined by others, including a good item on wheel performance by Kraig Willett at Bike Tech Review. In that item, Kraig runs through the physics and demonstrates how (in)significant a difference in wheel rotational inertia during accelerations is, relative to the other primary resistance forces encountered on a bike. In another, more simplified look, Tom Anhalt also examined this and illustrates the same finding in this article on Slowtwitch.
So one can reasonably ignore the difference in moments of inertia when considering overall acceleration performance. But for those who still care, the equations of motion for a cyclist have been developed, thoroughly tested and do include the moment of inertia. I'll get back this this soon.

So, back to weight v aero - the classic prize fight.

First let's consider the relative energy demands of the various resistance forces encountered when cycling, primarily:
  1. air resistance (bike and rider's aerodynamics, speed and wind)
  2. gravity (weight of bike and rider, and gradient)
  3. rolling resistance (tyres and road surface)
  4. drive-train friction losses
  5. changes in kinetic energy (accelerations)
We can examine the difference in relative energy demand of the various resistance forces a rider encounters when riding at steady state speed on roads of various gradients. An example is shown in the chart below:

In this example, we can see the relative importance of each resistance force, as gradient changes from flat terrain (0% slope) to very steep (10% slope). As the road gets steeper, the influence of gravity takes over, and as the road flattens, then air resistance is the dominant force.
Our speed when climbing steeper gradients is directly and almost linearly proportional to our power to mass ratio. Hence why weight is a primary consideration when the road tilts upwards. Lose 2% mass for same power, as you'll go nearly 2% faster. Pretty simple.
However when the terrain is flatter, then it's not so simple as the relationship between speed and power is not pseudo-linear, but rather a cubic relationship with relative air speed, meaning that to sustain a speed that's 2% faster (1.02 times), you'll need nearly 8% 1.02^3 or approximately 6% more power*. Ouch. Talk about diminishing returns. That's why aerodynamics matters so much.

* when you really account for all the forces correctly, then the increase in power demand for an increase in sustained speed from say 40.0 to 40.8km/h (a 2% speed increase) is more like 5.5%, and you can use a exponent of 2.7 rather than 3 as a slightly better ROT.

But what about accelerations?

Well the power required to accelerate is directly proportional to the mass and the rate of acceleration. Of course there will also be a power demand to overcome the varying air and rolling resistances at those varying speeds, as well as deal with gravity for any hill we might be climbing at the time.
So it all starts to get a little more complicated. Bear with me...

Back in the 1990s, a group of bright sparks did a lot of testing to develop and validate a mathematical model for the physics of road cycling, and have that published in the peer reviewed scientific world. The model was developed after extensive testing at highly variable wind speeds and yaw angles, and has been tested against real world data collected using SRM power meters. It's since been adapted and validated for velodrome track scenarios including standing start accelerations by world class track sprinters. I don't have a website link for the paper, but here's the reference details. OK, I found the website link to the paper:



Some of you might recognise a few of the author's names. In summary, the equations look like this (as per a slide from one of Dr Coggan's powerpoint presentations):

At first glance it looks a bit OTT, but really it's not that bad once you break it down to its constituent parts.

What this fancy pants maths enables us to do is something called forward integration - which is a way of being able to predict the second by second speed of a rider if we know their second by second power, and a handful of key variables like their aerodynamic drag coefficient, weight, tyre rolling resistance, gradient, wind and so on.
Now there are some websites that have been doing this stuff for years, and the best example I can think of is Tom Compton's analyticcycling.com. Check it out, Tom does some cool modelling.
For a bit of fun though, I thought I'd examine two acceleration scenarios using the forward integration technique to examine the performance trade off between a wheel set that's more aerodynamic versus one that's a bit lighter.
Here are the two scenarios.
Scenario 1:
A rider accelerates from a standing start with an average power of 1000 watts for 10 seconds.
Scenario 2:
A rider accelerates from 30km/h with an average power of 1000 watts for 10 seconds.

