Why do my bass drivers shake violently listening to vinyl

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.

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 ...