Pencil and Paper Power Estimates

I've never owned a power meter. But, I've always been curious how much power I produce on any given ride.

A great resource that motivated some of the analysis here is Bicycling Science. That book is just full of really interesting analyses, simplified models and equations related to cycling. If you're a cyclist that just happens also to be an engineering or math nut, you'll love it.

With some simple measurements, it is possible to use a basic potential energy analysis to compute a lower bound on the amount of power draw on the body for a given climb.

Its easiest to do this with a moderately long climb of nearly constant gradient. For riders in Livermore area, a good climb for this purpose is Del Valle. It is about a 1.6 mile climb of roughly constant 7.3% gradient. Some Garmin data for one of my rides up it is pictured below.

Elevation profile and ride data for Del Valle climb

Rider + Bike, Force through a Distance

The combined weight of my body plus my bike is about 200 lbs or 90.72 kg.

In the picture above, you can see it took me 14:03 to climb from an elevation of 768 feet (234.1 meters) to an elevation of 1385 feet (422.1 meters). That's a change in elevation of 188 meters in 843 seconds.

Now, potential energy is force through a distance. So, I need to know how much force my body + bike exerts in Earth's gravitational field. To get that I multiply the mass, 90.72 kg, times Earth's gravitational constant of 9.8 m/s/s yielding = 889 Newtons.

So, for the this climb, the total potential energy realized is 188 m * 889 N = 167132 Joules. Since this potential energy was realized over a period of time of 843 seconds, the power output was 167132 J / 843 s = 198 Watts.

Now, that is a lower bound on my power output. My actual power output was very likely a bit higher due to a number of losses including overcoming wind, flexing of components in the bike as well as other biological activities of my body such as breathing, heart pumping, sweating, shifting around on the bike, etc. Most important though, it does not include the power involved in simply moving the mass of my legs around on the cranks.

Power due to Moving Legs

To compute an estimate of the amount of power required simply to move the mass of my legs around the cranks, we need to know how much my legs weigh, how long they are, how long my cranks are and how fast I was turning the cranks.

My Garmin data also gives me cadence information and so I know during this climb I averaged about 55 rpms on the cranks.

But, short of cutting my legs off and weighing them, how do I compute how much my legs weigh? In particular, I need to know how much my upper and lower legs weigh and how long each is individually. Turns out De Leva (1996) has tabulated a statistical average for both weight and length of all limbs for a large population of men and women as percentages of there overall weight and height (I wonder if this data is somewhat skewed by the trend towards obesity in the average population).

Next, we need a very simplified model for the mechanical system representing my upper leg, lower leg, foot, and crank arms. To really simplify things, I have decided to actually decompose the leg motion into two parts we can superimpose (e.g. add) to compute a total power. This simplified model is surely not an accurate depiction of a real pedaling motion. But, its close enough for purposes of this power estimate.

Simplified mechanical model for moving legs while pedaling. The motion is decomposed into
two parts; one for the upper leg, rotating about the hip and one for the lower leg going straight
up and down above the pedal.

Power due to lower leg motion

In this simplified model, the lower leg moves straight up and down a distance equal to twice the crank length. On my bike, the cranks are 175mm. So, the total distance the lower leg in this model moves is 350mm. Next, in this potential energy analysis, we are concerned only with the part of the motion that works against gravity. Now, based on my weight of 180 lbs (81.65 kg), using De Leva's tables, my lower leg (shank + foot) weighs about 5.7% of my total weight or 4.65 kg. Again, we really need the force of the weight of my leg, so we need to multiply it by the gravitational constant, 9.8 m/s/s. So, the force of weight of my leg is 4.65 kg * 9.8 m/s/s = 45.57 Newtons.

So, the potential energy gain on the up-stroke of the crank is 0.35 m * 45.57 = 15.95 Joules. Next, at a cadence of 55 rpms, this potential energy gain happens 55 times in 60 seconds yielding a power output of 15.95 * 55 / 60 = 14.62 Watts. But, that is for just one of the two legs turning the cranks. So, the total power from keeping the lower legs in motion is 14.62 * 2 = 29.24 Watts.

Power due to upper leg motion

For the upper leg, one end is moving while the other end rotates around the fixed hip joint. The upper leg moves from almost vertical to almost horizontal, say between 70 degrees and 20 degrees relative to the horizontal. 

Near the bottom of the stroke, the amount of force required to rotate the free end of the upper leg is near zero because the motion is almost perpendicular to the direction of gravity. Near the top of the stroke, the opposite is true. Some simple statics equilibrium analysis should yield the forces involved.

For any given angle, in a statics equilibrium analysis, the sum of the forces is equal to zero and the sum of the moments is equal to zero. The only downward force is due to gravity which we can apply in our model at the center of the leg length. The motive force to move the leg is simply to overcome gravity. For any given angle, theta, the force F I think is 1/2*{leg force}*cos(theta). It is near zero at the bottom of the pedal stroke because the upper leg is just about hanging from the hip joint. It is near 1/2 the leg’s weight near the top of the pedal stroke.

To compute the total work in moving the leg, I believe the right answer is to integrate 1/2*{leg force}*cos(x)dx for x=0…pi/2. The result is 1/2*{leg force}*1/2(pi-2)*{length of leg}.

Using de Leva’s data, the upper leg is 14.16% of total body weight. So, for me at 180 lbs, the upper leg weighs 25.49 lbs (11.56 kg). Again, to get the force of the weight of my leg, we multiple by the gravitational constant yielding 11.56 * 9.8 m/s/s = 113.3 Newtons. Next, my upper leg length is about 2.4 feet or 0.732 meters

So, for one leg, we get a total work of 1/2 * 113.3 * 1/2 *(pi-2) * 0.732 meters = 9.34 joules. Again, for my cadence on this climb, this leg motion happens 55 times in 60 seconds. So, the total power is 2 * 9.34 * 55 / 60 = 17.2 Watts

Total power estimate for the climb

In this very simplified model for leg motion, we simply add the power contributions from both the upper and lower leg motions and arrive at 29.24 + 17.2 = 46.44 Watts.

So, the total power estimate for this climb is 198 + 46.44 = 244.44 Watts.

Although I don't ride with a power meter, I have done a little bit of time on a friend's CompuTrainer configured to impose resistence equivalent to powers in the range of 200-350 watts. On the trainer, 250 Watts felt like the kind of resistance I am accustomed to on a moderate length and gradient climb such as the one analyized here. So, that some emperical evidence to suggest the analysis here may yield a decent estimate of power.

It would be nice to be able to include power due to overcoming wind. In this case, since I was on a climb, my speed really wasn't very high. Going through still air at that speed there just isn't that much resistence due to wind. But, in a later post, I'd like to investigate a similar power estimate for my size (frontal area) to overcome wind drag while on a flat course at a higher speed.

A final disclaimer

This analysis is not aimed at computing a measurment similar to what you would get with a PowerTap power meter. A PowerTap power meter measures the power actually produced by your work at the rear hub. This analysis is aimed more or less at the power draw on your body. So, I think this means that this analysis is likely to yield a higher number than you might see from your PowerTap. On the other hand, this analysis, because it is using a basic potential energy model, is neglecting to include a number of factors (flexing, compression/expansion and even heating of bike and body components, etc) that add to the power output. That means, that this analysis represents a lower bound on the power draw on your body.