Why do my bass drivers shake violently listening to vinyl

That's very interesting.  You mentioned you are using a 1200GR turntable.  Do you own or can you borrow a different TT, just to see if the problem is the same with a different source TT?  I am wondering whether there is some sort of defect in your 1200GR.  That seems very unlikely, given that Technics make thousands of them and have vast experience building TTs with a very low error rate, but unlikely events do occur.  If you own a stethoscope, place the bell on the plinth and listen for excessive noise; that might be an indicator of a problem. (It's also interesting that a Pangea shelf is not as stable as an IKEA product not even made to support a TT.)

Sorry noob that you had to read through all the bad advice before finally lewm helped you out. It is not the record or the turntable or arm or cartridge or anything to do with compliance or any of that. Sorry you have been so misinformed. What you are seeing is the mechanical vibration being transmitted up from the floor, rack, and shelf into the turntable. 

This happens because the signal cut into the vinyl goes through RIAA equalization that turns the lowest frequencies down 20dB. That is a lot. When playing back the phono stage has to amplify that 20dB back to sound flat. Only problem, no way of knowing what low frequency vibration is signal and what is noise. Even the tiniest vibration amplified 20dB is gonna be quite loud. Which is exactly what you are seeing. Only it is too low frequency for your speakers to reproduce audibly, but it is there and so you see it. This is the grain of truth in some of the bad advice above.

The solution lewm has in mind is to put the turntable on something more stable. There are several ways of doing this, which one will work best in your situation is hard to say. But the good news is these are tried and true simple and easy and cheap to do.

Easiest/cheapest will probably be Nobsound springs. Put the correct number of springs under each footer, will be a big improvement and might just solve your problem in one fell swoop. For $30. If not no worries they will be an improvement for sure you just might need to go further.

A nice thick butcher block shelf works well. Another one is to build a sand box. Anything the right size you can fill with an inch or two of sand. Mix the sand with just enough mineral oil to eliminate dust, pack it down, put the butcher block on top, turntable on top of that. Nobsound between butcher block and turntable. Premium isolation for cheap.

Any one or two of these will probably do the trick, and not just eliminate your woofer pumping but improve sound quality a lot in the process.

I have posted this many times already but I will do it again.  I tried everything MC and other describe here, nothing worked on my VPI Prime.  I tried it on a 4 inch thick maple platform, I bolted a shelf to the wall, I moved the TT all over the room, tried isolation cones, etc.

Out of desperation, I bought the KAB filter and it was the best decision I made.  The filter is stuffed with audiophile grade parts and it does its job without altering the sound.  I run mine through a processor loop on my preamp so it is only used when I play vinyl.   Do yourself a big favor and get the KAB rumble filter.  I believe it is around $179.00.

Thanks @stereo5 I think that is the route I will go down. For my own understanding, is this a turntable feedback loop? The low frequencies being put out by the speaker are traveling along the floor, into the rack, picked up and amplified by the cartridge and phono pre and sent back out the speakers further exacerbating the issue?

I really appreciate everyone's responses. This has been a big help....

+3 On The KAB rumble filter.