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I work in railway noise and vibration mitigation, and @[email protected] has given you a great starting point. When we build rails and want to mitigate ground-borne noise and vibration (typically up to ~200 Hz), we generally mount the rails on soft pads and add extra mass to isolate the rails from the surroundings. The exact same approach will work at your computer. We don't typically use tuned mass dampers for ground-borne vibration, so I think that will be overkill for you, but you can try if you like.

I wanted to suggest that, in addition to the feet/foam/plywood, you can also add a big chunk of something heavy to help with isolation. Like put a heavy rock on top of the foam, and your computer on the rock. The trick is this: if k is the stiffness of your foam, and m is the mass of everything on top of the foam, then your isolating frequency is at √(2k/m). All frequencies above the isolating frequency will by mitigated (the further above, the more they're mitigated), while all frequencies below will be amplified.

(Quick aside if you actually want to calculate frequency with √(2k/m): check that your units for k and m are compatible, you should end up with a result in units of 1/s, which is actually radians per second, then multiply by 2π radians per cycle to convert to Hertz).

When it comes to measuring results, since your problem is in low frequencies, you can probably use your phone's accelerometer assuming it reads fast enough (the sample rate must be at least double the highest frequency you care about). Mount it as rigidly as you can to your computer, since if the connection is soft, the phone will be in its own isolating system. The quickest way to test your isolator would be to hit close to the base with a hammer; impacts excite a wide range of frequencies equally, so in the frequency domain you should see vibration amplitudes following a shape something like these.

But as @[email protected] notes below, you don't really care about your isolator's response, you care about what trains are doing to your computer. However, he said one thing I disagree with: it's not the amplitude of the acceleration that you care about, it's the amplitude of energy, and therefore velocity. This article gives a good introduction to ways you could analyze that. But now we're getting way in to the weeds on what should be a simple project!

One last aside: if the vibrations in your building are bad enough, you could raise it as an issue with the metro operator. The US Federal Transit Administration sets standards that are commonly followed even outside of the US (see Table 8-1 in their Noise and Vibration Manual); if your measurements show vibration exceeding those limits then they might pay me to fix it :D.

Honestly, I think your approach using the MEMS accelerometer in your smartphone is fine - just make sure to tape the back of your phone firmly to the board to get a rigid connection. This will be of particular importance if you want to do any kind of tuning, as you might measure spurious frequencies if the device is not properly attached.

Smartphone accelerometers are actually used in civil engineering / industrial applications to determine frequencies of e. g. bridges or check for bearing wear. If you are interested, here are some papers:

https://www.spiedigitallibrary.org/conference-proceedings-of-spie/9804/98040C/Identification-of-the-operational-frequencies-of-300-bridges-using-smartphones/10.1117/12.2222097.short#_=_

https://www.scielo.br/j/lajss/a/ZnWZ8T86HHBLFvdksCh7g9s/?lang=en

https://www.mdpi.com/1424-8220/19/14/3143

https://www.mdpi.com/1424-8220/15/2/2980

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9824767/

That being said, the accuracy of the frequency readings is not super important for your application, as what you are after is pretty much only a reduction in amplitude. I would assume spinning hard drives show different responses to different vibrational frequencies, but I did not have the time to research this myself. However, here are two papers that explore what you are trying to do, which I will link since they also mention a few related papers that show the impact of vibrations on hard drives:

https://link.springer.com/article/10.1007/s00542-012-1592-z

https://content.iospress.com/articles/shock-and-vibration/sav00458

This would in theory incentivise to optimize dampening certain frequencies, but I suspect you will quickly get into the realm of overengineering / premature optimization, as the dampening might be good enough to tune out all relevant frequencies without simulations or tuning etc. However, it's still certainly a worthwhile effort for educational purposes though.

As for a practical approach, I'd probably simply start out with some super cheap foam obtained at the nearest home improvement or crafts store and see how a sandwich using that affects signal amplitude. You could even introduce artifical vibrations using a DC motor with a weight mounted off-center on the shaft, which you can get ready-made for next to nothing on eBay ("vibration motor", ~ $2).

If you want to get into tuning / experiment / analysis territory, I would like to include additional motivation. Not only might proper decoupling increase the lifetime of your drives, but it could also improve performance. I will include the following video as humorous proof of that:

https://www.youtube.com/watch?v=tDacjrSCeq4

P.S.: I firmly believe that research should be freely accessible, so I feel obligated to mention that all papers linked above are, if not available for free on their respective webpages, obtainable via a certain scientific hub.

HDDs latency will go up from external vibrations (even yelling at your system will do the trick). This would be a good direct source to see if the vibrations are negatively impacting it.