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  1. #1
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    You know, it's possible to estimate CdA and Crr without a power meter

    Y'all probably know that it's possible to estimate CdA and Crr with field tests using a power meter. It's also possible to do it without a power meter -- it's more hassle but it's still possible.

    There are several ways to do this but if you happen to have a way to record speed accurately and precisely then it's a bit less of a hassle than it used to be. And, many riders now have GPS units on their handlebars that can record speed. That's the point of this post: even if you don't have a power meter but have a way to record speed you can do this.

    You'll still need a place to test that's protected from the wind and traffic. And, you'll get better results if you have one of those dedicated wheel sensors for speed rather than relying on the GPS signal -- that's especially true if your test venue relies on tall trees to help protect it from the wind. Don't use "smart" recording -- set your GPS to record second-by-second.

    Here are speed data from two coast downs I did a couple of days ago. The speeds are in km/h at one second intervals. As I said, these were coast downs so power, of course, was zero. I forgot to weigh myself and the bike when I got home so I'm guessing the all-everything mass to be around 86 kg. I didn't measure air density but I'm thinking it must be around 1.17 kg/m^3. And the total drop from entry to exit of the test section was about 5 meters. Besides not measuring my mass or the air density, there was a tiny amount of wind in my face during the test runs -- but let's ignore all those problems and for this example let's simply assume wind was zero and everything was exactly as noted above.

    Run 1:
    15.2 15.8 16.3 16.9 16.9 17.4 17.7 18.1 18.5 18.9 18.6 17.9 17.9 18.6 19.7 20.0 20.9 21.6 22.5 22.5 23.4 23.7 23.5 24.3 25.1 25.7 26.0 25.6 25.3 24.6 24.2 23.9 23.5 23.1 22.9 22.6 22.3 22.3 22.0 21.9 21.7 21.8 21.4 20.7 20.9

    Run 2:
    26.6 26.6 26.6 26.5 26.6 26.8 26.1 25.4 25.6 26.3 26.8 27.3 27.9 28.4 29.1 28.7 29.6 29.8 30.6 30.3 29.7 29.2 28.7 28.3 27.4 27.2 26.7 26.1 25.8 25.5 25.3 25.0 24.5 24.2 23.9

    Importantly, notice that the initial speeds are different for the two coast downs.

    Here's the challenge: based on these data, estimate my CdA and Crr. Show your work, and state any additional assumptions you may need.

    This was on my commuter bike with heavy duty tires and I was wearing street clothes so don't give me crap about my drag numbers.

    Bonus question: it's easy to figure out the average slope over the test section. What was the maximum slope (nearest 0.1% is okay)?

    Bonus question #2: why is it important to start the coast downs at different speeds?
    Last edited by RChung; 10-21-11 at 06:40 AM.

  2. #2
    fuggitivo solitario echappist's Avatar
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    nah, i'll just use aerolab

    you don't expect your students not to cheat after you have given them an easy way out, do you?

  3. #3
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    Quote Originally Posted by echappist View Post
    nah, i'll just use aerolab

    you don't expect your students not to cheat after you have given them an easy way out, do you?
    That's an acceptable method. I don't expect them to do square roots or logarithms with paper and pencil, so using Aerolab is fine. The main point is that if you have a way to record speed you can do coastdowns, get the data into Aerolab, and fiddle with it. Plus, if you do a coastdown you don't have to worry about the accuracy of your power meter -- you know the power was zero.

    So, what's your answer?

  4. #4
    fuggitivo solitario echappist's Avatar
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    Quote Originally Posted by RChung View Post
    That's an acceptable method. I don't expect them to do square roots or logarithms with paper and pencil, so using Aerolab is fine. The main point is that if you have a way to record speed you can do coastdowns, get the data into Aerolab, and fiddle with it. Plus, if you do a coastdown you don't have to worry about the accuracy of your power meter -- you know the power was zero.

    So, what's your answer?
    my answer is that i have a powermeter so i'll take it out and use it on a relatively windless day

    i mean, energy cost on rider is what, Crr component (fixed, dependent only on mass), CdA component (speed dependent), and gravitational (doable if you know the grade of road). Cda being dependent on the medium through which you are moving, relative ground speed, and speed at which medium is moving. So assume zero wind, it's just ground speed and the density of air (which you provided).

    The true answer is that i'm too lazy, so there

  5. #5
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    Fair enough. I'm usually too lazy to do it that way, too: pure coastdowns are more hassle than being able to do (powered) loops.

    However, if you did solve the challenge, one of the important lessons that applies even if you have a power meter is from bonus question #2: vary the speed of your loops.

    One more tip if you don't have a power meter and want to do coastdowns: either make sure your legs are always in the same position for each coastdown or (better) "spin" your legs slowly but without putting any power into the pedals so that you'll get a truer reading of your aero drag when you're pedaling.

