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  1. #1
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    Extracting Cd from accelerometer data

    Do any of the accelerometer altimeters compute drag coefficient or have a post-flight utility to do so? I checked the awesome Altimeter Guide sticky and did not notice this feature listed.

    I want something like this to compare to simulations, like Chuck Rogers posted:

    http://rasaero.com/hidden/img/fc-Vio...eementSS_4.jpg

    I am a numerical simulation guy and can probably figure out how to crunch the data. However, I don't need to reinvent the wheel if somebody already has a tool to share.

    Thanks!


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    OK, I'm curious. How do you calculate the Cd from acceleration? What is the formula?


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    Quote Originally Posted by sooner.boomer View Post
    OK, I'm curious. How do you calculate the Cd from acceleration? What is the formula?
    F=ma and Cd=drag/(0.5*rho*A*V**2)

    Where F is comprised of thrust, drag, and weight. Need some calculus to integrate acceleration to velocity and to distance. Then isolate drag force from the other forces. All of this is varying over time and space. Need numerical methods to solve. I don't think there are any closed form solutions to all this. There is an Apogee newsletter that describes some simpler ways to get drag coefficient from acclerometers, but I am looking for the complete solution.

    Where I get a little confused is the coordinate system of the acclerometer altimeter. What is recorded - one axis, two axis, resultant only? I have only used baro altimeters to this point, but looking to invest in an accel unit.
    Last edited by Buckeye; 19th April 2015 at 08:36 PM.

  4. #4
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    The Marsa54(L) and its associated connection software will do this for you. It will also plot the thrust curve of your motor.
    John Derimiggio NAR/TRA L3
    MarsaSystems

  5. #5
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    Quote Originally Posted by Buckeye View Post
    F=ma and Cd=drag/(0.5*rho*A*V**2)

    Where F is comprised of thrust, drag, and weight. Need some calculus to integrate acceleration to velocity and to distance. Then isolate drag force from the other forces. All of this is varying over time and space. Need numerical methods to solve. I don't think there are any closed form solutions to all this. There is an Apogee newsletter that describes some simpler ways to get drag coefficient from acclerometers, but I am looking for the complete solution.

    Where I get a little confused is the coordinate system of the acclerometer altimeter. What is recorded - one axis, two axis, resultant only? I have only used baro altimeters to this point, but looking to invest in an accel unit.
    Right. So to solve the equation, not only do you need to know the value measured by the accelerometer, you also need to know the cross sectional area and the density of air - which varies in a non-linear value with altitude. It takes knowing variables you can't know.

  6. #6
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    Quote Originally Posted by sooner.boomer View Post
    Right. So to solve the equation, not only do you need to know the value measured by the accelerometer, you also need to know the cross sectional area and the density of air - which varies in a non-linear value with altitude. It takes knowing variables you can't know.
    Altimeters measure air pressure and the corresponding air density can be calculated from the standard atm model. So all the parameters are available for calculating cd.
    John Derimiggio NAR/TRA L3
    MarsaSystems

  7. #7
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    Quote Originally Posted by sooner.boomer View Post
    Right. So to solve the equation, not only do you need to know the value measured by the accelerometer, you also need to know the cross sectional area and the density of air - which varies in a non-linear value with altitude. It takes knowing variables you can't know.
    Sure you do. Cross sectional area - easy. Air density is also easily computed from the 1977 Standard Atmosphere (or similar). Yes, all the variables are interconnected and non-linear. That's why numerical integration is needed.
    Last edited by Buckeye; 20th April 2015 at 02:30 AM.

  8. #8
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    Quote Originally Posted by jderimig View Post
    The Marsa54(L) and its associated connection software will do this for you. It will also plot the thrust curve of your motor.
    OK, cool.

  9. #9
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    Isn't the acceleration seen by the altimeter the net of thrust-drag? How can you back out the drag without knowing the motor thrust? You could use the published thrust curves, but wouldn't you have considerable error?
    Charles McGonegal
    Ciderwright at AeppelTreow Winery & Distillery
    www.appletrue.com
    NAR #103560 L1 6/25/17 Estes Leviathan CTI H175-SS
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  10. #10
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    During the COAST phase (burnout to apogee) the rocket is only slowed by drag and gravity.
    Subtract out gravity and you get drag.
    Watch the velocity drop and you get drag as a function of velocity....
    Fred Azinger

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    That should have been obvious. Thanks for supplying the clue.
    Charles McGonegal
    Ciderwright at AeppelTreow Winery & Distillery
    www.appletrue.com
    NAR #103560 L1 6/25/17 Estes Leviathan CTI H175-SS
    Ad Astra Tabernamque!

