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Bicycle Crank Power Meters and Round and Non-Round Chainrings

  1. Oct 3, 2014 #1
    One of the debates in part of the bicycling community is whether elliptical or non round chain rings cause a crank based power meter to over report power beyond the power meters accuracy rate when velocity is only measured once per revolution. Average and weight average of the velocity per revolution plays apart on this position.

    The counter argument non round chain rings are just an increase/decrease in the lever and power is coming from your foot on the pedal which is on a circular radius going around a circular axis.

    The accuracy of some power crank based power meter is report as +/- 2%. SRM +/-1% accuracy is considered the gold standard in bicycle power meter. The strain gauges are located between the pedal and the chain rings. Velocity is measured by a simple magnet and reed switch once per revolution.


    From SRM power meter manufacture http://www.srm.de/:
    We have had an increasing number of customers use Q-rings with their SRM Power Meter. There is no difference in wattage accuracy between using a regular chain ring or an elliptical chain ring. The important thing is that the SRM power meter is calibrated with a regular round chain ring to determine the slope of the power meter, which is exclusive to that power meter. Once this is determined, any type of chain ring can be used and wattage accuracy is maintained. Think of it in these terms, the elliptical chain ring only adds more leverage, which in turn can allow for more torque per pedal revolution. This is no more different than changing the crank arm length, which has no effect on the accuracy of the power meter. Torque is torque; and if you can spin the same cadence with the Q-rings, then in theory you will produce more power. We have found from reports of our customers using the q-rings that their cadence tends to drop about 5-10 rpm at a given power output over using regular round chain rings.

    This makes the most sense to me as the only power to propel the bike forward is coming from your foot and there are a number of levers and opposing forces that can change cadence.

    The best opposing argument comes from a Mechanical Engineer Tom Anhalt http://bikeblather.blogspot.com.au/2013/01/whats-up-with-those-funky-rings.html There is a picture of one manufacture’s of these non round ring (Osymetric chainrings). He does a comparison experiment towards the end of the blog.

    Osymetric Chainrings with SRM Powermeter

    I have a number of questions about his blog but one in particular is on how he calibrated. “Calibration shows the Round ring (mounted on the outer position) reads 1% high and the Osymetric (mounted on the inner position) reads 1% low and the torque slope was set to the average value between the 2 rings. Power values from the Quarq reported below were corrected based on the calibration.”

    All end user calibration spreadsheets are based on round rings to figure out the lever length so he doesn’t explain how he determined the lever length of the Osymetric chainrings. Also just doing the calibration with the round chainring 3 times, and averaging the values should be sufficient enough for both rings. SRM points out. His way seems to allow more room for error and question.

    The funny thing is that these non round rings are sold as: “because of their pseudo-elliptical shape, Osymetric rings concentrate your pedaling power where your force is at a maximum, while effectively reducing the load where your power input is at a minimum.” “It works on the objective of minimizing the time spent in dead spots, while maximizing the time spent within the radii of efficient power exertion -- horizontal. Basically, this means that, as the operator, you can apply more force while spending less energy. In fact, Osymetric's design minimizes torque and effort by creating a nearly constant angular velocity.”

    “apply more force while spending less energy” is a red flag for me. And than nearly constant angular velocity, I thought the problem was non round ring had a varied velocity compared to round rings during each revolution.
    There also seems to be the assumption that with round rings the velocity of the pedal/crank is fairly constant or maybe they know that.

    My take is that non round rings just move force and velocity around and that work is work.

    Thanks for your comments and insights.

    Typical power distribution on round chainrings
  2. jcsd
  3. Oct 9, 2014 #2
    SRM's explanation looks sound to me.

    The torque does not depend on the size or shape of the chain ring. I suspect, however, that the speed of revolution is more or less irregular with one or the other - from the text you quote, the elliptical rings have the purpose to make the speed of revolution more constant, by increasing the variation in torque over the different phases/angles of rotation.

    To get a more precise power reading you should therefore determine the speed of revolution more often than once per revolution - ideally synchronized with the torque readings. This could be done by adding more magnets, such that you get not just one pulse per revolution, but several.
  4. Oct 9, 2014 #3
    Thanks, I think this is key. I agree with you that the purpose of these non round chain rings is to make the speed of revolution more constant. That this what they advertise- smoother pedaling. If anything it is pedaling the round chain rings that are going to have a more changes in speed and torque.

