Why do my bass drivers shake violently listening to vinyl

The flopping is due to things you can't hear, and the OP should be grateful that he didn't have the volume up any higher. :)

Those signals could be in the 5-10Hz range, quite easily.  Removing them wont' make it sound less incredible. :)

I generally prefer to go from the simple to the complex, in trying to solve any problem.  Simple is to either move your turntable in your listening room or suspend it, or both.  Complex is to add a subsonic filter.  If you add a subsonic filter, then a steep slope at a frequency below 20Hz is desireable, but 24db/octave is to be preferred over 18db/octave, because of phase anomalies that may affect upper frequencies. (6db/octave is also good but much too gradual for slope in this case. 12db/octave and 18db/octave are less "good" compared to 24db/octave.)  I will bet that moving the TT will solve your problem or at least ameliorate it sufficiently so as not to be a bother.

+1 erik, problem most likely subsonic 5Hz or so. Use a 24dB/octave filter, the steeper the slope the better.

@lewm

I generally prefer to go from the simple to the complex, in trying to solve any problem. Simple is to either move your turntable in your listening room or suspend it, or both. Complex is to add a subsonic filter

I agree wholeheartedly with this approach

FYI: Japanese cartridge specs are rated at 100hz not 10hz. You have to multiply by 1.7 and the resonance frequency is not 11.12 it is 8.53. That is a fine figure and will not cause the problem of woofer pumping!