Fan blade Physics and a Peek inside C2's Black Box

[Nomath]

This is not a usual Forum topic, but I cannot think of another place to find readers who might be interested in how the C2 erg works and how the exercise performance is calculated. I hope that there are some readers who have an education in physics or an engineering background because I love to get feedback on my thoughts and measurements. I know that there are at least two frequent visitors to this forum, Carl Watts and JonW (John Wilson?) who know what is inside C2's black box. I hope that they will correct me if I get it terribly wrong.

Just a teasing question to start with for all readers who got this far. Do you know how long the fan keeps spinning after your last stroke? Please make your choice : 5 sec - 10 sec - 20 sec - 60 sec - 90 sec -120 sec. You will find the correct answer below.

The provisional title of this topic was 'Flywheel physics and a peek inside C2's black box'. The rotor of a C2 erg is not a typical flywheel. Flywheels are used to store energy and air is an unwanted drain. Many flywheels operate in a vacuum. The rotor of the C2 erg is designed to lose energy in a controlled way by fanning air. Thus 'fan blade' appears a better term, although I will mostly keep using 'flywheel'.

Three issues motivated this topic :
1. How does C2 measure the flywheel speed ?
2. Do the data confirm the physics of flywheels as explained in the well-known Physics of Ergometers ?
3. How might C2 calculate power from the flywheel speed ?

In this post I will only deal with the first question. The second and third question will be discussed in later posts. This is because each questions requires a long answer. I have found out that I can only attach a limited number of pictures and graphs to each post. Moreover, a few hours after posting, a post gets locked for further editing. So I chose for divide and rule.



The connector from the flywheel sensor to the Performance Monitor is a 2.5 mm stereo jack with 3 contacts. I had no idea which contact is transmitting what signal. There is also confusion about the type of sensor that C2 uses in their model D (mine is from around 2010). Obviously the sensor needs to respond to a moving magnetic field. A search in the forum contents most often comes up with a Hall effect sensor. This is a device that needs an external supply voltage of several volts. So, in case of a Hall sensor just measuring the voltage over the contacts will not give useful signal. Another type of magnetic sensor is a reed switch. This also requires an external voltage supply. Somewhere I read that C2 uses a coil to pickup the magnetic field. This is similar to what a moving-magnet pickup element in a classic turntable does. The output voltage is in the millivolt range and needs an amplifier.

So it was a big surprise that when I connected the bottom contact and the tip contact of the stereo jack with my DATAQ DI-145 data logger, which has an input impedance of 1 MOhm and can handle signals between -10V and +10V, that I got the result below. The data logger takes 240 samples per second. I started from standstil flywheel. After about 10 sec and did some 10 strokes. The last stroke was roughly at 40 sec from the start. I finished the recording when the fan got to a standstill, after about 170 sec.



Clearly the sensor signal was overloading my measuring device. The solution is a voltage divider. I connected the contacts on the jack by two 10 KOhm resistors in series and measured the voltage over 1 resistor. This halves the signal. The graph below shows that the signal was now inside the range of my data logger.



There are several remarkable features in the signal that merit a better view, because they tell us something about the sensor. First about what happens after the start from standstill. The graph below shows the first 15 seconds. The signal starts with some random noise, probably standstill, then there are some odd oscillations. After a few seconds a very dense set of pulses is generated that probably come from the passing magnets on the flywheel.



For this post I want to skip what happens during the strokes. This will be the topic of my 3th post. In this post I want to concentrate on the signal when there is no drive, but the flywheel is freely spinning and pumping air. Below is the signal during only 1 second in this rundown phase. We count about 35 peaks. Note that when the handle drives the flywheel, there is a relation between handle speed and flywheel speed. The chain has 1/4 inch elements and runs over a 14-teeth sprocket wheel. So one flywheel revolution corresponds to a chain displacement of 14*2.54/4 = 8.89 cm. A chain speed of 1 m/sec thus corresponds to 11.25 rotations per second. The flywheel has 3 magnets. So this signal can well be interpretated as a flywheel speed of 35/3 rps, i.e. roughly 11 rps. Note that the distance between the peaks is important ; their magnitude is irrelevant.



The final part of the recording is when the flywheel stops. This is detailed in the next graph. Note that the voltage scale is now very reduced and comes near to the resolution of my measuring device, about 20 mV.



As explained above, the information about the flywheel speed is contained in the spacing of the peaks. I will not go into how the exact position of each peak is determined, because this gets very technical. The plot below is a result of the data reduction process : flywheel speed as a function of time.
These data allow us to test various physical models and ultimately whether the power calculations make sense.