    Just yesterday, I asked just this question on the General Physics forum titled: Measuring varied power per revolution of a bicycle crank/chainring. 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.

    Interestingly there are a couple of crank power meter companies that have accelerometers in their power meters. All choose to average velocity and average force before figuring out power per revolution. One company's power meter has both a magnet/reed switch and accelerometer. They say that the one magnet is more accurate than the accelerometer. The accelerometer is there for bikes where a magnet can not be positioned to be picked up by the reed switch.
  5. Oct 9, 2014 #4
    I was assuming that the reed switch is on the frame and the magnet on the spinning part. It seem to be the other way around. In that case it would be easier to add more reed switches - but that would mean opening the thingy and fiddling around with the insides. Also with a commercial product like this it is probably difficult to modify the firmware to account for the larger number of pulses per revolution.

    May I ask why you are interested in this? From a pure user's perspective, or are you trying to design your own torque/power meter?
  6. Oct 9, 2014 #5
    I am a bicyclist training with power meters, pure user's perspective, and an interest in physics. My father was a theoretical physicist. He is dead or I would ask him.

    The software modification would be a problem. I believe data transmission can happen at up to 240 hz on ANT+. I believe all the power meter companies are trying to improve accuracy but most claim +/-2 %. SRM claims +/- 1%. They only use 1 magnet. I have to check the number of strain gauges they use. I suspect there greater accuracy has to do more with their calibration process -144 points rotational.
  7. Oct 9, 2014 #6
    OK, so unfortunately we cannot fiddle with the instrumentation then :)

    I have just posted my ideas about improving accuracy in the other thread.

    The other factor that affects the accuracy of the device is of course the accuracy of the strain gauge(s) themselves.
  8. Oct 20, 2014 #7
    All the chain and chainring do is hold the spider still when a known torque is applied to the spider via the crank arm. The only lever length you need to know is the crank arm length, as that combined with the mass hung from the pedal spindle of a horizontal crank arm tells you the torque being applied to the crank spider. What chain ring happens to be on the bike is irrelevant. You could take the chain and ring off the bike and simply secure the spider with a hook through the bolt hole and a cable if you liked.

    The fact that Tom saw a variance in measurement between the inner and out chainrings is simply a factor of the strain gauges reacting to the slightly different lateral forces when the spider secured by the different ring placement (inner v outer). Better spider design and materials minimises such measurement differences.
  9. Oct 20, 2014 #8
    Most modern SRMs use 8 strain gauges, SRM Science uses 20. They also mostly use 2 reed switches.
  10. Oct 20, 2014 #9
    SRM's website explanation however is incorrect.

    Once the assumption of constant angular velocity during a pedal stroke is violated, then the resulting power measurement will be based on an uneven weighting of torque signal during the crank rotation.

    As you rightly point out, the only way to accurately measure power when you have variable crank velocity is to measure both crank velocity and torque with higher frequency.
  11. Dec 20, 2014 #10
    I just saw these posts. Wait Alex, so you are saying that the chain and chain ring are holding the spider still? No it being attached to the crank arm attached to the bottom bracket?

    There are multiple ways to hold the crank arm horizontal with the chain rings on. There is the other crank arm on the other side.
    Last edited: Dec 20, 2014
  12. Dec 20, 2014 #11
    That is a big assumption even with round chain rings.

    "Using a circular chainring, the instantaneous velocity of the crank varies ±22% for an average pedalling cadence of 90 rpm” Hull ML, Kautz S, Beard A. An angular velocity profile in cycling derived from mechanical energy analysis. J Biomech 1991;24(7):577-86, from Rodrigo R. Bini and Frederico Dagnese in 2012
  13. Dec 21, 2014 #12
    Hi Mark

    I'm afraid that you have both misquoted and misunderstood the paper by Hull et al.

    They did not measure and find such variations in actual crank velocity. What they specifically noted about pedalling with a circular chainring is that crank velocity, when pedalling at a steady cadence, is nearly constant during a pedal stroke.

    The variable intra-crank rotation velocity they report is a calculated theoretical velocity required to minimise the intra-crank rotation variations in total energy (potential + kinetic) of the legs. It was not what actually happens, but rather an indication of the nature of mechanical changes you'd need to make to the system (e.g. via an eccentric chainring design) in order to minimise the leg kinetic + potential energy variations.

