While I agree with the comments about using a more rigid stand, I want to try and address your stated concern about what sounds like a harmonic vibration at 800 RPM. While I am no expert, well to be honest, probably not even qualified to be a novice but I will not let that stop me from sharing an opinion
. To change a harmonic vibration you have to change the mass distribution of the system. In other words, adding weight, removing weight, moving the weight ... I assume you get the vibrations even when not cutting. Have you tried a different chuck? Remove the Jaws, does it still vibrate? have you measured the run-out, perhaps there is bearing issue. If it is a harmonic vibration, you probably can dampen it to some extent by bolting the lathe to a solid platform on the floor but you may still get accuracy issues at the harmonic frequency until you find and fix the root issue.