At a first glance, the flywheel speed (rps) follows an 'exponential decay'. This is what many physical systems do. But a quick test shows that the behaviour is not exactly exponential. For an exponential decay, the logarithm of the rps versus time should be a straight line. This is not the case for the full range.



The decay of speed of the free-running flywheel is used to calculate the drag factor. This is done in real time, which means during a few seconds. In the next graph I have plotted the data in a time interval shortly after the last drive. There is more scatter in the individual data points. I also fitted a 4th degree polynomial to the data (least squares). The fit (blue curve) is quite good. A polynomial is a very convenient expression to calculate the change in speed, as is required for testing the physics in the follow-up post (I have exhausted my allowance of graphs in one post).



What I have learned so far :
- the C2 sensor probably includes a power generator that amplifies the voltage of the (Hall?, coil?) element ; it takes a few seconds to start up.
- at high rotation speeds (corresponding to 1.5-2.5 m/sec handle speed) the sensor pulses tend to overlap, which makes exact location more difficult ;
- the accuracy of the data is sufficient to test the physics.

[faach1]

Wow Nomath, thank you for posting this. It was very informative, I look forward to your subsequent posts!

[Carl Watts]

The flywheel takes close to 2 minutes to stop if your last pull was at 2:00 pace and the Erg is mechanically like new and on a 135 drag factor.

As the rpms' drop the air resistance begins to drop (probably an exponential decay) so it just keeps going for ages at a low rpm. This is coupled with the rotational inertia or energy stored in the steel flywheel to keep it spinning.

Personally I wouldn't spend a lot of time delving into the physics, it kind of reminds me way back in electronics delving into the transistor at an atomic level and what the electrons were doing to make it work in silicon with N doping and P doping. Interesting but really of no practical value for the rest of my life and is probably only useful to know for a handful of people on the planet. What you really need to know is a transistor from an external point of view in the real world and how to use it in different types of circuits.

Its the same for the Concept 2 monitor really, you need to know the functionality and how to setup the drag factor in particular. You don't want to speculate to much on how Concept 2 made it work, just that it works. Even the earliest PM1 monitor could do the math in 1986, its basically like just an old fashioned "calculator" using an 8bit NEC micro inside. The newer monitors have got smarter and faster and the calculation is more precise after shifting from 3 magnets to a 12 pole magnet.

I will add that I spent a bit of time trying to "Crack" the monitors and managed some pretty amazing verified results. The reason why got kind of lost on people but its sufficient to say that the conclusion is that a PM5 running ErgData to get a verified result that is then uploaded to your C2 Logbook and is then made a "Public" visible result for all to see that includes all the graphs, including preferably a heartrate on a near second by second is all but impossible to fake except for a handful of people on the planet with the knowhow and they are probably not rowers anyway.

[Nomath]

Its the same for the Concept 2 monitor really, you need to know the functionality and how to setup the drag factor in particular. You don't want to speculate to much on how Concept 2 made it work, just that it works. Even the earliest PM1 monitor could do the math in 1986, its basically like just an old fashioned "calculator" using an 8bit NEC micro inside. The newer monitors have got smarter and faster and the calculation is more precise after shifting from 3 magnets to a 12 pole magnet.

Thanks Carl, for your reply. It won't discourage me to think further on how the Concept2 performance evaluator works. I think it is high time that we get more transparency about C2's black box. Competitions and championships are based on the results displayed on the PM. It is not only me who is interested in its accuracy. At the University of Ulm a group of scientists is now working on this issue. They are using additional instruments to measure power, such as a force sensor and a speed sensor on de handlebar. This is not the first time that such instruments are used, but Ulm combines it with a motorized drive, which is new and eliminates the main irreproducible element in the test chain : the human rower. I eagerly await their results. Their study will probably not disqualify the C2 data, but it will add a more objective and independent certification and calibration.

As a hobby scientist, my means are much more limited. However, in another topic I have shown that even with limited means you can add new and interesting observations. For example, the speed of the handle during the drive. The graph of some 50 consecutive drives that I measured, see below, gives a much more realistic picture than any speed curve I found on the internet. It shows that after the catch the speed of the handle, which almost equals the speed of the rower's body because arms are fully extended, increases very rapidly until the handle speed matches the speed of the flywheel. At that point, at a distance of about 17 cm from the catch (my movements ; everybody is different), there is a distinct kink in the speed curve. It is as if you hit an invisible wall: before this point you couldn't add power to the flywheel because your speed was too low ; from that point on you have to apply force on the handle unless you get stuck. Compare this graph with the curves shown for the C2 static erg by Valery Kleshnev, a highly respected authority on indoor and outdoor rowing. You have to conclude that his measurements are much more sketchy. No doubt, Kleshnev is a world-class authority and I eargerly await the new edition of his book The Biomechanics of Rowing.



