Why do my bass drivers shake violently listening to vinyl

You need a rumble filter, a low frequency filter specifically created to solve these problems with vinyl playback.

What's going on is that you are getting a strong signal below the resonance filter of the port, so there's no longer an air spring there to hold the driver back.

If you can conveniently plug the port you should try and watch again. :)

Still, you need a rumble filter to reduce the strain on your amp and speaker, and yes it's dangerous, either from excess excursion or overheating of the coils.

Of course, you may also need better isolation of your turntable. On a springy wooden floor a simple stand will not be good enough.

What arm and cartridge are you using!  Put the information in this calculator and you will get the resonance frequency number!
http://www.mh-audio.nl/Calculators/RF.html

@erik_squires Excellent post!! 

I suspected it was dangerous. The excursion or cone travel is pretty extreme and it seems to be as you said with no spring holding back the driver. 

And, yes, I have a springy wooden floor in a second floor bedroom and a cheap Pangea audio rack.

I could try plugging the port although with the Treo's they are located on the bottom. I will absolutely look into getting a rumble filter.

Thanks all!!

@yogiboy Technics SL1210GR arm and AT VM760SLC cart. 

Arm = 12 g
Cart = 8 g
Fastener Mass = .5 g (guessed)
Compliance = 10 x 10-6 cm/Dyne  

Tonearm Resonant Frequency = 11.12 Hz

Erik is 100% right. Back in the day, rumble filters were a feature on many receivers as it was often needed for the very problem you are experiencing. I had the issue myself sometimes.