Accelerometer Question (Caution! On Topic!)

In candor,
I don't care about Brad Guth's credentials.
I don't care what liberals believe.
I don't care if Daniel Min goes to hell.

Riddle me this, Batman!
Consider a rocket with a single-axis accelerometer
as a payload. As the rocket sits horizontally on its
pre-launch rack, the instrument registers 0 g. As the
rocket sits on the pad oriented before launch, the
accelerometer registers +1g, indicating that it thinks
the rocket is accelerating upward at this rate. Actually,
the reading is caused by gravity.
The rocket ascends vertically under power. While still
vertically oriented, it burns out. A few seconds after
burnout, it is still vertically oriented, and the
accelerometer registers -2g.
Questions:
1) How much of the -2g is gravity and how much is drag?
2) What is the rocket's rate of acceleration?
-Larry Curcio
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Larry Curcio wrote:

All of the 2g is due to drag. After burnout with no air the rocket would be in free-fall and an accelerometer would register 0 g.

With respect to the Earth, the rocket is slowing down at 3g.
- Robert Galejs
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Larry Curcio wrote: > ... As the > rocket sits on the pad oriented before launch, the > accelerometer registers +1g, indicating that it thinks > the rocket is accelerating upward at this rate.
This is misleading. The accelerometer has been calibrated to read "+1g" when upright, but that doesn't mean it is accelerating. In fact, a reading of "+1g" means the rocket is not accelerating.
The reading of "0g" when the single-axis accelerometer is horizontal just shows the problem using a single-axis accelerometer--information (orientation) is missing and thus the reading is "bogus". A reading of "0g" should mean acceleration of 1g downward (or -1g), but of course the rocket is not moving, just lying on its side. Any angle from vertical produces a bogus reading.
> The rocket ascends vertically under power. While still > vertically oriented, it burns out. A few seconds after > burnout, it is still vertically oriented, and the > accelerometer registers -2g. > > Questions: > > 1) How much of the -2g is gravity and how much is drag? > 2) What is the rocket's rate of acceleration?
It may be less confusing to think about this with respect to the forces involved, then compute the acceleration.
The total forces acting on the rocket at various times (but not all at the same time) are T (thrust), D (drag), W (weight) and P (upward push from the pad). Before launch, T and D are 0 and you have
W + P = m * a
The rocket isn't accelerating, so a = 0 and thus W = -P. But the accelerometer is biased by +1g, so to get the true acceleration you subtract 1g from the reading.
After burnout, the forces on the rocket are D and W.
D + W = m * a or a = D/m + W/m
Again, to get the true acceleration you need to subtract the bias from the measured acceleration (called "a" below):
a = "a" - 1g = -2g - 1g = -3g = D/m + W/m
Both drag and weight are contributing to the -3g acceleration.
But we know W = -mg, so
-3g = D/m - g, or -2g = D/m
So -2g worth of the total acceleration is from drag, as Robert said.
--
Steve Humphrey
(replace "spambait" with "merlinus" to respond directly to me)
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Very good! Two correct answers out of two!
Interesting that you bring up the off-vertical issue, because I've been playing with it for awhile. I have been able to tease information on the sine of the angle with the horizontal from combined accelerometer/altimeter data. Some datasets are too noisy, but others work out rather well. I posted a graph on alt.binaries.models.rockets. (FWIW The post was clean when I put it there.)
Data were first smoothed with a Savitsky-Golay filter. Order = 3; Window Size = 61 Data are from an AltAcc altimeter. Sampling interval = .0625 seconds. They were processed by the following method:
(According to traditional numeric differentiation)
VerticalVelocity(i) = [Alt(i+1) - Alt(i-1)]/[2*DeltaT]
ObliqueVel(i) = ObviqueVel(i-1) + 0.5*[Acc(i) + Acc(i-1)]*DeltaT
Acc(i) = AccReading(i) - g*SinTheta(i)
And
SinTheta(i) = VerticalVelocity(i)/ObliqueVelocity(i)
Sin of Theta was then obtained as a quadratic equation. (I erroneously said, "In velocitiy" on the binaries site. In fact, have been sick for some days, and my head is mush.)
Notes: Launch rod was oriented vertically The initial dip is present in all data. It is not real. On fast flights, it can be quite deep. There is typically a brief rebound afterwards. Sines in the rebound region can exceed unity. It appears to be an aerodynamic anomaly with the altimeter. It makes powered flight useless for such analysis. I believe it makes the analysis best applicable to flights that start out vertical If the rocket does not begin to veer significantly until coast, one can approximate the orientation in powered flight by vertical flight.
Note that sin(76 Degrees) = .97, so if the rocket is visibly vertical in powered flight, error is very small. Note also that drag measurement itself is orientation-independent, although the associated velocity measurement (for Cd) is not.
Had been trying to account for short impulses derived from accelerometer flights. These do not appear to be caused by off-vertical orientation, as far as I can tell. Had hoped I'd find a bug in my rendering of drag. Thought I had found it. Nope. My head is just mush.
Anyway, Kudos to Robert and Steve. -Larry Curcio

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Larry Curcio wrote:

Jeez Louise, this was a test? I would have worn my hat if I knew it was a test.

Barometric altimeter lag?
--
Steve Humphrey
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Lag? Maybe, but I tried displacing the altimeter data every which way. No avail.
Coupla unrelated things I meant to mention:
1) When fully corrected for orientation, the doubly-integrated altimeter no longer registers altitude. It registers the length of the curve, which exceeds altitude. The altitude in this analysis actually comes entirely from the altimeter - it's reconstructed from the smoothed vertical velocity.
2) If a rocket fishtails going up, it is indistinguishable from a rocket dipping repeatedly to one side. Thus, in some cases, only flight notes or intuition can tell you how to interpret the data.
-Larry (And no. It ain't a composite thrust curve :-) C.

rebound
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Larry Curcio wrote:

Let me see if I understand this correctly. You are deriving a velocity estimate from the pressure data and comparing that with the velocity derived by integrating acceleration. From that comparison you infer an angle from vertical.
Earlier you were asking for data ("Accelerometer/Altimeter Instruments" of 13 March), I presume in connection with this effort, so I am curious why you used AltAcc data which you claimed was "terrible". I pointed you to some of my RDAS data which should have worked.
Some of my RDAS data sets also include three axis magnetic data which could be processed to produce a measured angle to compare with the inferred value. I worked through the math to do the calibration a few years ago but never finished the task. It is non-trivial. :-)
--
David W. Schultz
http://home.earthlink.net/~david.schultz /
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