I am intrigued by your comment that C2 shifted from 3 magnets to a 12 pole magnet. The measurements on my model D (year 2010-2013) suggest that there are 3 magnets on the flywheel. What do you mean by a 12-pole magnet ?

One more question: for my next post on the physics of ergometer rowing I like to know the moment of inertia (MoI) of the flywheel. I know this is the most important characteristic to make all C2-ergs behave identical and C2 has very narrow tolerances on MoI. Searching in the forum topics, I found a value of 0.1001 kg m² that was confirmed by c2jonW . In a recent article in a science journal by Stephen Tullis from the Department of Mechanical Engineering at McMaster University in Hamilton, Canada, the MoI was quoted as 0.2 kg m², a factor of 2 higher! The value given by c2jonW seems more plausible to me because the flywheel's radius is about 0.2 m and if all mass was concentrated on its circumference, the weight would be 0.1/(0.2)² = 2.5 kg. Clearly the mass of the fan blades is not concentrated on the circumference, so the estimated flywheel weight would be more like 4 kg, which seems reasonable.
I know how I could measure the MoI myself, but it would require me to disassemble the flywheel, which I am very reluctant to do.

[Carl Watts]

I wouldn't question the accuracy of the monitor, I have run two off the same rower with two pickups on a Model C and they are close enough not to matter. The longer the row the closer the result. There were fractional differences between the PM2 and PM3 but who is to say which is more "Accurate"

Wouldn't waste your time trying to delve into the black box to far, its Concept 2 intellectual property and trust me thats a brick wall. Concept 2 will not even give you a full electrical schematic to help you repair one, not even for the PM2 which is not that hard to reverse engineer into a schematic anyway and that ceased production in 2003. You have to respect that because the IP used in the Concept 2 rower is unique and makes every rowing machine a level playing field. Its pretty rare to have that condition in a sport where equipment is required.

You would be better of focusing on more universal adoption of the verification system which is the last remaining weakness when it comes to "Accuracy" of any results.

The continuous magnetic ring on the model D is a 12 pole magnet. 6 poles in one direction with 6 poles in the opposite direction alternating. The pitch of the poles no doubt lines up with the iron cores in the pickup for optimum effect.

The Model C is just 3 tiny magnets embedded in the flywheel. Its just simple magnetic theory of a magnetic field passing a coil of wire that induces a Voltage into it. You get pulses of about 0.6V peak.

The final mathematical formula that Concept 2 use may be rocket science but its not really rocket science to conceptualize what variables are needed. You can simply drive that flywheel at different speeds with an ac induction motor to obtain the power required for any given rpm at a range of drag factors which is either a calculation in itself or simply measured and put in a lookup table. Probably a lookup table in the early PM1 and this moved to a calculation in later models of monitor as it now had the grunt and speed to do it. The drag factor calculation comes from the rate of flywheel deceleration. The higher the drag factor the faster the flywheel rpms fall during the recovery. The way it does the calculation most definitely changed from the PM2 to the PM3 onwards as I discovered that in testing the limitations of the monitors. The Model D must have 6 or 12 pulses per rpm, I have never really bothered to measure it, regardless it has loads of pulses in terms of changes in frequency to run the calculations on and more precision on the transitions from drive to recovery and at the catch to the drive.

Yeah its all interesting stuff for Nerds but most people just want to get on it and yank the handle with as much power as possible for as long as possible and that works for me.

[Nomath]

The continuous magnetic ring on the model D is a 12 pole magnet. 6 poles in one direction with 6 poles in the opposite direction alternating. The pitch of the poles no doubt lines up with the iron cores in the pickup for optimum effect.

The Model C is just 3 tiny magnets embedded in the flywheel. Its just simple magnetic theory of a magnetic field passing a coil of wire that induces a Voltage into it. You get pulses of about 0.6V peak.

So how many magnets are there on the flywheel of the Model D?
I am almost certain that the number is 3. I counted the number of pulses per second after the last drive. From my own handle speed measurements, I know that at the end of the drive my handle speed is between 1.5 - 2 m/s. There is a fixed relationship between handle speed and the rotation speed of the flywheel, as explained in my first post. At a handle speed of 1.00 m/s the flywheel speed is 11.25 rps. So at the end of my last drive the flywheel speed is between 17-23 rps. I counted the number of pulses in intervals of 1 sec. It decayed from 61 pulses/sec in the first interval after the last drive to 35 pulses/sec after 10 sec free spinning. These numbers only match if there are 3 magnets on the flywheel.

