## 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:

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.