Measuring speed, RPM, prop slip via Doppler shift

[petiaccja]

Hi everyone,

I don't have a GPS tracker, so I decided to measure my boat's speed using the Doppler effect. I found the science behind it pretty cool, so I thought I'd share it, and of course the results.

The setup was very simple. I put my phone down on the pier and started sound recording, then drove past the phone in a straight line. The boat should be able to reach a constant top speed a little before passing by the phone. High quality illustration:

experimental_setup.jpg

Then I went home, and analyzed the recordings. First I took a look at the spectrogram, here is a screenshot from the spectrogram visualizer of the foobar2000 music player:

spectrogram_annotated.jpg

If you are not familiar with spectrograms, they answer questions like "how loud I heard a 433 Hz sound at 1.26 seconds into the recording". Instead of 433 Hz, you can pick anything by moving up and down on the above image (high frequencies up, lows down), and instead of 1.26 seconds you can pick any moment by moving left or right on the image (left happened earlier, right happened later). The brighter the spot on the image, the louder that frequency was at that time. So for example, on this image, you can tell you could hear the 433 Hz, 866 Hz, etc. frequencies pretty loud at the same time for a good second or so. As a sidenote, the scale on this spectrogram is not linear.

The recording turned out to be very good with barely any noise. On the spectrogram there are very clear lines which correspond to the rotating fequency of the motor. You can also see how the frequencies change, and the pitch of the sound lowers as it passes by the phone. Then the lower frequencies are stable again as the boat is going away. As expected, there are lines for the harmonics (multiples) of the rotating frequency as well.

The spectrogram is just for illustration, I needed the exact values for the incoming and outgoing frequencies. I loaded the recordings in Audacity and did a spectrum analysis using Fourier transforms. Here is a screenshot of the power spectrum of the recording:

fft_harmonics.jpg

If you are not familiar with spectrums, it shows how loud any frequency was at one single point in time. So while a spectrogram shows all time points, the spectrum focuses only on one. Both spectrograms and spectrums show the whole frequency range.

I marked the peaks with a red dot. The numerical values for these peaks are 433 Hz, 866 Hz, etc. They correspond to the bright lines that we saw on the spectrogram. The lowest one, 433 Hz, is the frequency of the rotation of the motor. So 433 Hz equals 25980 RPM, right? Let's not get ahead so much, the 433 Hz is Doppler shifted, it's not the real frequency of the motor. In fact, I analyzed the spectrum for the part of the recording where the boat was going away, and the frequency turned out to be 388 Hz.

I'm not gonna bore you with the math, but if you hear 433 Hz as the boat approaches you, and hear 388 Hz as the boat is leaving, the boat in reality was making a 409 Hz noise, provided that the speed of sound was about 343m/s. In other words,
the loaded RPM was 24556/min, or about 81% of the unloaded RPM of 30344/min (@12.35V, 2457 kV).

From these two frequencies and the speed of sound, we can also tell
the speed of the boat was 68 km/h (42 mph).

I was using a 36mm prop, with a pitch of 1.9, so at 24556 RPM the speed should've been 101 km/h (63 mph) if it wasn't for prop slippage. Reverse this, and we can tell
the prop slippage was 32.84%.

Some interesting observations:

• On the spectrum, the base frequency of the motor (433 Hz), is significantly quieter than the first harmonic of 866 Hz. This is probably because I'm using a two-blade prop, leading to two beats on the water for every one turn of the engine. Similarly, harmonic #3 is also quieter than #4. Most of the sound is probably coming from the blade (or the knock the blade makes on the body as it hits the water), not from the engine.
• The sound of the boat approaching was much louder than the sound of the boat leaving. I suspect this is because the front of the prop is heavily cavitating and mixing air, creating sound waves in the air, while the back of the prop is tight against the water, making sound waves only in the water but not the air.
• And most importantly my setup is #&@%&, 33% slippage sounds very bad for a hydro. Or... I might have been dragging seaweed, as summer turned the river into more of a swamp.

[Fluid]

A very interesting study. One explanation for the high “slippage” could be that the boat had not actually reached top speed and was still accelerating. It is prop slippage which provides thrust for acceleration and from SAW racing experience I know that it can take up to 100 feet or more for the boat to top out even at peak rpm. This may, or may not, explain at least part of the high slippage value.

The sound attenuation is more likely due to the roostertail absorbing/blocking the sound from the rear.



.



.

[NativePaul]

Using doppler to estimate speed in model aeroplanes is common practice, but I have never heard of anyone using it for model boats before. Well done on your out of the box thinking.

[petiaccja]

A very interesting study. One explanation for the high “slippage” could be that the boat had not actually reached top speed and was still accelerating. It is prop slippage which provides thrust for acceleration and from SAW racing experience I know that it can take up to 100 feet or more for the boat to top out even at peak rpm. This may, or may not, explain at least part of the high slippage value.

The sound attenuation is more likely due to the roostertail absorbing/blocking the sound from the rear.

Thanks for the explanation on the slippage. Unfortunately, I did not have so much space, so it really might have been that the boat was not at full speed just yet. I'm gonna have to give it another try! I'm also suspecting improper balance, my boat is a little bit tail-heavy, causing drag at the back. Also, due to the boat not passing right over the phone, but a few meters away, the speed is somewhat underestimated, while the RPM should still be pretty accurate, overall leading to a higher slippage value. Still, 33% is quite a lot higher than the guideline of 20% on OSE's speed calculator.

I haven't thought of the rooster tail's sound absorption, but that's a pretty good explanation!

[petiaccja]

Using doppler to estimate speed in model aeroplanes is common practice, but I have never heard of anyone using it for model boats before. Well done on your out of the box thinking.

As long as they are fast enough to see the frequency shift... Glad you found it interesting!

[larryrose11]

Pretty cool. and I taught there was only a 40% chance of Science today.