Oscillibations

A common question that arises for over-caffeinated physicists is, why does carrying a mug of coffee make it slosh over the rim? Ever since Marika got us our Apple Watches, I've been wanting to use the accelerometers in the watch, which allow it to measure your arm's orientation and motion, to record how my hand moves when walking with a mug. Sadly, I couldn't find an app that would allow me to download the data... until now! I recently decided to have another look, and found HemiPhysioData, designed to help people recovering from injuries track their progress regaining movement.

Accelerometers are a type of sensor that measure acceleration in a particular direction. A simple example is a weight on a scale: This measures a force, which is a mass multiplied by an acceleration. Typically these are used with just the acceleration due to gravity, but if you lift or lower a scale suddenly, you can increase or decrease the acceleration it reads. Inside the watch is a 3-axis accelerometer, which measures the acceleration through the face, top, and side of the watch. Using these measurements, it estimates a direction for gravity, since 9.8 m/s^2 is (hopefully) much more than an average person experiences otherwise. Subtracting that from the total acceleration gives just the wearer's contribution from moving around. We can also use it to find the orientation of the watch. All these measurements are spit out by the app as columns of a file:

1. ID columns, giving info about the run
2. Timestamp, measured at 100 Hz
3. Roll/Pitch/Yaw Euler rotations
4. Rotation vector x/y/z
5. Estimated gravity x/y/z
6. User acceleration x/y/z (total accel. minus gravity)
7. Quaternion rotation w/x/y/z
8. Raw acceleration x/y/z

The Euler, vector, and quaternion rotations are all methods for expressing the orientation of the watch. We can use these to rotate the user acceleration into the wearer's reference frame, rather than the watch's frame.

I decided to try comparing two runs: carrying an empty mug with a normal walking pace, and carrying a full mug being careful not to spill. Here's the output of the sensors for those two runs:

There's a clear periodicity to both datasets, but the differences between the two aren't clear. Instead of looking at the time-domain, we can look at the frequency spectra:

Now we can see that the empty mug has a few spikes between 4-7 Hz. If you look at Figure 5b in the paper I linked at the top, you can see this is the upper end of the frequencies that most excited the liquid in their mug. The paper points out that changing the radius of the mug will shift the resonance frequency, so the difference could be explained by the size of the mug.

The paper suggests a few methods for decreasing the risk of spilling, including dividing the cup into many small tubes, adding foam, or using a "claw grip", but I'll leave you with their comments on the suggestion of walking backwards to prevent spillage:

Of course, walking backwards may be less of a practical method to prevent coffee spilling than a mere physical speculation. A few trials will soon reveal that walking backwards, much more than suppressing resonance, drastically increases the chances of tripping on a stone or crashing into a passing by colleague who may also be walking backwards (this would most definitely lead to spillage).