Why do my bass drivers shake violently listening to vinyl

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.

I suspect it's a matter of low frequency feedback, otherwise known as rumble. Some of the low frequency rumble may be actually coming from certain discs. I have experienced both types of rumble sources in the past until I upgraded to a phono preamp with a rumble filter. 
If you have a rumble filter in your preamp then the source is definitely coming from some sort of mechanical feedback that the rumble filter can't totally filter out.  
Unfortunately this sort of audible issue can be tricky to troubleshoot, if the rumble filter doesn't work. Looks like most everyone agrees on a "rumble filter"