How is power on PM5 calculated?

[EarthRower]

Power is work/time. PM5 outputs a number for power after each stroke. Is that for work/(time of drive) or work/(time of drive + time of recovery)?

[Carl Watts]

Area under the graph you can see on the drive for the stroke and recovery so its an average for drive+recovery time.

The actual power peak during the drive is massive compared to the average, the lower the rating the higher it gets.

Just as a simplistic example to make it easy, you can see that say 20spm is 1 sec drive and 2 sec recovery and at 2:00 pace thats "200W" average so the drive is 600W average to give you the "200W" average.

Peak power is significantly higher than 600W because the body cannot generate a nice square wave, its more sinusoidal in nature.

The Erg is not an efficient motion for the body, 200W on a bike is easier than 200W on the Erg. The peak power loading due to nothing going in on the recovery and the fact you have to move your whole body mass up and down the slide as well is a significant waste of power.

Having switched to the bike and doing the occasional row now, 180W on the Erg is like 250W on the Bike in terms of perceived effort and heartrate. My weight is a killer on the rower but I'm not penalised on the bike in terms of raw power generated.

That's a simplistic guide to how it works anyway from a user point of view.

[Citroen]

http://eodg.atm.ox.ac.uk/user/dudhia/ro ... meter.html
has the complete picture of how the maths works in the PM2/PM3/PM4/PM5.

[JaapvanE]

http://eodg.atm.ox.ac.uk/user/dudhia/ro ... meter.html
has the complete picture of how the maths works in the PM2/PM3/PM4/PM5.

This document leaves a lot to be desired when it really comes to how stuff can and actually is calculated, especially power (as I experienced first hand), as most is just a theoretical excercise but omits testing the algorithms completely and actually contains some ambiguity on this specific subject. I've written down how OpenRowingMonitor calculates it at https://github.com/JaapvanEkris/openrow ... monitor.md , which is accurate within 0.05% of the PM5 results.

To make a story short, it is "work/(time of drive + time of recovery)".

[Nomath]

Power is work/time. PM5 outputs a number for power after each stroke. Is that for work/(time of drive) or work/(time of drive + time of recovery)?

It is the latter : power = (work in drive)/(time of drive + recovery).

[EarthRower]

Very interesting information. Thank you all.

So lower rating is more efficient, because less work is wasted on accelerating the body up and down the rail. On the other hand, lower rating demands much higher power during the drive phase to generate the same average power throughout the drive. For a short and light rower like me, it seems that I will benefit from higher rating (at relatively low DF) than the larger rowers, because the absolute power during the drive that I can generate is smaller and my shorter drive length minimize the penalty from moving my body mass up and down. I guess this holds true up to a sweet spot rating. I wonder if there is a formula to derive the optimal rating based on drive length and ability to generate power.

[Carl Watts]

Correct, lower ratings are harder on the body and hence heartrate is higher for a given pace. Below 20spm it gets pretty ugly fast because the percentage change in strokes begins to change fast.

Although the losses are lower in terms on moving the body u and down the slide, the much higher peak power requirements off set this make it harder.

Therefore there is a sweet spot depending on a number of factors for each individual to find.

Don't think you will find a formula, too many variable in the individual, its not just height and weight. Some people are just better endurance athletes others are better at power for shorter durations. Heart size, lung capacity muscle types all come into play.

Some things you cannot change but obviously the "Ideal" comes to the top in any elite sport. For rowing its tall like 6ft 5 and like 95kg and all the internals optimised.

[jamesg]

Is that for work/(time of drive) or work/(time of drive + time of recovery)?

It's Work done by the fan on the air it pumps, over the entire stroke.

The difference in fan-flywheel speeds at release and next catch gives the Work (say Z) done on air by the flywheel as it slows.

The average Power loss during recovery is then = Z/Rest time. This power loss holds for the entire stroke, at constant pace, since it depends only on fan characteristics.

So to calculate average power the PM needs to know:

Flywheel inertia
Flywheel speed at Catch
Flywheel speed at Release
Rest time.

Peak power delivered is much higher, since it only happens during pull; so if rest time = 3 x pull time, average pull Watts = 4 x stroke average. If the pull curve is a half sinewave, the peak would be almost 6x average power shown by PM.

