Re: ceiling fan installation
Many of the boxes that originally had a lightweight light fixture on them are not solid enough to support a fan. If holding up on a non rotating part of the fan reduces the vibrations and noise, that is the problem.
If you have access to the area above the ceiling, you may be able to run a brace from one joist to another. If not, check with where you bought the fan and see if they have anything meant to support retrofit installations. At least they should have a big plate, escutcheon, to cover the big hole you may have to cut in the ceiling to add more support.