    Anyone interested can download a pdf of the paper via this link:
    http://www.researchgate.net/profile/Maury_Hull/publication/21276126_An_angular_velocity_profile_in_cycling_derived_from_mechanical_energy_analysis/links/02e7e535ebcb2abf2c000000 [Broken]

    Cheers, Alex
    Last edited by a moderator: May 7, 2017
  14. Dec 21, 2014 #13
    What I'm saying is that if you wish to test the torque readings of a crank spider based power meter by comparing it to the known torque applied when hanging an accurately known mass from the pedal spindle of a horizontal crank arm of precisely know length while it's in the forward position (either crank arm, it doesn't matter), then naturally you need to secure the crank spider to prevent it from rotating, else the torque applied will simply see the mass fall to the floor.

    You can easily do that via securing the rear wheel so that it cannot rotate, and then the chain which is attached to the rear cogs and the chainring and via that, the crank spider will prevent the spider from rotating, which is all that you need do.

    Here's an example of how you might do that. This is not my picture, I think it's from John Verhuel. I do a similar thing, except that I have special "pedals" I screw into the crank arm which enable me to slide weight plates on/off easily without need to raise the bike up to give room for the weights. If you are testing a crank spider based meter (not the hub based meter in the picture), then the size of the chainring is not relevant, it's only role is being means (along with the chain and cogs and rear wheel brake) to prevent the crank from rotating while under the load being applied.


    However if you remove the power meter from the bike and clamp it in a vice and hang a mass from a chain wrapped around the chainring, then knowing the precise radius of the chainring at the vertical tangent point the mass is hanging from is needed in order to calculated the applied torque value accurately. This radius gets harder to precisely measure with a non-circular chainring. With a circular ring it can be calculated fairly easily as tooth size determines chainring radius. In this method the crank arm is removed from the process and the spider is secured from rotating via the BB spindle being held in a vice.


    I prefer the method of doing it while the unit is installed on the bike, one because it's more convenient, secondly because it's a measurement in situ, thirdly because you can test the torque values while applying the torque to either drive and non drive side crank arms, and finally because it removes the need to know the chainring radius, all you need to know is the crank arm length (which is easy), hence you can test with whatever chainring you like, circular or non-circular.
  15. Jan 7, 2015 #14
    So a lot of conversation on this topic happened on another more bicycle specific forum. I will be hopefully balanced.

    I actually didn't quote or misquote the paper. The quote comes from Rodrigo R. Bini and Frederico Dagnese in 2012
    "Noncircular chainrings and pedal to crank interface in cycling: a literature review"
    “Using a circular chainring, the instantaneous velocity of the crank varies ±22% for an average pedalling cadence of 90 rpm" in review of the Hull et al, 1991 paper Alex has linked to.

    Dr Bini is a currently active bicycle pedal researcher.

    My summation of the Hull et al. paper:
    Just a short summation of the study: The researchers came up with two reference angular velocity profile. One was from the five subjects: "Using data recorded from five subjects, this procedure was used to determine a reference profile for an average equivalent cadence of 90 rpm." The other was "A five-bar linkage model (thigh, shank, foot, crank and frame) of seated (fixed hip) cycling served for the derivation of the equations to compute potential and kinetic energies of the leg segments over a complete crank cycle. With experimentally collected pedal angle data as input, these equations were used to compute the total combined mechanical energy (sum of potential and kinetic energies of the segments of both legs) for constant angular velocity pedalling at 90rpm." Notice constant not near constant for this second model, this will come up in the other Hull paper response. Then subject MH using "experimentally determined pedal angle data (Subject MH)" shown in Fig 4 was compared against the two models in order to see how you could minimize internal work.

    If you bring in five people to sent on an erg, you got to have experimental determined data not theoretical or what is the point? More so if you are going to compare it to two models, it doesn't make since to compare theoretical to theoretical.

    This study and another by Hull where "near constant angular velocity" comes from became moot. Hull paper 1992:
    “To date, there have been no human performance studies that test the validity of optimization analyses. The human performance studies cited earlier had deficits in that subject populations were generally small, equipment simulations were unrepresentative, the cyclists were untrained, or the power outputs and cadences were uncharacteristic of high performance cycling. Given these limitations, no firm conclusions can be drawn and more experimental work is needed to validate the different designs described in the literature.”

