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Measuring varied power per revolution of a bicycle crank/chainring

  1. Oct 8, 2014 #1
    A bicyclist applies varied force and velocity on their pedals as it goes around one revolution. The power distribution over one revolution is like two waves with the higher power being made on the downstrokes. The push through the bottom/top produces the lowest power. The question is does measuring velocity 1 per revolution and averaging the force gives you an accurate measurement or a weighted average velocity matched to force every quarter rotation gives you a better power reading per revolution?

    One problem with the second is the timing of those quarter sampling in a varied force velocity application can give you a range of different results.

    A simplified example:

    1, 0.25sec, 62rpm, 10 torque units (e.g. left leg across top of stroke)
    2, 0.25sec, 58rpm, 50 torque units (left leg downstroke)
    3, 0.25sec, 62rpm, 10 torque units (right leg across top of stroke)
    4, 0.25sec, 58rpm, 50 torque units (right leg downstroke)

    Average velocity = 60rpm

    Magnet/reed switch calculation:
    Average torque = (10 + 50 + 10 + 50) / 4 = 30TU

    Power = 60rpm x 30TU = 1,800 power units

    However, if you calculate the power for each time sample (hence account for the different velocity recorded during each sample), then:

    Power = (62x10 + 58x50 + 62x10 + 58x50) / 4 = 1,760 power units

    But the timing of sampling is arbitrary and it could measure have started measuring .135 sec earlier in .25 sec samplings so you could get.

    Power= (60x30 + 60x30 + 60x30 + 60x30)/4 = 1,800 power units

    I you moved the .25 sec sampling from this point to the first .25 sec sampling you can get power units from 1,760 to 1,800.

    Which is the correct measuring of power per revolution? and why?

  2. jcsd
  3. Oct 9, 2014 #2


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    What do you mean by "averaging the force"? If you have numerous torque samples per revolution and can produce a reasonably accurate average then that sounds like the way to go.

    I think velocity/rpm is unlikely to change much in one revolution. If the rider is applying any pressure on the pedals the rpm will depend on bike speed. Would that vary much from one revolution to the next?
  4. Oct 9, 2014 #3
    If I understand correctly, you are interest in the average power (let's call it "<P>") over some time (let's call it "t") that is a lot longer than one turn.

    The best way to do this is to calculate the energy ("E") spent during the time t, and then divide this value by t.


    The energy is the integral of the instantaneous power over time.

    The integral can be approximated by a sum over the readings taken at many points. If you take readings at regular intervals of 0.25 seconds, then

    The instantaneous power is the instantaneous torque ([itex]\tau(t')[/itex]) multiplied by the instantaneous angular frequency ([itex] \omega(t')[/itex], which you measure in rpm).

    [itex] E = \int\limits_0^t P(t') \mathrm{d}t' = \int\limits_0^t \tau(t') \cdot \omega_i(t') \mathrm{d}t' [/itex]

    [itex] E \approx 0.25 s \cdot \sum_i P_i = 0.25 s \cdot sum_i (\tau_i \cdot \omega_i) [\itex]

    so in short, if you want to calculate the average power over one second from 4 readings taken at intervals of 0.25 seconds, then this is the right formula:

    Given the speed, you probably want to take readings at a much higher frequency than 4 Hz, and average over longer times, like 60 seconds or so.
  5. Oct 9, 2014 #4


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    It strikes me that this is where a bit of sampling theory comes in. If you sample at more than twice the frequency of the crank rotation, you have enough information to describe the power variation with time (i.e. reconstruct the original waveform). Make it four times and you will definitely have enough data. The phase of the rotation relative to the sampling would not matter.
  6. Oct 9, 2014 #5
    Only if you assume that the wave form is a nice sine.
  7. Oct 9, 2014 #6


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    You have a point there. Sampling needs to be a at least twice the highest frequency involved. There could. I suppose, be significant components at four times the crank frequency so sample at eight times and that would see you OK.
    Is there a problem with sampling at this pedestrian (sorry for the pun) rate?
  8. Oct 9, 2014 #7
    Great you found this thread. I am just going to repeat what I wrote in the previous for continuity.

    I think there is a problem in that the positioning and number of strain gauges do have an influence on the resulting power number per revolution. You are weight averaging each portion of a continues application of a varied velocity and force. You can have different results depending on what slice of the pie is included in each measured average.

    This "Power = (62x10 + 58x50 + 62x10 + 58x50) / 4 = 1,760 power units" I think this is only "right" because of the position of the four strain gauges and the slice of pie averaged. Had they been positioned differently you could get any number of averages up to 1,800 power units. So what I get from what you are saying so far is that it is important to position the strain gauges so that you measure the highest and lowest possible torques?

