## 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:

- A standing start acceleration lasting 10-seconds on flat windless road,
- A rolling start acceleration lasting 10-seconds from an initial speed of 30km/h
- 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!

### Keep it steady son!

OK so the point of all this was to demonstrate the relative unimportance of wheel rim mass and why aerodynamics matters *even when accelerating, *as dynamic scenarios are somewhat harder to calculate than steady state cycling scenarios such as time trialling, or hill climbing where accelerations are very small and changes in rotational kinetic energy are zero, or so tiny as to be completely negligible.

For steady state cycling, well the heavier but more aero wheel still wins in just about every scenario.

For the rider in this set of examples, at 300W the bike/rider with the heavier but more aero wheel still climbs faster on gradients of up to 8%, and of course will descend more quickly as well. At 9% gradient it's line ball and once you go steeper than that, well the lighter rim is quicker.