  12. #12
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    Quote Originally Posted by FredA View Post
    During the COAST phase (burnout to apogee) the rocket is only slowed by drag and gravity.
    Subtract out gravity and you get drag.
    Watch the velocity drop and you get drag as a function of velocity....
    When I was in college we had an engineering project to design bicycle fairings, this is how we calculated the Cd. We took it to a really flat area, ran it up to a known velocity, then timed how long it took to coast to a stop, with and without the fairing. We added a sandbag without the fairing so the weight and rotational friction were the same, the difference was due solely to aerodynamic drag, or at least it should have been close.

  13. #13
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    OK, since you know altitude at known time intervals, why can't you calculate the Cd from the coast time data from a baro altimeter since you can determine the speed and rate of deceleration based on the data?
    Handeman

    TRA #09903 L3 :D 3/29/2015

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  14. #14
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    Quote Originally Posted by cerving View Post
    When I was in college we had an engineering project to design bicycle fairings, this is how we calculated the Cd. We took it to a really flat area, ran it up to a known velocity, then timed how long it took to coast to a stop, with and without the fairing. We added a sandbag without the fairing so the weight and rotational friction were the same, the difference was due solely to aerodynamic drag, or at least it should have been close.
    Yep, same concept.

    In the automotive biz, we call this "coast down testing" (imagine that!) for fuel economy certification stuff. However, it is not a very precise measure of aerodynamic drag. Official Cd's are quantified in the wind tunnel or CFD.

  15. #15
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    Quote Originally Posted by Handeman View Post
    OK, since you know altitude at known time intervals, why can't you calculate the Cd from the coast time data from a baro altimeter since you can determine the speed and rate of deceleration based on the data?
    To go from distance to velocity to acceleration requires numerical differentiation, which is inherently noisy and inaccurate. Going from acceleration to velocity to distance uses numerical integration which is much more accurate.

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    Quote Originally Posted by jderimig View Post
    Altimeters measure air pressure and the corresponding air density can be calculated from the standard atm model. So all the parameters are available for calculating cd.
    You've got to know a lot more about the atmosphere than just the pressure to calculate the density. Humidity (and other gas composition) is one factor, along with temperature. And the value of these variables at the surface is not their value at any other altitude.

  17. #17
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    Quote Originally Posted by FredA View Post
    During the COAST phase (burnout to apogee) the rocket is only slowed by drag and gravity.
    Subtract out gravity and you get drag.
    Watch the velocity drop and you get drag as a function of velocity....
    You get drag as a function of gravity, but you do not get an accurate measurement of the Cd. You also get the density of air as a function of gravity because at that point, the other variables (cross sectional area, Cd, are constants (assuming you are below trans/super-sonic)).

  18. #18
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    Quote Originally Posted by Buckeye View Post
    To go from distance to velocity to acceleration requires numerical differentiation, which is inherently noisy and inaccurate. Going from acceleration to velocity to distance uses numerical integration which is much more accurate.
    Differentiation and integration are inverse functions of each other. I've found it's a lot easier to do instantaneous calculations of instantaneous differentials a lot easier than integrals because of that "C" offset value in the calculation in the summation series. You can compensate for it, but it's usually done iteratively in post.

  19. #19
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    You want a fun calculation for drag?

    It's Spring time, and storms bring hail.

    Calculate the curve for wind velocities to suspend a hailstone from a diameter ranging from 1/8" to 3". Use metric if it makes it easier. Standard values for (pure) water/ice densities.

  20. #20
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    Sooner:

    Why are you fighting the facts? Extracting Cd from acceleration data is well-known, as shown in examples in the above posts. People do it.

    No, drag is not a function of gravity. Yes, one can compute density from the standard atmosphere model. Lastly, you are confusing well-behaved analytical integration/differentiation with numerical. Take your barometric altitude data and try differentiating it twice to get acceleration. What do you get? Junk.

    Do you how Rocksim, OpenRocket, RASP, or RASAero actually work? You input Cd and other stuff, out comes acceleration. In this case, acceleration is measured from the altimeter, and you back out Cd. Same process, kinda reversed.