    I think you are right here but I don't know why. I can understand taking readings at a much higher frequency than 4 hz especially if you are interested in a shorter period of time like how a bicyclist makes power during each revolution. The calculation of power done with software calculate power tell you the max amount of continuous average power at different time lengths over your whole ride or you can pick any segment and length. Getting back to the above quote is there any reason that length of time greater than a revolution like 60 seconds is going to be more accurate average?
  9. Oct 9, 2014 #8
    You lost me there. I thought that there is only one strain gauge that rotates with the pedal(s)/chain ring and measures the torque between them.

    I have no clue how one could position 4 strain gauges at fixed positions that would then measure the torque.

    About the 60 seconds: I just pulled that number out of my hat, based on the two following hand-waving arguments:

    To properly account for the variation of torque during a single turn you should take several readings per turn. If you have enough readings then where exactly you take them should not matter anymore, so your resulting energy/turn will be accurate and robust.

    For a cyclist, the power spend on a single turn is probably not that interesting. If you start to calculate averages over very short times with, say 1.3 turns or 2.2 turns or whatever (if the readings are based on constant time intervals and not constant pedal position!), then the average will depend on the pedal angle where the measurement was started. To get a meaningful average you should average over many turns - if you want to be accurate to 1% and the power varies by 10% during a turn, then you should average over at least 10 turns, i.e. the one possible incomplete turn is no more than 10% of the total turns, with 10% variation during the turn that makes 10% x 10% = 1% possible error.
  10. Oct 9, 2014 #9


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    If this is a thought experiment, so far, I could suggest a modification. Why not use a single strain gauge against the wheel spindle, which would measure the chain tension continuously. You could sample that as fast as you like.

    A number of gauges, on each end of the spindle and at different angles could cancel force vectors other than the chain tension, even when you go over a bump.
  11. Oct 9, 2014 #10
    Sorry, I didn't mean to leave that out. I made a mistake in implying it in the example. Currently crank base power meters are made with 4, 5, and 8 strain gauges. But I figure above 4 strain gauges help in only certain times like stopping.

    So I think I figured out what these power meter companies know. While I don't have the time or really inclination to build a power meter, I can build one in my mind and rotate it around. You comment on more strain gauges and reed/switches got me thinking. We build this power meter with all the strain gauges and reed switch or magnets that we can fit. You can make it infinite or a lot, whatever doesn't matter. Keeping this simple to start with one revolution looks like two waves identical waves like the original example you just have more samplings.

    1, 0.25sec, 62rpm, 10 torque units (e.g. left leg across top of stroke)
    2, 0.25sec, 58rpm, 50 torque units (left leg downstroke)
    3, 0.25sec, 62rpm, 10 torque units (right leg across top of stroke)
    4, 0.25sec, 58rpm, 50 torque units (right leg downstroke)

    Now this wave has two lowest points at bottom of the two troughs and two high points at the peaks of the wave per revolution. Now draw a line through the waves exactly between the lowest and high points or 12.5 seconds back in the above example and this gives you four points of power sampling. The other multi/infinite-samplings both higher and lower than these points will cancel or average each other out leaving you with these four points which in the above example gives you:

    Power= (60x30 + 60x30 + 60x30 + 60x30)/4 = 1,800 power units

    This continues to work if even if peaks goes higher or troughs go lower you simply draw the line different for each wave going for the middle. Where you run into possible errors is when a person stops pedaling, or there is great amount of force applied in a very short period of time, a spike (speculation on the second point).
  12. Oct 10, 2014 #11
    There was actually a chain tension power meter invented and in production mid 1990s to about mid 2000s. Invented by the Cote brothers and sold by polar. It sensed chain tension acoustically like a guitar pickup, a chain speed sensor pulley on the back. I will show pictures below.

    They have built power meters just about everywhere you can put them. Pedal, sole of shoe, cleat between pedal and shoe, crank, spider, chain, rear wheel hub, and a power meter that calculates power by measuring resistive forces http://www.ibikesports.com/index.php/product/how-the-ibike-works/ [Broken]. I think one of the most interesting is the laser spoke power meter http://bicycledesign.net/2012/05/laser-spoke-power-meter/. It use the whole rear wheel as a strain gauge. What is really interesting is that it can tell when your rear wheel is getting worn out. The whole concept open up other possibilities of making power meter between components and being able to tell when a part is wearing out. They are now making bikes with electronic shifters to shift the chain and that opens possibilities because now a computer knows what gears you are in, the speed of the bike, RPMs, it does take much more to make a power meter.

    I am just an end user bicyclist and long time user of power meters but part of what I like about bicycling is all the physics, math, and married to physiology. I just keep on learning.

