Remembering our physics we know that a = F/m.
With that in mind, I figured one of the best ways to reduce vibration (which is proportional to a), would be to increase the mass of the APM. To that end, I taped together a stack of 4 US quarters and sticky-taped that to the underside of the APM.
So now I have my APM supported by 4 pillars of 1/2" Dubro with a stack of quarters stuck to the APM as a mass damper. Judging by the raw logs, this seems to have smoothed things out quite nicely.
Can anyone see any drawbacks to this approach?
You are correct.
When it comes to motors and propellers this is one of the often misunderstood parts of multicopter design. Mounting your motors on soft rubber feet will perhaps at best isolate some of the vibration from the rest of the frame, but the stress will play havoc on the motor ball bearings and propellers. On the other hard by hard mounting the motors to a well designed rigid frame (solid mass), you to a large degree stop the vibrations from a curing in the first place. Then you isolate the autopilot sensors from the leftover vibrations using gels/foam padding.
In this regard you solution definitively has merit. By adding mass to the autopilot you make it harder for the vibrations in the frame to traverse the padding and travel into the sensors. In fact more expensive professional autopilots usally have the sensors build into a very sturdy and relatively heavy metal housing just for this reason. The only drawback I can see is that by adding weight, you are reducing flight time. But the effect should be negligible with the amount of weight we are talking about here.
Three years back I was trying to soundproof a club running in a buildings basement. The best material on the market was a sandwich made of rubber, lead foil and foam.
Probably would work for our FCs as well :)