Why do my bass drivers shake violently listening to vinyl

@sbank No. None of that is happening. And this isn't the first time I've seen this. It happened with a completely different set of speakers and different cart, on the same table. I thought it was something particular to those other speakers. 

@audionoobie

RE:   But the main source is feedback exciting the natural stylus/arm resonance between 7 and 12 hz.

If this is the case try placing a small amount of bluetac or silly putty on the heard/arm and recalibrate the stylus down force to the recommended amount for the cartridge. This will change the overall mass, which may reduce, eliminate or exacerbate the problem. Either way you will know the cause of the problem. 

Do the bearings in the arm have any play in them?

Regards

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.