ArduinoMania

The speed-of-sound lab we did on 30 Dec 2011 went pretty well after all.

Coil (about 0.5H) with refrigerator magnet on top. Magnet is stuck to the core of the coil just by its own magnetism.

I used a setup inspired by the one in the Chapter 4 Lecture 3 video at http://courses.ncsu.edu/py581/common/podcasts/.  That is, a long metal bar tapped with a metal striker at one end.  A clock is started when the tap is made (a simple electrical connection), and stopped when the wave is detected at the other end.  As mentioned in my earlier post More on the slinky and the speed of sound, I used an electromagnet and a small refrigerator magnet to detect the sound wave.  The coil has a  68.3 Ω resistance and a laminated iron or steel core, and I estimated the inductance at about 0.5 Henry (the estimate  may easily be off by a factor of 2—I should measure it better some day when it matters). When I rested a piece of aluminum bar stock on the magnet and tapped the other end, I got a signal of about 0.3 v, which I could see clearly on my oscilloscope.  (Note: the analog view of the signal was not done in the lab with the students, because we were pressed for time and moving the analog oscilloscope out to the room worked in and setting it up would have taken too long.)

The signal is not large enough to be measured by a digital input on the Arduino that we used for timing, and the Arduino analog-to-digital converter (accessed with analogRead()) is a very slow one, that would limit our time resolution to about 100 µsec, rather than 4 µsec as we can get with the “micros()” function call. I happened to have an LM311 comparator chip from about 30 years ago, so I made a comparator circuit to convert the analog signal to a clean digital signal.

Comparator circuit used to convert the small electrical signal induced in the coil to digital levels for input to the Arduino. The pair of 15kΩ resistors serve as a voltage divider to set the bias voltage for the inputs to about 2.5 volts, in the middle of the range. The output pull-up resistor provides a load for the comparator. The two capacitors filter out high frequency signals picked up by the coil—they were not part of the circuit provided in the LM311 datasheet, but turned out to be essential.

I did have to modify the circuit a little from the one for a magnetic pickup given in the data sheet, as the output remained high with that circuit.  Adding small capacitors to the input and output seemed to fix things.  I arbitrarily used 47000pF capacitors, because I happened to have several of them, but I also experimented with some other sizes (560 000 pF and 1000pF) which did not work.  One effect of the capacitor on the input is to make a resonant circuit that rings with a period of about 800 µsec (eyeballed from the oscilloscope trace), which would make the inductance of the coil about 0.34H.  This ringing has a couple of consequences: 1) if the magnet is the wrong way around, the polarity of the impulse is reversed and the comparator will detect the impulse half a cycle later, adding about 400 µsec to the reading, and 2) the signal ramps up slowly in response to an impulse, and the delay in the comparator circuit is dependent on the magnitude of the input signal.  This will add noise to the timing measurements.

Because the magnet we used had residues of paper and glue on one side, it was easy to check that it was oriented correctly.  For one set of measurements, the coil and magnet had to be moved, and the magnet may have been upside down for that set of measurements, resulting in a different offset for those measurements.

To keep the noise from differences in amplitude to a minimum, we took several measurements (generally 10 or 20) of each time, discarding obviously bogus numbers (like 8 µsec when the comparator had already detected something before the strike, or > 10 msec, when an impulse had been missed).

On the Arduino, the following program was used measure times.  Pin 2 of the Arduino was connected with a long wire to a metal striker, with the object being struck connected to Arduino’s GND with another wire.  Contact between the two metal objects pulled pin 2 down (overpowering the 20kΩ pull-up in the Arduino), recording the time in start_1. As soon as the comparator detects the sound, pin 3 is pulled down, and the time is recorded on stop_1.  The resolution of the timer is about 4 µsec, but the repeatability of the times varied more, depending in part on how consistently the strikes were made (one of us appeared to have much more consistent technique than the others, and the times from his strikes had lower spread—we did not record who did the strikes on our data log, and so we can’t quantify this observation).

void setup()
{
  //  put a 20k pullup resistor on pin 3 (sound detector)
  pinMode(3,INPUT);
  digitalWrite(3, HIGH);
  //  put a 20k pullup resistor on pin 2 (striker)
  pinMode(2,INPUT);
  digitalWrite(2, HIGH);

  Serial.begin(115200);
}

void loop()
{
  Serial.println("Ready");

  // wait for pin 2 to go low (contact with striker)
  while (digitalRead(2)>0) {}
  long start_1=micros();

  // wait for pin 3 to go low (sound detected)
  while (digitalRead(3)>0){}
  long stop_1=micros();
  long diff = stop_1-start_1;
  if (diff > 10000)
  {   Serial.print(F("rejecting large delay: "));
  }
  Serial.print(diff);
  Serial.println(F(" microseconds"));
  delay(200);  // wait 200 msec

  // wait some more if the striker still in contact
  while (digitalRead(2)==0) {}

}

The vertical rod is the one being measured. The sensor is on the floor and the rod is struck at the top end. The breadboard has the comparator circuit connected to the Arduino, which in turn is connected to a laptop (not in the photo).

The setup for most measurements was simple: the coil was put on the floor with the magnet resting on top. A rod was held vertically on top of the magnet and struck at the far end with another rod. Initially, we used a small screwdriver as the striker, but this turned out to be hard to hold, and so we switched to using a foot-long piece of ¼” steel rod, which was also used for some of the timing tests.

