Gaussamer Threads

At my lab's group meeting this week, one of my colleagues was showing off the tungsten cube he had purchased to be used in a satellite designed to measure the Earth's mass distribution, similar to the GRACE mission I discussed earlier.

(May contain Infinity Stone)

Since we want the cube to be only affected by gravity, one of the steps in fabrication is degaussing, or removing any residual magnetism. I was curious about this process, since it didn't align with my previous context for degaussing: In high school, I worked for the IT department during the summers, and I was once assigned the task of erasing a collection of video tapes that had been used for a media class. This was done using a degausser, which was essentially an electromagnet that I ran over the surface of the tapes. However, this would put all the magnetic fields pointing in the same direction, not zero them.

One technique I found for driving the field to zero is to apply a large external field, then repeatedly reorient it while decreasing the magnitude. I decided to try this in 2D, similar to the Ising model, but more classical: The magnetic spins can point any direction in the plane, and experience a torque from the surrounding spins and external field. The animation below shows each spin's direction in black, the average direction in red, and the total magnitude of the field as time progresses in the lower plot. The external field is shown on the outer edges.

The way I applied the external fields I think results in the diagonal bias you can see near the end, but overall I'm impressed I was able to reduce the field to 25% of the original value – Not nearly enough for the sensitivity we need though!

An Honest PUC

Back in January, my brother Nate got several BirdWeather PUCs, small outdoor devices that listen for bird songs, and identify the species responsible, similar to the popular Merlin phone app. He gave one to our parents, one to Marika and me, and kept one for himself. If you follow those links, you can see each PUC uploads its observations to a central server, which can be queried through an API. Nate used this to set up a daily email giving the previous day's species counts for each of our stations, and I was curious if I could apply some of the data analysis techniques I've learned to the results.

The first thing I wanted to try was identifying trends in the time of day each bird is heard. To do this, we can use an idea called data folding. This is similar to the idea of a Fourier transform, where we're dealing with periodic data, but we're only interested in a single period: 1 day. If we split our months of data into single-day segments and stack them on top of each other, we can get better statistics about when each bird is heard. There are too many individual species to look at all of them, so I considered finding a way to group them. One idea was to use their taxonomy, but depending on the level I chose, I'd get either one big group, or a group for every bird. Going back to the better statistics idea, I decided to just plot the ones with more than 200 total detections:

This is called a violin plot, often used to show statistical distributions like these. The end caps show the max/min values, and the bulges show the more frequent times. You can see that most peak around dawn hours, but a few are heard throughout the day.

In the daily emails Nate set up, I noticed that for a long time I only saw the Carolina wren show up in Florida, but then it started popping up in Massachusetts as well. I wondered if I was seeing a spring migration, so I got the data from all 3 of our stations, and looked at the number of wrens for each day:

This seems to show it was just my imagination: The wrens are much more frequent in Florida overall, and there doesn't appear to be a trend toward MA over time.

In the first plot I showed, you may have noticed the "Engine" entry. It turns out the PUC has several non-bird sounds it recognizes, and it picked up the highway traffic next door to us. I was curious to see what other non-bird detections it had made. The reported data gets a "confidence" rank based on both how close the sound was to the model, and how likely it is to hear that sound in the location. I split up the detections below on those rankings:

It's reassuring to see the few "gun" detections don't rise above uncertain, though the siren counts are quite high (and I'm assuming that refers to the emergency-vehicle type, not luring-sailors-to-doom type). Seeing these makes me curious what other sounds it can recognize. Thanks for this cool gift, Nate!

[Edit: I noticed this post is getting lots of traffic from the BirdWeather community. Maybe someone *cough*Nate*cough* shared this post there, but anyway, here's my code!]

Circumferential Evidence

Shortly before starting our move to Florida last year, we took our car to have all 4 tires replaced for the 1000 mile trip. On our way back from the dealer, the car's tire pressure monitoring system went off, warning that one or more tires were not properly filled. We checked each one after we got home and all were fine – Consulting the manual told us the system often gives false positives after tire replacements, and needed to be reset. Clearly the system was not directly measuring the pressure, so I was curious how these work. I found a nice explanation from Bridgestone Tire: The system measures the revolution speed of the tire and compares to the speed over the road, in effect measuring the tires' circumference. What I wanted to know is, what sort of conversion is needed to get from circumference to pressure?

The effect of lower pressure will be for the tires to ride lower to the ground, flattening against the road. Here's how I imagined it:

Here the tire has inner radius r, outer radius R, and the axel is a height h above the road. Initially, I tried to figure out the pressure directly: Each tire is applying a force to hold the car up, and force is equal to pressure times the area of the contact patch. That was a bit of a mess, but I realized to detect an under inflated tire, we can instead look at the volume of air inside, which is related to the pressure through the ideal gas law. That turns this into a geometry problem: We want to find the perimeter and area of a circle with a wedge replaced by a triangle. Skipping some intermediate steps, the perimeter is

and the area is

To get the volume, we can just multiply by the width, w, of the tire. To put some numbers to this though, we have to enter the arcane realm of tire sizing.

Our car uses tires of size P235/55R19. The first number tells us the width in millimeters. The number after the slash is the "aspect ratio", which is a percentage given by (R-r)/w. Things get even weirder from there, since the third number is the wheel diameter, equal to 2r, in inches! We can combine these measurements (after converting to a single system of units) to get r, R, and w, then plug those in along with a couple values for h. We can then look at how the tire circumference changes with volume of air lost from the tire:

This plot covers the range from fully inflated to riding on the rims, so I was surprised it stayed so linear. While thinking about this, I remembered an episode of Mythbusters where they looked at improved gas mileage from over-inflated tires – Here's a similar analysis from Popular Mechanics. That article attributes the effect to lower rolling resistance, but my results make me wonder if part of it is getting increased circumference, resulting in the same number of revolutions taking you a greater distance: For the two ends of the plot above, each mile could require 693 to 718 revolutions.