Why do my bass drivers shake violently listening to vinyl

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.

Miller is right in this matter,  you need better feets..I like Solid tech feets of silence ( there is plenty of other doing the same thing) isonoe is OK but spring based solutions are better . Then it is the question of the tonearm mounted on 1200gr !!! I removed it from my deck after 2 months and fitted audiomods series 6 arm , that's that  ...
No need for rumble filter anymore..
This is what I did with my 1200gr , but this is your ocean to navigate ... 

Lots of helpful thoughts, but so far no-one has asked the right question.

Do the bass driver cones flap the same all the time, whatever you play, or do they flap in line with the timing and intensity of the bass in the music.

If the former then don't listen to Miller who says your home is being destroyed by invisible vibrations - stand on the floor in your room in silence - can you feel them?  Probably not.  Yes, you need to look at your turntable rig.  Try playing an unmodulated LP if you have one - i.e with a groove but no music. Do you get the flapping now?
You may need a rumble filter or to clean and re-oil the bearing.  Or to check the bearings in your arm can move freely but do not have any jitter.  Or to check the tightness of the cartridge fixings (go easy here though). Or to check the arm and cartridge are well matched and not causing resonance.  Or to upgrade your turntable and arm.

But if the flapping changes in time with the music then I am inclined to say this is just your analogue system doing its job.  You say you like the sound better than your digital sources, so what's to worry about?

Enjoy.

Rumble (sub-sonic) in any system is not good unless it is designed for it.

I have spent considerable time at Masterdisk in NYC.  Previously owned by Bob Ludwig and currently by Scott Hull.  They would always check for
rumble in the master disk they cut. The turntable is in a different room and here are the speakers they use:  Hartley 24" subs which will play down to 7hz.  They don't flutter but you can sure feel it!
https://reverb.com/item/4252846-hartley-sw-24-speakers-bob-ludwig-mastering-subwoofers

This has been a public service announcement.

BTW, I own a pair of Hartley Reference which is the same cabinet and woofers along with a full complement of drivers.
Regards,
barts

Get a KAB rumble filter. Or move the turntable to a different room.