    "The main sensor unit houses two sensors: a cadence and a chain tension sensor. An LED next to the Polar logo flashes red and green(initially) to confirm that a signal is being received by the speed and cadence sensor, respectively. Yes, the cadence magnet is mounted on the right crank. The speed sensor is mounted on the left chainstay (partially hidden here). A third wire (seen here) going to the chain speed sensor mounted on the lower derailleur pulley."

    Picture and text pulled from this website: http://www.u.arizona.edu/~sandiway/bike/s710/
    Last edited by a moderator: May 7, 2017
  13. Oct 10, 2014 #12
    From the picture posted in the other thread (just look at the list of Mark's post, there are not many), it appears that the device measures the strain between the front axis that carries the pedals, and the chain ring(s). If I had to implement that I would use one, maybe 2 strain gauges that could be sampled continuously. This is cleaner (encapsulated) and easier to implement than a chain tension sensor, and it is probably also more accurate.

    Having said that I only know the pictures from the other thread. I have never seen the device in real life, let alone opened it and looked inside.

    I am not sure if the additional sensors you propose would help in better measuring the power developed by the cyclist - which is the ultimate purpose of the whole thing.

    BTW, data rate should not be a problem because the first sampling and averaging could easily be done in the device, and only the statistics and averages communicated to the host device. On the other hand, using a smart phone and bluetooth as the host controller there are basically no limits on the data rate.
  14. Oct 26, 2014 #13
    I need to correct my above post for the sake of giving correct information and that this thread shows up to high in search engines when searching for topics along these lines. The main correction is there are eight points of power sampling you have to worry about to get a correct average power per revolution instead of four that the above post gives the impression of.

    Now this wave has two lowest points at bottom of the two troughs and two high points at the peaks of the wave. Now draw a line through the waves exactly between the lowest and high points or 12.5 seconds back in the above example and this gives you four points of power sampling. The other multi-samplings will cancel or average each other out leaving you with these eight points, which in the above example gives you:

    Power = (62x10 + 60x30 + 58x50 + 60x30 + 62x10 + 60x30 + 58x50 + 60x30) / 8 = (620 + 1800 + 2900 + 1800 + 620 + 1800 + 2900 + 1800)/8= 1780 power units

    To prove that adding more samplings give you the same number we go to sixteenths. (I am skipping figuring out the torque/velocity of all the points because it depends on what the identical waves look like, but remember they are identical in this simple example.)

    (620 + 1210 + 1800 + 2350 + 2900 + 2350 + 1800 + 1210 + 620 + 1210 + 1800 + 2350 + 2900 + 2350 + 1800 + 1210)/16 = 1780 power units

    Adding 32 points and so forth gives you the same average so you don’t need any more points other than the eight points.

    This continues to work if even if peaks goes higher or troughs go lower or the waves are not identical you simply draw the line differently for each wave going for the middle. Where you run into possible errors is when a person stops or starts pedaling, or there is great amount of force applied in a very short period of time, a spike (speculation on the second point).

    Accuracy: 1780 power units falls right in the middle of 1800 and 1760 power units. Since I hate the high power unit numbers because they psychologically have greater power unit differences but percentage wise it will be the same, I will chop off the end zeros and we have 178 pu in between a high of 180 pu and 176 pu. The 2 pu difference looks not so bad and the percentage accuracy is 1% in Alex simple example. The accuracy further improves because most of the time we pedal above 60 RPM and the more sampling.

    Crank/pedal based power meters only have to worry about measuring these eight points, starting and stopping, and data handling. Now how well they do that is another question entirely.
  15. Oct 26, 2014 #14


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    There are so many ways of skinning a cat.
    All you need, basically, is the tension in the chain. Multiply that by the speed of the chain and that gives the power delivered to the wheels. Chain tension could be difficult to measure so I just thought that the force on the bearing should suffice. Measuring forces in various directions and doing some vector addition would / could eliminate error and the static forces and give the effective force (torque) from the pedals. I suspect that the stretch in the chain (or the difference in angle of crank and sprocket) could be used to measure the tension. A bit of laser interferometry would do that for you.
    There is no significant limit to the rate that you can sample this and the calculation is (Arduino - level) trivial. Taking an average over a number of pedal strokes would give a good measure of average power. Using an optical sensor would make it easy to identify the rotation period.
  16. Oct 26, 2014 #15
    The ellipsoid shape of the chain ring means that the instantaneous speed of rotation of the front and rear gears are different. That is the whole point of the thing, to provide constant torque at the rear gear while the torque delivered by the legs varies depending on position.

