I recently got a guitar with a floating bridge and whammy. The guitar itself is great, and sounds good. (especially for the price)

The only issue is that if I strum hard, the bridge sort of hums from the vibration, and it is picked up clearly through the amp. This kind of ruins recording with it.

Any ideas on how to fix it?

Thanks in advance.
make sure its the actual bridge and not thte springs. alot of guitars have similar problems with the springs. if its the springs, take som e cotton and tape it down in the cavity
Also the bar itself, does it do it with the bar off?

I've tried foam etc in the back but it tends to get caught in the springs and throws the tuning out, I'm looking at getting some plastic tubing to fit over the springs on mine. Mine doesn't come through the amp but just bugs me anyway.