    You may want to read up on the workings of altimeters, simulation software, differential equations, and numerical methods.
    Last edited by Buckeye; 21st April 2015 at 12:36 PM. Reason: more words

  21. #21
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    Quote Originally Posted by sooner.boomer View Post
    You get drag as a function of gravity, but you do not get an accurate measurement of the Cd. You also get the density of air as a function of gravity because at that point, the other variables (cross sectional area, Cd, are constants (assuming you are below trans/super-sonic)).
    Out of curiosity, when you say that "you do not get an accurate measurement of the Cd", how far is it off as a percentage? 5%, 10%, 25%?

    Also, do you have side-by-side comparisons showing one method vis-a-vis the other?

    Again, just curious.

    Greg

  22. #22
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    Quote Originally Posted by Buckeye View Post
    Sooner:

    Why are you fighting the facts? Extracting Cd from acceleration data is well-known, as shown in examples in the above posts. People do it.

    No, drag is not a function of gravity. Yes, one can compute density from the standard atmosphere model. Lastly, you are confusing well-behaved analytical integration/differentiation with numerical. Take your barometric altitude data and try differentiating it twice to get acceleration. What do you get? Junk.

    Do you how Rocksim, OpenRocket, RASP, or RASAero actually work? You input Cd and other stuff, out comes acceleration. In this case, acceleration is measured from the altimeter, and you back out Cd. Same process, kinda reversed.

    You may want to read up on the workings of altimeters, simulation software, differential equations, and numerical methods.
    I'm very familiar with the methods and math. My job for many years was designing and building instrumentation to measure human performance. We were some of the first people to measure human performance using accelerometers. A number of papers and research projects have been published. I personally did a poster presentation in '94 at a conference in Denver on this.

    Back to the original post. The graph http://rasaero.com/hidden/img/fc-Vio...eementSS_4.jpg was mentioned. Look at that graph. Between 0 and ~Mach 1 there should be a straight line. The value of the Cd is a constant and should not change. Yet the graph shows change.

    Second, and I probably should have been more clear about this, you cannot empirically derive the Cd with (only) a measurement of acceleration (actually I think were looking at the way the body slows down under drag, so it might be more accurate to say deceleration or negative acceleration). This (deceleration) is affected by air density. In the equation for drag, is is shown as a constant, but in reality, it is not a constant value. You can get "close" to the value of the actual Cd. How close? There are many things that will contribute errors, so it's hard to say. Is "close" good enough for you? Why do you need the data, and what will you do with it?

    I'm not trying to be a pedant, but at least this discussion (I hadn't intended for it to be an argument) has at least gotten some of you thinking about what goes on.

  23. #23
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    Quote Originally Posted by GregGleason View Post
    Out of curiosity, when you say that "you do not get an accurate measurement of the Cd", how far is it off as a percentage? 5%, 10%, 25%?

    Also, do you have side-by-side comparisons showing one method vis-a-vis the other?

    Again, just curious.

    Greg
    A very good question! Also, how accurate are the accelerometers? As previously posted, what are you trying to do with the data? Is "close" good enough?

  24. #24
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    Quote Originally Posted by FredA View Post
    During the COAST phase (burnout to apogee) the rocket is only slowed by drag and gravity.
    Subtract out gravity and you get drag.
    Watch the velocity drop and you get drag as a function of velocity....
    Long story. Accelerometers donít really sense gravity at all. Yes yes. I know they register 1g when the rocket is vertically oriented on the launch pad. That is actually a measure of thrust. What thrust? The thrust the pad imposes on the bottom of the rocket Ė and itís equal to the weight of the rocket. The instrument thinks itís in free space (It doesnít sense gravity, remember?), and in free space, that much thrust would accelerate the rocket at the rate of 1g. You can replace the launch pad by a motor thrusting upward with thrust equal to weight and you get the same reading. Itís thrust.

    What does this mean? It means that in the coast phase, the entire accelerometer reading is drag acceleration. This is true REGARDLESS of the orientation of the rocket with respect to the vertical.

    Does that mean that the orientation of the rocket doesnít affect your Cd reading? No. gravity doesnít affect accelerometer reading, but it does affect the motion of the rocket. In this case, it affects the velocity that you associate with the drag acceleration to tease out Cd.

