Why do my bass drivers shake violently listening to vinyl

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.

Before purchasing a rumble filter, I would first see whether changing the location of your turntable in relation to the speakers has any effect on the phenomenon you are observing. Also, the addition of some isolation to the turntable base might help the problem. You might try to suspend the Technics turntable in someway. There are “feet” available for that. I would put all this ahead of adding a rumble filter in the signal path, only because the rumble filter is not without its sonic penalties. There is no free lunch in Audio. The only other thing I would add to the discussion is that someone mentioned DC voltage as the possible culprit. That is impossible since DC by definition has no frequency. Significant DC voltage delivered to the speaker voice coil could damage it, but it is not causing the piston like movements you are observing.