Regarding Stabilizing the plates.

Please reply to the list,not to me.

Fix aveforce can be used on a set of boundary atoms (e.g. your plate)

to keep them coherent. If they are still moving up/down that is likely
due to the forces the plate is experiencing from other atoms, which
means it is real. If you want to turn that off so that the plate
moves exactly as you prescribe, then you can look at fix move.