Why do my bass drivers shake violently listening to vinyl

Lots of helpful thoughts, but so far no-one has asked the right question.

Do the bass driver cones flap the same all the time, whatever you play, or do they flap in line with the timing and intensity of the bass in the music.

If the former then don't listen to Miller who says your home is being destroyed by invisible vibrations - stand on the floor in your room in silence - can you feel them?  Probably not.  Yes, you need to look at your turntable rig.  Try playing an unmodulated LP if you have one - i.e with a groove but no music. Do you get the flapping now?
You may need a rumble filter or to clean and re-oil the bearing.  Or to check the bearings in your arm can move freely but do not have any jitter.  Or to check the tightness of the cartridge fixings (go easy here though). Or to check the arm and cartridge are well matched and not causing resonance.  Or to upgrade your turntable and arm.

But if the flapping changes in time with the music then I am inclined to say this is just your analogue system doing its job.  You say you like the sound better than your digital sources, so what's to worry about?

Enjoy.

Rumble (sub-sonic) in any system is not good unless it is designed for it.

I have spent considerable time at Masterdisk in NYC.  Previously owned by Bob Ludwig and currently by Scott Hull.  They would always check for
rumble in the master disk they cut. The turntable is in a different room and here are the speakers they use:  Hartley 24" subs which will play down to 7hz.  They don't flutter but you can sure feel it!
https://reverb.com/item/4252846-hartley-sw-24-speakers-bob-ludwig-mastering-subwoofers

This has been a public service announcement.

BTW, I own a pair of Hartley Reference which is the same cabinet and woofers along with a full complement of drivers.
Regards,
barts

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.