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Bifilar ballistic pendulum and accelerometer
#1
Shocked 
Cool  one more experiment...

For bifilar ballistic pendulum (the geometry presented in most textbooks) it is possible to use an accelerometer for graphical visualization of its oscillations upon a bullet impact and, thus, determination of the bullet speed. The sensor does not rotate and its measurements of the horizontal acceleration are not affected by the gravitation. Another advantage of the bifilar pendulum is independence of the result from the exact location of the bullet impact.

The photo shows the pendulum (a carton tube with double-V suspension to a support), the children spring gun and the projectiles. The sensor tag CC2650STK of Texas Instruments is visible inside the pendulum bob at the left. The projectiles are captured at the right.

The given phyphox program isolates the acceleration measurements after the impact and determines the acceleration amplitude and the pendulum frequency. Using the equation shown in the final graph (Python) it calculates the bullet speed.

The result is verified by a slightly modified version of the phyphox acoustic stopwatch which measures the time between the shot and the impact to a target placed at 3 m from the gun. The accordance is reasonable.

There is A PROBLEM: phyphox does not have a fitting block and FFT is used to determine the oscillation frequency. The result is not very precise even if a lot of oscillations are registered. To the contrary, Python fit by a sinusoidal function can determine the frequency just from some swings with a very high precision (the last figure).

More details can be found in a following paper. Any comments are welcome as usually.

               

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Attached Files
.phyphox   Ballistic pendulum 3.phyphox (Size: 15.56 KB / Downloads: 24)
.phyphox   Velocity using Acoustic Stopwatch.phyphox (Size: 69.1 KB / Downloads: 22)
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#2
Excellent timing: we have just discussed such a pendulum a few days ago when analysing students' setups for the “pendulum” experiment. Smile

However, we have not really pursued the math. As far as I know, this is Ackley's ballistic pendulum. Are you aware of references that go beyond the energy conservation part? If the length of the linkage is (and stays) identical, it should perhaps behave like a simple gravity pendulum even for unequal mass distribution thanks to all the constraints…

The width of the FFT “signal” appears to be rather wide that might be an issue here. The algorithm for the autocorrelation graph of the “pendulum” experiment should work better: take the distance between the first two maxima (in the debug tab) as an estimate, set windows for the next maxima, and determine the right most maximum. Then divide the elapsed time by the multiplicity of it.

Taking the zeroes would be more precise – possibly going beyond the capabilities of phyphox' analysis tool.

p.s. the second FFT “signal” could be attributed to the damping?
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#3
Thank you for comments and suggestions, Jens.

I will look for Ackley's pendulum. Yes, such a pendulum really behave as a simple gravity pendulum. I found a nice demonstration of this, but Ackley was not mentioned.

I thought about the autocorrelation, but with the rathe small noise it shoud not be better FFT or even simple direct measuring the signal period. (??) Fit gives much higher precision.
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#4
I am sorry for the misunderstanding: I meant to refer just to the algorithm for the autocorrelation graph (rather than autocorrelation itself) utilized to the top graph in your debug tab. With ten periods, the accuracy should be sufficiently high.

(nevertheless agreeing on regression/fit being an enhancement – unfortunately like many other things…)
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