So in training we keep the peak power high, otherwise we won't be training at all: rowing does need a certain amount of strength, using which produces endurance. If you can pull say 500N at peak speed 2m/s, that's 1kW.

[JaapvanE]

The difference in fan-flywheel speeds at release and next catch gives the Work (say Z) done on air by the flywheel as it slows.

This is not how power is calculated.

There is the precise calculation, and there is the approximation.

The precise calculation to calculate the work done (see the Physics of Ergometers, formula 8.2):
∆E= I ( ∆ω / ∆t ) ∆θ + k ω^2 ∆θ
where I is the flywheel inertia, k the drag, ω is the angular velocity and θ is angular distance.

and then formula 8.3 tells us:
P = E / t

Please note that Dudhia does not explain what this t actually should be, and how it relates to the ∆t in formula 8.2.

This is in fact the approach Dave Venrooy has taken with his monitor, and that can be found in early prototypes of OpenRowingMonitor as well as we initially used his physics engine.

Huge disadvantage of this approach is the presence of many instantaneous ω's and small ∆t's, making this calculation mathematically volatile. What I mean with "volatile" is that small measurement errors tend to propogate and even get enlarged throughout the calculation. So using this approach is extremely challenging, ESPECIALLY on a C2 RowErg, where a systematic magnet pattern is present.

However, in formula 9.2 another formula is presented
P = k ω^3 = c u^3

This is NOT a correct formula, but a mere approximation. In fact, P = k ω^3 is what you get close to when you combine formula 8.1 and 8.2, and assume that ∆ω = 0 across the stroke. Dave Venrooy indicates that using the average ω across the stroke gets you within 5% error.

An average ω is mathematically much more robust, so it is preferred. On OpenRowingMonitor we switched to this approach because of that property, and the fact that C2 seems to use the same formula.

The University of Ulm indicates that the C2 power calculation is problematic as the power generated in an inconsistent stroke is missed. My perception is that the PM5 does not account for a ∆ω across the stroke, introducing this behaviour. So an alternating stroke when the catch isn't optimal (i.e. a missed catch leading to a weak stroke, followed by a heavy stroke due to the slow flywheel, followed again by a weak catch...) is in fact missed by the PM5 and power is underreported.

So to be effective on a PM5, you need stroke-by-stroke consistency first.

[JaapvanE]

So lower rating is more efficient, because less work is wasted on accelerating the body up and down the rail. On the other hand, lower rating demands much higher power during the drive phase to generate the same average power throughout the drive.

From a pure energetic perspective, keeping the boat/flywheel velocity as stable as possible is the most effective. Accelerating the boat to slow it down again requires velocity peaks, and due to cube law, it isn't that optimal. So keeping a boat speed steady is much more effective.

So aiming for peak power might not be optimal here. A more broader power curve might catch more energy. So instead of aiming for a high peak power (i.e. a narrow but high power curve), you could maximize average power, which could also be achieved by a broad but less high power curve.

[EarthRower]

So lower rating is more efficient, because less work is wasted on accelerating the body up and down the rail. On the other hand, lower rating demands much higher power during the drive phase to generate the same average power throughout the drive.

From a pure energetic perspective, keeping the boat/flywheel velocity as stable as possible is the most effective. Accelerating the boat to slow it down again requires velocity peaks, and due to cube law, it isn't that optimal. So keeping a boat speed steady is much more effective.

So aiming for peak power might not be optimal here. A more broader power curve might catch more energy. So instead of aiming for a high peak power (i.e. a narrow but high power curve), you could maximize average power, which could also be achieved by a broad but less high power curve.

Does C2 mimic the physics of the real boat. After all, C2 converts certain numbers into distance. Is that number the total revolutions of the flywheel, or the total work, or combination of some other parameters?

[JaapvanE]

Does C2 mimic the physics of the real boat. After all, C2 converts certain numbers into distance. Is that number the total revolutions of the flywheel, or the total work, or combination of some other parameters?