My guess is that the 12 pole magnetic ring on the model D serves a different purpose, probably as a power generator to amplify the signal of the true sensor (Hall element or coil). My measurements on the connector that goes into the PM show that the signal had an amplitude of about 16V. This is without external power supplied to the sensor. The jack was disconnected from the PM and its batteries.

In guessing the content of black boxes, do you know the South-Asian parable of the elephant and the 6 blind men?
There are many versions. I took this from Wikipedia :
A group of blind men heard that a strange animal, called an elephant, had been brought to the town, but none of them were aware of its shape and form. Out of curiosity, they said: "We must inspect and know it by touch, of which we are capable". So, they sought it out, and when they found it they groped about it. The first person, whose hand landed on the trunk, said, "This being is like a thick snake". For another one whose hand reached its ear, it seemed like a kind of fan. As for another person, whose hand was upon its leg, said, the elephant is a pillar like a tree-trunk. The blind man who placed his hand upon its side said the elephant, "is a wall". Another who felt its tail, described it as a rope. The last felt its tusk, stating the elephant is that which is hard, smooth and like a spear.
In one version it came to a fight between the men, in another they agreed to disagree. There is also one version in which they collaborated to piece their different experiences together.

[Nomath]

This is a continuation on the story of a blind man investigating an elephant.....

I am glad that I took a week or so for a better inquiry on how many pulses C2's black box generates for each revolution of the flywheel. In my last post I concluded that the number of magnets on the model-D flywheel is 3, so that the sensor generates 3 pulses per revolution. I have to correct this. I found out that the flywheel generates 6 pulses per revolution. Whether this involves magnets or not, I am now less sure.


I removed the flywheel cage and gave the flywheel a spin by hand. I stopped it after a precise number of revolutions also by hand. What I got is 6 pulses per revolution. Below is the sensor signal for 5 revolutions : it clearly shows 30 peaks.





The sensor signal was measured at the contacts of the 2.5 mm connector to the PM. The connector was not inserted in the PM. I used a 100K resistor to 'shorten' the contacts (the data logger has 1M-ohm input resistance). The resistor had a significant effect on stabilizing and zero-ing the signal.

The result means that I get a more accurate quantification of the properties that determine the speed of the flywheel when it is not externally driven, e.g. during the recovery. This will be in next post "Do the data confirm the physics of flywheels as explained in the well-known Physics of Ergometers ?'

[Carl Watts]

its a 12 pole magnet. 6 facing N-S and 6 facing S-N alternating.

Easily checked if you pull the flywheel and have a small magnet in your hand that you just run round the magnetic "Ring" with. 6 attractions, 6 deflections in a 360 Degree pass.

[Nomath]

Thanks, Carl. I am poorly educated in electronics, so I have little notion what a 12-pole magnet means. However, for my check on the physics it is crucial to know the relationship between number of pulses generated by the sensor and the flywheel speed. Thanks to your comment that the Model-D has not a simple 3-magnets system as for Model-C, I checked the relation by counting pulses. I am reluctant to pull the flywheel.

[Nomath]

Three issues motivated this topic :
1. How does C2 measure the flywheel speed ?
2. Do the data confirm the physics of flywheels as explained in the well-known Physics of Ergometers ?
3. How might C2 calculate power from the flywheel speed ?

In this post I will only deal with the first question. The second and third question will be discussed in later posts.

This is Part 2 dealing with second question. Part 3 will follow soon. I am glad I took some weeks to combine the physics with more experimental data. I believe I have come closer than anybody outside Concept2 in understanding the physics, although some finesses still elude me. I tried to get not very technical, but this is about physics and it requires some mathematics.

Basic Equation
I will not repeat the Physics of Ergometers. The essence is capped in a note titled Behind the Ergometer Display written by Prof. Van Holst in 2008. It is the only public document I know that tries to retro-engineer the calculations done by C2.
The basic equation is :
P(rower) = C1 * ω³ + ω * MoI * dω/dt
The first term (C1 * ω³) is the power needed to maintain the flywheel speed. The constant C1 is related to the drag factor. The angular velocity ω is related to the rotation period Tr of the flywheel by ω = 2π/Tr).
The second term (ω * MoI * dω/dt) is the power used to accelerate the flywheel. dω/dt is the angular acceleration. MoI is the moment of inertia. In order to ensure that all C2 ergometers (models C, D and E) behave identical, C2 takes great care to keep the MoI the same within tight margins. In another topic I found that MoI = 0.1001 kg m². I hope this is correct, because it matters for the quantitative results below.
The main assumption in this equation is that resistances from bearings are negligible. Such resistances would add a linear in ω . We will test this assumption by analysing the coast-down behaviour of the flywheel when no external power is applied.