    So while Dr Bini is right in his review the data is off.

    There is a study that shows a +/- 1.5 to 3% angular velocity variance but it use a Powerbeam Pro http://www.cycleops.com/product/powerbeam electronic braking trainer that has a +/- 5% accuracy and low torque (power numbers between 100 -200).
    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
    pdf here:

    The main point of the study was not to research pedal angular velocity, but as a comparison of a proposed small wireless motion sensor units system with a motion capture device. The data only had to be synced not accurate as both device got the same information at the same time.

    Last edited by a moderator: May 7, 2017
  16. Jan 7, 2015 #15
    One of the concerns that people had was inertial load effects on crank velocity especially non-circular chain rings. While the two studies below are with round chain rings, inertial load is not a problem with non-round rings as I will explain in a subsequent post.

    I don’t have access to the full text of this research. Two studies, the second one repeats the first study but with trained cyclist and comes to the same conclusions. “However, the results presented here are mostly consistent with previous work in suggesting that crank inertial load has little direct effect on either physiology or propulsion biomechanics during steady-state cycling, at least when cadence is controlled.” The high inertia load was 94.2 kg in the second study.

    Bold in Abstracts are by me to emphasize.

    Crank inertial load has little effect on steady-state pedaling coordination

    Benjamin J. Fregly§, Felix E. Zaja, Christine A. Dairaghi; Journal of Biomechanics; Published December 2006
    Inertial load can affect the control of a dynamic system whenever parts of the system are accelerated or decelerated. During steady-state pedaling, because within-cycle variations in crank angular acceleration still exist, the amount of crank inertia present (which varies widely with road-riding gear ratio) may affect the within-cycle coordination of muscles. However, the effect of inertial load on steady-state pedaling coordination is almost always assumed to be negligible, since the net mechanical energy per cycle developed by muscles only depends on the constant cadence and workload. This study tests the hypothesis that under steady-state conditions, the net joint torques produced by muscles at the hip, knee, and ankle are unaffected by crank inertial load. To perform the investigation, we constructed a pedaling apparatus which could emulate the low inertial load of a standard ergometer or the high inertial load of a road bicycle in high gear. Crank angle and bilateral pedal force and angle data were collected from ten subjects instructed to pedal steadily (i.e. constant speed across cycles) and smoothly (i.e. constant speed within a cycle) against both inertias at a constant workload. Virtually no statistically significant changes were found in the net hip and knee muscle joint torques calculated from an inverse dynamics analysis. Though the net ankle muscle joint torque, as well as the one- and two-legged crank torque, showed statistically significant increases at the higher inertia, the changes were small. In contrast, large statistically significant reductions were found in crank kinematic variability both within a cycle and between cycles (i.e. cadence), primarily because a larger inertial load means a slower crank dynamic response. Nonetheless, the reduction in cadence variability was somewhat attenuated by a large statistically significant increase in one-legged crank torque variability. We suggest, therefore, that muscle coordination during steady-state pedaling is largely unaffected, though less well regulated, when crank inertial load is increased.

    Me: I should be noted that subjects were “instructed to pedal steadily (i.e. constant speed across cycles) and smoothly (i.e. constant speed within a cycle) against both inertias at a constant workload.” IE they worked at pedaling smoothly not necessarily their normal pedal velocity variation within a cycle. They were asked to work on keeping it this way.

    The effect of crank inertial load on the physiological and biomechanical responses of trained cyclists

    Lindsay M. Edwardsa*, Simon A. Jobsona, Simon R. Georgea, Stephen H. Daya & Alan M. Nevilla; Journal of Sports Sciences, Volume 25, Issue 11, 2007; Published online: 19 Jul 2007
    The existing literature suggests that crank inertial load has little effect on the responses of untrained cyclists. However, it would be useful to be aware of any possible effect in the trained population, particularly considering the many laboratory-based studies that are conducted using relatively low-inertia ergometers. Ten competitive cyclists (mean [Vdot]O2max = 62.7 ml • kg−1 • min−1, s = 6.1) attended the human performance laboratories at the University of Wolverhampton. Each cyclist completed two 7-min trials, at two separate inertial loads, in a counterbalanced order. The inertial loads used were 94.2 kg • m2 (high-inertia trial) and 2.4 kg • m2 (low-inertia trial). Several physiological and biomechanical measures were undertaken. There were no differences between inertial loads for mean peak torque, mean minimum torque, oxygen uptake, blood lactate concentration or perceived exertion. Several measures showed intra-individual variability with blood lactate concentration and mean minimum torque, demonstrating coefficients of variation > 10%. However, the results presented here are mostly consistent with previous work in suggesting that crank inertial load has little direct effect on either physiology or propulsion biomechanics during steady-state cycling, at least when cadence is controlled.

