Why do my bass drivers shake violently listening to vinyl

Millercarbon, so far you have been totally and completely off base twice in this thread. The OPs problem has nothing to do with the turntable's location. Isolating it will do absolutely nothing. If there is a cartridge mismatch improving that will help otherwise his system is appropriately amplifying the signal that the cartridge is picking up. With smaller woofers this causes excess movement. Next. Stopping that excess movement results in a large improvement in sound quality because of the marked decrease in several forms of distortion. It also results in greater headroom because power is not being wasted. The improvement a subsonic filter will make will be far in excess of any problems it might cause. 
@lewm, subsonic filters are not complicated issues in the digital world, you just program one in. I know this rubs you the wrong way but personally I do not like being stuck in the past. Digital reproduction has advantages that can not be matched in the analog world and the advantages outweigh the disadvantages. I am dead certain you would agree if you could hear it in action. 

I agree with those who say before you do anything, try another turntable in your system at the same location. Whether or not you get the woofer excursion with the second table will determine what to do next. Before you can solve your problem, you have to know what the problem is.

You either need a better turntable with a much better bearing or a very good rumble/subsonic filter on your amplifier.

The OP is experiencing an undamped medium mass arm with a too-compliant cartridge in combination with vented speakers. A stiffer suspension cart, sealed box speakers, a viscous damped arm…any will stop excess sub-bass from causing woofer oscillation. Eric Squires explained how driving a vented speaker below its cutoff frequency is part of the picture, and Yogiboy provided a calculator for arm/ cartridge resonance. Getting reliable data for that from manufacturers specs…good luck! The results speak for themselves. IF OP’s preamp has a processor loop or tape monitor then a high pass filter is the simplest solution but beware unwanted loss of subtle quality cues if the HPF isn’t as good as your excellent Herron phonostage. Maybe they can add an HPF to their circuit??

One of the problems with a ported speaker is that below the port resonance the woofer is literally floating in the breeze with nothing to stop it(unlike a closed box where the compression of the air brakes the driver). If there is a low frequency resonance the driver can go in and out totally uncontrollably which can damage a driver. A poorly matched arm/cartridge resonance can be a source of this kind of low frequency energy. Look at controlling this resonance and/or use a rumble filter.