Coast down behaviour
Coast down is my name for the decrease in angular speed with time when no external power is applied. The above relation then reduces to :
0 = C1 * ω³ + ω * MoI * dω/dt, hence dω/dt = - C1/MoI * ω² .
The test whether the erg complies with above physical model is that a plot of the decrease in angular velocity versus the square of angular velocity is linear and this line should go through the origin.
Now comes a trick of mathematical derivatives : d(1/ω) = -1/ω² * dω/dt. Substituting the above relation then results in d(1/ω)/dt = C1/MoI. In other words: the test described above is identical to a test whether a plot of 1/ω versus time is linear! Note that 1/ω = Tr/2π ; Tr is the rotation period. So the test is whether Tr increases linear with time. The slope of that line is then equal to 2π*C1/MoI.

Measurements
A recap for the casual reader : the Model-D rower has a magnetic sensor close to the flywheel to measure the flywheel speed. The sensor's signal is fed into the PM3/4/5 by a wire with a 2.5 mm stereo plug. If you take this plug out of the PM and connect the bottom contact and the tip contact to an oscilloscope or a data logger, you can observe a complex signal that consists of pulses. Each pulse is for one magnet passing the sensor. I have found that there are probably 6 pulses for each rotation. During a normal rowing session, the flywheel rotates between 10-30 rps. So the signal contains 60-180 pulses per second. The information about power, pace and the force applied by the rower is in the time interval between two pulses. The precise timing of these pulses puts considerable demands on the speed of the data logger and on the software to determine the position of the pulses. I have a data logger that can handle a maximum of 240 samples/second.

The graph below shows the result of this measurement. I did 5 strokes at a pace of about 2:00 (a guess ; PM disconnected). The Drag Factor was set at 80 (before the PM was disconnected). With some imagination the 5 strokes can be distinguished, but the data are very scattered. However, a few seconds after the last stroke the data line up. The period between successive pulses increases regularly.





Keeping in mind that one flywheel revolution causes 6 pulses , the second graph can be converted in a plot of rotation period Tr vs Time. A second degree polynomial gives a very good fit (blue line in above graph). There is quadratic term (slight curvature), but it is fairly small. The slope near the start equals 0.00457. So we estimate that the ratio C1/MoI = 0.00457/2π =0.00073. If indeed MoI = 0.1, the constant C1 (drag coefficient) = 0.000073. Very small, but influential, as we will see later in the computer simulations.

I have measured the flywheel sensor signal for 5 different drag factor settings : 80 - 100 - 125 - 150 - 200. The graph below shows the coast down results.



What could be expected is indeed observed : the rotation period increases faster with a higher drag factor setting. The different slopes reflect the different drag coefficients C1. The graph below shows the correlation between the ratio C1/MoI and the drag factor setting. The correlation is quite good.



Concept2 doesn't give physical units for the displayed drag factor. Van Holst already observed that it is much too high for identifying it with the coefficient C1. Somewhere I have read that the display drag factor equals C1*10^6. This comes closer too my observation.
Assuming MoI=0.1 kg m², the ratio I found is : DF = 1.2 * C1 * 10^6. The units of C1 are [W.s³] or [N.m.s²].

It is a pity that I couldn't measure the flywheel speed during the drive. This is probably caused by the limited sampling rate of my data logger: 240/sec. With 6 pulses per revolution and 10-30 rps, the frequency is too high. For an accurate timing of each pulse I need roughly 4 samples per pulse, hence 240-720 samples/sec. More modern data loggers than my DATAQ DI-145 can do this, e.g. the DI-1100. The DI-1100 has no driver for Windows-XP, so I would have to update data logger and computer.

