Why do my bass drivers shake violently listening to vinyl

A rumble filter will definitely eliminate woofer flutter. But it will do nothing to improve sound quality. Springs, sand box, etc will definitely improve sound quality, and also in most cases eliminate woofer pumping. You can spend your time and money on improvements, or fixes. 

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?

No. That is different. There are two main forms of vibration to deal with, both vibrations but completely different in effect. Your original question is about subsonic vibration. Speakers can't audibly output, you can't hear it, you're concerned with something you can see but not hear. 

Then there are vibrations from speakers in the audible frequency range you can hear. This range of vibration will smear detail and color instrumental timbre making individual instruments sound less distinctive. Putting speakers on springs like Nobsound will greatly improve this. As will putting the turntable on springs. They will not eliminate the large amplitude low frequency pumping you are seeing. They will however greatly improve clarity, detail, dynamics, imaging, etc.

This is why I recommend the things I do. Springs, mass, etc work together to solve both problems. Solve. Eliminate. Not patch over with a band-aid. While simultaneously improving sound quality. None of which you get with a filter.

@audionoobie
I kindly ask that you post your solution after you get this problem solved. I'm just curious

As a long shot remove the dustcover!

I have a TT and cart combo that will blow a normal sub 80 hz < but with servo plate amps and servo correction, it’s a tame, deep, very clear bass signal to 25 hz. I’ve always used a spring suspension and gummy puffers for isolation.. A heavy ply plinth, rubber line and spring load the bottom with springs or an inner tube. I use Thoren TD124s and Russco.

The main speakers can’t pump because the bass is cut from 280< and from 100-300hz are bass columns. My system CAN’T pump!! You can throw the TT in the middle of the room... Servo is pretty cool.. ;-)

I use up to 6 12" OB HE drivers. Maybe 100 watts to each driver @ 8 ohms. 100hz < no TT problems..:-)

Regards

Millercarbon, so far you have been totally and completely off base twice in this thread. The OPs problem has nothing to do with the turntable's location. Isolating it will do absolutely nothing. If there is a cartridge mismatch improving that will help otherwise his system is appropriately amplifying the signal that the cartridge is picking up. With smaller woofers this causes excess movement. Next. Stopping that excess movement results in a large improvement in sound quality because of the marked decrease in several forms of distortion. It also results in greater headroom because power is not being wasted. The improvement a subsonic filter will make will be far in excess of any problems it might cause. 
@lewm, subsonic filters are not complicated issues in the digital world, you just program one in. I know this rubs you the wrong way but personally I do not like being stuck in the past. Digital reproduction has advantages that can not be matched in the analog world and the advantages outweigh the disadvantages. I am dead certain you would agree if you could hear it in action.