    (See my 2007 NARAM R&D report for further information.)

  25. #25
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    Quote Originally Posted by sooner.boomer View Post
    You've got to know a lot more about the atmosphere than just the pressure to calculate the density. Humidity (and other gas composition) is one factor, along with temperature. And the value of these variables at the surface is not their value at any other altitude.
    pv=nRT. n/v = RT/p v ~ 1/density. At any altitude during ascent p is measureable, if you know the atmospheric make-up including humidty n can also be estimated with reasonable accuracy, T is also measureable but often not. T can be assumed from the standard atmospheric model. There will be some error because of deviation from the model but the same error will also be equally evident is altitude readings but no seems to be overly concerned about that.....

    More on the calcuation of air density with available known measureable parameters here http://en.wikipedia.org/wiki/Density_of_air

    The value of the Cd is a constant and should not change
    .
    Incorrect, Cd is not constant. It varies with Reynold's number of the flow.

    Out of curiosity, when you say that "you do not get an accurate measurement of the Cd", how far is it off as a percentage? 5%, 10%, 25%?

    The same calculations to estimate Cd are used to estimate motor thrust. In my experience I have been able to produce thrust curves estimates that are within 10% of the published motor thrust test stand data. So I estimate that Cd can be calculated with 10% of true value. Most of that error can attributed to deviations from the S.A.M. and acceleration errors from verticallity assumption.
    Last edited by jderimig; 21st April 2015 at 07:15 PM.
    John Derimiggio NAR/TRA L3
    MarsaSystems

  26. #26
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    Quote Originally Posted by jderimig View Post

    ... Incorrect, Cd is not constant ...
    The curve below bears out the above statement. The Cd changes with velocity.

    Click image for larger version. 

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    Click image for larger version. 

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    Greg

  27. #27
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    Quote Originally Posted by sooner.boomer View Post
    Second, and I probably should have been more clear about this, you cannot empirically derive the Cd with (only) a measurement of acceleration.... In the equation for drag, is is shown as a constant, but in reality, it is not a constant value.
    You are being much too literal and over-simplifying the equations. Nobody ever said you can simply convert an acceleration into drag with a one-variable formula. The entire flight trajectory must be computed, as I said here:

    Quote Originally Posted by Buckeye View Post
    ...All of this is varying over time and space. Need numerical methods to solve....
    Anyway, back to my original post. I am not asking about the accuracy of anything. I am looking for a worksheet, code, snippet, or program that backs out CD vs. Mach from the flight data and other user-supplied inputs. So far, only jderimig replied with a useful, on-topic answer - the MARSA54L.

  28. #28
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    Clarification. The Marsa54L software gives you an Cd coefficient not a Cd versus velocity curve. You select a point in time after motor burnout at the Cd coefficient is estimated based on the velocity at that point. Usually for subsonic flights the Cd is relatively unchanged but you can sample more points to get an idea of the curve..
    John Derimiggio NAR/TRA L3
    MarsaSystems

  29. #29
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    As part of a 4 week short course I helped teach out of the Air Force Test Pilot School from 2000 through 2005, the students flew Caliber Isp high power rockets with Black Sky AltAcc accelerometers/barometric altimeters. Using the onboard acceleration data and barometric altimeter data, and a Matlab program run by the students, the students backed out power-off CD versus Mach number and the thrust curve of the rocket motor from their flights. Each team presented a briefing with their flight data and CD and thrust data a few days after the flights.

    The slides from the class which include the AltAcc instrumentation and the equations and calculation techniques for backing out CD and motor thrust are attached.


    Chuck Rogers
    Attached Files Attached Files

  30. #30
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    Quote Originally Posted by Chuck Rogers View Post
    As part of a 4 week short course I helped teach out of the Air Force Test Pilot School from 2000 through 2005, the students flew Caliber Isp high power rockets with Black Sky AltAcc accelerometers/barometric altimeters. Using the onboard acceleration data and barometric altimeter data, and a Matlab program run by the students, the students backed out power-off CD versus Mach number and the thrust curve of the rocket motor from their flights. Each team presented a briefing with their flight data and CD and thrust data a few days after the flights.

    The slides from the class which include the AltAcc instrumentation and the equations and calculation techniques for backing out CD and motor thrust are attached.


    Chuck Rogers
    Nice summary, Chuck. Though, I can't say I like your choice of units!

    Thanks.


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