Computer simulation of a stroke
Fortunately, it is possible to simulate a rowing stroke using the differential equations if good estimates for the physical constants are available. What you do is to 'play' the equations in small discrete time steps and keep the changes. For example, how much the flywheel speed increases by applying a certain force. Note that during the actual drive the handle speed and the flywheel speed (rps) are related : V-handle [m/s] = 11.25 * rps.
In the simulation below I used the following input that I took from my DIY handle speed measurements:
- the initial flywheel speed corresponds to a handle speed of 1.5 m/s, i.e. 16.9 rps ;
- after the catch (time=0), the handle speed increases from zero with a constant acceleration of 8 m/s² until it matches the flywheel speed ;
- moment of inertia of the flywheel = 0.1 kg m² ;
- drag coefficient C1 = 0.000107 (corresponding to a drag factor of 125) ;
- the force curve is parabolic with a maximum of 650N and a width of about 0.7 sec ;
The results are displayed below in a set of graphs



Top-left : the force curve
Top-right : the angular speed of the flywheel resulting from this force curve and the momentary handle speed. Note that it takes about 0.17 sec before the handle speed matches the speed of the flywheel. Only from this point on can power be delivered to the flywheel. It results in a sharp acceleration.
Bottom-left : the external power input . Note that this is the instantaneous power, not the average power in the stroke (displayed on the PM). The instantaneous power reaches a maximum of roughly 1200W.
Bottom-right : the work done during the drive. It amounts to 520 J.
Results not shown in these graphs : drive length, average handle speed, etc.

The next graph shows the rotation period of the flywheel.



Without external power, the rotation period increases linearly with time with a slope determined by the ratio C1/MoI. This can be observed for a short period after the catch and again when the force curve gets to zero. The graph shows that the initial flywheel speed is regained after 2.7 sec. At that moment the next drive could be started and the same data would get recycled continuously. The stroke rate is then 60/2.7 = 22 spm. The work in the drive divided by the duration of the stroke amounts to 520/2.7 = 193 W.

Although this force curve is just an arbitrary model based on a max force from ErgData, I found that this simulation (including drive length, drive speed, etc.) agrees fairly well with what is measured by the PM for my typical exercise. The advantage of a simulation is that some elusive variables can also be calculated, e.g. the mechanical impedance.

Conclusions
1. Coast-down measurements confirm that the main resistance is air drag on the flywheel.
2. The ratio of the air drag coefficient C1 and the moment of inertia MoI can be determined quite accurately from the coast-down data.
3. Computer simulations based on these properties agree surprisingly well with reality.

[Nomath]

I can no longer edit my above post, so here are two corrections:
The link to the note bij Van Holst doesn't work. This link should work.

The mathematical trick for a differential of an inverse function was incomplete. It should read: d(1/ω)/dt = -1/ω² * dω/dt.

[Zuman]

This is awesome work and VERY well done! It's fun to follow along with your adventure (somewhere between a blind man investigating an elephant and Shackleton's navigation from Elephant Island to South Georgia). Thanks!

[Nomath]

Three issues motivated this topic :
1. How does C2 measure the flywheel speed ?
2. Do the data confirm the physics of flywheels as explained in the well-known Physics of Ergometers ?
3. How might C2 calculate power from the flywheel speed ?

This is Part 3, dealing with the last question.

Earlier in this topic, Carl Watts commented that it is a waste of time to delve into C2’s black box because it is intellectual property and locked by a brick wall. It should then come as a surprise that someone emulated the PM2 using an Arduino computer in 2015, as can be seen on Youtube. The output is virtually identical. The Arduino programmer is Steven Aitken from Aberdeen (UK). However, I haven’t seen a force curve computed by the Arduino and finding out how C2 does that is my ultimate challenge. There is nothing against elucidating intellectual property ; you are only not allowed use it commercially.

I start again from the basic equation in the note by Prof. Van Holst :
P = C1*ω³ + ω*I*dω/dt [Eq.1]
ω is the angular velocity of the flywheel ; P is the momentary power input by the rower to the flywheel.

Please note that P differs from the average power during a stroke. To avoid confusion I will use the symbol Pa for the latter. P can reach well over 1000 W during the drive, while Pa is more likely between 150-400 W.
C1 is the drag coefficient; I is the moment of inertia of the flywheel. Because of the mathematical identity d(1/ω)/dt = - 1/ω2 * dω/dt, this expression can be converted in another one that I find more convenient for analysis :
P = C1*ω³ - ω³ * I/2π * d(Tr)/dt = ω³ * (C1 - I/2π * d(Tr)/dt) [Eq.2]
Tr is the rotation period of the flywheel, the time for a full turn. It is the inverse of the angular velocity : Tr = 2π/ω.