    Me: Why inertial loads do not matter is something called gears. Gear selection.

  17. Jan 7, 2015 #16
    My take on measuring non-round chainrings

    The change in chain distance that matters is where the chain engages the top of the chain ring and the chain is pulled, transferring force. On non-circular chain rings your leverage is changing and you are trading force for speed or speed for force as this distance goes up and down. You are the only one providing energy into the system to propel the bike forward. Since the amount of power is set as you push your pedals around ie. power does not increase or decrease magically. All that increase and decrease is force to velocity relationship to equal that power you made
    The crank power meter measures velocity and force (power) before the leverage change at the top of the chain ring. The PowerTap (powermeter located in the rear wheel hub) measures power after the leverage change but the power is still the same minus any lost in the drivetrain. Crank based power meters have the same challenges measuring power whether the chain ring is circular or non-circular.

    Can changes in leverage change your ability to produce power? Yes, if you have been caught in the wrong gear you know it. But people select gear or change gears (leverage) so that they can produce the amount of repeatable force necessary for the time needed. If the greatest repeatable force need is at the point where leverage is least on a non-circular chain rings than you would set your rear cog to accommodate that. Inertia of the rider and bike’s mass is the “baseline” resistive force that you gear yourself to and than as other resistive forces change you change gears in response. Inertia is constant.

    Can a change in leverage or other resistive forces change the velocity of the your foot on the crank (cadence)? Yes, but so can you. There is always a give and take in you producing power on any gearing or terrain. Power numbers are jumpy (of course not on an erg). But it is you that are producing the power.

  18. Jan 13, 2015 #17
    I'll let others who understand some basic physics to check the following thought experiment:

    80kg bike (8kg) + rider (72kg) riding along at a steady state at 36km/h (10m/s) on flat road, pedalling at 90rpm using circular chainrings. Nothing particularly out of the ordinary about that.

    Since the chainrings are circular, the crank rotational velocity and bike's translational velocity will be proportional. If there is any chain stretch, it will be minor.

    Mark seems to think that the crank's rotational velocity during a pedal stroke can vary quite a lot, and quoted values of +/-22%.

    So let's assume for a moment that Bini's quote is not incorrect. What does that mean in terms of the power demand required for such a variation in velocity during a pedal stroke?

    If it were true, then we'd have a rider that accelerates from 10-2.2 = 7.8m/s to 10+2.2 =12.2m/s twice each pedal stroke.

    At 90 rpm, one revolution = 0.6667 seconds

    We have two velocity peaks and two velocity lows per revolution. So the time between each velocity low and peak is a quarter of 0.6667 seconds = 0.1667 seconds.

    Ignoring the small amount of rotational inertia from wheels and other rotational parts (it's tiny), the translational kinetic energy of an 80kg bike and rider at 7.8m/s is 0.5 x mass x velocity^2 = 0.5 x 80 x 7.8 x 7.8 = 2433.6J

    At 12.2 m/s, rider's KE = 5953.6J

    So over 0.1667 seconds, we have an increase in the rider's KE of 5953.6 - 2433.6 = 3520J

    That's an average power of 3520 / 0.1667 = 21120 watts for that 0.1667 seconds.

    Then there is an equivalent deceleration over the next 0.1667 seconds. Which means an equivalent braking power is required. Keep in mind that the total resistance forces on a road race bike rider at 36km/h on a flat road requires ~ 250W give or take to maintain speed.

    No human on the planet is physiologically capable of such high power outputs as 21kW, let alone instantaneously on a pedal crank with a single leg. We are talking about applying an average force for that quarter pedal stroke equivalent to >16 times their body mass and a power to mass ratio of 264 W/kg! The very best elite world level track cycling sprinters and BMX riders could manage at best ~40W/kg for the equivalent time period.

