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Math Help - Calculating Cd and Crr coefficient experimentally

  1. #1
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    Calculating Cd and Crr coefficient experimentally

    On this site I found how to calculate Cd and Crr coefficient from experimantal data:
    Measure the drag coefficient of your car


    I am trying to implement it in a web page and in a smartphone application, so I must "replace" Excel Solver by an algorithm, but I'm very in trouble.
    I figured out how to calculate coefficients of fitting polynomial, so I now have the equation which approximates the speed curve, in form of (1) y = ax^2+bx+c


    But now how do I correlate this equation to real data to obtain Cd and Crr?


    The excel sheet relies on this formula:
    v1 = v0 - a*t
    where:
    a = F/m
    and
    F = air drag + roll friction
    air drag = 0.5 * r *A * Cd * v^2
    roll friction = m * g * Crr


    so I think it is:


    v1 = v0 - 0.5 * r * A * Cd * v0^2 / m - g * Crr


    if this is correct, I can "zip" it to:


    (2) v1 = v0 + K v0^2 + H


    and


    K v0^2 + H = v1-v0


    My idea is to use the next sample to build an equation couple which I can solve for K and H, and then get Cd and Crr from them... but it looks like it does not work, what am I doing wrong?


    K v0^2 + H = v1-v0
    K v1^2 + H = v2-v1

    ==> K, H ==> Cd, Crr



    Any idea?
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Calculating Cd and Crr coefficient experimentally

    Quote Originally Posted by jumpjack View Post
    so I think it is:

    v1 = v0 - 0.5 * r * A * Cd * v0^2 / m - g * Crr
    You have left time out of the equation - it should be:

    v_1 = v_0 - \frac {\rho AC_dv_o^2t}{2m} - g C_{rr} t

    Note that I changed your "r" to  \rho.

    The rest looks fine. However, be aware that since air drag is a function of velocity, by using v_o^2 in the above you are making just an estimate of the actual drag that occurs as velocity changes from v_0 to v_1. Consequently the values for the constant that you calcuate will change with each set of data. One suggestion: instead of using v_0 in the drag formula use the average of v_0 and v_1. Also use small values of t so that the difference between v_0 and v_1 is minimized.
    Last edited by ebaines; February 25th 2014 at 01:41 PM.
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    Re: Calculating Cd and Crr coefficient experimentally

    Quote Originally Posted by ebaines View Post
    be aware that since air drag is a function of velocity
    Indeed this is what is driving me mad...


    Anyway I didn't understand which method you are suggesting.

    Does it exist a tool which allows performing symbolic calculation? It would be error-proof, unlike me...
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  4. #4
    MHF Contributor ebaines's Avatar
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    Re: Calculating Cd and Crr coefficient experimentally

    Quote Originally Posted by jumpjack View Post
    Anyway I didn't understand which method you are suggesting.

    Does it exist a tool which allows performing symbolic calculation? It would be error-proof, unlike me...
    I thought you had a method already figured out. I assume you have a set of data: v_0, v_1, v_2, etc for times t_o, t_1, t_2, etc. You can set up two equations - one using v_0 and v_1 for t_0 and t_1, and the other using some other pair of velocities, such as v_7 and v_8 at t_7 and t_8 - and then solve for the two unknowns C_d and C_rr:

     K v_0^2 + H = \frac {v_0 -v_1}{t_0-t_1}

     K v_7^2 + H = \frac {v_8-v_7}{t_8-t_7}

    To solve for K and H start by subtracting the 2nd equation from the first to get:

    \frac {v_0-v_1}{t_0-t_1 } - \frac {v_8-v_7}{t_8 -t_7} = K (v_0^2-v_7^2)

    From this you can determine the value of K, and hence C_d. To get the value of H substitute the known value of K into either of the two initial equations and solve for H, and from there determine C_rr.
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    Re: Calculating Cd and Crr coefficient experimentally

    Quote Originally Posted by ebaines View Post
    I thought you had a method already figured out. I assume you have a set of data: v_0, v_1, v_2, etc for times t_o, t_1, t_2, etc. You can set up two equations - one using v_0 and v_1 for t_0 and t_1, and the other using some other pair of velocities, such as v_7 and v_8 at t_7 and t_8 - and then solve for the two unknowns C_d and C_rr:

     K v_0^2 + H = \frac {v_0 -v_1}{t_0-t_1}

     K v_7^2 + H = \frac {v_8-v_7}{t_8-t_7}
    As time intervals are constant and = 10, I included them in K and H, considering them like DeltaT rather than t. Maybe this is my mistake?? I thought the excel sheet considered DeltaT=10 rather than different valuse of t, but maybe I got it wrong.


