Why do my bass drivers shake violently listening to vinyl

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.

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.