A C2 really does mimic a boat in many ways, both in movement (i.e. speeding up at the drive, slowing down in the recovery) and drag behaviour (i.e. the faster you go, the more drag you get working against you). In fact, the balance between flywheel Inertia (i.e. resistance to speed change) and drag (i.e. speed dependent resistance) are very good approximations of real boats. As the C2 owners are former Olympic and world cup rowers, they really knew how the C2 should feel to mimic a boat. To be honest, that is the biggest mechanical difference with many competitors: most competitors don't seem to have any clue how a rower should feel to mimic a boat, so the catch feels weirdly heavy/light, or the stroke itself feels weirdly heavy/light.

Looking at the distance/speed calculation, most competitors (including the much higher priced NordicTracks) effectively count revolutions. Some (especicially C2, RowPerfect and if I recall correctly SmartRow) actually do the math pretty decently. Basically, they determine angular position, angular velocity and angular acceleration, as well as the drag factor dynamically, and use this to calculate the power. They use that power to calculate the Linear Velocity.

As formula 9.2 says P = k * ω^3 = c * u^3, where u is the linear velocity. So one can deduct that u = (k/c)^1/3 * ω
In this formula, k is the drag for the stroke (which varies a little bit from stroke to stroke) and c is a constant (2.8 voor Concept2). So there is a linear relation between flywheel speed and boat speed, and a cubic relation between velocity and power.

[jamesg]

This is not how power is calculated.

Right, that's what it is, and how it could be calculaed with the right data.

[Nomath]

... I wonder if there is a formula to derive the optimal rating based on drive length and ability to generate power.

I don't know such a formula. However, I expect that if a formula for optimal stroke rate exists, the key parameter is not drive length but body mass.
As CarlWatts mentioned above, all or some of the work in moving the body back and forth on the slide is not accounted in the power displayed by the performance monitor. The higher the stroke rate, the more an athlete is punished by unaccounted work. The total physical effort is the sum of the power displayed by the PM and the unaccounted work.
So I hypothesize a graph as sketched below in which the optimum stroke rate decreases roughly linear with body weight.


To put numbers in this qualitative graph, the following points are relevant :

• Power in the drive is directly linked to the duration of the drive : higher power → shorter drive. So higher power at constant stroke rate implies a longer recovery.

• There are sound arguments that the work of moving the body in the drive is not lost. The kinetic energy of the body developed after the catch is used in the second part of the drive to pull the handle. So this work is accounted in the power displayed by the performance monitor.

• Most of the work for moving the body in the recovery is probably lost. A study suggest that there is an 'elastic bounce' when the moving body slows down to the catch position, so a fraction of the kinetic energy developed in the recovery might transfer to the drive. However, it is hard to find quantitive data for this elastic transfer.

• The mass accelerated in the recovery is not the total body mass. The lower legs hardly move because they are attached to the foot stretchers.

• The distance over which the body is accelerated is much shorter than the drive length. Roughly, it equals the travel distance of the seat.

• It seems fair to assume that the travel distance of the seat increases with body height. For the same recovery time and body weight a taller body implies a higher average speed, hence more unaccounted work (as suggested by the two parallel blue lines in the figure).

It probably requires a computer simulation with real anthropometric data to quantify these effects.

[jamesg]

Does C2 mimic the physics of the real boat. After all, C2 converts certain numbers into distance. Is that number the total revolutions of the flywheel, or the total work, or combination of some other parameters?

Erger Power is the average energy loss rate of the fan when not driven, as observed during the erger's rest time. Power is then converted to speed in m/s using W = 2.8 V³, which gives 2k times mostly between six (480W) and ten minutes (100W).

if there is a formula to derive the optimal rating

No formula needed, just pull a good stroke, long and hard but sustainable for the duration. For training, stay between 18 and 23, which will let you pull that good stroke for a time sufficient to get fit. Then use that stroke or better at any higher rate for tapering and racing, ie 27 to 35.

Nominally, our stroke might be of length 60% of height (Ergdata shows me 1.15 at height 1.85) and force 50-60% of nominal weight, or whatever you can hold for say 500 strokes at training rates. These conditions are reached if we use the legs with standard rowing style. This has to be learnt in order to be effective and also avoid injury from repeated action.