Why do my bass drivers shake violently listening to vinyl

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!

Well if the OP's tt is not broken, and if moving the tt in his listening room or improving tt isolation do not have any effect, up or down, and if his LPs are not warped, then he does need a subsonic filter, preferably a good one.

Another word for rumble filter is subsonic filter. I have a Technics 1200G and this problem was eliminated by activating the subsonic filter on my phono stage.

It is the nature of the vinyl format to have this rumble.

Not sure it helped in any way but I kept the subsonic filter off when breaking in my Tannoys.

I also ran my Salk speakers for 10 years with the subsonic filter off. Nothing bad ever happened. Woofers were always ’excited’ during the space between LP tracks.

For me, activating the filter didn’t seem to help/hurt the sonics in any real way. 

It is the nature of the vinyl format to have this rumble.

Never seen anything like that on any of my high efficient drivers using vinyl only format for over 25 years without any subsonic filers! 

Not sure it helped in any way but I kept the subsonic filter off when breaking in my Tannoys.

Right, it’s absolutely useless with a proper drivers like Tannoy Dual Concentric (I have 15 inch and 8 inch Tannoy speakers ) or any other full range drivers like Zu and related.