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A note on driver control (aka Newton's third law): When the driver moves to produce sound, the driver itself and the baffle plate to which it is attached vibrates in the opposite direction. This is the whole equal and opposite reaction thing. Things that vibrate the cup (either external factors or even the sound from the driver itself) will sneak back towards the driver as well. Now obviously, we don't want the driver to move at all. The more stationary it is, the more "pure" the sound that will be produced. There are three ways to accomplish this: mass, stiffness, and absorption.
Mass is literally that. Make the driver and baffle heavier, and it will move less from any applied force. Absorption is acknowledging that vibrations are there, but trying to squash them as quickly as possible. Stiffness is sort of this in between that affects the both. So let's address a few common things:
1) dynamat: It adds a tiny bit of mass, but primarily this is a shear based dampener. What that means is it has a layer of sticky goopy stuff, and a stiff metallic layer on top. When you stick it to something that vibrates, those vibrations move the goopy stuff, but the stiff layer on top doesn't move. This causes viscous shear in the goop which absorbs the vibrations.
2) blue tack, plasticine, etc: Common stuffing for baffle plates (especially Fostexen) to make them heavier. That's it. You get minimal absorption from this. Some people try attaching metal bits to the drivers to make them heavier, but that's generally difficult to do
3) stiffness: Things that have better bracing generally want to vibrate/resonate less. If you have a way to brace or stiffen a cup, it will move less. Increasing clamping pressure physically couples a headphone harder to your head. This is somewhat related to stiffness, and effectively connects the headphone to the mass of your head, and again heavier things will vibrate less (no jokes about being dense please!).
Annnnd, I think I had something else but I'm procrastinating from work and need to get back. Cheers y'all.