Because the delay in the comparator is unknown, but likely to be substantial compared to the time of flight, we tried to measure the same material at different lengths, and do a straight-line fit of the data to estimate the offset. By doing many measurements at each length, we could average out a lot of the noise. We could also see how well the data fitted a model that assumed that the time delay for the sound arriving would be proportional to length—that is, does the speed-of-sound model make sense for this data?

The simplest set of data was for 3 rods ¼” in diameter, made of hot-rolled weldable steel (that’s all the information about the material that the hardware store had on the tag). I’ve put the data in a page on the blog: Steel rod speed of sound lab data.

We also measured one aluminum bar (Aluminum bar speed of sound lab data), one long copper tube (Copper tube speed of sound lab data), and two wood dowels (Wood speed of sound lab data).  Because wood is not conductive, we added a washer on the striking end of the dowel, to provide a conductive contact. The copper tube was about 3m (10′) long and very soft, so we laid it on the floor and duct-taped the sensor to one end.

Perhaps the most interesting data set comes from an aluminum ladder (Aluminum ladder speed of sound lab data).  We removed one foot from the ladder and duct-taped and bungey-corded the sensor to the bottom of the ladder.  We then struck the ladder inside the hollow rungs, providing a nicely spaced series of different ten different lengths.  Because of the difficulty in getting the duct tape to stick to the ladder, the magnet fell off and was replaced a few times in setting up the sensor.  The final orientation was not checked, but I believe that it was backwards, so that the delays of the sensor were substantially larger for the ladder than for the other measurements.  Luckily, there are enough different lengths that we can get a very good linear fit even with a large offset.

I did a fit for the steel-rod data using the following gnuplot script:

unset key

set title "Time of flight for compression wave in 1/4\" hot-rolled steel rod"
set xlabel "length (meters)"
set ylabel "time (seconds)"

set xrange [0:*]
set yrange [0:*]

fit a*x+c 'steel-rod.txt' using (0.01*$1):(1e-6*$2) via a,c

print "velocity=", 1/a, "m/s"

plot a*x+c, 'steel-rod.txt' using (0.01*($1+2*rand(0))):(1e-6*$2)

which produced an estimate of 5292 m/s for the speed of sound in the steel rod. I was noticing on the web that the speed of sound in thin rods may be a bit different from the speed of a planar wave in bulk steel, so I’m not sure what the “right” value is for this measurement, but it sure seems reasonably close to reported values around 5000–6000 m/s.  I also get from the same fit an estimate of the delay in the sensor and comparator of 33.7 µsec, which I can use for the aluminum bar and copper tube data. How does the fit look?  See for yourself:

The linear model seems like a pretty good fit for the data on the steel rod, but the scatter on the data is a little high. Note: small random jitter was added to the x values, in order to spread the points out.

If I remove outliers (the two largest and two smallest measurements from each length), I get a tighter fit (naturally), but one with is also likely to be more accurate, as outliers have a large influence in linear regression:

After eliminating the outliers, the estimated speed of sound is 5267 m/s and the sensor delay is 31 µsec.

Using a similar script, but with a fixed 31 µsec offset, for the aluminum bar data and the copper tube data (again eliminating the two largest and two smallest measurements), we get 4412 m/sec for the aluminum bar and 3857 m/s for the copper tube. (Of course, we don’t have anywhere near 4 significant figures, so I should probably round these to 4400 m/s for aluminum and 3900 m/s for copper.)

The ladder data used essentially the same script as the steel-rod, but even after censoring the data had a pretty wide spread:

The ladder data, after removing the two largest and two smallest times at each data point, got an estimated speed of sound in aluminum of 3955 m/s. The offset was 256.6 µsec, confirming that the magnet had been reversed. (It also implies that one period of the ringing is about 450 µsec, which is shorter than the period I thought I saw on the oscilloscope.)

If I fit the data from the wood dowel using the 31 µsec offset, the line does not fit the data at all well. If I fit with a 2-parameter model, I get an offset of 162.6 µsec (between the ones for the steel rod data and the ladder data), and a velocity of 6150 m/s, which is unusually high for wood. It is a very light wood, so perhaps the number is reasonable, but I’d be more comfortable if we had had more different lengths to test. The difference in the delay introduced by the sensor suggests to me that I should have used multiple lengths for each of the materials, despite the inconvenience (I only had one piece of aluminum bar and did not want to cut it—similarly, I did not want to cut the copper tube.)

Comparing our speeds with typical speeds from engineeringtoolbox.com:

The best match is for the copper, with our aluminum alloys having a much lower speed of sound than typical for pure aluminum (does our alloy have higher mass? lower stiffness?), and our wood dowel having a much higher speed of sound than typical for wood. The steel rod was a little low, but within the range of reported speeds of sound in steel.

This report took me several hours to write, in part because producing the graphics took a while, and in part because I fussed around a lot with seeing if removing outliers from the data helped get better results.  My son produced a substantially similar lab report, with graphs but no pictures or schematic of the comparator, in 2–3 hours.  He did not play with removing outliers.  He also correctly reported the velocities with only 2 significant figures (in cm/µsec).