Why is this expression more convenient? Well, if there is no power input to the flywheel, the expression becomes very simple : C1 = I/2π * d(Tr)/dt or d(Tr)/dt = 2π*C1/I. Because the moment of inertia of the flywheel is manufactured by C2 within narrow margins, this expression states that the rotation period increases linearly with time ; the proportionality is only determined by the drag coefficient.

Detecting and quantifying a linear trend is one of the easiest things to do with a microprocessor. The Model-D sensor generates 6 pulses per turn. The flywheel easily reaches 15-25 turns/sec at the end of the drive. The recovery lasts roughly 1 sec or more. This is enough data for measuring the drag coefficient C1 accurately for each stroke.

Another driving option is that the power input P is just enough to keep the angular speed and the rotation period constant. The equation then simplifies to : P = C1*ω³ . This is indicated in the graph below by the horizontal arrow.

To explain how C2 might calculate power I took a computer simulation of a stroke. My measuring tools are inadequate to measure the flywheel speed during the drive. This is because the flywheel sensor produces 6 pulses per turn, which is too much for my data logger. So I have no curve from real data ; however in the simulation I used realistic values for the relevant characteristics.




The input characteristics are : moment of inertia I = 0.1 kg m2 ; drag coefficient C1 = 0.000120 kg m2 ; acceleration of the rower’s upper body mass at the catch 8 m/s² . The rotation period at the catch was taken as equivalent to a handle speed of 1.5 m/s : Tr(0) = 1/(1.5*11.25) = 0.0593 s. The factor 11.25 rotations/m is the gearing constant (¼ inch chain driving a 14-teeth sprocket wheel).

The time axis starts at the catch, the start of the drive. Note that I define the catch as the point where the handle is closest to the flywheel. By definition the handle speed is zero at the catch. Also note that at this moment in time, the erg’s PM is unaware what the rower is doing. He might slide forth or back or pause ; it doesn’t affect the flywheel speed. So from the PM’s point of view, the drive starts when the speed of handle matches the current speed of the flywheel. Several posters and also some authors of scientific papers mix up these two points of view, with lots of confusion and sometimes painful errors.

Sticking to my definition of the drive, there comes a moment when the speed of the handle matches the speed of the flywheel. From this point on the rower powers the flywheel, resulting in a sharp kink in the curve. I will note this point as t1 and the associated angular velocity as ω1. As the drive continues, the rotation period drops further. The drop depends on the force curve. Near the end of the drive the power decreases to the point where it can no longer increase the flywheel speed, but just enough to main the current speed. This results in a minimum in the rotation period. This point t2 and associated ω2 is less sharp. The Tr-curve after the minimum soon continues as a straight line with the known slope. Note that t2 is not the end of the drive, but near to it. The end of the drive is more difficult to pinpoint exactly. The work in the drive can be determined by integrating the power input over the drive time. For this we best return to [Eq.1] because the second term can be more simply written as d(½I*ω²)/dt. After integration this term simplifies as a difference in kinetic energy before and after the drive. The time t3 is taken early in the recovery ; where precisely is uncritical.

Work W = ʃ C1*ω³ dt + ½I *(ω3² – ω1²) ; the integral runs from t1 to t3 [Eq.3]

Note that for any two points on the recovery line, the power used for blowing air (i.e. ʃ C1*ω³ dt ) equals the loss in kinetic energy (i,e. ½I *∆ω² ). The further t3 is moved into the recovery, the larger the integral term becomes, but the smaller the second term in [Eq.3]. The two terms are communicating vessels; the effects cancel.

Once the work in the drive is calculated, the average power Pa in the stroke depends on the duration of the recovery. Van Holst already noticed that the PM display is refreshed immediately after the finish. This implies that the PM uses the drag factor and the time of the previous recovery in the calculation of average power. Therefore I have started the simulation with a recovery from a high angular speed, corresponding to 2 m/s handle speed and lasting 2 seconds. The catch is exactly at 2 seconds. See graphs below.





There is one final finesse in calculating the average power Pa : if I would simply take Pa = W/(t3 – 0), I would ignore that the flywheel does not have exactly the same speed at t3 as at t=0. This requires a correction term. However, this correction term can be omitted if I take t0 as the point where the flywheel has the same angular speed as at t3. Because the slopes of the recovery lines are identical, it can be derived that t0 = I/C1*(1/ω3 – 1/ω0). Hence

Pa = W/(t3 – t0) [Eq.4]

In this simulation t0 = 0.091 sec (where the blue horizontal line intersects the red curve near time=0). In this simulation, time=3.000 sec is the exact end of the drive. The correction is minor. I explained that the choice of t3 is uncritical as long as it is in the recovery. If I would take t3 and t0 as the points where the green horizontal line intersects the simulated curve, I would obtain the same time difference because the two lines are parallel. But also the same average power Pa !

