Why do my bass drivers shake violently listening to vinyl

Microphonics are real. I used to have a phonograph preamp that you could smack and have it ring through the speakers, but the very very first problem you face is the cartridge itself. It’s essentially a microphone with a needle and the LP acting as a diaphragm.

The microphonic pick up of modern electronics however tends to be very small to none, and much higher in frequency. 

The flopping is due to things you can't hear, and the OP should be grateful that he didn't have the volume up any higher. :)

Those signals could be in the 5-10Hz range, quite easily.  Removing them wont' make it sound less incredible. :)

Another way to read "ultra long throw" driver is:

This speaker will play louder before you hear distortion or compression than if it wasn't ultra-long-throw

I copied this from the Vandersteen site . Notice " Ultra-Long-Throw" IMO, that is what you are experiencing, an ultra long throw of the woofer

Your opinion in this case is not actually correct. :)

The phrase "ultra-long-throw" or "long throw" have no specific meaning, but are understood to mean that the driver MAY travel further without distortion than conventional drivers of the same size. Typically, larger drivers have longer throws.  This is specified by manufacturers as Xmax, and is usually in millimeters.  So you'd read it something like this:

Xmax : 10mm (the distance the driver can travel from rest is 10mm before distortion sets in).

That doesn't mean they flap back and forth more than any other driver of the same size and output.  The symptoms from the OP are definitely being caused by excess, and inaudible, bass in the signal which could probably be replicated by jumping on the floor. :)

You don't need an "ultra-long-throw" woofer to flap back and forth, just too much rumble in a ported enclosure.

You need a rumble filter, a low frequency filter specifically created to solve these problems with vinyl playback.

What's going on is that you are getting a strong signal below the resonance filter of the port, so there's no longer an air spring there to hold the driver back.

If you can conveniently plug the port you should try and watch again. :)

Still, you need a rumble filter to reduce the strain on your amp and speaker, and yes it's dangerous, either from excess excursion or overheating of the coils.

Of course, you may also need better isolation of your turntable. On a springy wooden floor a simple stand will not be good enough.