Why do my bass drivers shake violently listening to vinyl

Get a KAB rumble filter. Or move the turntable to a different room.

I forgot to mention one other major factor that makes this problem worse,
the RIAA curve. When a lacquer is cut the high frequencies are boosted and the low frequencies are cut. On playback the high frequencies are cut which reduces high frequency record noise just like Dolby noise reduction. Unfortunately, low frequencies are boosted making low frequency record noise worse. Only 3 dB means the woofer cones move twice as far. At higher volumes that becomes a huge problem. IMHO subsonic filters are mandatory for people who like to play above 80 dB.

I think dynamic and crusty have very good ideas about what might be going on. I was an early fan of transmission line woofers. Specifically I bought Irving M Fried (IMF) speakers in the early 1970s, because I thought they had unusually natural low bass response, and they were darn good above those frequencies as well. When I was once in the presence of IMF himself, he pontificated that all preamplifiers should have a subsonic filter built-in. This is because the transmission line is a ported design in which the woofer is completely free at low frequencies. And on occasion my woofers iused to pump visibly in those days. I ignored it. I still use transmission line woofers to support the low bass response of my Beveridge 2SW speaker panels. (80Hz cutoff). The TL cabinets I am using I built in something like 1971. Because I am aware of this pumping tendency, I am often staring at the KEF B139 woofers built into my transmission lines, while I am listening to music. They are pretty much static at all times, visually, from my listening seat. No pumping has been observed yet. Could be because this system is in my basement, with concrete flooring over earth, or it could be because my turntables are well isolated and quiet. And I don’t mess with warped LPs unless the warp is minuscule and I love the particular recording.

Hello,
Do you own an old mouse pad you can cut up or bubble wrap you can put under the feet? This should tell you if it is a vibration problem from the supporting table or base you are using. I have heard this was a frequency problem. Do you have a different cart/ head shell you can try really quick. If not can you switch to another cart to see if it is a frequency problem?
If it is and you like the sound of the cart then you will need to buy the device to remove the rumble. If it were the excursion of the woofer it would happen while streaming too. I had this happen to me when I used an aftermarket head shell. If yours is removable you might try a different head shell. That should change the frequency. I hope this helped. 

I had this exact problem about which I posted a month or 2 ago.

In my case, not only did I have the wild woofer pumping, but also not enough gain from my phono stage.

I tried isolation first, no improvement.

Then I tried a much better and more expensive turntable, arm, cartridge solution.  Still, the woofer pumping persisted.

Ultimately, I fixed the issue with a new photo stage, both with more gain AND a rumble filter.  The latter is inaudible BTW, other than resolving the woofer dance while spinning vinyl.

Hope this helps!