I'm going to use the following as assumptions on the differences in key variables:
Bike set up A: CdA of 0.320m^2 and mass of 80.0kg (lighter but less aero)
Bike set up B: CdA of 0.297m^2 and mass of 80.5kg (heavier but better aero)

and the following assumptions apply to both bike set ups:
Air density: 1.2kg/m^3
Crr: 0.005
Drivetrain efficiency: 100%
No wind

I chose that difference in CdA as it is representative of a real world difference I have measured between two rear wheels (one a low profile light-ish 32 spoke wheel, the other a wheel designed solely for aerodynamic performance), although for the purpose of this exercise, I have exaggerated the mass difference.
By using the equations of motion, and the technique of forward integration, in this case using a time interval of 0.1 seconds, we can show what happens when we accelerate from a standing start. Here is the speed plot for those 10 seconds for each bike set up:

Well, the lines pretty much overlap, but as you get closer to the end of the acceleration  we can see that the heavier, but more aero set up results in a higher top speed after 10 seconds. But does that mean they are ahead? If they were initially slower in the early phases of the acceleration, will they catch up? Well to examine that, we simply inspect the difference in cumulative distance travelled at each time point:

So, now we can see that initially after starting together, the rider with the heavier but more aero wheel falls behind slightly in the opening seconds and the distance grows initially until they lose a maximum of 4.6cm on their "rival" after 4.3 seconds. But after that point, the rider on the heavier and more aero wheel begins to catch up, eventually overtake his rival after 7 seconds, and wins the 10-second sprint by 17cm, or about 1/4 of a wheel. For even just a half lap track sprint, that's way more than enough to justify the aero option over the weight penalty. But if your speciality events lasts less than 6 seconds from a standing start, then go for the lighter rim.

OK, but what about accelerating from a rolling start?
Well let's examine the same scenario, with the only change being that we start at 30km/h, then apply an average of 1000W for 10 seconds. Speed difference plot:
Again we can see that the speed lines are closely matched, except now the top speed reached after 10 seconds is higher and the top speed difference of 0.5km/h between each set up is larger than the top speed difference in the standing start scenario. And the gap in distance? 
Well this time the lighter/less aero wheel loses out straight away and never gains an advantage. The guy with the heavier but more aero wheel wins the 10-second sprint by 60cm - nearly a full wheel width.

OK, so if flattish terrain is your thing, and regular accelerations are part of the game, then perhaps a re-think about the relative merits of aerodynamics and weight when considering which wheels to use. And keep in mind that for the purpose of this exercise I over exaggerated the typical mass difference, while using a fairly typical improvement in aerodynamics attainable from using a deep section aero wheel set over a lighter low profile wheel.

For my next trick, I will examine the shape of a typical power curve during such accelerations, and apply that variable power supply to the models, since nobody really accelerates with a flat power curve. Look out for Part II.

And as they say in the trade, YMMV.

4 comments:

KenGS said...

Slight correction I think in the discussion on power speed relationship n the flat. In a cunic relationship to go 2x faster you need 8x the power but to go 2% faster you need 1.02 cubed or 6% more power - not 8%. Or did I miss something?

Alex Simmons said...

Hi Ken, nope, you're right. I'll amend that bit, thanks.

In real life when you account for all the forces on flat windless ground, it's more like an increase in power of 5.5% to sustain a 2% increase in speed from say 40.0 to 40.8 km/h.

The cube is just a rule of thumb.

Steve Mansfield said...

Hi. So I'm watching a MythBusters and they explain how a dimpled golf ball goes further than smooth due to the dimples reducing net drag

http://math.ucr.edu/home/baez/physics/General/golf.html

So... should TT / Tri frames be dimpled?

Alex Simmons said...

Unlikely that dimples on a frame will make sense (but you don't rule it out unless tested).

Golf balls are (i) a different shape and size, (ii) are spinning, and (iii) travelling at a far greater velocity, and so the Reynolds numbers are significantly different to that representing air flow around a bike frame.

Bike frames are made with aerofoil shapes - which are exceptionally slippery - far more aerodynamic than a cylinder or a sphere ever will be.

There is some merit in the idea of strategically placed trips on certain components or parts of the body (e.g. upper arms, lower legs), so that the boundary layer remains attached for longer and reduces the for-aft pressure differential but such things on clothing are not permitted under UCI rules.