    If we perform the same calculation for a 60kg bike + rider riding at 90rpm along at a modest 24km/h (6.667m/s), then the power demand for the full quarter pedal stroke if such a velocity variation occurred is still 3200W or ~62W/kg. That's still greater than the what the world's best track sprint cyclists can manage to repeat for just a few pedal strokes before fatiguing badly.

    It's just physical and physiological nonsense to believe such high pedal stroke crank velocity variations occur when riding along at a steady state. Small variations, yes. As crank inertial load reduces, then the velocity variations increase. Which is exactly what is found when pedalling a bike on a trainer, with a much reduced crank inertial load. A few percent.

    The Bini quote is just wrong. That's doesn't mean Bini is wrong, just that the quote is.
  19. Jan 13, 2015 #18
    No, that is not what I think. As stated above Hull paper 1992 said previous data was bad.
    “To date, there have been no human performance studies that test the validity of optimization analyses. The human performance studies cited earlier had deficits in that subject populations were generally small, equipment simulations were unrepresentative, the cyclists were untrained, or the power outputs and cadences were uncharacteristic of high performance cycling. Given these limitations, no firm conclusions can be drawn and more experimental work is needed to validate the different designs described in the literature.”

    Reading other researcher's paper it sounds like they modified the equipment to take the measurements and the modifications probably led to unnatural pedaling. The whole point is moot.

    I agree velocity fluctuates per revolution in a sine wave, so does torque, so does power. Most of the torque and power we create is the when we push down on the pedal. There is no deceleration or equivalent braking as the rear hub has a freewheel. The problem you are having in your calculations is that you are thinking of torque and the resulting power as constant throughout a pedal revolution. It is not. We are not like a motor.

    Your acceleration calculations apply to both circular and non-circular chain rings and if they were the reality we wouldn't be able to ride a bike. But we have gears and choose our RPM (cadence) for the amount of force we can sustain over the period of time we need to. When it becomes to hard we change to an easier gear. If we are on a steep incline and the gearing is not easy enough for the amount of repetitive force we can sustain, we stop.

    A bicyclist gears for inertial crank load as the two studies above explain: “However, the results presented here are mostly consistent with previous work in suggesting that crank inertial load has little direct effect on either physiology or propulsion biomechanics during steady-state cycling, at least when cadence is controlled.” Inertial load is a constant.

    Yes, the data from the study was bad. The quote on the data is bad. This was all explained earlier in this thread.

  20. Jan 14, 2015 #19


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    You are correct, with toe clips and cleats it is possible to provide close to constant torque throughout the circulation of the pedals, without any need for an asymmetric chain wheel. The asymmetric chain wheel can really only be an advantage when the rider is applying their weight to one pedal only at the time, they then do no work while the pedals are at the top or bottom.

    There are many ways of avoiding the dead spot at TDC and BDC. What we really need invented is three pedals so we can have three phase power; but that may require three legs. I don't think two phases in quadrature would outperform the differential single phase typically seen with humans on bicycles today. Toe clips and cleats make a big difference, far more than an asymmetric chain wheel possibly can.
  21. Jan 14, 2015 #20
    Then why did you quote it and use that quote to suggest there were large fluctuations in crank rotational velocity?

    OK, glad to see that you now agree that crank rotational velocity variations are generally pretty small.

    Yes, that's correct.

    On the contrary, indeed I have modelled this sinusoidal application of power during a pedal stroke, for both steady state cycling and for non-steady state scenario, i.e. during a hard acceleration, and examined the impact to crank rotational velocity variations.

    You can read about the impact of this sinusoidal like application of power for the steady state cycling scenario here:

    and here for the dynamic hard acceleration scenario here:

    No, they don't, because with non-circular rings, crank rotational velocity is no longer directly proportional to bike translational velocity - because the gearing is constantly changing due to the variable radius of the chain ring.

    That's the whole point of non-circular chain rings - to vary the crank rotational velocity far more than otherwise would be the case with a circular chain ring.

    Sure, but that's irrelevant to the discussion about crank velocity variations within a pedal stroke.

    You are conflating the impact of crank inertial load on our ability to generate power with the impact of crank inertial load on the variability of crank rotational velocity. IOW this is also irrelevant to the discussion about crank velocity variations within a pedal stroke.
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