Why do my bass drivers shake violently listening to vinyl

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"

Some really interesting information from several members.

Subsonic frequencies have direct effect on the hormonal system, they affect the biorhythm of the internal organs, vision, balance. They are an essential part of perceiving the hall acoustics of a concert hall. 
If it's part of the recording, then you hit the real deal. The ability to recreate those frequencies literally is a paradigm shift.
If it happens only on certain records (which have that frequency range recorded), you have arrived home. 
In case it's on every record then there's an issue.