  6. #6
    Wheelsuck Fat Boy's Avatar
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    OK, let me take a stab.

    Assumptions: All inertia is linear.

    convert speeds to m/s for convenience.

    The runs are at 2 speeds are so you can separate rolling resistance from aero resistance.

    The max slope is calculated by atan(max accel)*g, then convert to slope from angle

    ((kinetic + potential energy at the start) - (kinetic energy at the finish)) / Time = power @ average speed

    Run2-Run1 = aero power

    Aero drag = aero power / average run speed

    Dynamic pressure ~= 1/2 *air density * average speed^2

    Cd = aero drag / (surface area * dynamic pressure)

    Rolling resistance force = run 1 power / average run speed

    Crr = Rolling resistance force / (mass * g)

    ----------------------------------------------------------------

    Am I on the right track?

  7. #7
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    Quote Originally Posted by Fat Boy View Post
    OK, let me take a stab. [...] Am I on the right track?
    Ah, you're definitely on the right track. Here's a hint: integrate the power equation over time to convert it into a work equation.

    The runs are at 2 speeds are so you can separate rolling resistance from aero resistance.
    Exactly.

    The max slope is calculated by atan(max accel)*g, then convert to slope from angle
    Not quite. That's close to true for the max change in slope but not the slope itself.

  8. #8
    Resident Alien Racer Ex's Avatar
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    Quote Originally Posted by RChung View Post
    I'm guessing the all-everything mass to be around 86 kg
    More training, less calculating. I'd like to see you at 67 kilos by February, and yes that's just you.

    Also, define the protocol for a "coast down".

  9. #9
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    Quote Originally Posted by Racer Ex View Post
    More training, less calculating. I'd like to see you at 67 kilos by February, and yes that's just you.
    Hmmm. I'd like to see me at 67 kg, too. I'd also like to see me younger, taller, better-looking, and flanked by nubile young women. Oh, wait. I've already got that last bit covered.

    Also, define the protocol for a "coast down".
    Let's not make this too complicated. Cyclists in general, but racers in particular, don't have especially long attention spans if you know what I mean and I'm sure you do so it's good to keep things simple.

  10. #10
    Resident Alien Racer Ex's Avatar
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    Quote Originally Posted by RChung View Post
    Hmmm. I'd like to see me at 67 kg, too. I'd also like to see me younger, taller, better-looking, and flanked by nubile young women. Oh, wait. I've already got that last bit covered.
    I like "nubile young vixens", both as a term and as bookends.

    That said we all know you're an engineer. Googling the "Girls of MIT' produced mostly financial resource applications and gender pay equality studies. For my local community college I had to agree that I was over 18 before I could even see the search results.

    Quote Originally Posted by RChung View Post
    Let's not make this too complicated. Cyclists in general, but racers in particular, don't have especially long attention spans if you know what I mean and I'm sure you do so it's good to keep things simple.
    If you're measuring MV (max velocity) as an absolute, then "coast down" could be an inverse term for what you're actually doing. And some folks use "coast down" as a measured distance instead of using velocity.

    And I was serious about the protocol you were using because depending on the hill and run out, MV might not be affected by start speed.

    Besides, the only guys you're going to get to play here have long attention spans.

  11. #11
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    I get Crr = 0.0064 and CdA = 0.378 m^2. Basically you do as above and write out the energy budget. It looks like

    alpha * Crr + beta * CdA = gamma

    where

    alpha = -mg*integral(v)
    beta = -0.5*rho*integral(v^3)
    gamma = 0.5 m(vf^2-vi^2) - mgh

    Doing this for each run you gives you a 2x2 set of linear equations that can be solved for CdA and Crr.

    I'd say the two runs are needed to disentangle CdA, Crr, and the unknown elevation profile. If you could just ride your bike on flat ground, or use the altimeter data, then only one run is needed since you could do a nonlinear fit to the expected v(t).

    In principle, now that you have CdA and Crr, you can calculate the force of gravity as the balance of all the other forces (aero, rolling, net). The peak would tell you when you hit max slope. But I don't think you have the time resolution to calculate dv/dt cleanly enough for that to work well. Then again I didn't try it.

    Do I get a cookie?

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    Resident Alien Racer Ex's Avatar
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    Quote Originally Posted by stedalus View Post

    Do I get a cookie?
    I don't know. But I have to go in for an IRS audit and would like you to come along.

  13. #13
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    Quote Originally Posted by Racer Ex View Post
    I don't know. But I have to go in for an IRS audit and would like you to come along.
    Sorry, I wouldn't be of much help. Clearly the laws of mathematics stop working when you put a $ in front of the number.