So here is how I think Concept2 calculates power :
1. Early in a recovery, the time t0 and angular velocity ω0 are taken as reference values.
2. The drag coefficient C1 is determined during the recovery using the linear relationship between the period Tr = 2π/ω and time.
3. For the PM the drive starts when there is a sharp kink in the speed/period curve : (t1,ω1)
4. During the drive the integral ʃ C1*ω³ dt is steadily updated as a summation in small time steps.
5. Once the period starts to rise again and the slope of Tr vs Time becomes linear, the drive is finished. This time t3 is uncritical. The difference ½I *(ω3² – ω1²) is calculated and the work W in the drive by [Eq.3]
6. Average power in the stroke is calculated by [Eq.4]

A few additional comments :
This calculation is quite intricate. It involves logic about the detection of a significant change in the rotation period. The topic How C2 measures Power has been discussed here before. Two posters on that topic conjectured about how C2 does the calculation. Compared to the above their ideas are too simplistic, I am afraid.

An interesting questions is how you could simplify the calculation and still get close to the accurate result. Suppose you could find a characteristic angular velocity ωc that would reduce the integral ʃ C1*ω³ dt , running from t1 to t3, to the simple term C1*ωc³*(t2 – t1), where t2 is the time where the period curve has a minimum. The second term in [Eq.4], ½I *(ω2² – ω1²) , now includes the angular speed at t2. Since the integral contains the third power of ω, it is obvious that ωc should be much closer to ω2 than to ω1. Intuitively I took ωc = (2* ω2 + ω1)/3. It turned out that this is a very good choice, as the results for several simulations were within 1% of the accurate result. Only very sharp force curves caused larger deviations. I couldn’t find a mathematical justification, but it looks like a very effective simplification.

In the note by Van Holst on page 7 in his equation 5 the second term is 0.5*J*(ω3² – ω1²), where ω3 is the angular velocity at the end of the cycle (=start recovery), and ω1 is the angular velocity at the start of the cycle. Although I hesitate to correct a recognized authority in the field, I think this is a mistake and ω1 should be replaced by ω2,, the angular velocity at the catch.


The force curve
The PM measures power input. But how can it measures the force applied on the handle? Power is force x handle speed, so it seems that other measurements are needed. The answer is that during the drive the handle speed is not a free variable. It is related to the flywheel speed by the gearing constant: one meter of handle displacement corresponds to 11.25 flywheel turns. Therefore [Eq.2] for power input can be converted in an expression for force :

F = P/v-handle = ω² * 11.25 * 2π * (C1 - I/2π * d(Tr)/dt) [Eq.5]

The dynamic element in this equation is d(Tr)/dt, the change in the period of the flywheel with time. It changes from positive during the recovery to strongly negative during the drive, as indicated by a downward arrow in the first graph. The precise measurement of a derivative is tricky. Small errors are enlarged. But with 6 pulses per turn, a high sampling rate and a precise internal clock it can be done.

Other performance data
The PM does only monitor the flywheel speed. It cannot ‘see’ our movements unless the handle speed matches that of the flywheel. So several data displayed by the PM and on the ErgData display are subject to this limitation. This applies to drive time, drive distance and drive speed. The real drive time, taken as the time from catch to finish, is substantially longer than displayed by ErgData ; the difference amounts to about 0.2 sec, depending on how fast we accelerate our body after the catch. The same applies for drive distance, which may typically be 20 cm longer. The real drive speed will be correspondingly slower. In the acceleration directly after the catch, the rower applies a high force on the foot stretchers. For an upper body mass of 60 kg and acceleration of 9.8 m/s² (=1g) the required force is about 600N. In an estimated ‘catch slack’ of 0.15 sec, the rower has already produced 68 J (450W) before the PM took notice! This energy is not lost. Well before the finish of the drive, the main part of the upper body has come to standstill and only the arms produce the final power input. The upper body has come to a standstill because its energy is used to pull the handle. Fortunately the energy for moving the body on the slide is not lost!

[faach1]

Great read Mathieu, is there a reason why ErgData/the PM cannot give a watts reading for splits below 1:00/500m?

[Citroen]

Great read Mathieu, is there a reason why ErgData/the PM cannot give a watts reading for splits below 1:00/500m?

Because they're still using a single byte integer in the PM code and you're going below zero at 1:00/500m (that was done in the past to save RAM in the monitor) so they picked an arbitrary low limit.