Why do my bass drivers shake violently listening to vinyl

It sounds incredible! I'm just concerned that I may be doing damage to my speakers

Rarely does a problem allow the sound to be good, much less incredible. What kind of music do you notice this with?
Some music causes this phenomenon of very active woofers. Heavy metal and hard rock is one of them . Plus your speakers have a a " 8” Carbon-Loaded Cellulose Flat-Cone Woofer with Ultra-Long-Throw Motor Assembly" according to their literature (specs) I doubt you have a problem. unless the woofer moves without being fed any signal

@williewonka how can I tell if the bearings have any play in them? Not sure what you're referring to.

@artemus_5 I noticed this any time any record is playing. It is not genre specific. And like I said, there is no discernible audio issue. Just the bass drivers going in and out like mad. 

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.

What arm and cartridge are you using!  Put the information in this calculator and you will get the resonance frequency number!
http://www.mh-audio.nl/Calculators/RF.html