  14. #14
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    Quote Originally Posted by stedalus View Post
    I get Crr = 0.0064 and CdA = 0.378 m^2. Basically you do as above and write out the energy budget. [...] Do I get a cookie?
    How about a hot dog cuz, ladies and germs, we got a WEINER. You can get max slope by plugging the Crr and CdA back into the power equation and solving for the slope. Overall average slope is -1.8%, but the slope isn't constant. It maxes out about two-thirds of the way through the test section at -4.1%.

  15. #15
    Draught jwible's Avatar
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    I was lost at Nubile Young Vixens.

  16. #16
    Wheelsuck Fat Boy's Avatar
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    Quote Originally Posted by stedalus View Post
    Do I get a cookie?
    I think I mentioned elsewhere that I'm not a strong mathematically as I should be. Well done, stedalus!

  17. #17
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    Quote Originally Posted by RChung View Post
    You can get max slope by plugging the Crr and CdA back into the power equation and solving for the slope. Overall average slope is -1.8%, but the slope isn't constant. It maxes out about two-thirds of the way through the test section at -4.1%.
    Like I said, you can do that calculation, but the output is going to be really noisy because of the dv/dt term. Probably at best you can say -4.1 +/- 1.

  18. #18
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    So, here's the main point: if you have a way to record second-by-second speed (and more riders are buying devices that can do this every day) then you can do coastdowns that will let you estimate both CdA and Crr.

    1. You must roll over the same section of road (at least) twice, at different speeds.
    2. Do this on a calm day, in a place free of traffic.
    3. Weigh your total all-inclusive weight.
    4. Go to weatherunderground.com or some other weather site and figure out the air density.
    5. Do your very best to find out the true elevation change between the start and end of your test section.

    This is more hassle than using a power meter but it's not impossibly hard.

  19. #19
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    Quote Originally Posted by stedalus View Post
    Like I said, you can do that calculation, but the output is going to be really noisy because of the dv/dt term. Probably at best you can say -4.1 +/- 1.
    Oh, I don't know 'bout that.
    max-slope.png

  20. #20
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    Quote Originally Posted by RChung View Post
    Oh, I don't know 'bout that.
    max-slope.png
    Yes, an interpolated elevation profile will appear nice and smooth. Differentiate it (ie, figure out the grade) and you get this: (red and blue is each run)

    coast.png

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    Quote Originally Posted by stedalus View Post
    Yes, an interpolated elevation profile will appear nice and smooth. Differentiate it (ie, figure out the grade) and you get this: (red and blue is each run)

    coast.png
    Ah, we must be calculating the dv/dt slightly differently. Here are my slopes:

    slopes.png

  22. #22
    fuggitivo solitario echappist's Avatar
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    i'm gonna throw this a bit OT

    now that we got rchung on the board, i'd like to ask him how well the virtual elevation should fit the real elevation for the data to have meaning. Would something like the two examples above be usable or not?

    Also, something about the way elevation is calculated. Assuming that temperature and pressure stays pretty constant, how trustworthy is it to use a computer that calculates altitude by using a barometer (e.g. Joule 2.0)?

  23. #23
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    Quote Originally Posted by RChung View Post
    Ah, we must be calculating the dv/dt slightly differently. Here are my slopes:

    slopes.png
    If you can look at that graph and think that you know the grade down to 0.1%, more power to you. (Pun not intended. I'll quit here.)

  24. #24
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    Quote Originally Posted by stedalus View Post
    If you can look at that graph and think that you know the grade down to 0.1%, more power to you. (Pun not intended. I'll quit here.)
    Yeah, maybe 0.1% is a tad optimistic -- but there's something wrong with your dv/dt. I'm guessing you were just using a simple difference? Real roads don't change slopes as drastically as you're estimating.

  25. #25
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    Quote Originally Posted by echappist View Post
    i'm gonna throw this a bit OT

    now that we got rchung on the board, i'd like to ask him how well the virtual elevation should fit the real elevation for the data to have meaning. Would something like the two examples above be usable or not?
    Well, as I said, there was a tiny amount of wind that we assumed away. The real reason not to simply solve the work conservation equation and stop is because we only had two equations for two unknowns (CdA and Crr) so there isn't a simple way to determine "goodness of fit." That's why I usually recommend doing a VE profile: it provides a diagnostic about areas of fit and misfit.

    Also, something about the way elevation is calculated. Assuming that temperature and pressure stays pretty constant, how trustworthy is it to use a computer that calculates altitude by using a barometer (e.g. Joule 2.0)?
    I don't have any experience with the Joule 2.0's altimeter. In theory, for short test distances it should be quite good (though they often have a reported resolution of a meter or so -- internally a barometric sensor has much higher precision than that).
    http://jasperga.blogspot.com/2009/11...s-no-joke.html

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