    Measuring the angular position of the front crank shaft or pedal or gear is trivial with a few optical sensors or reed relays.
    The best way to measure the torque that I can see is to have a force sensor between the chain ring and the pedal, at a fixed known distance from the center of rotation. You could have more than one, e.g. if the chain ring is fixed at several positions you might want a sensor at each one. Each one of these sensors can be read as fast as you want, and the average should give only the torque plus maybe a fixed offset.
  17. Oct 26, 2014 #16


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    I really find it hard to trudge through large numerical examples and I may have missed something but it seems to me that all you are saying is that the time (angle?) variation over a cycle is pretty much sinusoidal with only a small amount of second harmonic - which is why higher sampling rate makes little difference. But just to be sure - these are measured values, are they? If they're not then they don't prove anything other than that your calculations are self-consistent and that basic Sampling Theory works.

    What I don't understand is why there is any problem with grossly oversampling when you realise just how fast measurements and calculations can be made.
  18. Nov 1, 2014 #17
    You have it right, except I was just trying to get within the error rate with least amount of sampling and measuring rotation only once a pedal revolution.

    There is no problem with grossly oversampling except outside of a lab it has not been done that I know of. Since most cyclist bike outside it would be more useful information. Throw in a freewheel and an uneven road as most roads are and you could have small phases of the pedal revolution where your pedal and wheel speed become disengaged from each other.

    Currently the best power meter on the market for outside use has a eight strain gauges and measure pedal velocity once per revolution (like my example above). There are some pedal base power that use a torque tube or strain gauges and an accelerometer with higher sampling but apparently there is some difficulty in measuring outside the lab so they also default to velocity once per revolution. Gross sampling outside a lab is easier said than done.
  19. Nov 1, 2014 #18


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    Sampling the rate at which the spokes pass an optical shutter would be very easy and would give you 'continuous' wheel speed; counting the revs of the crank seems a strange choice. I don't know of an ADC that has problems sampling at a hundreds of Hz (in all sorts of adverse physical conditions). The only practical difficulty - and I know that's not trivial - is measuring forces somewhere to give the torque on the sprocket. What do all the strain gauges do on the commercially available meter? Presumably the sprockets are circular even if the crank ring is not and it would be trivial to factor in the gear you're in.
    There must be reasons for some of the practical realisation details that I clearly haven't grasped; for instance, why put the strain gauges at the crank end? It would be useful to know the crank angle, of course, but that would be relatively simple.
    Electromechanical systems are very dependent on the transducers but the electronics to deal with loads of data wouldn't be a problem.
  20. Nov 20, 2014 #19
    Sorry for the very delayed reply. Just very busy. First you probably know more than me about measuring power in general on all sorts of devices.

    As far as measuring the continuous wheel speed, one thing you need to realize is that most bicycles have a free wheel so that wheel speed doesn't match crank speed all the time. It may not match even with in a rotation ( a question of mine going on in the back of my head). Any time the wheel speed is greater than the crank speed than the free wheel will disengage from the rear cog/hub.

    There is a Powertap power meter http://www.powertap.com/ that measure power from the rear hub. It use a torque tube to measure force.

    Having a computer know what gear your in will be easier in the future as more electronic shifting becomes use and hopefully tracks the shifting.

    There is about a 1 to 2 % loss of power in the drive train.

    The strain gauges (anywhere from 4 to 8 depending on the model) on the crank measure force around a round axis. The only thing on a non round crank traveling in anything but round is the chain. The purpose of these non round rings are to lengthen the time in the power down stroke and shorten the time in the bottom/top part of the stroke often called the dead zone. What seems to happen is that a least the velocity of the chain slows in the down stroke where force is applied and the velocity increases in the dead zone.

    Why the interest to put strain gauges at the crank, I guess to reduce error being closer to where power is being made. Why an interest in finding out how people make power per revolution? To see if there are differences in people in how they make power. If there are more efficient ways physiological/mechanical. Why some people are better climbers or better on a flat course.

    One question I have is what drives velocity? I would think it would be the foot as all power/energy to propel the bike forward comes from the bicyclist. If we say all resistive forces are constant and the only change is the force application on the pedal and the chain traveling around an oval with a like 10% increase and decrease in gearing is the wheel speed changing up and down 10% per revolution or is there to much inertia of the whole mass? One other thing the wheel is actually a spring with the spokes winding and unwinding with each power change. This is how the laser spoke power meter measures power by measuring the fluctuation of distance and speed between the hub and the rim. How much depends on how it tight it was built in the first place and how old it is now. The tire also has some spring action. Hopefully I explained that well. Maybe I should start another thread with that question.
  21. Nov 20, 2014 #20


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    Any time the wheel rotational speed is greater than the sprockets, NO power is being transferred. So that scenario is a 'special' or of no interest.

    If you measure the flex of the spokes (difference between angle near the hub and angle near the rim) then that will tell you the driving torque. That value times the rotational rate will tell you the power. That's yet another way of measuring power and another way of skinning the cat.
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