Why do my bass drivers shake violently listening to vinyl

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.

...or try a sealed box woofer

@chakster, it helps if you have a driver that actually makes bass and have an amplifier with the power to make it happen. But, no bass is better than bad bass. This is a solution we used for decades until decent sub bass drivers can along in the early 80s. You use subwoofers for two reasons, to make real bass below 80 Hz and to take that part of the spectrum away from your main speakers because it screws them up. 
It was a tough sell back in the days of analog bass management. Now it is a no brainer.