    (1) v_1 = v_0 - at =
    (2) v_0 - \frac{F}{m} t =
    (3) v_0 - \frac{\frac{1}{2} \rho AC_dv_0^2 + m g C_r_r}{m} t
    (4) v_0 - (\frac{1}{2} \frac{\rho AC_dv_0^2 }{m})t - gC_r_rt

    Different grouping:

    (5) v_1 = v_0 + (- \frac{1}{2}\frac{\rho AC_dt}{m})v_0^2 + ( - gC_r_rt)

    (6)  K = -  \frac{1}{2} \frac{\rho AC_dt}{m}


    (7) H = -gC_r_r t

    (8) v_1 = v_0  + Kv_0^2 + H
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: Calculating Cd and Crr coefficient experimentally

    Looks good. Let us know how it works out.
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    Re: Calculating Cd and Crr coefficient experimentally

    It works out very bad, that's why I am here! :-(

    But maybe Mathics can help me being sure about calculation steps...
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  8. #8
    MHF Contributor ebaines's Avatar
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    Re: Calculating Cd and Crr coefficient experimentally

    What you've done is correct. Tell you what - share some of your data - various v's at various t's - and your calculations of resulting C_d and C_rr and we'll see if we get the same values.
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    Re: Calculating Cd and Crr coefficient experimentally

    Samples fot time 0, 10, 20,... 70 secs:
    19,44
    16,76
    14,31
    12,08
    10,35
    8,92
    7,55
    6,18


    Excel Solver results:
    Cd = 0,538210351
    Crr = 0,01055849

    My results:
    0,373384937
    0,016317342


    My "alternative" results:
    0,38092961
    0,015870614

    (sorry for commas but I use Italian version of Excel...)
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  10. #10
    MHF Contributor ebaines's Avatar
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    Re: Calculating Cd and Crr coefficient experimentally

    Couple of points:

    1. First, what vaues are you using for rho, A and m?
    2. Next, because of the built in uncertainty in your data, I find that calculations for both K and H vary wildly. This is because the data is not perfect - some variability is expected as the points don't all fit on a single curve thatfits your mathemtaical model. I find that depending on which points you pick the calculated value for K can vary from 0.00024 to 0.00085, and H can vary from -0.049 to -0.178, or by almost a factor of 4! So you now need to decide how to handle the variability in results - for example you can decide to average the results, which yields K = -0.00039 and H = 0.118, whihc gives a Crr of 0.012.
    3. The consistency of your results will be improved greatly by taking data for many runs and averaging the results. I suggest using data from at least ten runs and see what happens.
    Last edited by ebaines; February 26th 2014 at 01:37 PM.
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  11. #11
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    Re: Calculating Cd and Crr coefficient experimentally

    As I said in first post, I'm using data in excel sheets linked in this page:
    Measure the drag coefficient of your car

    direct link:
    http://www.iwilltry.org/b/wp-content...oefficient.xls

    Data:
    time V1 V2 V3 V4 V5 V6
    sec kph kph kph kph kph kph
    0 70 70 70 70 70 70
    10 61 60 60 60 61 60
    20 52 52 51 51 52 51
    30 44 44 43 43 43 44
    40 37 37 38 37 37 37,5
    50 32 32 32,5 32
    60 27 27,5 27
    70 22 22,5

    Constants:
    rho 1,22 kg/m^3
    g 9,81 m/s^2

    I used several values for A and m; last run:

    A 2,2 m^2
    M 1500 kg

    Results:

    Cd: 0,53788
    Crr: 0,01056


    I also tried manual tuning of Cd and Crr, by visually looking at resulting chart for $y_1=y_0+Ky_0^2+H$ ; when I have the best fitting curve, I have:
    Cd = 0,63
    Crr = 0,00967

    Results:
    x y
    0 19,3896 0%
    10 16,7446 +1%
    20 14,3596 +1%
    30 12,2346 0%
    40 10,3696 -1%
    50 8,7646 -2%
    60 7,4196 -1%
    70 6,3346 +3%
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  12. #12
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    Re: Calculating Cd and Crr coefficient experimentally

    Sorry, results in last post are wrong, here are right results:
    0 19,44 0%
    10 16,36130078 -1%
    20 13,90385368 -2%
    30 11,86561391 -3%
    40 10,12342241 -3%
    50 8,597156386 -3%
    60 7,231936874 -3%
    70 5,988521019 -3%


    The others were results for y=ax^2+bx+c with a=0,0013 , b=-0,2775 and c = 19,386 , repeated here:

    0 19,3896 0%
    10 16,7446 1%
    20 14,3596 1%
    30 12,2346 0%
    40 10,3696 -1%
    50 8,7646 -2%
    60 7,4196 -1%
    70 6,3346 3%
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